an introduction to automorphic representations

Jayce R. Getz, Heekyoung Hahn

An Introduction to AutomorphicRepresentations

with a view toward Trace Formulae

August 26, 2019

Springer

To Angela and Adsila, with love

Preface

The instructor of an introductory course on automorphic representations, thestudent beginning their foray into the subject, and the experienced mathe-matical researcher in another discipline trying to obtain conversational knowl-edge in the field are all faced with a formidable task. The prerequisites areessentially infinite, as the theory of automorphic representations draws on,and makes substantial contributions to, much of algebraic geometry, har-monic analysis, number theory, and representation theory. Moreover, it is asubject in which one can spend a career moving in any direction one chooses.

The student should not be overly discouraged, however, because there is noone alive who has the necessary prerequisites to really understand the entiresubject (and probably no one in the future will either). This is meant as nodisrespect to the luminaries in the field that, either through their work or per-sonal contact, have made a profound impact on the mathematical communityand the authors of this book. It is simply that broad and deep a subject ofresearch. Extremely few important results that were proven by one individual(or even by a group working on a single project). Most of the important re-sults in the field have been obtained by individuals or small teams with verydifferent specialties working on various pieces of a larger whole, sometimesover the course of decades. We say this to encourage the reader; one need notknow everything in order to get started and make important contributions.

In the authors’ experience, after scrounging around the literature onelearns enough to begin discovering mathematics connected to their partic-ular point of view. The goal of this book is to make this process easier. Themore particular aim is to develop enough of the language and machinery ofthe subject that after reading this book one can start reading original researchpapers. We have also tried to be concise enough that the reader doesn’t getmired in so many details that they are unable to see the overarching themesand goals of the subject. This means that we have not made any systematicattempt to prove every result that we use in the book. However, we havemade an effort to provide precise references where the reader can find moredetails.

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This book is written so that it can be used as a text for a semester oryear-long course, a resource for self-study of more advanced topics, and asa reference for researchers. We hope that our efforts to provide definitions,proofs or precise references for proofs will be helpful to the last group. Wetreat general reductive groups over arbitrary global fields.

We now describe several outlines for possible courses. The sections we havenot included have valuable information that may be substituted or added asdesired. Here is an outline for the sections one might cover for a semester-longintroduction to automorphic representation theory:

Algebraic groups §1.2, §1.3, §1.5, §1.7, §1.8Adeles §2.1, §2.2, §2.3, §2.4, §2.6Automorphic Representations §3.1, §3.2, §3.3, §3.4, §3.5, §3.6Archimedean Representation Theory §4.2, §4.3, §4.4, §4.7Representations of Totally Disconnected Groups §5.1, §5.2, §5.3, §5.4, §5.5,§5.6, §5.7Automorphic Forms §6.1, §6.2, §6.3, §6.4, §6.5Unramified Representations §7.1, §7.2, §7.3, §7.5, §7.6, §7.7

It is useful to say in words what these chapters cover. Given a global field Fand a reductive algebraic group G over F , an automorphic representation ofG in the L2-sense is an admissible representation G(AF ) which is isomorphicto a subquotient of L2(G(F )\G(AF )). The first goal of the course outlineabove is to make sense of this definition. The second goal, realized in Chapter6, is to give a refinement of it that is technically very useful and make theconnection with classical automorphic forms transparent. Finally in Chapter7 one finds a rough form of the Langlands functoriality conjecture.

For a second semester or an advanced class one has more flexibility. Chap-ter 8 covers deeper topics in nonarchimedean representation theory, includingthe crucial related notions of Jacquet modules and supercuspidal representa-tions. It also includes a statement of the Bernstein-Zelevinsky classification(see §8.4). These topics, though important, come up only in passing later inthe book, and could be omitted. Chapter 9 on the cuspidal spectrum gives acomplete proof of the fact that the cuspidal spectrum decomposes discretely.It is a basic fact that is hard to extract from the literature, but it could easilybe treated as a black box if desired. Chapter 10 outlines the theory of Eisen-stein series. This foundational theory lies at the heart of many constructions(and analytic difficulties) in the theory of automorphic representations. How-ever, the book is written so that Chapter 10 can be skipped without muchloss of continuity.

With these comments in mind, a second course that does not cover simplerelative trace formulae might have an outline as follows:

Rankin-Selberg L-functions §11.2, §11.3, §11.4, §11.5, §11.6, §11.7, §11.9Langlands Functoriality §12.1, §12.2, §12.3, §12.4, §12.5, §12.6, §12.7Known Cases of Langlands Functoriality §13.3 §13.4, §13.5, §13.6, §13.7,§13.8

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Distinction and Period Integrals §14.2, §14.3, §14.4, §14.5The Cohomology of Locally Symmetric Spaces §15.2, §15.3, §15.4, §15.5,§15.6, §15.8

The last three chapters of the book together give a proof of a simple ver-sion of the (relative) trace formula. This implies the trace formula, twistedtrace formula, etc. in simple cases. In a course these three chapters shouldbe covered together, and they rely in particular on §14.3. If covering the sim-ple relative trace formula is a priority for the instructor, they may considerdropping Chapter 15 or postponing it to the end of the semester.

The spectral side of the trace formula is considered in §16.1 and §16.2. Thegeometric side is the focus of Chapter 17. Chapter 9 on the discreteness ofthe cuspidal spectrum and Chapter 17 are easily the most technical chaptersof the book. The first requires some analysis and the second some algebraicgeometry. In some sense they are the also the most important, because theygive complete proofs of results that are very difficult to extract from theliterature, if they exist at all. That said, just as much of Chapter 9 can betaken as a black box, the same is true of Chapter 17.

In a course it might make sense to treat §17.1 and §17.3 only superficiallyand focus on the definition of the local relative orbital integrals in §17.4 andthe definition of global relative orbital integrals in §17.7. Moving forwardthe key result is in Theorem 17.7.2. After this preparatory work the generalsimple relative trace formula in §18.2 follows easily. The sections §18.3, §18.4§18.5 provide specializations of the simple relative trace formula that areimportant for illustrating what the formula means. Readers interested inanalytic number theory should peruse §18.7, §18.8, and §18.9 to obtain acomplete derivation of the famous Kuznetsov formula in a form suitable forapplications to analytic number theory. After all the hard work required toderive (even simple versions) of relative trace formulae, it would be remissnot to discuss some applications. These are contained in Chapter 19. Themost important sections are §19.2 and §19.3, but to really understand thetheory in §19.3 the examples and clarifications in the remaining sections ofthat chapter are invaluable. Some suggestions for further work are containedin §19.6.

Most of the material in this book has appeared in one form or anotherin the literature. We have made an honest attempt to give accurate attribu-tions for both results and expository treatments. We would appreciate beinginformed of any serious errors we have made either in mathematics or in at-tributing results. We have made an effort to work as much as possible witharbitrary reductive groups over arbitrary global fields. In particular, we haveallowed global fields of positive characteristic. We have also refrained fromassuming that the affine algebraic groups under consideration are reductiveuntil necessary. This both emphasizes the results that really depend on thefact that G is reductive and develops the preliminaries necessary to under-stand, for example, unitary representations of nonreductive groups. This is

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not vacuous, as the theory of Jacobi forms already plays a role in automorphicforms [BS98].

The results on orbital integrals in Chapter 17 and the simple relativetrace formula in Chapter 18 are based partially on work of the second author[Hah09] and both authors [GH15]. However they have not appeared previ-ously in the generality in which we prove them here. In particular we treatarbitrary reductive groups over arbitrary global fields. Thus we allow positivecharacteristic.

In the interest of full disclosure, we should say a word about set theoreticfoundations. In §17.5 we make use of some more serious algebraic geometryinvolving sheaves of sets on a site, and this puts us up against the usualset-theoretic difficulties. To handle these we assume the universe axiom andassume (implicitly) that all our categories are small and contained in someuniverse. For more details we refer to [Poo17, Appendix A]. It is likely that thereliance on the universe axiom could be removed at the cost of complicatingthe definition of the sets of torsors we consider.

Durham, NC, Jayce R. GetzAugust, 2019 Heekyoung Hahn

Acknowledgements

The authors thank Francesc Castella, Andrew Fiori and Cameron Franc fortypsetting the first draft of the lecture notes that formed the germ of thisbook, and thank Brian Conrad, Tasho Kaletha, Minhyong Kim, Jason Polak,Leslie Saper, and Chad Schoen for many useful corrections and comments.Many people have been generous in answering questions as this book waswritten, including Avner Ash, David Ginzburg, Rahul Krishna, Erez Lapid,Freydoon Shahidi, Micha l Zydor, and any others who we may have forgotten.The authors give special thanks to Alex Youcis for his very careful readingof the first two chapters, which lead to the clarification of many details. Theauthors also thank Ken Ono for suggesting this book project. The first authorwishes to thank the National Science Foundation for support at various timesduring the preparation of this book. Part of this book was written while bothauthors were members at the Institute for Advanced Study in the spring of2018, we thank the institute for its hospitality and providing us with excellentworking conditions. Finally, we truly appreciate the encouragement of MadhiAsgari, Peter Sarnak and many graduate students.

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Contents

1 Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Extension and restriction of scalars . . . . . . . . . . . . . . . . . . . . . . . 81.5 Reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Root data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.9 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Adelic points of affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3 Relationship with restricted direct products . . . . . . . . . . . . . . . . 372.4 Hyperspecial subgroups and models . . . . . . . . . . . . . . . . . . . . . . . 392.5 Approximation in algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . 422.6 The adelic quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.7 Reduction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Automorphic Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1 Representations of locally compact groups . . . . . . . . . . . . . . . . . 533.2 Haar measures on locally compact groups . . . . . . . . . . . . . . . . . . 563.3 Convolution algebras of test functions . . . . . . . . . . . . . . . . . . . . . 583.4 Haar measures on local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Haar measures on the points of algebraic groups . . . . . . . . . . . . 603.6 Automorphic representations in the L2-sense . . . . . . . . . . . . . . . 623.7 Decomposition of representations . . . . . . . . . . . . . . . . . . . . . . . . . 643.8 The Fell topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

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3.9 Type I groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.10 Why affine groups? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Archimedean Representation Theory . . . . . . . . . . . . . . . . . . . . . 734.1 The passage between analysis and algebra . . . . . . . . . . . . . . . . . 734.2 Smooth vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Restriction to compact subgroups . . . . . . . . . . . . . . . . . . . . . . . . 774.4 (g,K)-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Hecke algebras with K-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.6 Infinitesimal characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.7 Classification of (g,K)-modules for GL2R . . . . . . . . . . . . . . . . . . 894.8 Matrix coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.9 The Langlands classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Representations of Totally Disconnected Groups . . . . . . . . . . 975.1 Totally disconnected groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 Smooth functions in td groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3 Smooth and admissible representations . . . . . . . . . . . . . . . . . . . . 1005.4 Contragredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.5 The unramified Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.6 Restricted tensor products of modules . . . . . . . . . . . . . . . . . . . . . 1065.7 Flath’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.1 Smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 Classical automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3 Adelic automorphic forms over number fields . . . . . . . . . . . . . . . 1166.4 Adelic automorphic forms over function fields . . . . . . . . . . . . . . 1196.5 The cuspidal subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.6 From modular forms to automorphic forms . . . . . . . . . . . . . . . . 1226.7 Ramanujan’s ∆-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Unramified Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.1 Unramified representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Satake isomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.3 The Langlands dual group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.4 Parabolic subgroups of L-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.5 The Satake isomorphism for quasi-split groups . . . . . . . . . . . . . 1367.6 The Principal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.7 Weak global L-packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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8 Nonarchimedean Representation Theory . . . . . . . . . . . . . . . . . . 1498.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.2 Parabolic induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.3 Jacquet modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.4 The Bernstein-Zelevinsky classification . . . . . . . . . . . . . . . . . . . . 1578.5 Traces, characters, coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.6 Parabolic descent of representations . . . . . . . . . . . . . . . . . . . . . . . 1638.7 Parabolic descent of orbital integrals . . . . . . . . . . . . . . . . . . . . . . 166Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

9 The Cuspidal Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739.2 The cuspidal subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.3 Deduction of the discreteness of the spectrum . . . . . . . . . . . . . . 1799.4 The function field case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1829.5 Estimates for the Fourier transform of the kernel . . . . . . . . . . . 1849.6 The basic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899.7 Rapidly decreasing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1909.8 Cuspidal automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

10 Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19510.1 Induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19510.2 Intertwining operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19710.3 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910.4 Decomposition of the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.5 Local preparation for isobaric representations . . . . . . . . . . . . . . 20110.6 Isobaric representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.7 A theorem of Moeglin and Waldspurger . . . . . . . . . . . . . . . . . . . 206Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

11 Rankin-Selberg L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21111.1 Paths to the construction of automorphic L-functions . . . . . . . 21111.2 Generic characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21211.3 Generic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21311.4 Formulae for Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . 21711.5 Local Rankin-Selberg L-functions . . . . . . . . . . . . . . . . . . . . . . . . . 21811.6 The unramified Rankin-Selberg L-functions . . . . . . . . . . . . . . . . 22111.7 Global Rankin-Selberg L-functions . . . . . . . . . . . . . . . . . . . . . . . . 22211.8 The nongeneric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22511.9 Converse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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12 Langlands Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22912.1 The Weil group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22912.2 The Weil-Deligne group and L-parameters . . . . . . . . . . . . . . . . . 23212.3 The archimedean Langlands correspondence . . . . . . . . . . . . . . . 23712.4 Local Langlands for GLn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23912.5 The local Langlands conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 24012.6 Global Langlands functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . 24412.7 Langlands L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24912.8 Algebraic representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

13 Known Cases of Global Langlands Functoriality . . . . . . . . . . 25513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25513.2 Parabolic induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25613.3 L-maps into general linear groups . . . . . . . . . . . . . . . . . . . . . . . . 25613.4 The strong Artin conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25713.5 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26013.6 The Langlands-Shahidi method . . . . . . . . . . . . . . . . . . . . . . . . . . 26213.7 Functoriality for the classical groups . . . . . . . . . . . . . . . . . . . . . . 26413.8 Endoscopic classification of representations . . . . . . . . . . . . . . . . 26913.9 The function field case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

14 Distinction and Period Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 27514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27514.2 Distinction in the local setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 27614.3 Global distinction and period integrals . . . . . . . . . . . . . . . . . . . . 27714.4 Spherical subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28114.5 Symmetric subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28314.6 Relationship with the endoscopic classification . . . . . . . . . . . . . 28614.7 Gan-Gross-Prasad type period integrals . . . . . . . . . . . . . . . . . . . 28814.8 Necessary conditions for distinction . . . . . . . . . . . . . . . . . . . . . . . 292Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

15 The Cohomology of Locally Symmetric Spaces . . . . . . . . . . . . 29515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29515.2 Locally symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29615.3 Local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29915.4 (g,K∞)-cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30215.5 The cohomology of Shimura manifolds . . . . . . . . . . . . . . . . . . . . 30315.6 The relation to distinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30515.7 More on (g,K∞)-cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30715.8 Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Contents xvii

16 Spectral Sides of Trace Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 31516.1 The automorphic kernel function . . . . . . . . . . . . . . . . . . . . . . . . . 31516.2 Relative traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31816.3 The full expansion of the automorphic kernel . . . . . . . . . . . . . . 32216.4 Functions with cuspidal image . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

17 Orbital Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32717.1 Group actions, orbits and stabilizers . . . . . . . . . . . . . . . . . . . . . . 32717.2 Luna’s slice theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33617.3 Local geometric classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34117.4 Local relative orbital integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34317.5 Torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34517.6 Adelic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34817.7 Global relative orbital integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 355Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

18 Simple Trace Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36318.1 A brief history of trace formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 36318.2 A general simple relative trace formula . . . . . . . . . . . . . . . . . . . . 36518.3 Products of subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37118.4 The simple trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37218.5 The simple twisted trace formula . . . . . . . . . . . . . . . . . . . . . . . . . 37318.6 A variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37518.7 The Petersson-Bruggeman-Kuznetsov formula . . . . . . . . . . . . . . 37618.8 Kloosterman integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37918.9 Sums of Whittaker coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 385Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

19 Applications of Trace Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 39119.1 Existence and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39119.2 The Weyl law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39219.3 The comparison strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39319.4 Jacquet-Langlands transfer and base change . . . . . . . . . . . . . . . 39619.5 Twisted endoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39919.6 The interplay of distinction and twisted endoscopy . . . . . . . . . . 400Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

A The Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407A.2 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408A.3 The dynamic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410A.4 Proof of Theorem A.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412A.5 An addendum in the hyperspecial case . . . . . . . . . . . . . . . . . . . . 414

xviii Contents

B Poisson Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415B.1 The standard additive characters . . . . . . . . . . . . . . . . . . . . . . . . . 415B.2 Local Schwartz spaces and Fourier transforms . . . . . . . . . . . . . . 417B.3 Global Schwartz spaces and Poisson summation . . . . . . . . . . . . 417Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Hints to selected exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Chapter 1

Algebraic Groups

Before Genesis, God gave thedevil free reign to construct muchof creation, but dealt withcertain matters himself, includingsemi-simple groups.

attributed to Chevalley byHarish-Chandra

Abstract In this chapter we briefly recall some of the basic notions relatedto affine algebraic groups. If the reader knows the definition of a reductivealgebraic group (which can be found in §1.5) then the rest of the chapter canprobably be skipped and then referred to later as needed.

1.1 Introduction

Someone beginning to learn the theory of automorphic representations doesnot need to internalize the content of this chapter. For the most part, onecan get by with much less. For example, one can think of GLn, which isreally an affine group scheme over the integers, as a formal notational devicealready familiar from an elementary course in algebra: for commutative ringsR the notation GLn(R) refers to the invertible matrices with coefficientsin R. Likewise, one knows that, e.g., Sp2n(R) is the group of matrices withcoefficients in R that fix the standard symplectic form on R2n. One can thinkof an reductive group G as some gadget of this type that assigns a group toevery ring of some type (say Q-algebras) and behaves like GLn. In fact, fora first reading of this book, whenever one sees the assumption that G is areductive group one could assume that G = GLn for some integer n and notlose a lot of the flavor.

1

2 1 Algebraic Groups

That said, at some point it is not a bad idea for any student of automor-phic representation theory (and really, any mathematician) to come to gripswith the “modern” (e.g. post 1960s) terminology of algebraic geometry andalgebraic groups. It is useful, beautiful, and despite its reputation not reallythat counterintuitive or difficult. Thus we will use it in this chapter, mostlyas a means of fixing notation and conventions, and make minor use of it laterin the book.

Much of the material in this chapter can be found in one form or another inthe references in algebraic groups [Bor91, Hum75, Spr09]. Unfortunately theywere all written in the archaic language of Weil used before Grothendieck’sprofound reimagining of the foundations of algebraic geometry. The mostcomplete reference is [DG74], but it is hard to penetrate if one doesn’t havesolid preparation in algebraic geometry. The reference [Wat79] is more acces-sible, but covers less. Fortunately J. Milne has reworked the three standardreferences in modern language. His book [Mil17] was instrumental in thepreparation of this chapter.

1.2 Affine schemes

Throughout this chapter we let k be a commutative Noetherian ring withidentity. Let Algk denote the category of commutative k-algebras with iden-tity. For a k algebra A, one obtains a functor

Spec(A) : Algk −→ Set (1.1)

R 7−→ HomAlgk(A,R).

Moreover, if φ : A→ B is a morphism of k-algebras, then we obtain for eachk-algebra R a map

Spec(B)(R) −→ Spec(A)(R) (1.2)

ϕ 7−→ ϕ φ .

This collection of maps is an example of what is known as a natural transfor-mation of functors. More precisely, to give a natural transformation X → Yof set-valued functors on Algk is the same as giving for each k-algebra R amap

X(R) −→ Y (R) (1.3)

such that for all morphisms of k-algebras R → R′ the following diagramcommutes:

1.2 Affine schemes 3

X(R) −−−−→ Y (R)y yX(R′) −−−−→ Y (R′).

(1.4)

One can check that (1.2) is a natural transformation of functors.

Definition 1.1. An affine scheme over k is a functor on the category of k-algebras of the form Spec(A). A morphism of affine schemes is a naturaltransformation of functors.

This definition gives us a category AffSchk of affine schemes over k,equipped with a contravariant anti-equivalence of categories

Spec : Algk −→ AffSchk

(see Exercise 1.3). In other words, the category of affine schemes over k isthe category of k-algebras “with the arrows reversed.” If k is understood weoften omit explicit mention of it. By way of terminology, if X is an affinescheme then

X(R)

is called its R-valued points.

Definition 1.2. A functor S : Algk −→ Set is representable by a ring Aif S = Spec(A). In this case we write

O(S) := A

and refer to it as the coordinate ring of S.

To ease comparison with other references, we note that we are definingschemes using their functors of points, whereas the usual approach is to defineschemes as a topological space with a sheaf of rings on it satisfying certaindesiderata and then associate a functor of points to the scheme. The twoapproaches are equivalent. The usual approach is desirable for many purposes,but the approach via the functor of points is more suitable for the study ofalgebraic groups. For more details on the usual approach and on the schemetheoretic concepts mentioned below we point the reader to [Har77, EH00,Mum99, GW10]. The functor of points approach is used in [DG74].

In order to work with affine schemes in practice one must usually imposeadditional restrictions on the representing ring A.

Definition 1.3. An affine scheme Spec(A) is of finite type if A is a finitelygenerated k-algebra.

Therefore Spec(A) is of finite type if and only if

A ∼= k[t1, . . . , tn]/(f1, . . . , fm)


for some (finite) set of indeterminates t1, . . . , tn and finite set of polynomialsf1, . . . , fm. An important example of a scheme of finite type is affine n-space:

Gna := Spec(k[t1, . . . , tn]).

This is also denoted by An, but we avoid this notation because we will usethe symbol A for the adeles (which will be defined in Definition 2.5).

Here are two other nice properties that the schemes of interest to us willoften enjoy:

Definition 1.4. An affine scheme X is reduced if O(X) has no nilpotentelements and irreducible if O(X) has a unique minimal prime ideal, or,equivalently, if its nilradical is prime.

It is important to note that an affine scheme can be both reduced and irre-ducible (Exercise 1.4).

Assume for the moment that k is a field. An affine scheme Spec(A) of finitetype over k is smooth if the coordinate ring A is formally smooth. Here ak-algebra A is said to be formally smooth if for every k-algebra B with idealI ≤ B of square zero and any k-algebra homomorphism A→ B/I one has amorphism A→ B such that the diagram

A //

!!

B/I

k

OO

// B

OO

commutes.For example, an affine scheme is smooth if it is isomorphic to an affine

scheme

Spec(k[t1, . . . , tn]/(f1, . . . , fn−d)

)(1.5)

where the ideal in k[t1, . . . , tn] generated by the fi and all the (n−d)×(n−d)minors of the matrix of derivatives (∂fi/∂xi) is the whole ring k[t1, . . . , tn].We have not and will not define open subschemes and Zariski covers ofschemes, but for those familiar with this language we remark that one canalways cover a smooth affine scheme by open affine subschemes of the form(1.5).

We now discuss closed subschemes.

Definition 1.5. A morphism of affine schemes

X −→ Y

is a closed immersion if the associated map O(Y ) −→ O(X) is surjective.

Remark 1.1. If X → Y is a closed immersion then X(R)→ Y (R) is injectivefor all k-algebras R (see Exercise 1.5).

1.2 Affine schemes 5

If X → Y is a closed immersion then O(X) ∼= O(Y )/I for some ideal Iof O(Y ). Motivated by this observation, one defines a closed subschemeof the scheme Y to be a scheme of the form Spec(O(Y )/I) for some idealI ⊆ O(Y ). Thus schemes of finite type over k are all closed subschemes ofGna for some n. One thinks of the closed subscheme k[t1, . . . , tn]/(f1, . . . , fm)as being the zero locus of f1, . . . , fm, although some care is required whenapplying this intuition when k is not an algebraically closed field, or even afield for that matter.

The definition of a closed subscheme can also be given from a topologi-cal perspective as we now explain. There is a topological space attached toSpec(A) which is also denoted by Spec(A). It is the collection of all primesp ⊆ A equipped with the Zariski topology. To define this topology, forevery subset S ⊆ A we let

V (S) = p ∈ Spec(A) : p ⊇ S.

One checks that the sets V (S) are closed under infinite intersections andfinite unions. Thus we can define the Zariski topology on Spec(A) to be thetopology with V (S) as its closed sets. A morphism of schemes induces amorphism of the associated topological spaces, and the image of an injectivemorphism

Spec(A/I) −→ Spec(A)

is V (I) (see Exercise 1.1).We close this section with the notion of fiber products. This is a simple

and useful method to create new schemes from existing ones. Let

X = SpecA, Y = SpecB, Z = SpecC

be k-schemes equipped with morphisms f : X → Y and g : Z → Y . We canthen form the fiber product

X ×Y Z := Spec(A⊗B C)

where A and C are regarded as B-algebras via the maps B → A and B → Cinduced by f and g, respectively. This scheme has the property that for k-algebras R one has

(X ×Y Z)(R) = X(R)×Y (R) Z(R)

where the fiber product on the right is that in the category of sets:

X(R)×Y (R) Z(R) := (x, z) ∈ X(R)× Z(R) : f(x) = g(z). (1.6)

By far the case that will be used the most frequently in the sequel is the so-called absolute product in the category of k-schemes. To define it note thatevery k-scheme X = SpecA comes equipped with a morphism X → Spec k,


namely the morphism induced by the ring homomorphism k → A that givesA the structure of a k-algebra. This morphism X → Spec k is called thestructure morphism. Thus for any pair of k-schemes X,Z we can form theabsolute product

X × Z := X ×k Z := X ×Spec k Z.

It follows from (1.6) that for k-algebras R,

(X × Z)(R) = X(R)× Z(R).

1.3 Group schemes

Definition 1.6. An affine group scheme over k is a functor

Algk −→ Group

representable by a k-algebra. A morphism of affine group schemes H → Gis a natural transformation of functors from H to G.

Here a natural transformation of functors is defined as in the previous section,though this time we require the associated maps

X(R) −→ Y (R)

to be group homomorphisms, not just maps of sets. It is clear that an affinegroup scheme over k is in particular an affine scheme over k. In this book wewill only be interested in affine group schemes (as opposed to elliptic curves,for instance), so we will often omit the word “affine.”

The most basic examples of affine group schemes are the additive andmultiplicative groups:

Example 1.1. The additive group Ga is the functor assigning to each k-algebra R its additive group,

Ga(R) := (R,+).

It is represented by the polynomial algebra k[x]:

Homk (k[x], R) = R.

Example 1.2. The multiplicative group Gm is the functor assigning to eacheach k-algebra R its multiplicative group,

Gm(R) = R×.

1.3 Group schemes 7

It is represented by k[x, y]/(xy − 1).

These affine group schemes are both abelian in the sense that their R-valued points are always abelian. The most basic example of a nonabeliangroup scheme is the general linear group GLn. It is the functor taking a k-algebra R to the group of n×n invertible matrices (xij) with coefficients xijin R. The representing ring is

k[xij ][y]/(det(xij) · y − 1).

Note that GL1 = Gm. If one wishes to be coordinate free, then for any finiterank free k-module V one can define

GLV (R) := R-module automorphisms V → V .

A choice of isomorphism V ∼= kn induces an isomorphism GLV ∼= GLn.We isolate a particularly important class of morphisms with the following

definition:

Definition 1.7. A representation of an affine group scheme G is amorphism of affine group schemes r : G→ GLV . It is faithful if it is a closedembedding.

Assume that k is a field and r is injective in the sense that r : G(R) →GLV (R) is injective for all k-algebras R. Then r is faithful [Mil17, Corollary3.35].

Definition 1.8. An affine group scheme G is said to be linear if it admitsa faithful representation G→ GLV for some V .

We will usually be concerned with linear group schemes. We shall see belowin Theorem 1.5.1 that this is not much loss of generality if k is a field.

Much of the basic theory of abstract groups goes through for affine al-gebraic groups over a field without essential change, but the proofs are farmore complicated. We will not develop the theory; we refer the reader to[Mil17] for details. However, it is useful to define injections and quotient inthis category:

Definition 1.9. Let k be a field. A morphism H → G of affine algebraicgroups over k is a monomorphism (resp. quotient map) if the corre-sponding map O(G)→ O(H) is surjective (resp. injective).

If H → G is a monomorphism then it is easy to verify that H(R)→ G(R)is injective for all k-algebras R (see Exercise 1.5). However, if H → G issurjective, then H(R) → G(R) need not be surjective for a given k-algebraR. An trivial example is the map Gm → Gm of group schemes over Q givenon points by x 7→ xn for an integer |n| > 1. It is true, however, that if H → Gis a quotient map then H(ksep)→ G(ksep) is surjective and a converse holdswhenever G is smooth [Mil17, Proposition 5.32].


Definition 1.10. Let G be a group scheme over k. A subgroup schemeH ≤ G is a subscheme of G such that H(R) ≤ G(R) is a subgroup for allk-algebras R.

It is known that if k is a field and H and G are affine groups schemes offinite type over with H ≤ G then H is automatically a closed subscheme ofG [Mil17, Proposition 1.41].

1.4 Extension and restriction of scalars

Let k → k′ be a homomorphism of rings. Given a k-algebra R, one obtainsa k′-algebra R ⊗k k′. Moreover, given a k′ algebra R′, one can view it as ak-algebra in the tautological manner. This gives rise to a pair of functors

Algk −→ Algk′ and Algk′ −→ Algk

known as base change and restriction of scalars, respectively. These func-tors are useful in that they allow us to change the base ring k.

For affine schemes we similarly have a base change functor

k′ : AffSchk −→ AffSchk′

given by Xk′(R′) = X(R′); the ring representing Xk is simply O(X) ⊗k k′.

In fact this is a special case of the fiber product construction of §1.2: in thenotation of that section one takes Y = Spec k and Z = Spec k′ and let usequipped with the map Z → Y induced by k → k′.

The functor in the opposite direction is a little more subtle. One can alwaysdefine a set valued functor

Resk′/kX′(R) := X ′(k′ ⊗k R)

called the Weil restriction of scalars. A priori this is just a set valuedfunctor on Algk, but if k′/k satisfies certain conditions then it is representable,and hence we obtain a functor

Resk′/k : AffSchk′ −→ AffSchk.

For example, it is enough to assume that k′/k is a field extension of finitedegree, or more generally that k′/k is finite and locally free [BLR90, Theorem4, §7.6].

Example 1.3. The Deligne torus is

S := ResC/RGL1.

1.5 Reductive groups 9

One has S(R) = C× and S(C) = C××C×. If V is a real vector space, then togive a representation S → GL(V ) is equivalent to giving a Hodge structureon V [Mil05].

Example 1.4. Let d be a square free integer and L = Q(√d). If we view L as

a two-dimensional vector-space over Q one obtains an embedding of L× intoGL2(Q). This can be refined into an embedding

ResL/Q(Gm) −→ GL2

of reductive groups over Q. One can choose the embedding so the R-valuedpoints of its image is (

a dbb a

)∈ GL2(R)

for Q-algebras R.

1.5 Reductive groups

We now assume that k is a field and let ksep ≤ k be a separable (resp. alge-braic) closure of k. Much of the theory simplifies in this case.

One has the following definition:

Definition 1.11. An (affine) algebraic group over k is an affine groupscheme of finite type over k.

In other words G is algebraic if G is represented by a quotient of k[x1, . . . , xn]for some n (namely O(G)). There is a natural action of G on O(G); if wethink of O(G) as functions on G, then the action is that induced by righttranslation. For any k-subspace of O(G) invariant under G we obtain a rep-resentation of G, and this representation can be chosen to be faithful. If onefills in the details of this discussion one obtains the following theorem [Wat79,§3.4]:

Theorem 1.5.1 An algebraic group admits a faithful representation. ut

Thus algebraic groups are linear. Another nice feature of algebraic groups isthat they are often smooth (see [Mil17, Proposition 1.26, Theorem 3.23]):

Theorem 1.5.2 An algebraic group G over a field k is smooth if k has char-acteristic zero or if Gk is reduced. ut

We will also require the notion of connectedness. An affine scheme is con-nected if the only idempotents in O(X) are 0 and 1. A group scheme isconnected if its underlying topological space is connected. Though it is nottrue for general affine schemes, a group scheme is connected if and only if Gkis irreducible [Wat79, §6.6]. Using this fact one shows that if k is a subfield


of C, then G is connected if and only if G(C) is connected in the analytictopology. Every algebraic group has a normal subgroup G ≤ G that is themaximal connected subgroup of G containing the identity. It is called theneutral component of G.

We now discuss the Jordan decomposition. Let Mn denote the scheme ofn by n matrices over k. One reference for the following is [Wat79, Chapter9].

Definition 1.12. Let k be a perfect field. An element x ∈Mn(k) is said to besemisimple if there exists g ∈ GLn(k) such that g−1xg is diagonal, nilpo-tent if there exists a positive integer n such that xn = 0, and unipotentif (x− I) is nilpotent. For an arbitrary linear group we say that an elementx ∈ G(k) is semisimple, (resp. unipotent) if r(x) is so for some faithfulrepresentation r : G→ GLV .

It turns out that if x ∈ G(k) has the property that r(x) is semisimple(resp. unipotent) for some faithful representation r then it is semisimple(resp. unipotent) for any faithful representation r.

Theorem 1.5.3 (Jordan decomposition) Let G be an algebraic groupover a perfect field k. Given x ∈ G(k) there exist unique xs, xu ∈ G(k)such that xs is semisimple, xu is unipotent x = xsxu = xuxs. ut

This leads us to the notion of a unipotent group. An algebraic groupover k is unipotent if every representation of G has a fixed vector. If G issmooth then G(k) consists of unipotent elements if and only if G is unipotent[Mil17, Corollary 14.12]. We remark that unipotent groups are always upper-triangularizable. More precisely, an algebraic group G is unipotent if and onlyif there exists a faithful representation r : G→ GLn such that the image of Gis a closed subgroup of the group of upper triangular matrices in GLn [Mil17,Theorem 14.5].

To define the weaker notion of a solvable group we recall that the derivedsubgroup Gder := DG of an algebraic group G is the intersection of allnormal subgroups N ≤ G such that G/N is commutative. If G is connectedthen Gder is connected. One has

Gder(k) = xyx−1y−1 : x, y ∈ G(k).

In analogy with the case of abstract groups, we define

DnG := D(Dn−1G)

inductively for n ≥ 1 and say that G is solvable if DnG is the trivial groupfor n sufficiently large.

Definition 1.13. Let G be a smooth algebraic group. The unipotent rad-ical Ru(G) of G is the maximal connected unipotent normal subgroup of G.

1.5 Reductive groups 11

The (solvable) radical is the maximal connected normal solvable subgroupof G.

We remark that since a unipotent group is always solvable we have Ru(G) ≤R(G).

Definition 1.14. A smooth connected algebraic group G is said to be re-ductive if Ru(Gk) = 1 and semisimple if R(Gk) = 1.

If k is a perfect field, then Ru(Gk) = Ru(G)k. However, this is false in generalfor nonperfect fields. For more details see [CGP10].

For n > 1 the group of upper triangular matrices in GLn is not reductive.The most basic example of a reductive group is GLn. It is not semisimple sinceZGLn = Ru(GLn). Here ZG denotes the center of the algebraic group G. Thesubgroup SLn ≤ GLn is the derived group of GLn and is semisimple (whichimplies reductive). In fact for any reductive group G one has ZG = Ru(G)[Mil17, Corollary 17.62] and Gder is semisimple [Mil17, Proposition 19.21].

Suppose that G is a reductive group. Then

G = ZGGder (1.7)

[Mil17, §21.f]. We note thatZG ∩Gder

is the (finite) center of Gder. For example, when G = GLn the derived sub-group is SLn and the center ZG consists of the diagonal matrices of determi-nant 1.

We warn the reader that one has to be a bit careful with the statementthatG = ZGG

der, and with similar statements involving products of algebraicgroups in a larger group below. The statement that G = ZGG

der means thatthe product map ZG ×Gder → G is a quotient map. In particular,

ZG(k)×Gder(k)→ G(k)

is surjective, butZG(k)×Gder(k)→ G(k)

need not be. This is false even for k = Q and G = GL2.We close this section by recalling the following theorem of Mostow [Mos56],

which states that we can always break an algebraic group in characteristiczero into a reductive and unipotent part:

Theorem 1.5.4 Let G be an algebraic group over a characteristic zero field.Then there is a subgroup M ≤ G such that M is reductive and

G = MRu(G).

All such subgroups M are conjugate under Ru(G). ut


The decomposition G = MRu(G) is called a Levi decomposition and M iscalled a Levi subgroup. The Levi decomposition is often written G = MN ,where N is the unipotent radical.

1.6 Lie Algebras

Now that we have defined reductive groups, we could ask for a classification ofthem, or more generally for a classification of morphisms H → G of reductivegroups. The first step in this process is to linearize the problem using objectsknown as Lie algebras. We will return to the question of classification in §1.7and Theorem 1.8.3 below.

Definition 1.15. Let k be a ring. A Lie algebra (over k) is a free k-moduleg of finite rank together with a bilinear pairing

[·, ·] : g× g −→ g

satisfying the following assumptions:

(a) [X,X] = 0 for all X ∈ g.(b) For X,Y, Z ∈ g,

[[X,Y ], Z] + [[Y,Z], X] + [[Z,X], Y ] = 0. (1.8)

Morphisms of Lie algebras are simply k-module maps preserving [·, ·].

The pairing [·, ·] is known as the Lie bracket of the Lie algebra g andthe identity (1.8) is known as the Jacobi identity. We remark that if R isa k-algebra then g ⊗k R inherits the structure of a Lie algebra over R in anatural manner.

Let LAGk denote the category of linear algebraic groups over k and letLieAlgk denote the category of Lie algebras over k. There exists a functor

Lie : LAGk −→ LieAlgk

defined by

Lie G = ker(G(k[t]/t2)→ G(k)).

From the definition just given it is not clear that LieG is a Lie algebra,or even a k-algebra, for that matter. At the moment it is only a set-valuedfunctor. There is one important special case where it is easy to deduce a Liealgebra structure. Let G = GLV for a free finite rank k-module V . In thiscase it is not hard to see that

LieG = IV +Xt : X ∈ Endk-linear(V )−→Endk-linear(V )

1.6 Lie Algebras 13

IV +Xt 7−→ X.

This Lie algebra is traditionally denoted glV , or simply gln when V = Rn.One defines the bracket via

[X,Y ] = X Y − Y X

and verifies that it satisfies the axioms for a Lie bracket.In order to deduce that Lie G is a Lie algebra in general we essentially just

embed Lie G in glV , but in a functorial way. The first step is to give Lie G ak-algebra structure. Let

ε : O(G) −→ k

be the coidentity, i.e. the homomorphism corresponding to the identity ele-ment of G(k).

Lemma 1.6.1 There is a canonical bijection

Lie G−→Homk-linear(ker(ε)/ ker(ε)2, k).

In particular Lie is a functor from LAGk to the category of free k-modulesof finite rank.

Using the k-algebra structure of g given above, one checks that for anyk-algebra R

LieG(R) := (Lie G)⊗k R

Proof. An element of LieG is a k-algebra homomorphism

ϕ : O(G) −→ k[t]/t2

such that the composite with the natural map k[t]/t2 → k is ε. Thus ϕmaps the ideal ker(ε) into t and hence factors through O(G)/ ker(ε). SinceO(G)/ ker(ε)2 = k ⊕ ker(ε)/ ker(ε)2 (see Exercise 1.9) we obtain a bijection

Lie G −→ Homk-linear(ker(ε)/ ker(ε)2, k)

ϕ 7−→ ϕ|ker(ε)/ ker(ε)2

as claimed. We leave it to the reader to check the functoriality assertion. ut

The group G always admits a representation

Ad : G −→ AutLieG.

To define it, note that for any k-algebra R the morphism R −→ R[t]/t2 givingR[t]/t2 its R-algebra structure gives rise to a map G(R) −→ G(R[t]/t2). Thusthe conjugation action of G(R) on itself gives rise to a conjugation action ofG(R) on G(R[t]/t2) which preserves LieG(R). This is the action denoted Adand it is known as the adjoint representation.


We finally give the reader a definition of the bracket on Lie G. For this wenote that by functoriality of Lie as a k-module valued functor the morphismAd gives rise to a morphism of free k-modules of finite rank

ad : Lie G −→ Lie AutLie G = glLieG.

We then define[A,X] = ad(A)(X).

One checks that this is indeed a Lie bracket and that when G = GLV thisrecovers the earlier definition of the bracket.

In practice, to compute the Lie algebra of a linear algebraic group G onejust chooses an embedding G −→ GLV and computes LieG in terms of theconditions that cut the group G from GLV . One then obtains the bracket forfree; it is just the restriction of the bracket on GLV .

Example 1.5. If G = SLn, the special linear group of matrices in GLn ofdeterminant 1, then

det(In +Xt) = 1 + tr(X)t (mod t2)

by Taylor expansion, so

sln := Lie SLn = X ∈ gln : tr(X) = 0.

Notice how the use of the Taylor expansion amounted to differentiatingsome equation once and then taking the zero term; this method of determin-ing Lie algebras works for at least all the classical groups, which are roughlysubgroups of GLV defined as the subgroup fixing various perfect pairings onV (see Exercise 1.9).

1.7 Tori

Throughout this section we assume that k is a field with separable closureksep and algebraic closure k. Set

Galk := Gal(ksep/k).

Definition 1.16. An algebraic torus or simply torus is a linear algebraicgroup T over k such that Tk

∼= Gnm for some n. The integer n is called therank of torus.

We remark that if T is a torus of rank n over k then there is a finiteseparable extension L/k over which T splits, in other words, TL ∼= Gnm[Con14, Lemma B.1.5].

1.7 Tori 15

Definition 1.17. A character of an algebraic group G is an element of

X∗(G) = Hom(G,Gm).

A co-character is an element of

X∗(G) = Hom(Gm, G).

For k-algebras k′ one usually abbreviates X∗(G)k′ := X∗(Gk′), etc. We warnthe reader that usually the notation X∗(G) is used for what we would callX∗(G)k. Therefore, if there is danger of confusion we will often write X∗(G)kfor X∗(G), though strictly speaking this is redundant.

One indication of the utility of the notion of characters is the followingtheorem [Wat79, §7.3]:

Theorem 1.7.1 The association

T 7−→ X∗(T )ksep

defines a contravariant equivalence of categories between the category of al-gebraic tori defined over k and finite rank Z-torsion free continuous Z[Galk]-modules. ut

Here a Z[Galk]-module V is continuous if the associated map

Galk −→ GLV (Z)

is continuous where we give Galk the usual profinite topology and GLV (Z)the discrete topology.

We now record a few examples of tori.

Example 1.6. We define a special orthogonal group SO2 ≤ GL2 by stipulatingthat for Q-algebras R one has

SO2(R) =(

a b−b a

): a, b ∈ R and a2 + b2 = 1

.

Over any field containing a square root of −1 one has

12 ( 1 i

i 1 )(a b−b a

) (1 −i−i 1

)=(a−bi

a+bi

). (1.9)

It follows that SO2Q(i)∼= Gm.

Example 1.7. Let L/k be a separable extension and let

NL/k : ResL/kGm −→ Gm

be the norm map; it is given on points by x 7→∏τ∈Homk(L,k) τ(x). Then the

kernel of NL/k is an algebraic torus. When L = Q(i) and k = Q this torus isisomorphic to the group SO2 of the previous example.


Example 1.8. If L/k is any (separable) field extension then ResL/k(Gm) is analgebraic torus. Moreover one can show that:

X∗(ResL/k(Gm))L '⊕τ

Zτ

where the summation runs over the embeddings τ : L → ksep of L into a k.In the special case where L/k is Galois the left hand side admits a naturalaction of Gal(L/k), and as a representation of Gal(L/k)

X∗(ResL/k(Gm))L ⊗Z C

is isomorphic to the induced representation IndGal(L/k)1 (1), where 1 denotes

the trivial representation (of the trivial group).

The examples above illustrate that though an algebraic torus T satisfiesTksep ∼= Gnm, it may not be the case that T ∼= Gnm. This motivates the followingdefinition:

Definition 1.18. An algebraic torus T over a field k is said to be split ifT ' Gnm over k or if equivalently X∗(T )k ∼= Zrank(T ). An algebraic torus Tis said to be anisotropic if X∗(T )k = id.

Any torus T can be decomposed as T = T aniT spl where T spl is the maximalsplit subtorus, T ani is the maximal anisotropic subtorus, and T ani ∩ T spl isfinite.

Definition 1.19. A torus T ≤ G is maximal if Tk is maximal among alltori of Gk.

Theorem 1.7.2 Every connected algebraic group admits a maximal torus.All maximal tori in Gk are conjugate under G(k). ut

Proof. In the generality we are considering the first result is due to Grothendieck(see, e.g. [Con14, Appendix A]). The second is [Bor91, IV.11.3]. ut

In view of the second assertion of Theorem 1.7.2, the rank of a maximal torusof G is an invariant of G; it is known as the rank of G.

For the remainder of this section G is a reductive group and T ≤ G is atorus. Let NG(T ) be the normalizer of T in G and ZG(T ) is the centralizer ofT in G. The torus T is maximal if and only if ZG(T ) = T [Mil17, Corollary17.84].

Definition 1.20. The Weyl Group of T in G is W (G,T ) := NG(T )/ZG(T ).

This brief definition hides some important subtleties as we now explain. Apossible reference is [Con14, §3]. The basic reference for quotients by reductivegroups in the affine case is [MFK94, §0]. The group schemes NG(T ), ZG(T )

1.7 Tori 17

are algebraic subgroups of G and W (G,T ) the GIT quotient of NG(T ) byZG(T ). This is explained in some detail in §17.1.

It turns out that W (G,T ) is again a smooth (even etale) group scheme offinite type over k. Moreover one has

W (G,T )(ksep) = NG(T )(ksep)/ZG(T )(ksep). (1.10)

To describe W (G,T )(R) for general k-algebras R is involved, but at leastfor subfields k ≤ L ≤ ksep we can describe it as follows. Since NG(T ) andZG(T ) are schemes over k the set NG(T )(k) comes (equipped with an actionof Galk preserving ZG(T )(ksep); hence we also obtain an action of Galk onW (G,T )(ksep). Then

W (G,T )(L) = W (G,T )(ksep)Gal(ksep/L).

A nice discussion of this point, which is a special case of what is knownas Galois descent, is contained in [Mum99, §II.4]. We wish to discuss theseconcepts in the special case where G = GLn. Before beginning we record thefollowing definition:

Definition 1.21. The group G is said to be split if there exists a maximaltorus of G that is split.

If T ≤ GLn is the torus of diagonal matrices then T is a split max-imal torus, and hence GLn is split. For any field k′/k the Weyl groupW (T,GLn)(k′) can be identified with the group of permutation matrices andhence in this case W (T,GLn)(k′) is Sn, the symmetric group on n letters.More generally, if T ≤ G is a split maximal torus one has

W (G,T )(k′) = NG(k′)/ZG(k′) (1.11)

[Mil17, Proposition 21.1].In general, if L/k is an etale k-algebra of rank n (for example a field

extension of degree n) then choosing a basis for k we obtain an embedding

ResL/kGm −→ GLn.

In particular if L/k is Galois then W (GLn, T )(k) ∼= Gal(L/k). Every maximaltorus in GLn arises in this manner for some L/k (compare Exercise 1.11).

After spending all this time discussing tori we give one important moti-vation for introducing them. Suppose that G is a split reductive group andthat T ≤ G is a split maximal torus. Given any representation

r : G −→ GLV

it follows from the fact that T is abelian and reductive that we can decomposeV as


V = ⊕α∈X∗(T )Vα (1.12)

where Vα = v ∈ V : r(t).v = α(t).v. The α occurring in this decompositionare known as weights and the Vα are known as weight spaces.

One has the following basic theorem:

Theorem 1.7.3 The collection of weights α together with the weight multi-plicities dimVα determine the representation V up to isomorphism. ut

It is not hard to see that the set of weights is invariant under the naturalaction of W (T,G)(k) on X∗(T ) which leaves the multiplicities of the weightspaces dimVα fixed.

In fact, there is an partial ordering one can put on the space X∗(T ) suchthat each representation of G admits a highest weight with respect to the or-dering, unique up to the action of W (T,G)(k). Moreover, one can characterizethe set of highest weights that can occur. This is known as Cartan-Weylhighest weight theory.

One key fact one can take from this discussion is that the representa-tion theory of G is determined by the restriction of the representation to asufficiently large subgroup (in this case a maximal torus). Happily, the rep-resentation theory of a maximal torus is fairly simple and can be studied vialinear algebra. The idea of studying representations by studying their restric-tion to subgroups is an incredibly useful one; it will come up in various guisesthroughout this book.

1.8 Root data

Let G be a split reductive group over a perfect field k and let T ≤ G be asplit maximal torus. Our next goal is to associate to such a pair (G,T ) a rootdatum Ψ(G,T ) = (X,Y, Φ, Φ∨). The root datum is a refinement of the relatednotion of a root system (also defined below) that Demazure introduced tosystematically keep track of the central torus of G [DG74, Expose XXII]. Theroot datum characterizes G, and in fact Demazure proves that it characterizes(split) G even in the case where k is replaced by Z. This in some sensecompleted work of Chevalley who proved the analogous statement for semi-simple groups over fields [Che58].

Let g denote the Lie algebra of G. As we saw in §1.6 one has an adjointrepresentation

Ad : G −→ GL(g). (1.13)

For example, when G = GLn this is the usual action of GLn on the space ofn× n matrices gln by conjugation.

Since Ad(T ) consists of commuting semi-simple elements the action of Ton g is diagonalizable. For a character α ∈ X∗(T ), let

1.8 Root data 19

gα := X ∈ g | Ad(t)X = α(t)X for all t ∈ T (k). (1.14)

Definition 1.22. The nonzero α ∈ X∗(T ) such that gα 6= 0 are called theroots of T in G. We let Φ(G,T ) be the (finite) set of all such roots α, andcall the corresponding gα root spaces.

Just as in (1.12) we have a decomposition

g = t⊕⊕

α∈Φ(G,T )

gα

where t := LieT . It turns out that each of the root spaces gα are one dimen-sional.

One turns the set of roots Φ(G,T ) into a combinatorial gadget as follows.Let

V := 〈Φ(G,T )〉 ⊗Z R,

where 〈Φ(G,T )〉 ⊂ X∗(T )(k) denotes the (Z-linear) span of Φ(G,T ). The pair(Φ(G,T ), V ) satisfies remarkable symmetry properties that are axiomatizedin the following:

Definition 1.23. Let V be a finite dimensional R-vector space, and Φ asubset of V . We say that (Φ, V ) is a root system if the following threeconditions are satisfied:

(a) Φ is finite, does not contain 0, and spans V ;(b) For each α ∈ Φ there exists a reflection sα relative to α (i.e. an involution

of V with sα(α) = −α and restricting to the identity on a subspace of Vof codimension 1) such that sα(Φ) = Φ;

(c) For every α, β ∈ Φ, sα(β)− β is an integer multiple of α.

A root system (Φ, V ) is said to be of rank dimRV , and is said to be reducedif for each α ∈ Φ, ±α are the only multiples of α in Φ.

We will not need the notion until §1.9, but a subset

∆ ⊂ Φ (1.15)

is a base if it is a basis of V and each α ∈ Φ can be uniquely expressed as

∑i=1

ciαi

where the ci’s are all integers that have the same sign. We define

Φ+ ⊂ Φ (resp. Φ− ⊂ Φ)

to be the set of roots expressible as a sum of positive (resp. negative) linearcombinations of the elements in the base. Then Φ = Φ+ q Φ−. One can also


define a system of positive roots a priori and use it to define bases; see, forexample, [Hum78, §10].

The Weyl group of (Φ, V ) is the subgroup of GL(V ) generated by thereflections sα:

W (Φ, V ) := 〈sα : α ∈ Φ〉 ⊆ GL(V ).

The following is [Mil17, Corollary 21.38]:

Proposition 1.8.1 If (Φ, V ) is the root system associated with the split max-imal torus T ≤ G then (Φ, V ) is reduced and

W (Φ, V ) ∼= W (G,T )(k).

ut

If Φ ⊂ V is a finite spanning set not containing 0 then (Φ, V ) is a rootsystem if and only if there is a map

Φ −→ V ∨ := Hom(V,R)

α 7−→ α∨

such that α∨(α) = 2, α∨(Φ) ⊂ Z, and the reflection

sα(x) := α∨(x)α

preserves Φ. If such a map Φ → V ∨ exists it is unique [Mil17, C.14]. Theimage of the map Φ→ V ∨ is denoted Φ∨ and called the set of coroots.

Lemma 1.8.2 The pair (Φ∨, V ∨) is a root system. ut

A fundamental result (Theorem 1.8.3) is that the quadruple

Ψ(G,T ) = (X∗(T ), X∗(T ), Φ, Φ∨)

attached to T ⊂ G contains enough information to characterize G, at leastover k. To be more precise suppose that we have a quadruple (X,Y, Φ, Φ∨)consisting of a pair of free abelian groups X, Y with a perfect pairing 〈 , 〉 :X × Y → Z and a bijective correspondence

Φ −→ Φ∨

α 7−→ α∨.

For each α ∈ Φ we let sα : X → X and sα∨ : Y → Y be the endomorphismgiven by

sα(x) := x− 〈x, α∨〉αsα∨(y) := y − 〈y, α〉α∨.

Definition 1.24. The quadruple (X,Y, Φ, Φ∨) is a root datum if

1.8 Root data 21

(a) 〈α, α∨〉 = 2, and(b) if for each α ∈ Φ, then sα(Φ) ⊂ Φ, and the group 〈sα | α ∈ Φ〉 generated

by sα is finite.

We say that a root datum is reduced if α ∈ Φ only if 2α /∈ Φ.

An isomorphism of root data (X,Y, Φ, Φ∨)∼−→ (X ′, Y ′, Φ′, Φ′∨) is a group

isomorphism X∼−→ X ′ sending Φ to Φ′ whose dual Y ′

∼−→ Y sends Φ′∨ to Φ∨.We note that an isomorphism of reductive groups over k gives rise to an iso-morphism of their underlying root data. Indeed, if G→ G′ is an isomorphismof reductive groups over k, then upon precomposing the isomorphism with aninner automorphism of G and postcomposing with an inner automorphism ofG′ we can assume that the isomorphism maps a given maximal torus T of Gto a given maximal torus T ′ of G′, and hence induces an isomorphism of theunderlying root data. The following result not only implies that every rootdatum comes from an algebraic group, it implies that every isomorphism ofroot data arises in this manner:

Theorem 1.8.3 (Chevalley, Demazure) Assume k = k. The mapisomorphism classes ofreductive groups over k

−→

isomorphism classes of

reduced root data

G 7−→ Ψ(G,T )

is bijective. Moreover, every isomorphism of root data Ψ(G,T )→Ψ(G′, T ′) isinduced by an isomorphism G→G′ sending T to T ′, unique up to the conju-gation actions of T (k) and T ′(k). ut

This theorem tells us that if we can classify root data, then we can classifysplit reductive groups. In fact, Killing obtained a classification of the under-lying root systems (up to some mistakes) in [Kil88, Kil90]. Killing’s work wasrevisited and corrected in E. Cartan’s thesis.

If (X,Y, Φ, Φ∨) is a root datum, then so is (Y,X,Φ∨, Φ). The associated

reductive algebraic group over C is denoted G and is called the complexdual of G. We note that there is an isomorphism

W (G,T )k−→W (G, T )(C) (1.16)

sα 7−→ sα∨ .

The complex dual G is a key component in the definition of Langlands dualgroup (see §7.3 ).

One might ask if one could define in a natural way a morphism of root data,and thereby use root data to classify morphisms between reductive groups.If such a definition exists, we do not know it, and there are reasons to bepessimistic. However, it is the case that a great deal of information aboutmorphisms between reductive groups can be deduced by considering root


data. A systematic account of this for classical groups is given in Dynkin’swork [Dyn52].

We now record arguably the most basic example of a root datum:

Example 1.9. Let G = GLn. The group of diagonal matrices

T (R) :=

( t1. . .

tn

)| ti ∈ R×

is a maximal torus in G. The groups of characters and of cocharacters of Tare both isomorphic to Zn via

(k1, . . . , kn) 7−→(( t1

. . .tn

)7→ tk1

1 · · · tknn)

and

(k1, . . . , kn) 7−→

(t 7→

(tk1

. . .

tkn

)),

respectively. Note that with these identifications, the natural pairing 〈 , 〉 :X∗(T )×X∗(T )→ Z corresponds to the standard “inner product” in Zn. Theroots of G relative to T are the characters

eij :

( t1. . .

tn

)7→ tit

−1j

for every pair of integers (i, j), 1 ≤ i, j ≤ n with i 6= j, and the correspondingroot spaces gln,eij are the linear span of the n×n matrix with all entries zeroexcept the (i, j)-th component. The coroot e∨ij associated with eij is the map

sending t to the diagonal matrix with t in the ith entry and t−1 in the jthentry and 1 in all other entries.

1.9 Parabolic subgroups

We assume in this subsection that G is a reductive group over a perfect fieldk with algebraic closure k.

Definition 1.25. A closed subgroup B ≤ G is a Borel subgroup if Bk isa maximal smooth connected solvable subgroup of Gk. A smooth subgroupP ≤ G is a parabolic subgroup if Pk contains a Borel subgroup of Gk.

The group Gk trivially has Borel subgroups, and hence G always has atleast one parabolic subgroup, namely G itself. A proper parabolic subgroupis a parabolic subgroup of G not equal to G. In general, a reductive groupneed not have proper parabolic subgroups. For example, if D is a division

1.9 Parabolic subgroups 23

algebra with center k of dimension bigger than 1 over k and G is the algebraicgroup defined by

G(R) = (D ⊗k R)×, (1.17)

then G does not have a proper parabolic subgroup (Exercise 1.12).It is useful to isolate a weakening of the notion of a split reductive group:

Definition 1.26. A reductive group G is said to be quasi split if it containsa Borel subgroup.

Note that G is split only if it is quasi split, but that the converse is nottrue. Indeed, take G to be the unitary group U(1, 1) over the real numbersdefined by

G(R) =g ∈ GL2(C⊗R R) : gt

( −1−1

)g =

( −1−1

)where the bar denotes complex conjugation. Then the subgroup of uppertriangular matrices in G is a Borel subgroup of G. Thus G is quasi split. Itis not, however, split.

From the optic of finite-dimensional representation theory the behavior ofa split reductive group is essentially as simple as a group over an algebraicallyclosed field (compare §1.7). Quasi-split groups are a little more technical tohandle, but the existence of the Borel subgroup makes the theory not muchmore difficult. The fact that a general reductive group does not have a Borelsubgroup (over the base field) creates more substantial problems. In thiscase one has to do with a minimal parabolic subgroup. Of course, in thequasi-split case, a minimal parabolic subgroup is simply a Borel subgroup.

Basic facts about parabolic subgroups come up constantly in the theory ofautomorphic representations. For example, they are used to understand thestructure of adelic quotients at infinity (see §2.7) and are used to describethe representation theory of a reductive group inductively (see §4.9, §8.2 andChapter 10). Thus we record some of the basic structural facts about the setof parabolic subgroups of G in this section. One reference is [Bor91, §20-21].

It is convenient to start with split tori in G. If there is no split torus con-tained in G then G is said to be anisotropic. In this case the only parabolicsubgroup of G is G itself. Otherwise G is said to be isotropic. If G is isotropicthen there exists a maximal split torus T ≤ G unique up to conjugation. Justas in the case when G is split (discussed in §1.8), we can decompose g underthe adjoint action (1.13) into eigenspaces under T :

g := m⊕⊕

α∈Φ(G,T )

gα (1.18)

where gα is defined as in (1.14). Here Φ(G,T ) is set of nonzero weights occur-ring in the decomposition above, and m is the 0 eigenspace. The set Φ(G,T )is usually referred to as the set of relative roots (or more precisely, roots


relative to k as opposed to k), but we will avoid this terminology because wewill later discuss relative trace formulae, which have nothing to do with thisnotion of relative. The set Φ(G,T ) ⊂ X∗(T )⊗Z R is a root system, but it isnot in general reduced.

Just as all the maximal split tori in G are conjugate under G(k), theminimal parabolic subgroups are also conjugate under G(k). Moreover, everyminimal parabolic subgroup contains the centralizer Z(T ) of a maximal splittorus T ≤ G. Thus to describe all minimal parabolic subgroups it suffices tofix a maximal split torus T ≤ G and describe all minimal parabolic subgroupscontaining Z(T ). Let ∆ ≤ Φ(G,T ) be a base. Then we can consider the setof positive roots Φ+ with respect to ∆. Then there is a unique minimalparabolic subgroup P0 ≥ Z(T ) whose unipotent radical N0 ≤ P0 satisfies

LieN0 =⊕α∈Φ+

gα.

Theorem 1.9.1 The correspondence described above defines a bijection

bases ∆ ⊆ Φ(G,T ) −→P0 ≥ Z(T ) : P0 minimal parabolic subgroups . ut

If we fix a minimal parabolic subgroup P0 then the parabolic subgroupscontaining P0 are called standard. Every parabolic subgroup is conjugateunder G(k) to a unique standard parabolic subgroup, and two standardparabolic subgroups are G(k)-conjugate if and only if they are equal. Thusto describe all parabolic subgroups of G it suffices to fix a minimal parabolicsubgroup P0 of G and describe all standard parabolic subgroups (with respectto P0).

To accomplish this, let J ≤ ∆ be a subset. Let

Φ(J) := ZJ ∩ Φ

There is a unique parabolic subgroup PJ ≥ P0 with unipotent radical NJsuch

LieNJ =⊕

α∈Φ+−(Φ(J)∩Φ+)

gα.

For the proof of the following theorem see [Bor91, Proposition 21.12]:

Theorem 1.9.2 There is a bijective correspondence

J ⊂ ∆−→standard parabolic subgroups of GJ 7−→ PJ .

The two bijections in theorems 1.9.1 and 1.9.2 allow us to define the no-tion of a parabolic subgroup opposite to a given parabolic subgroup. Moreprecisely, suppose we are given a standard parabolic subgroup P ≤ G with


unipotent radical N containing the centralizer of a maximal split torus T ofG: Z(T ) ≤ P ≤ G. Then one has an opposite parabolic P− constructed asfollows: If P = P0 is minimal, then we take P−0 to be the minimal parabolicsubgroup attached to the base

−∆ := α ∈ Φ : −α ∈ ∆.

If P = PJ , then we define P− to be the parabolic subgroup containing theminimal parabolic subgroup P−0 that is attached to the subset

−J := α ∈ Φ : −α ∈ J ⊆ −∆.

It turns out thatP ∩ P− = M,

the Levi subgroup of P . The unipotent radical of P− is usually denoted N−.

Example 1.10. The subgroup B ≤ GLn of upper triangular matrices is aBorel subgroup. Throughout this book when we speak of standard parabolicsubgroups in GLn we will mean parabolic subgroups that are standard withrespect to this choice of Borel subgroup. The base of Φ(G,T ) and set ofpositive roots associated to B are

∆ : = ei,i+1 : 1 ≤ i ≤ n− 1Φ+ : = ei,j : i < j

respectively. In particular there is a bijection

subsets of ∆−→n1, . . . , nd ∈ Z>0 :

d∑i=1

ni = n

where n1 is the first index such that en1,n1+1 is not an element of the subset,n2 is the second index such that en2,n2+1 is not an element of the subset, etc.

Unwinding these equivalences, we see that the standard parabolic sub-groups of GLn correspond bijectively to ordered tuples of positive integersn1, . . . , nd such that

∑di=1 ni = n. The corresponding parabolic subgroup is

the product of B and the block diagonal matrices of the form

M(R) :=

x1

. . .

xd

: xi ∈ GLni(R)

We refer to this parabolic subgroup as the (standard) parabolic subgroup oftype (n1, . . . , nd). The subgroup M(R) is referred to as the standard Levisubgroup of type (n1, . . . , nd).


Finally, it is sometimes useful to generalize the notion of roots still further.If T is any split torus in G, we can decompose g into eigenspaces as before.We let Φ(G,T ) be the set of nonzero eigenspaces. Suppose that P ≤ G is aparabolic subgroup and that T ≤ P is a maximal split torus in the centerof P . In this case, the set of roots need not be a root system [Kna86, §V.5].However, we can still define a partition Φ+ q Φ− = Φ(G,T ) such that Φ+ isthe set of roots contained in the unipotent radical of P . This is the set ofpositive roots defined by P . We will also require the notion of a positiveWeyl chamber in this context. Let Λ ⊂ X∗(T )⊗Z R denote the union of thehyperplanes on which an element of Φ(G,T ) vanishes. A Weyl chamber isa connected component of X∗(T )⊗ZR\Λ; they are permuted transitively byW (G,T ). The positive Weyl chamber attached to P is the unique Weylchamber whose elements are positive R-linear combinations of the positiveroots.

Exercises

1.1. Prove that a morphism of affine schemes induces a continuous map ofthe underlying topological spaces. Prove that if

Spec(A/I) −→ Spec(A)

is an embedding then the image of the topological space of Spec(A/I) inSpec(A) is V (I).

1.2. Assume that X = Spec(C[t1, . . . , tn]/(f1, . . . , fn−d)) is a smooth schemeover the complex numbers. Show that X(C) may be identified with a smoothcomplex manifold, namely the zero locus of the fi.

1.3. Let X and Y be affine schemes over the ring k. Prove that giving anatural transformation of functors X → Y is equivalent to giving a morphismof k-algebras O(Y ) → O(Y ). Deduce that Spec induces an equivalence ofcategories from the category of k-algebras to the category of affine k-schemes.

1.4. Give examples of affine schemes of finite type over C that are nonreduced,reducible, and reduced and irreducible.

1.5. Prove that if X → Y is a closed immersion of affine schemes over k thenX(R)→ Y (R) is injective for all k-algebras R.

1.6. Let X,Y, Z be affine k-schemes equipped with morphisms f : X → Yand g : Z → Y . Prove that

X ×Y Z(R) = X(R)×Y (R) ×Z(R)

for k-algebras R.


1.7. Let G be an affine scheme over k. Let Id : G −→ G denote the identitymorphism, let

pi : G×G −→ G

denote the two projections, and let

diag : G −→ G×G

denote the diagonal map. We say that G is a group object in the categoryof k-schemes if there exist morphisms of k-schemes

m : G×G −→ G

e : Spec k −→ G

inv : G −→ G

such that the following diagrams commute:

G×G×G

m×Id

Id×m // G

m

G

m // G

Spec k ×G e×Id //

''

G×G

m

G× Spec kId×eoo

wwG

G

(inv,Id) // G×G

m

G(Id,inv)oo

Spec k

e // G Spec keoo

Prove that G is a group scheme if and only if it is a group object in thecategory of k schemes.

1.8. For R-algebras R define

Un(R) := g ∈ GLn(C⊗R R) : ggt = In

where the bar denotes the action of complex conjugation. Show that Un isan algebraic group over R, that Un(R) is compact, and that

UnC ∼= GLnC.

The group Un is called the definite unitary group over R.

1.9. Prove that


O(G)/ ker(ε)2 = k ⊕ ker(ε)/ ker(ε)2

as k-algebras.

1.10. Assume k is a field of characteristic not 2 and J ∈ GLn(k) is an invert-ible symmetric or skew-symmetric matrix. For k algebra R, if

G(R) := g ∈ GLn(R) : gtJg = J

thenLie G = X ∈ gln : XtJ + JX = 0.

1.11. Let k be a perfect field. Prove that the set of conjugacy classes ofmaximal tori T ≤ GLn/k is in natural bijection with etale k-algebras ofdegree n.

1.12. If D is a division algebra over a field k and G is the algebraic groupdefined by

G(R) = (D ⊗k R)×,

then G has no proper parabolic subgroups (over k).

1.13. Show that for a perfect field k the set of GLn(k)-conjugacy classes ofparabolic subgroups of GLn is in bijective correspondence with the partitionsof n. Here a partition of n is a nonincreasing set of positive integers whosesum is n.

1.14. Let H ≤ G be affine algebraic groups over a perfect field k. Show thatthere is an exact sequence of pointed sets

1→ H(k)→ G(k)→ G/H(k)→ ker(H1(k,H)→ H1(k,G)

).

Chapter 2

Adeles

Adeles make life possible.

J. Arthur

Abstract Automorphic representations are defined as certain representationsof the adelic points of reductive algebraic groups. In this chapter, we introducethe adele ring and recall the basic properties of the adelic points of reductivegroups. They play a central role in the arithmetic theory of algebraic groupsand the class field theory.

2.1 Adeles

The arithmetic objects of interest in this book are constructed using globalfields and their adele rings. We recall the construction of the adeles in thissection; references include [CF86, Neu99, RV99].

Definition 2.1. A global field F is a field which is a finite extension of Qor of Fp(t) for some prime p. Global fields over Q are called number fieldswhile global fields over Fp(t) are called function fields.

To each global field F one can associate an adele ring AF . Before definingthis ring, we recall the related notions of a valuation (or a finite place) of aglobal field.

Definition 2.2. Let F be a field. A (nonarchimedean) valuation on F is amap

v : F −→ R ∪∞

such that for all a, b ∈ F , v satisfies the following:

(a) v(a) =∞ if and only if a = 0.(b) v(a) + v(b) = v(ab).

29

30 2 Adeles

(c) v(a+ b) ≥ min(v(a), v(b)).

These axioms are designed so that if one picks 0 < α < 1 then

| · |v : F −→ R≥0 (2.1)

x 7−→ αv(x)

is a nonarchimedean absolute value on F in the sense of the following defini-tion:

Definition 2.3. Let F be a field. An absolute value

| · |v : F −→ R≥0

is a function satisfying the following axioms:

(a) |a|v = 0 if and only if a = 0.(b) |ab|v = |a|v|b|v.(c) |a+ b|v ≤ |a|v + |b|v.

It is nonarchimedean if it satisfies the following strengthening of (c):

(c’)|a+ b|v ≤ max(|a|v, |b|v).

If | · |v satisfies (c) but not (c’) we say | · |v is archimedean.

We point out that (c’) implies (c), but not conversely. We often implicitlyexclude the trivial absolute value given by |a|v = 1 for all a ∈ F×.

An absolute value | · |v induces a metric on F known as the v-adic metric.The completion of F with respect to this metric is denoted Fv. This comple-tion is a local field, that is, a field equipped with a nontrivial absolute valuethat is locally compact with respect to the induced metric. All local fieldsarise as the completion of some global field [Lor08, §25, Theorem 2].

Definition 2.4. A place of a global field F is an equivalence class of ab-solute values, where two absolute values are said to be equivalent if theyinduce the same topology on F . A place is (non)archimedean if it consists of(non)archimedean absolute values.

We now describe these places and fix representative absolute values in eachplace. These representative absolute values are said to be the normalizedabsolute values. The places of a global field F fall into two categories: thefinite and infinite places.

The finite places are in bijection with the prime ideals of OF . The placev associated to a prime p of OF is the equivalence class of an absolute valueattached to the valuation

v(x) := maxk ∈ Z : x ∈ pkOF .

The normalized absolute value in this equivalence class is

2.1 Adeles 31

|x|v = q−v(x)v

where qv := |OF /p|. These valuations are all nonarchimedean.The infinite places of a number field are all archimedean. They are indexed

by embeddings τ : F → C up to conjugation; the associated normalizedabsolute value is

|x|v :=

|τ(x)| if τ(F ) ≤ Rτ(x)τ(x) if τ(F ) 6≤ R.

Here on the right the | · | denotes the usual absolute value on R and thebar denotes complex conjugation. Notice that in the complex case (i.e. whereτ(F ) 6≤ R) this is the square of the usual absolute value.

The infinite places of a function field are those attached to extensions ofthe absolute value ∣∣∣∣f(t)

g(t)

∣∣∣∣∞

:= pdegg−degf

on Fp(t). If Fv is the completion of F with respect to such a valuation, theassociated normalized absolute value is

|x|v := |NFv/Fp((t))((x))|1/[Fv :F((t))]∞ .

These valuations are archimedean, in contrast to the archimedean case.Henceforth we will always take | · |v to be the unique normalized absolutevalue attached to v.

We note that the set of infinite places of F is often denoted by ∞, andone often writes v|∞ or v - ∞ as shorthand for “v is an infinite place of F”and “v is a finite place of F ,” respectively.

For any place v, write Fv for the completion of F with respect to somechoice of absolute value associated to v. Since the absolute values correspond-ing to v all induce the same topology on F , the field Fv is independent ofthis choice. If v is finite, then the ring of integers of Fv is

OFv = x ∈ Fv : |x|v ≤ 1;

it is a local ring with a unique maximal ideal

$vOFv := x ∈ Fv : |x|v < 1.

Here $v is a uniformizer for Fv, that is, a generator for the maximal ideal ofOFv .

Let us make these constructions explicit when F = Q. If p ∈ Z is a prime,then completing Q at the p-adic absolute value gives rise to the local fieldQp. Its ring of integers is Zp and the maximal ideal is pZp. The residue fieldis Zp/pZp ∼= Z/pZ ∼= Fp, so the normalized absolute is just the usual p-adicnorm. There is only one infinite place of Q, denoted ∞, and Q∞ ∼= R. Thenormalized archimedean norm | · |∞ is the usual Euclidean norm on R.

32 2 Adeles

The normalization allows one to prove that the following product for-mula holds:

Proposition 2.1.1 For x ∈ F× one has∏v

|x|v = 1

where the product is over all places v of F . ut

Definition 2.5. Let F be a global field. The ring of adeles of F , denotedby AF , is the restricted direct product of the completions Fv with respect tothe rings of integers OFv . In other words,

AF =

(xv) ∈

∏v

Fv : xv ∈ OFv for all but finitely many places v

. (2.2)

The restricted product is usually denoted by a prime:

AF =∏′

v

Fv.

Note that AF is a subring of the full product∏v Fv. If S is a finite set of

places of F then we write

ASF =∏′

v 6∈S

Fv

: =

(xv) ∈∏v 6∈S

Fv : xv ∈ OFv for all but finitely many places v

,

FS := AF,S =∏v∈S

Fv.

Thus we may identify FS × ASF = AF . We write ∞ for the set of infiniteplaces of F . Instead of writing v ∈ ∞ it is traditional to write v | ∞ andsimilarly v -∞ instead of v 6∈ ∞. For any finite sets of places S′ ⊆ S we set

FS′

S :=∏

v∈S−S′Fv. (2.3)

We endow AF with the restricted product topology. This is definedby stipulating that open sets are sets of the form

US ×∏v 6∈S

OFv

2.1 Adeles 33

where S is a finite set of places of F including the infinite places and US ⊆ FSis an open set. We give ASF = 0 × ASF ⊂ AF the subspace topology andFS =

∏v∈S Fv the product topology. Then the identification FS ×ASF = AF

becomes an isomorphism of topological rings. We will often use the followinguseful notation: if S is a set of places of F

OSF :=∏

finite v∈S

OFv . (2.4)

It agrees with the profinite completion of OSF , the ring of S-integers of F ,that is, elements of F that are integral outside of the finite places in S.

The topology on AF is not the same as the topology induced on AF byregarding it as a subset of the direct product

∏v Fv. While

∏v Fv is not

locally compact, for AF one has the following:

Proposition 2.1.2 The adele ring AF of a global field F is a locally compactHausdorff topological ring.

Proof. We prove that AF is locally compact and leave the other details tothe reader. For any finite set S of places F including the infinite places, thesubset ∏

v∈SFv × OSF (2.5)

is an open subring of AF for which the induced topology coincides withthe product topology. In particular (2.5) is locally compact by Tychonoff’stheorem. Every x ∈ AF is contained in a set of the form (2.5), which showsthat AF is locally compact. ut

There is a natural diagonal embedding F → AF given by sending anelement of F to all of its completions.

Lemma 2.1.3 The image of F in AF under the diagonal embedding is closedand discrete.

Proof. Choose x ∈ F×. We will construct a neighborhood of x in AF con-taining no other element of F . Since AF is a topological group under additionthis will suffice to complete the proof.

For each finite place v of F let nv = v(x), so that x ∈ $nvv but x 6∈ $nv+1

v

for all v. Note that nv = 0 for all but finitely many places. For each infiniteplace v let

Uv :=

y ∈ Fv : |y − x|v <∏v-∞

|x|−|∞|v

(here |∞| is the number of infinite places of S). Consider the open subset ofAF defined by

U =∏v|∞

Uv ×∏v-∞

$nvv .

34 2 Adeles

By construction, x ∈ U . Suppose that y ∈ U . Then |x − y|v ≤ |x|v for allfinite places v. Thus∏

v

|x− y|v ≤∏v-∞

|x|v ×∏v|∞

|x− y|v < 1.

since |x−y|v ≤ |x|v for all finite places v. By the product formula we concludethat x = y. ut

We often identify F with its image under the diagonal embedding. GivenLemma 2.1.3 the following theorem can be surprising the first time one seesit:

Theorem 2.1.4 (Strong Approximation) If S is any finite nonempty setof places of F then F is dense in ASF . ut

Thus omitting one place is enough to move F from being discrete to beingdense. The proof can be found in any standard reference, see [Cas67, §15] forexample.

We close this section by remarking that one can construct analogues ofthe finite adeles in more general situations, e.g. schemes of finite type overthe ring of integers of a global field [Hub91]. The construction is quite a bitmore involved than that given above.

2.2 Adelic points of affine schemes

Weil and Grothendieck both gave approaches (under different hypotheses) totopologizing the points of schemes of finite type over a topological ring (forexample, AF ). Conrad gave a beautiful exposition/elaboration in [Con12b].We lift Theorem 2.2.1 below and its proof from loc. cit. with little modifica-tion.

Theorem 2.2.1 Let R be a topological ring. There exists a unique way totopologize X(R) for all affine schemes X of finite type over R such that

(a) the topology is functorial in X; that is if X → Y is a morphism of affineschemes of finite type over R, then the induced map on points X(R) →Y (R) is continuous;

(b) the topology is compatible with fibre products; this means that if X → Yand Z → Y are morphisms of affine schemes, all of finite type over R,then the topology on X ×Y Z(R) is exactly the fibre product topology;

(c) closed immersions of schemes X → Y induce topological embeddingsX(R) → Y (R);

(d) if X = SpecR[t] then X(R) is homeomorphic with R under the naturalidentification X(R) ∼= R.

2.2 Adelic points of affine schemes 35

If R is Hausdorff or locally compact, then so is X(R). Moreover, if R is Haus-dorff then closed immersions induce closed embeddings, not just topologicalembeddings.

In the proof and throughout this book we take the convention that locallycompact spaces are Hausdorff.

If the reader is uncomfortable with fiber products then they can omit theassertions in the theorem involving them and their proof. The only conse-quence of these facts we really need in the sequel is that with the definitionof the topology on Gna(R) given above the natural bijection

Gna(R)−→Rn

is a homeomorphism if we give the right hand side the product topology.

Proof. Let X be an affine R-scheme. Pick an R-algebra isomorphism

A := O(X) ∼= R[t1, . . . , tn]/I (2.6)

for an ideal I, and identify X(R) with the subset of Rn on which the elementsof I (thought of as polynomials on Rn) vanish.

We start with uniqueness. By our assumption on compatibility with fiberproducts the natural bijection

SpecR[t1, . . . , tn](R)−→Rn

is a homeomorphism provided that we give the right hand side the producttopology. By assumption, this induces a topological embedding X(R) → Rn.This completes the proof of uniqueness and also shows that X(R) is Hausdorffif R is Hausdorff. If R is Hausdorff, then 0 ∈ R is closed, so viewing X(R) asthe the vanishing locus of f ∈ I (viewed as polynomials on Rn) we see thatX(R) is closed. Thus if R is locally compact X(R) then is as well.

We now prove existence. Note that there is a canonical and tautologicalinjection

X(R) = Hom(A,R) −→ RA. (2.7)

We claim that the topology defined using (2.6) as above is the same as thesubspace topology defined by the canonical injection X(R) → RA, so itis independent of the choice of (2.6). Let a1, . . . , an ∈ A correspond to t1(mod I), . . . , tn (mod I) via (2.6), so the injection X(R) → Rn defined by(2.6) is the composition of the natural injection X(R) → RA and the mapRA → Rn given by projection to the factors indexed by (a1, . . . , an). There-fore every open set inX(R) is induced by an open set inRA becauseRA → Rn

is continuous. Since every element of A is an R-polynomial in a1, . . . , an andR is a topological ring, it follows that the map X(R)→ RA is also continuous.Thus X(R) has been given the subspace topology from RA. This completes

36 2 Adeles

the proof of the claim. It also implies that the formation of the topology onX(R) is functorial (i.e. morphisms of affine schemes induce continuous mapson R-points).

Consider a closed immersion

i : X := SpecA → X ′ := SpecA′

corresponding to a surjective R-algebra map h : A′ → A. The map

j : RA −→ RA′

(ra) 7−→ (rh(a′))

is visibly a topological embedding; it topologically identifies RA with the sub-set of RA

′cut out by a collection of equalities among components. Moreover

j is a closed embedding when R is Hausdorff. We have

X ′(R) ∩ j(RA) = j(X(R))

because a set theoretic map A → R is an R-algebra homomorphism if andonly if its composition with h is an R-algebra map. Hence i : X(R)→ X ′(R)is an embedding of topologival spaces and it is a closed embedding when R isHausdorff. By forming products of closed immersions into affine spaces Gna ,we see that

(X ×SpecR X′)(R)→ X(R)×X ′(R)

is a topological ismomorphism via reduction to the trivial special case whenX and X ′ are affine spaces.

Finally, we prove that for given maps X → Y and Z → Y between affineR-schemes the bijection (X ×Y Z)(R) → X(R) ×Y (R) Z(R) is a topologicalisomorphism. Consider the (tautological) isomorphism

X ×Y Z ∼= (X ×R Z)× Y ×R Y

and its topological counterpart. The product map

O(Y )⊗R O(Y ) −→ O(Y )

is surjective and henceY −→ Y ×R Y

is a closed immersion (i.e. affine schemes are separated).Since we have already checked compatibility with fiber products over

SpecR, we see that we are reduced to the case in which one of the mapdefining the fiber product is a closed immersion. We have already proventhat closed immersions yield topological embeddings, so we deduce compati-bility with fiber products. ut

2.3 Relationship with restricted direct products 37

2.3 Relationship with restricted direct products

In practice we will be interested in the topological space X(AF ) when X isan algebraic group. To explain how this works, consider first the case whereX = GLn. There is a closed immersion GLn → Mn ×Ga given on points ina Z-algebra R by

GLn(R) −→Mn(R)×Ga(R) (2.8)

g 7−→ (g,det g−1).

Here Mn is the affine scheme of n × n matrices. It is the affine schemewhose points in a Z-algebra are the R-linear endomorphisms of Rn; it can beidentified as a scheme with Gn2

a . Thus for any topological ring R the groupGLn(R) is endowed with the subspace topology if we view it as a subspaceof Mn(R)×Ga(R) via (2.8). For example, if v is an archimedean place of F ,then by this recipe GLn(Fv) acquires its usual topology.

To describe the topology for nonarchimedean v, we recall that to describea topology on a space it suffices to give a neighborhood base for every pointx, that is, a set of open neighborhoods

x ∈ Uα

such that every neighborhood of x contains some Uα. In a topological group,it suffices to just give a neighborhood base of the identity, since we can taketranslates of this neighborhood base to be neighborhood bases at every otherpoint.

Now if $v is a uniformizer for OFv then

(In +$kvMn(OFv ))× (1 +$k

vOFv ) : k ∈ Z≥1

forms a neighborhood base of the point (In, 1) ∈Mn(Fv)×Fv. It follows that

In +$kvMn(OFv ) : k ∈ Z≥1

forms a neighborhood base of the identity in GLn(Fv). Note that this isthe same topology we would obtain if we just gave GLn(Fv) ⊂ Mn(Fv) thesubspace topology.

On the other hand, if S is any finite set of places of F including the infiniteplaces then

(In +mMn(OSF ))× (1 +mOSF ) : m ⊂ OSF

forms a neighborhood basis for (In, 1) ∈ Mn(ASF ) × ASF . Here m runs overproper ideals of OSF , and

38 2 Adeles

mMn(OSF ) :=∏v 6∈S

$v(m)v Mn(OFv ).

If we intersect one of these neighborhoods with the image of GLn(ASF ) under(2.8) then we obtain∏

v 6∈Sv(m)6=0

(In +mMn(OFv ))×∏v 6∈S

v(m)=0

GLn(OFv ).

Here mMn(OFv ) = $v(m)v Mn(OFv ). This is not the same as the topology

obtained by giving GLn(ASF ) ⊂Mn(ASF ) the subset topology.Finally, for any set of places S of F including the infinite places it is not

hard to see by modifying this argument that one has a topological isomor-phism of locally compact groups

GLn(AF ) = GLn(FS)×GLn(ASF ).

If F is a local field and G is an algebraic group over F to define thetopology on G(F ) we choose an embedding G → GLnF and give G(F ) thesubspace topology. Theorem 2.2.1 tells us that this topology is independentof the choice of embedding G → GLnF . If F is global instead we similarlyobtain the topology on G(AF ).

A helpful way to organize these comments is to introduce the notion of arestricted direct product of topological spaces (we already encountered thisnotion in a special case when we defined the topology on AF in the previoussection). Let Xαα∈A be a set of locally compact topological spaces indexedby a countable set A, and for all α outside a finite subset S0 of A let Kα ⊆ Xα

be a collection of compact open subsets of Xα. Then the restricted directproduct of the Xα with respect to the Kα is the set

′∏α∈A

Xα :=

(xα) ∈

∏α∈A

Xα : xα ∈ Kα

with topology given by declaring a subset of X to be open if it is of the form

U ×∏

α∈A\S

Kα

where S is a finite subset of A including S0 and U ⊆ XS is an open subset.One verifies that this is indeed a topology and with respect to this topologyX is locally compact. Note that if the Xα are topological groups and the Kα

are topological subgroups then∏′α∈AXα is again a topological group. We

also note that the topology does not change if we replace S0 by any largerfinite subset of A.

2.4 Hyperspecial subgroups and models 39

Now assume that G is an algebraic group over the global field F . Choosea faithful representation

G −→ GLn.

Identify G with its image in GLn and define, for all v -∞,

Kv := G(Fv) ∩GLn(OFv ).

Then Kv is a compact open subgroup of G(Fv). Our discussion of the adelictopology on G(AF ) above can be summarized as follows:

Proposition 2.3.1 One has an isomorphism of topological groups

G(AF ) ∼=′∏v

G(Fv)

where the restricted direct product is defined with respect to the subgroupsKv. ut

In fact, in most references G(AF ) is defined using Proposition 2.3.1. Thishas the advantage of being concrete, but it makes it awkward to rigorouslyprove that the topology satisfies good functorial properties.

2.4 Hyperspecial subgroups and models

Let G be an algebraic group over a nonarchimedean local field F and letG→ GLn be a faithful representation. Identify G with its image in GLn. Inthe previous section we made use of the fact that K := G(F ) ∩GLn(OF ) isa compact open subgroup of G(F ). This construction is actually algebraic innature in the sense that K is the OF -points of an affine group scheme overOF . We explain this in more detail in this section and use it as motivation tointroduce the notion of a hyperspecial subgroup. The reader should feel freeto skip this section since the construction will only come up in passing laterin the book. Our presentation in this section was influenced by [Yu].

We start with a Dedekind domain O with fraction field F . For example, Ocould be the ring of S-integers OSF of a number field F for a finite of places Sof F including the infinite places or the ring of integers of a local field. Let Zbe an affine scheme over O. The generic fiber of Z is ZF . If F is local, thenO has a unique prime ideal $; in this case we let k = O/$ be the residuefield. The scheme Zk is known as the special fiber of Z.

Suppose we are given an affine scheme Y over F ; often it is useful considerschemes Y that have Y as their generic fiber and satisfy certain desiderata:

Definition 2.6. A model Y of Y over O is an affine scheme of finite typeover O of the form SpecA where A ≤ O(Y ) and A⊗O F = O(Y ).

40 2 Adeles

The assertion that an affine O-scheme SpecA is flat is equivalent to theassertion that A is flat as an O-module, which is equivalent to the assertionthat A is torsion-free since O is a Dedekind domain. Thus we see that modelsare flat schemes. Suppose conversely we are given an affine scheme of finitetype Y = SpecA over O together with an isomorphism YF ∼= Y . Assumemoreover that Y is flat, which is to say that A is flat as an O-module. Thusthe map A→ O(Y ) induced by the isomorphism YF ∼= Y is injective and weobtain a model of Y .

Sometimes there are obvious models for affine schemes. For example, wecan view GLn as an affine group scheme over O, and it is clearly a model ofits generic fiber GLnF . However, there are many other models of GLn (seeExercise 2.8 for an example).

We now describe a particular type of model of a reductive group. Assumethat F is a local nonarchimedean field.

Definition 2.7. A reductive group G over F is unramified if it is quasisplitand there is a finite degree unramified extension E/F such that GE is split.

The following theorem is foundational for the theory of automorphic rep-resentations. It is an amalgamation of several results of Bruhat and Tits, see[Mac17, §1.2] for precise references.

Theorem 2.4.1 The reductive group G is unramified if and only if thereexists a model G of G over OF such that the special fiber of G is reductive. IfG is model of G over OF such that the special fiber of G is reductive then thesubgroup G(OF ) ≤ G(F ) is a maximal compact subgroup of G(F ). ut

Definition 2.8. A subgroup of G(F ) of the form G(OF ) for a model G as inTheorem 2.4.1 is called a hyperspecial subgroup.

Example 2.1. When G = GLn, it is clear that GLn(OF ) is a hyperspecial sub-group of GLn(F ). It turns out that all maximal compact subgroups of GLn(F )are conjugate to GLn(OF ) [Ser06, Chapter IV, Appendix 1]. In loc. cit. onealso finds a proof that GLn(OF ) is maximal in GLn(F ).

We quickly discuss conjugacy classes of hyperspecial subgroups. The groupG/ZG(F ) acts on G(F ) by conjugation. In general orbits under this action arelarger than the orbits ofG(F ) on itself by conjugation. The set of hyperspecialsubgroups is permuted transitively by the action of G/ZG(F ) [Tit79, §2.5].However not all hyperspecial subgroups are conjugate under G(F ) in general.

Now suppose that we are given an unramified reductive group G. Onemight ask how one ought to go about finding a hyperspecial subgroup of G.One method proceeds via constructing schematic closures as we now explain.We again revert to the assumptions at the beginning of this section, so O isa Dedekind domain with fraction field F . Let Y be a scheme of finite typeover F and let Y be a model of Y . Suppose in addition we are given a closedimmersion of F -schemes

2.4 Hyperspecial subgroups and models 41

X −→ Y ; (2.9)

we identify X with a closed subscheme of Y . Let

A := Im(O(Y)→ O(Y )→ O(X)).

Since O(Y)→ O(Y ) is injective and (2.9) is a closed immersion we see thatX := SpecA comes equipped with a closed immersion X → Y and XF = X.We leave the proof of the following lemma as an exercise (see Exercise 2.4)

Lemma 2.4.2 The scheme X is a model of X and

X (O) = X(F ) ∩ Y(O).

For the proof of this lemma, see Exercise 2.5. The scheme X earns its monikeras a schematic closure via a universal property.

If Y = GLn, viewed as an affine group scheme over O, then the conditionthat O(GLn) → O(GLnF ) is injective is clearly satisfied. Thus given anyfaithful representation G −→ GLn, we can form the schematic closure G ofG; it is a scheme over O with generic fiber G. In the special case where F isa local field and O = OF is its ring of integers we obtain

G(OF ) := G(F ) ∩GLn(OF ),

which we denoted by K earlier. Thus in our construction of the restricteddirect topology from the previous section we implicitly used schematic clo-sures.

Now suppose that F is global and that we are given a faithful represen-tation G → GLn of a connected reductive group G. We can then take theschematic closure G of G in GLnOF . One has the following proposition:

Proposition 2.4.3 For almost all nonarchimedean places v of F the sub-group G(OFv ) is hyperspecial.

Thus taking a “global” schematic closure gives us hyperspecial subgroups atalmost all places. This proposition and variants of it are extremely useful inapplications, so we indicate the proof after giving some preparation.

Recall that an O-scheme Y is smooth if it is flat and of finite type andfor all algebraically closed fields k that admit a nonzero O-algebra morphismO → k the fiber Yk is smooth. The following lemma is a very special case of[GW10, Proposition 6.18]:

Lemma 2.4.4 Let S0 be a finite set of finite places of F (possibly empty),let Y be a smooth OS0

F -scheme and let X ≤ YF be a smooth subscheme of itsgeneric fiber. Let X be the schematic closure of X in Y. Then there is a finiteset S of finite places containing S0 such that XOSF is smooth over OSF . ut

We now indicate the proof of Proposition 2.4.3:

42 2 Adeles

Proof of Proposition 2.4.3: The schematic closure G is again a group schemeover OF (compare Exercise 2.6 below). It is also smooth over OSF for a suf-ficiently large finite set S of finite places of F . The result now follows from[Con14, Proposition 3.1.9]. 2

2.5 Approximation in algebraic groups

For a global field F and a nonempty finite set S of places of F , the image of Funder the diagonal embedding F → FS is dense. This is fairly easy to proveand can be viewed as a generalization of the Chinese remainder theorem. IfX is an affine scheme of finite type over F and S is a finite set of places ofF then one can ask if a similar phenomenon holds:

Definition 2.9. Let S be a nonempty finite set of places of F . The affinescheme X satisfies weak approximation with respect to S if the image ofthe diagonal embedding X(F ) → X(FS) is dense. It satisfies strong ap-proximation with respect to S if X(F ) is dense in X(ASF ).

Here when we speak of density we are of course using the canonical topologieson X(FS) and X(ASF ) afforded by Theorem 2.2.1.

Despite the relative ease of proving weak approximation when X = Ga,establishing whether or not weak approximation holds for a general affinescheme is very difficult. In general it is false. We refer to [Har04] for a moredetailed discussion.

Our goal in this section is to describe when weak and strong approximationhold in settings related to algebraic groups. To simplify the discussion weoften assume that F is a number field; additional complications come up inthe general case. Our primary reference is [PR94, Chapter 7]. We start withthe following proposition, the proof of which we leave as an exercise:

Proposition 2.5.1 Let G be a connected algebraic group over a number fieldF with Levi decomposition G = MN . Then G admits weak (resp. strong)approximation with respect to a finite set S of places of F if and only if Mdoes. ut

Thus in the number field case studying strong and weak approximation ofalgebraic groups is equivalent to studying it for the smaller set of reductivegroups. It turns out that under suitable restrictions on the set of places Sweak approximation always holds:

Theorem 2.5.2 Let G be a connected algebraic group over a number fieldF . There is a finite set S0 of finite places of F such that G has weak approx-imation for any finite set of places of F not containing S0. ut

2.5 Approximation in algebraic groups 43

We refer to [PR94, §7.3] for the proof.If we wish to guarantee that we can take S0 to be empty we need to assume

additional arithmetic conditions on the group G. To state them, recall thata central isogeny of connected algebraic groups G → H over a field k is asurjective homomorphism whose kernel is finite and contained in the centerof G. A semisimple group G is simply connected if any central isogenyH → G is an isomorphism, and adjoint if any central isogeny G→ H is anisomorphism.

If G → H is a central isogeny between two semisimple k groups then Gand H have the same root system, but not necessarily the same root datum.In general, simply connected groups have the largest center out of the set ofsemisimple groups with the same root system and adjoint groups have trivialcenter. For example, SLn is simply connected and PGLn is adjoint. Thequotient map SLn → PGLn is a central isogeny. See [Bor91, Hum75, Spr09].

For the following theorem we refer again to [PR94, §7.3]:

Theorem 2.5.3 A simply connected or adjoint semisimple group G over anumber field F admits weak approximation with respect to any set finite S ofplaces F . ut

We finish our discussion of weak approximation by considering the casewhere X is not an algebraic group, but a particular type of homogeneousspace. Let G be a reductive group over a number field F and let H ≤ G bea reductive subgroup. Then there is a natural action of G on

X := G/H := Spec(O(G)H).

For the definition of O(G)H we refer to (17.8). If F is an algebraic closure ofF and F ≤ L ≤ F is a subfield then

X(L) = (G(F )/H(F ))Gal(F/L)

with the natural Galois action (see Proposition 17.1.5). For more details onalgebraic group actions we refer to [MFK94] or §17.1.

To state weak approximation theorems in this context, we recall that analgebraic torus T is quasi trivial if X∗(T ) is a permutation Gal(F/F )-module. A reductive group G is quasi trivial if the torus G/Gder is quasitrivial and Gder is simply connected. We record the following theorem ofBorovoi [Bor09, 3.12], which generalizes Theorem 2.5.2:

Theorem 2.5.4 Let G be a reductive quasi trivial group and assume that His a connected subgroup. There is a finite set S0 of finite places of the numberfield F such that G/H has weak approximation for any finite set of places ofF not containing S0. ut

In particular, G/H always has strong approximation with respect to ∞.Similarly we have a generalization of Theorem 2.5.3:

44 2 Adeles

Theorem 2.5.5 Assume that G is a reductive quasi trivial group with aconnected subgroup H. Suppose that H/Hder splits over a cyclic extension ofF . Then G/H satisfies weak approximation for any finite set S of places ofF . ut

This is [Bor09, Corollary 3.14]. We note that Borovoi actually works in a moregeneral context where G is not necessarily reductive, but we have restrictedto the situation above for simplicity.

We now turn to a discussion of strong approximation. We recall that aconnected algebraic group over a global field F is almost simple if LieG isa simple Lie algebra, that is, a Lie algebra with no proper ideals. The basicresult on strong approximation is the following. For the proof see [Pra77]:

Theorem 2.5.6 Let G be a connected absolutely almost simple algebraicgroup over a global field F and let S be a finite set of places of F . If Gis simply connected and G(FS) is noncompact then G satisfies strong approx-imation relative to S. ut

At least in the number field case there is a converse to this theorem, see[PR94, Theorem 7.12]. We point out in particular that the semisimplicityassumption is necessary. Even in the case even in the case G = GL1 strongapproximation is false in general. Indeed,

F×\(A∞F )×/O×F

is the class group of F which is finite, but often nontrivial.Here is an analogue of Theorem 2.5.6 for certain homogeneous spaces

[Rap14, Proposition 2.4]:

Theorem 2.5.7 Assume that H ≤ G are almost simple algebraic groups thatare simply connected and semisimple. Then G/H satisfies strong approxima-tion relative to S if and only if (G/H)(FS) is noncompact. ut

2.6 The adelic quotient

LetG be an affine algebraic group over a global field F (so we do not assumeGto be reductive or connected). Then the subgroup G(F ) ≤ G(AF ) is discrete(compare Exercise 2.2) and we can consider the quotientG(F )\G(AF ). In thissection we recall basic facts on the topology of G(F )\G(AF ). More preciseresults will be recalled in the following section.

The first result we recall is the following (see [Con12a] for the proof):

Theorem 2.6.1 (Borel, Conrad, Oesterle, Prasad) For any finite setS of places of F containing the infinite places and for any compact opensubgroup KS ≤ G(ASF ) the quotient

2.6 The adelic quotient 45

G(F )\G(ASF )/KS

is finite. ut

We note that one does not even have to assume that G is smooth (al-though this is automatic in the characteristic zero case, see Theorem 1.5.2).Theorem 2.6.1 is known as finiteness of class numbers (for affine algebraic

groups). Indeed, in the special case G = GL1, K∞ := O×F the set above canbe identified with the class group of F , and hence the theorem implies thefiniteness of the class group.

It is not hard to see that in general the quotient G(F )\G(AF ) itself isinfinite. However, we could ask when the quotient is compact or finite volumewith respect to a suitable measure (see §3.5), and there is a complete answerto this question which we recall in Theorem 2.6.2 below. However, it is usefulto first eliminate a trivial obstruction to the quotient G(F )\G(AF ) beingfinite volume or compact.

The units F× of F embed into A×F diagonally as a discrete subspace. Sincethere the idelic norm | · |AF provides a continuous nontrivial homomorphism

| · |AF :=∏v

| · |v : F×\A×F −→ R>0,

we conclude that F×\A×F is noncompact and actually it has infinite volumewith respect to the Haar measure (for more on Haar measures see §3.2 below).An analogous phenomenon occurs for other groups, and this motivates thefollowing definition:

G(AF )1 :=⋂

χ∈X∗(G)

ker(| · | χ : G(AF )→ R>0). (2.10)

Here |x| := |x|AF =∏v |x|v. Note that G(F ) is contained G(AF )1 in virtue of

the product formula Proposition 2.1.1. Moreover, G(F ) is discrete in G(AF )1

(see Exercise 2.2).We now define a subgroup

AG ≤ G(F∞) ≤ G(AF ) (2.11)

such thatAGG(AF )1 = G(AF ),

the product being direct.When F is a number field we let AG be the identity component of the

R-points of the greatest Q-split torus in ResF/QZG. When F is a functionfield of characteristic p temporarily write Z for the largest Fp(t)-split torusin the center of

ResF/Fp(t)ZG.

46 2 Adeles

Then if Z has rank d there is an isomorphism Z(Fp(t)∞) ∼= (Fp((t))×)d.We let AG be the subgroup (tZ)d. We note that in either case when G isreductive the inclusion G(AF )1 → G(AF ) induces an isomorphism G(AF )1 ∼=AG\G(AF ). Thus it is largely a manner of taste as to whether one works withG(AF )1 or AG\G(AF ).

The adelic quotient of G is the quotient

[G] := G(F )\G(AF )1. (2.12)

The isomorphism G(AF )1−→AG\G(AF ) induces a canonical isomorphism

[G]−→G(F )AG\G(AF ) (2.13)

that intertwines the action of G(AF )1 on the left hand side and the actionof AG\G(AF ) on the left. We will therefore allow ourselves to let the symbol[G] denote either AGG(F )\G(AF ) or G(F )\G(AF )1. The group G(AF )1 isunimodular by Lemma 3.5.4 below, and hence admits a Haar measure (thenotion of a Haar measure and a unimodular group will be recalled in §3.2).

The basic topological and measure theoretic properties of [G] are summa-rized in the following theorem:

Theorem 2.6.2 (Borel, Conrad, Harder, Oesterle) Assume that G issmooth and connected. The quotient [G] defined in (2.12) has finite volumewith respect to the measure induced by a Haar measure on G(AF )1. The adelicquotient [G] is compact if and only if for every F -split torus T ≤ G one hasTF ≤ R(GF ).

Proof. The first assertion is [Con12a, Theorem 1.3.6]. See [Con12a, TheoremA.5.5] for a proof of the last assertion. ut

For comparison with the classical theory of automophic forms it is con-venient to relate the adelic quotient to locally symmetric spaces (compareChapter 6). For this purpose, if K∞ ≤ G(A∞F ) is a compact open subgrouplet

h := h(K∞) = |G(F )\G(A∞F )/K∞|.

By Theorem 2.6.1, h < ∞. Let t1, . . . , th denote a set of representatives forG(F )\G(A∞F )/K∞. We then have a homeomorphism

h∐i=1

Γi(K∞)\G(F∞) −→ G(F )\G(AF )/K∞ (2.14)

given on the ith component by

Γi(K∞)g∞ 7−→ G(F )g∞tiΓi(K

∞),

whereΓi(K

∞) := G(F ) ∩ ti ·G(F∞)K∞ · t−1i .

2.7 Reduction theory 47

Notice that the Γi(K∞) are discrete subgroups ofG(F ) and they are moreover

arithmetic in the following sense:

Definition 2.10. Let G ≤ GLn be a linear algebraic group. A subgroupΓ ≤ G(F ) is arithmetic if it is commensurable with G(F ) ∩GLn(Z).

The notion of arithmeticity does not depend on the choice of representationG ≤ GLn (see Exercise 2.13). The subgroups of G(F ) that are intersectionsof a compact open subgroup of G(A∞F ) and G(F ) are known as congruencesubgroups. Not every arithmetic group is a congruence subgroup in general,although for some groups G this is the case. For more information the readercan consult the literature on the so-called congruence subgroup problem.

We end this section by considering the special case where G = GL2/Q.

Then K∞ = GL2(Z) is a maximal compact open subgroup, where Z =∏p Zp

is the profinite completion of Z. Moreover, if we denote by

K0(N) : =

(a bc d

)∈ GL2(Z) : N |c

and

Γ0(N) : = K0(N) ∩GL2(Z)

thenΓ0(N)\GL2(R) = GL2(Q)\GL2(AQ)/K0(N).

If we let K∞ = SO2(R) then :

Γ0(N)\(C− R)→GL2(Q)\GL2(AQ)/K∞K0(N)

= GL2(Q)\(C− R)×GL2(A∞Q )/K0(N),

where on the left Γ0(N) acts via Mobius transformations:(a bc d

)· z :=

az + b

cz + d.

2.7 Reduction theory

Reduction theory allows us to control the adelic quotient [G] when it is non-compact. To be more precise about this assume that G is a reductive groupover a global field F . In the number field case this assumption can be weak-ened to allow for arbitrary connected affine groups of finite type over F atthe cost of introducing more notation [PR94, §4.3].

The following theorem provides what is known as the Iwasawa decom-position:

Theorem 2.7.1 Let G be a reductive group over a global field F and letP ≤ G be a parabolic subgroup. There exists a maximal compact subgroup

48 2 Adeles

K ≤ G(AF ) such that

G(AF ) = P (AF )K. (2.15)

We sketch the proof of this theorem in Appendix A.We now assume that P is a minimal parabolic subgroup of G with unipo-

tent radical N . Fix a Levi subgroup M ≤ P . Let T0 be a maximal split torusof Gder contained in M , then T := T0Z

G is a maximal split torus of G. As

recalled in §1.9 the set of nonzero weights of T acting on LieP is a base∆ ⊂ Φ(G,T ) for the set of roots of T in G.

A Siegel set in G(AF )1 is a set of the form

S(t) := SΩ(t) = ΩAT0(t)K, (2.16)

where Ω ⊂ N(AF )M(AF )1 is a relatively compact subset, t ∈ R>0, and

AT0(t) := x ∈ AT0

: |α(x)| ≥ t for all α ∈ ∆. (2.17)

Here | · | denotes the usual norm on R when F is a number field and thecanonical norm on Fp((t)) when F is a function field of characteristic p.The basic result of reduction theory is the following:

Theorem 2.7.2 (Reduction theory) There exists a Siegel set such that

G(F )AGS(t) = G(AF ).

Proof. See [Spr94]. To aid the reader we note that in loc. cit. the group G(F )acts on the right. Thus one has to take an inverse to move G(F ) to the left,and this changes the inequality in the analogue of AT0

(t) in loc. cit. to thatwe used in defining AT0(t). ut

To understand what is going on in Theorem 2.7.2 it is useful to considerwhat is arguably the simplest nontrivial case, that when G = GL2. In thiscase we can take K = O2(R)GL2(Z) and let B = P be the Borel subgroupof upper triangular matrices (since it is a Borel subgroup it is a minimalparabolic subgroup). Then ∆ consists of the single root

α : T (R) −→ R×(t1t2

)7−→ t1t

−12 .

In this setting reduction theory tells us that there is a compact subgroupΩ ≤ N(AF ) such that

GL2(AF ) = ΩAGL2AT0(t)K.

for small enough t ∈ R>0. This is the content of Exercise 2.15. We remarkthat it is this example that gives reduction theory its name. Indeed, a slightstrengthening of Theorem 2.7.2 in this case yields the familiar fact from the


reduction theory of positive definite binary quadratic forms over Q, namelythat every form of a given discriminant is representable by a reduced form.

We end this section by explaining how reduction theory is used in practice.Let ϕ : [G]→ C be a continuous function. Reduction theory implies that if wecan control the restriction of ϕ to AT0(t) then we can control the function onall of [G]. This observation motivates growth conditions we place on functionson [G] (see Lemma 6.3.1) and plays a key role in convergence arguments. See,for example, §9.4, §9.5 and the proof of Proposition 14.3.3.

Exercises

2.1. Prove Proposition 2.1.1.

2.2. Let R → R′ be a continuous map of topological rings and let X be anaffine scheme of finite type over R. Show that X(R) −→ X(R′) is continuous.Show moreover that if R→ R′ is a

(a) topological embedding(b) open topological embedding(c) closed topological embedding(d) topological embedding onto a discrete subset

then so is X(R)→ X(R′).

2.3. Let G be an algebraic group over a local field F , let ρ : G → GLn be afaithful representation, and let K ≤ GLn(F ) be a compact open subgroup.Prove that ρ(G(F )) ∩K is a compact open subgroup of ρ(G(F )).

2.4. Prove Lemma 2.4.2.

2.5. Let O be a Dedekind domain with fraction field F and let Y be a flatO-scheme. Suppose that X → Y := YF is a closed immersion. Let X be theschematic closure of X in Y. Then for any closed immersion Z → Y whosegeneric fiber is an isomorphism onto X there is a unique closed immersionX → Z such that the following diagram commutes:

X

Z // Y

2.6. Let O be a Dedekind domain with fraction field F . Let G/O be a flatgroup scheme of finite type with generic fiber G := GF . Let H be a groupscheme over F equipped with a morphism of group schemes H −→ G that isa closed immersion. Show that the schematic closure H of H in G is a groupscheme.

50 2 Adeles

2.7. Let S be a finite set of places of a global field F . Prove that F is densein FS .

2.8. Let m > 1 be an integer. Let

A := Z[xij , tij , y]1≤i,j≤n/(

det(xij)y − 1, (xij)− 1 +m(tij)).

Here we view (xij), (tij) as matrices in order to make sense out of the poly-nomial relations. Show that G := Spec(A) is a model of GLn over Z andthat

G(Z) := g ∈ GLn(Z) : g ≡ 1 (mod mMn(Z)).

2.9. Let X1, X2 be affine schemes of finite type over a global field F . Showthat X1 and X2 admit weak (resp. strong) approximation with respect to afinite set S of places F if and only if X1 ×X2 does. Deduce that if G is analgebraic group over a number field F then G is satisfies weak (resp. strong)approximation with respect to a finite set S of places F if and only if a Levisubgroup of G does.

2.10. Let F be a global field. Show that GLn admits weak approximationwith respect to any proper subset of the places of F .

2.11. Assume that the affine algebraic groupG over the global field F satisfiesstrong approximation with respect to a finite set S of places F . Show that

|G(F )\G(ASF )/KS | = 1

for any compact open subgroup KS ≤ G(ASF ).

2.12. Let F be a number field. Show that GLn does not admit strong ap-proximation with respect to the set of infinite places of F if the class numberof F is not 1.

2.13. Let ρi : G→ GLni , i ∈ 1, 2 be a pair of faithful representations of theaffine algebraic group G over the number field F . Let Gi be the schematic clo-sure of ρi(G) in GLniOF . Show that a subgroup Γ ≤ G(F ) is commensurablewith G1(OF ) if and only if it is commensurable with G2(OF ).

2.14. Let Bn ≤ GLn be the Borel subgroup of upper triangular matrices.Show that

GLn(R) = Bn(R)On(R)GLn(C) = Bn(C)Un(R)

and that for a nonarchimedean local field F

GLn(F ) = Bn(F )GLn(OF ).

Here for R-algebras R, On(R) := g ∈ GLn(R) : ggt = In and Un(R) :=g ∈ GLn(C⊗RR) : ggt = In. Deduce the adelic Iwasawa decomposition forGLn(AF ).


2.15. LetΩ = ( 1 t

1 ) : t ∈ [0, 1] .

Show that GL2(AQ) = GL2(Q)ΩAGL2AH (t)K for t sufficiently small.

2.16. Let G be a smooth algebraic group over a global field F with unipotentradical N . Prove that under the map

G(AF ) −→ G/N(AF )

the group AG is mapped isomorphically onto AG/N .

2.17. For each integer N ≥ 1 let Γ (N) ≤ GLn(Z) be the kernel of thereduction map

GLn(Z) −→ GLn(Z/N).

Prove that a subgroup of GLn(Q) is a congruence subgroup if and only if itcontains Γ (N) for some N as a subgroup of finite index.

Chapter 3

Automorphic Representations

In mathematics you don’tunderstand things. You just getused to them

J. von Neumann

Abstract In this chapter we give a definition of an automorphic represen-tation in the category of unitary representations of the adelic points of analgebraic group. This definition will subsequently be refined, by enlargingthe category of representations considered to the category of admissible rep-resentations, and then later by replacing representations of the archimedeancomponent of the adelic group by (g,K)-modules.

3.1 Representations of locally compact groups

Our goal in this section is to give a definition of an automorphic representa-tion that is completely natural from the point of view of trace formulae.

We will start by introducing some basic abstract representation theory(i.e. representation theory that requires no more structure than a locallycompact group). Our basic reference is [Fol95]. Throughout this chapter welet G be a locally compact (Hausdorff) topological group (for example G(R)for a locally compact topological ring R such as the adele ring AF ). Let Vbe a topological vector space over C and let

End(V )

be the space of all continuous linear maps from V to itself. We let GL(V ) ⊂End(V ) be the group of invertible continuous linear maps with continuousinverse.

53

54 3 Automorphic Representations

Often we will be concerned with the case where V is a Hilbert space,that is, a complex inner product space complete with respect to the metricinduced by the inner product that admits a countable orthonormal basis. Inparticular, Hilbert spaces will always be assumed to be separable. The innerproduct will be denoted ( , ) and the norm by ‖·‖2. In this case we let U(V )be the subgroup of GL(V ) preserving ( , ).

Definition 3.1. A representation (π, V ) of G is a continuous group homo-morphism

π : G −→ GL(V ).

If V is a Hilbert space and π maps G to the subspace of unitary operators,that is

(π(g)ϕ1, π(g)ϕ2) = (ϕ1, ϕ2)

for all ϕ1, ϕ2 ∈ V and g ∈ G, then π is said to be unitary.

Often one omits explicit mention of the space of π. If we wish to specify thatπ acts on a ?-space (e.g. Frechet, Hilbert, locally convex) we often to π as a?-representation.

A morphism of representations (or a G-intertwining map) is a morphismof topological vector spaces that is G-equivariant; it is an isomorphism ifthe underlying morphism of topological vector spaces is an isomorphism.If two representations are isomorphic then they are also called equivalent.Two unitary representations (π1, V1) and (π2, V2) are unitarily equivalentif there is a G-equivariant isomorphism of Hilbert spaces V1 → V2. Thusunitary equivalence implies equivalence, but not conversely.

A subrepresentation of (π, V ) is a closed subspace W ≤ V that is pre-served by π, a quotient of π is a topological vector space W equipped withan action of G and a G-equivariant continuous linear surjection V →W , anda subquotient is a subrepresentation of a quotient.

We will mostly be interested in unitary representations, and we need lan-guage to describe when a representation that is not originally presented asa unitary representation actually “is” unitary. First, we say that a repre-sentation on a Hilbert space is unitarizable if it is equivalent to a unitaryrepresentation. Sometimes the natural representations to study are not onHilbert spaces, or even complete spaces (see, for example, Chapter 5). How-ever, one can still discuss unitarity in this situation as follows. Recall that apre-Hilbert space is a complex inner product space that admits a countableorthonormal basis. Thus a Hilbert space is simply a pre-Hilbert space thatis complete with respect to the metric induced by the inner product. We canstill define U(V ) as before. A representation π of G on a pre-Hilbert space Vextends by continuity to the completion V of V (which is a Hilbert space).With this in mind we say that a representation π on a pre-Hilbert space Vis pre-unitary if π(G) ≤ U(V ). Thus for every pre-unitary representationof G we obtain, by continuity, a canonical unitary representation of G on V .Finally, we say that a representation is pre-unitarizable if it is equivalent

3.1 Representations of locally compact groups 55

to a pre-unitary representation. Of course, in practice, these distinctions areoften ignored, and one sometimes calls a unitarizable representation, or evena pre-unitarizable representation, simply a unitary representation.

We now discuss the basic means of constructing unitary representations.Assume one has a continuous right action

X ×G −→ X (3.1)

(x, g) 7−→ xg

where X is a locally compact Hausdorff space equipped with a G-invariantRadon measure dx. The Radon measure gives rise to an L2-space, namely thespace L2(X, dx) of square-integrable complex valued functions on X endowedwith the inner product

(ϕ1, ϕ2) :=

∫X

ϕ1(x)ϕ2(x)dx. (3.2)

We obtain a left action of G on L2(X, dx) via

π(g)ϕ(x) := ϕ(xg). (3.3)

Theorem 3.1.1 The action of G on L2(X, dx) defined as in (3.3) is contin-uous.

We refer to the action of G on L2(X, dx) just constructed as the naturalaction or the regular action.

For the proof we follow [Fol95, Proposition 2.41]:

Proof. LetCc(X) ≤ L2(X, dx)

be the subspace of compactly supported continuous functions. Let U ⊂ Gbe an open neighborhood of the identity 1 ∈ G with compact closure, letϕ ∈ Cc(X), and let W = supp(ϕ)U ⊂ X. Then W is relatively compact, andπ(g)ϕ is supported in W if g ∈ U . Thus

‖π(g)ϕ− ϕ‖2 ≤ vol(W )1/2‖π(g)ϕ− ϕ‖∞.

This goes to zero as g → 1 (see Exercise 3.1).Since X is locally compact and dx is a Radon measure Cc(X) is dense

in L2(X, dx), given ϕ1 ∈ L2(X, dx) we can choose ϕ2 ∈ Cc(X) such that‖ϕ1 − ϕ2‖2 < ε for any ε > 0. Then

‖π(g)ϕ1 − ϕ1‖2 ≤ ‖π(g)(ϕ1 − ϕ2)‖2 + ‖π(g)ϕ2 − ϕ2‖2 + ‖ϕ2 − ϕ1‖2≤ 2ε+ ‖π(g)ϕ2 − ϕ2‖2

which, by what we’ve already shown, goes to zero as g → 1. ut


3.2 Haar measures on locally compact groups

To give more concrete examples of representations it is useful to recall thenotion of Haar measure. If G is a locally compact group then there exists apositive Radon measure drg on G that is right invariant under the action ofG: ∫

G

f(gx)drg =

∫G

f(g)drg for all x ∈ G. (3.4)

[Fol95, Theorem 2.10].Moreover, this measure is unique up to multiplication by an element of

R>0. A right Haar measure is a choice of such a measure. There is alsoa left invariant positive Borel measure d`g = dr(g

−1), again unique up toscalars. Such a measure is known as a left Haar measure.

The existence of the right Haar measure gives via the recipe of (3.1) arepresentation of G on L2(G) = L2(G, drg) called the (right) regular rep-resentation.

Definition 3.2. A locally compact group G is unimodular if any right Haarmeasure is also a left Haar measure.

We note that an abelian group is trivially unimodular. Moreover a compact(Hausdorff) group is unimodular (see Exercise 3.2).

Example 3.1. The points of Borel subgroups are, in general, not unimodular.For example, if B ≤ GL2 is the Borel subgroup of upper triangular matricesthen for any local field F with normalized valuation | · | we can write

B(F ) =

(ab

)(1 t

1

): a, b ∈ F×, t ∈ F

.

With respect to this decomposition one can take

d`g :=dadbdt

|a||b|and drg :=

dadbdt

|b|2.

where da, db and dt are Haar measures on the additive group F .

The failure of a group to be unimodular is measured by an abelian qua-sicharacter called the modular quasicharacter. If G is a locally compactgroup with right Haar measure drg, then the modular quasi-character is thehomomorphism

δG : G −→ R>0

defined by:

dr(hg) = δG(h)drg. (3.5)

3.2 Haar measures on locally compact groups 57

We note that one can alternately define δG(g) via the relation

d`(g−1) = δG(g)d`g (3.6)

(compare Exercise 3.3).

Remark 3.1. Some references define the modular quasicharacter to be themultiplicative inverse of δG. In particular, in [Fol95], ∆(g) := δG(g)−1. Thechoice of normalization we use in this book seems to be consistent with mostof the literature in automorphic representation theory; one motivation todefine things this way is provided by Proposition 3.5.1.

The following proposition provides a useful means of decomposing Haarmeasures:

Proposition 3.2.1 Suppose that S and T are closed subgroups of G withcompact intersection and that the product map S × T → G is open withimage of full measure with respect to d`g. Then one can normalize the leftand right Haar measures on S and T so that∫

G

f(g)d`g =

∫S×T

f(st)δT (t)

δG(t)d`sd`t

=

∫S×T

f(st)

δG(t)d`sdrt.

In particular, if G is unimodular, then∫G

f(g)dg =

∫S×T

f(st)d`sdrt.

Proof. The first assertion is [Kna86, Theorem 8.32] in the case where G isa Lie group. The proof is the same in general. We also note that in thenotation of loc. cit. one has ∆G := δG, ∆S := δS , and ∆T := δT . For thesecond equality we use (3.6). ut

Sometimes we will require a slight generalization of the notion of a Haarmeasure. Let G be a locally compact group and let H ≤ G be a closedsubgroup. We will sometimes wish to integrate over the quotient H\G. Thefollowing theorem tells us precisely when this is possible (see, e.g. [Fol95,Theorem 2.49] for a proof):

Theorem 3.2.2 There is a right G-invariant Radon measure µ on H\G ifand only if δG|H = δH . In this case, µ is unique up to a constant factor, andfor a suitable choice of right Haar measures dg (resp. dh) on G (resp. on H)one has ∫

G

f(g)dg =

∫H\G

(∫H

f(hg)dh

)dµ(Hg)


for f ∈ Cc(G). ut

The measure dµ in the theorem will be denoted by dgdh . We remark that in

loc. cit. the author considers right invariant measures on G/H instead, butthe result above follows in the same manner, or from the version with H\Greplaced by G/H.

3.3 Convolution algebras of test functions

To any locally compact group G we can associate the space of integrablefunctions

L1(G) :=

drg-measurable f : G→ C :

∫G

|f(g)|drg <∞

and the subspace of compactly supported continuous functions Cc(G). Thisspace is endowed with the structure of a C-algebra with product

f1 ∗ f2(g) :=

∫G

f1(gh−1)f2(h)drh. (3.7)

This product is associative (Exercise 3.15).

Example 3.2. When G = R, the convolution f1 ∗ f2 is exactly the usual con-volution from Fourier analysis.

Given a representation π : G → GL(V ) of G, if f ∈ Cc(G) one obtains alinear map

π(f) : V −→ V

ϕ 7−→∫G

π(g)f(g)ϕdrg.

If π is unitary then this map is bounded, and in fact in the unitary case it iswell-defined for all f ∈ L1(G) (Exercise 3.16). The definition (3.7) is chosenso that

π(f1 ∗ f2) = π(f1) π(f2)

(see Exercise 3.17). Thus, in particular, representations π of G give rise toalgebra representations of Cc(G), that is, C-algebra homomorphisms

Cc(G) −→ End(V ).

3.4 Haar measures on local fields 59

3.4 Haar measures on local fields

Let F be a local field, that is, a field equipped with an absolute value that iscompact with respect to the topology induced by the absolute value (see §2.1).All infinite locally compact fields arise as the completion of some global field[Lor08, §25, Theorem 2] with respect to some absolute value. In this sectionwe define Haar measures on the additive group of F .

Assume first that F is archimedean, so F is isomorphic to R or C. Thestandard Haar measure on R is the usual Lebesgue measure, and twice thestandard Haar measure on C. If z = x + iy with x, y ∈ R, then this Haarmeasure is |dz ∧ dz| = 2dx ∧ dy where dx and dy are the Lebesgue measureon R.

Let F be a nonarchimedean field with ring of integers OF and let $ bea uniformizer. Since the additive group of F is a locally compact topologi-cal group, it admits a Haar measure dx. The set OF ⊂ F , being compact,has finite measure, so to specify the Haar measure it suffices to specify themeasure dx(OF ) of OF . We often assume dx(OF ) = 1, but sometimes othernormalizations are convenient.

Open sets in F of the form a + $kOF where a ∈ F and k ∈ Z>0 form aneighborhood base. Using additivity and invariance of the Haar measure onecan compute the measure of such sets:

Lemma 3.4.1 If dx is a Haar measure on F then

dx(a+$kOF ) = q−kdx(OF ).

ut

Thus for every local field F we have a Haar measure dx. Via the usualprocedure we then obtain a product measure on any F -vector space. It isagain an Haar measure with respect to addition.

Given a Haar measure dx on F we obtain a Haar measure

dx× :=dx

|x|

on the unit group F×. Similarly, we can define a Haar measure on GLn(F )by viewing it as an (open) subset of the n by n matrices gln(F ) and thentaking the measure

dX

|detX|n

where dX is the (additive) Haar measure on gln(F ) (see Exercise 3.11. Thisconstruction will be generalized to treat the F -points of affine algebraicgroups in the following section.


3.5 Haar measures on the points of algebraic groups

Let G be an (affine) algebraic group over a local or global field F . It isuseful to more explicitly describe the Haar measures on G(R) for a locallycompact topological F -algebras R using differential forms. We will do this inthe current section.

Let n = dimF G. Then there is a nonzero top-dimensional left-invariantdifferential form

ω` ∈ ∧ng

which is unique up to multiplication by an element of F×. For every place vof F we therefore obtain a differential form ω` := ω`v upon localizing. Thisallows us to define for each v a Radon measure

Cc(G(Fv)) −→ C

f 7−→∫G(Fv)

f(g)d|ω`|v(g).

Since ω` is left G(Fv)-invariant, the measure d|ω`|v is left G(Fv)-invariant,and hence is a left Haar measure. This is explained in more detail in thearchimedean case in [Kna86, Chapter VIII.2], and in the nonarchimedeancase in [Oes84, §2].

Proposition 3.5.1 One has

δG(Fv)(g) = |det(Ad(g) : g(Fv)→ g(Fv)

)|v. (3.8)

Proof. One hasAd(g)ω` = c(g)ω`

wherec(g) := det

(Ad(g) : g(Fv)→ g(Fv)

).

We write this in terms of Haar measures:

d|ω`|v(hg−1) = d|ω`|v(ghg−1) = |c(g)|vd|ω`|v(h).

On the other handd`(hg

−1) = δG(Fv)(g)d`(h)

for any left Haar measure d`h on G(Fv) by Exercise 3.3. We deduce that|c(g)| = δG(Fv)(g). ut

Corollary 3.5.2 If G is reductive, then G(Fv) is unimodular.

Proof. Let ZG be the center of G and Gder its derived group. We then havea commutative diagram

3.5 Haar measures on the points of algebraic groups 61

ZG ×Gder a−−−−→ Gm

b

y yId

Gc−−−−→ Gm

where a and c are given on points by

g 7→ det(Ad(g) : g→ g),

the left vertical map b is the product map, and the right vertical map is theidentity. The map a is trivial on ZG and it is trivial on Gder since X∗(Gder)is trivial. Since b is a quotient map, c is trivial as well, and we deduce thecorollary. ut

This provides us with a description of the left Haar measures dg` on G(Fv)for every v. We similarly obtain right Haar measures dr(g) := d`(g

−1). Onenow wishes to obtain a right Haar measure on G(AF ). To do this let S be afinite set of places of F including the archimedean places and let

KS =∏v 6∈S

Kv ≤ G(ASF )

be a maximal compact open subgroup. We then define a right (resp. left)Haar measure on G(AF ) via

d`g =∏v∈S

d`gv and drg =∏v∈S

drgv, (3.9)

where we assume that d`gv(Kv) = drgv(Kv) = 1 for all v 6∈ S. Then it iseasy to see that the measures d`g, drg so defined are left and right Haar mea-sures on G(AF ), respectively (see Exercise 3.6). If we wish, we can apply ananalogous procedure and obtain left and right Haar measures on AG\G(AF ).

We can now prove the following lemma:

Lemma 3.5.3 Any right Haar measure on G(AF ) is left invariant underG(F ).

Proof. Since the right Haar measure is constructed as the product of localHaar measures one has

δG(AF )(g) =∏v

δG(Fv)(gv) = |det(Ad(g) : g(AF )→ g(AF )

)|AF . (3.10)

By the product formula we deduce that δG(AF )(γ) = 1 for γ ∈ G(F ). ut

The proof here amounted to observing that δG(AF ) is trivial on G(F ).In fact there is a much larger subgroup of G(AF ) on which the modularquasicharacter is trivial, namely G(AF )1, defined as in §2.6:


Lemma 3.5.4 The group G(AF )1 is unimodular. If G is reductive, thenG(AF ) is unimodular.

Proof. The mapping

g 7→ det(Ad(g) : g→ g

)defines an element of X∗(G), the character group of G. Since the modularquasicharacter of G(AF ) is just this character followed by the adelic norm by(3.10) we deduce that G(AF )1 is unimodular.

Finally, if G is reductive, then the modular quasicharacter of G(AF ) is theproduct of the local modular quasicharacters by our construction of the Haarmeasure, so we deduce that G(AF ) is unimodular by Corollary 3.5.2. ut

Now G(F ) is a discrete subgroup of G(AF ) and G(AF )1. It is in particularclosed in either of these groups. Thus in view of Lemma 3.5.3 and Lemma3.5.4 drg induces a right G(AF )-invariant Radon measure on

G(F )\G(AF )

(see [Fol95, Theorem 2.49]). This quotient only has finite volume if G(AF ) =G(AF )1. Similarly any Haar measure on G(AF )1 induces a right G(AF )-invariant Radon measure on [G]. The latter quotient always has finite volume(see Theorem 2.6.2).

The way we obtained the Haar measure on G(AF ) is a little unnatural.It would be more natural to just take the product

∏v dωv directly. However,

this product is often not convergent. One can introduce a canonical family ofpositive real numbers λv such that∏

v

λvd|ω|v

is convergent, and in this way one obtains the so-called Tamagawa measureon G(AF ). Upon choosing a suitable measure on AG we obtain a measureon G(AF )1 and can thereby obtain the so-called Tamagawa measure on [G].The measure of [G] with respect to the Tamagawa measure is called theTamagawa number of G. This number is intimately connected with thearithmetic of G. We refer to the reader to [Kot88] and the references thereinfor more details.

3.6 Automorphic representations in the L2-sense

Now assume that G is an affine algebraic group over a global field F . Considerthe quotient

[G] := G(F )\G(AF )1

3.6 Automorphic representations in the L2-sense 63

as in (2.12).By Lemma 3.5.3 a right Haar measure on G(AF ) is left invariant under

G(F ) and with respect to it the space [G] has finite volume (compare Theorem2.6.2). Thus we have a well-defined Hilbert space

L2([G])

with inner product defined as in (3.2) (but with X replaced by [G] and dxreplaced by the right Haar measure drg).

Definition 3.3. An automorphic representation of G(AF )1 is an irre-ducible unitary representation π of G(AF )1 that is equivalent to a subquo-tient of L2([G]).

Here a subquotient is a subrepresentation of a quotient representation. Thereason we must consider subquotients, and not just subrepresentations, isdiscussed in §3.7 below. We mention a variant of the definition that we willalso use. When G is reductive, an automorphic representation of AG\G(AF )is an irreducible unitary representation that is equivalent to a subquotientof L2(G(F )AG\G(AF )). When G is reductive we can identify automorphicrepresentations of G(AF )1 and AG\G(AF ) via the commutative diagram

G(AF )1 × L2([G]) −−−−→ L2([G])y∼ y∼AG\G(AF ) −−−−→ L2(AGG(F )\G(AF ))

(3.11)

where the horizontal arrows are the action maps and the vertical arrows arethe isomorphisms induced by G(AF )1→AG\G(AF ).

Note that contrary to the usual convention we have not assumed that Gis reductive. Even without this assumption the preceding definition makessense. It is only when one starts talking about admissible representationsthat the assumption of reductivity is really necessary. A concrete example ofthis is given right after [BS98, Proposition 3.2.11]. The nonreductive case hasnot received as much attention as the reductive case, but see [Sli84].

We end this chapter with §3.7 through 3.10. Sections 3.7 through 3.9discuss how one can decompose unitary representations of locally compactgroups in general, and §3.10 discusses why we have restricted our attentionto affine algebraic groups instead of certain natural larger classes of groups.All of these sections can be omitted on a first reading.

Before delving into these sections we note that our definition of an auto-morphic representation leaves much to be desired in that it is unclear how onewould go about studying these objects and how they are fit into arithmeticand geometry. In the subsequent two chapters we will develop the basic al-gebra of convolution operators on automorphic representations that are usedto study them. It is a refinement of the algebra L1(G(AF )) discussed in §3.3above. For F a number field and the decomposition


G(F∞)×G(A∞F )

naturally gives rise to a decomposition of L1(G(AF )); the first factor is thearchimedean factor which will be discussed in Chapter 4 and the second factoris the nonarchimedean factor which will be discussed in chapters 5 and 8.

This gives us tools to study automorphic representations, but it still doesnot shed much light into how they appear in geometry and arithmetic. Thelink with geometry will be made precise in Chapter 6. The primary link witharithmetic is much more profound, subtle, and conjectural; it is the contentof the Langlands functoriality conjectures, explained in Chapter 12.

3.7 Decomposition of representations

If V is a finite dimensional complex representation of a finite group G, thenit is completely reducible, i.e. admits a direct sum decomposition

V = ⊕iVi

where the Vi are irreducible representations of G (see Exercise 3.13). Thusa subquotient of V in this case is just a subrepresentation. When G is nolonger a finite group, these sorts of decompositions may no longer hold (seeExercise 3.14).

However, there is a very general theory which guarantees some sort ofdecomposition exists if we restrict our attention to unitary representations.Let G be a locally compact (Hausdorff) second-countable group (for example,G(AF ) or AG\G(AF ) for some affine algebraic group G over a global fieldF ). Recall that a Borel space (a.k.a. measurable space) is called standard ifit is isomorphic in the category of measurable spaces to R (with the standardLebesgue measure), Z, or a finite set (with the counting measure). For anyunitary representation π : G→ GL(V ) there is a standard Borel space X witha measure µ and a measurable field of unitary representations (πx, Vx)x∈Xequipped with a G-equivariant Hilbert space isomorphism

V −→∫ ⊕X

Vxdµ(x). (3.12)

This is called a direct integral decomposition. We will not make these no-tions precise here; a nice introduction to these concepts (but without proofs)is given in [Fol95, §7], for proofs one can consult [Dix77]. Two motivatingexamples to keep in mind are the right regular representation of R on L2(R)and the representation L2(Z\R) induced by it. Fourier analysis tells us that

L2(Z\R) = ⊕n∈ZVn and L2(R) =

∫ ⊕t∈R

Vtdt. (3.13)

3.8 The Fell topology 65

Here Vt is the one-dimensional C-vector space on which R acts via t 7→ e2πint.The decomposition of L2(Z\R) is said to be discrete and the decompositionof L2(R) is said to be continuous; this will be discussed in greater generalityin §3.9 below.

Decompositions like the decomposition of L2(R) abound in the theory ofautomorphic forms (compare §10.4). This allows us to explain why one takesan automorphic representation of G(AF )1 to be a subquotient, not just asubrepresentation, of L2([G]). Indeed, one would like to consider all of the Vtoccurring in (3.13) as pieces of the representation on L2(R), but they are notsubrepresentations. They are only subquotients:

Lemma 3.7.1 For every t ∈ R there is a continuous linear R-intertwiningmap

L2(R) −→ Vt.

There is no R-intertwining continuous linear maps

Vt −→ L2(R).

Proof. For each t one has a continuous R-intertwining linear map

L2(R) −→ Vt

f 7−→∫Rf(x)e−2πitxdx

by Fourier theory.For the converse statement, note that a linear R-intertwining map

Vt −→ L2(R)

is either zero or has image contained in the C-span of x 7→ e2πitx. On the otherhand x 7→ e2πitx is not L2(R), so we deduce that any linear R-intertwiningmap Vt → L2(R) is zero. ut

Unfortunately the direct integral decomposition (3.12) is not unique andis fairly badly behaved for general locally compact topological groups. If onerestricts to the class of groups that are of type I then the situation is muchmore pleasant. We will discuss this in §3.9.

3.8 The Fell topology

To describe the refined decomposition that exists for type I groups G it isuseful to define the unitary dual G of G. Our basic reference is [Fol95,

§7.2]. As a set, G is the set of unitary equivalence classes of unitary rep-

resentations. The unitary dual is G equipped with the Fell topology. The


procedure for defining this topology is very similar to the procedure for defin-ing the Zariski topology as in §1.2. To define this topology, we recall from§3.3 every representation (π, V ) of G gives rise to an algebra homomorphismCc(G) −→ End(V ) which extends to L1(G) if π is unitary. We can extendthis homomorphism still further by completing. If we define

‖f‖∗ = sup[π]∈G‖π(f)‖1

then ‖f‖∗ ≤ ‖f‖1 and ‖·‖∗ is a seminorm (here [π] is the equivalence class ofπ). We define C∗(G) to be the completion of L1(G) with respect to ‖·‖∗. Theconvolution operation extends by continuity to C∗(G), and endows it withthe structure of an algebra, called the C∗-algebra of the group G. Thus foreach unitary representation (π, V ) of G we obtain a representation of algebras

C∗(G) −→ End(V ).

This is the motivating example of a C∗-algebra, but we will not give the(elementary) definition of a C∗-algebra, referring instead to [Fol95, §1.1]. Nowgiven a unitary representation π of G, we obtain a closed two-sided ideal

ker(π) = f ∈ C∗(G) : π(f) = 0.

If π is irreducible, this ideal is called a primitive ideal of C∗(G). In factit depends only on the equivalence class of π in the category of unitaryrepresentations. The set of all primitive ideals in C∗(G) is called Prim(G):

Prim(G) := ker(π) : [π] ∈ G.

If we think of this as the analogue of the prime spectrum of a ring, thenproceeding as follows is reasonable: For any subset S ⊂ Prim(G) we let

V (S) :=

p ∈ Prim(G) : p ⊃⋂q∈S

q

.

By convention V (∅) = ∅. Then one checks that the V (S) just defined form theclosed sets of a topology on Prim(G), called the hull-kernel or Jacobsontopology. Now we have a natural map

G −→ Prim(G)

[π] 7−→ ker(π)

and we can pull back the topology on Prim(G) to obtain a topology on G.

In other words the open sets in G are just the inverse images of open sets inPrim(G). This is the Fell topology. Since the Fell topology is a topology

the set of Borel sets on G is defined; this give G the structure of a Borel

3.9 Type I groups 67

space. When speaking of measures and measurable functions on G we willalways mean measures and measurable functions with respect to this Borelstructure (i.e. set of measurable sets). It is useful to point out that if G is oftype I then as a Borel space it is standard [Fol95, Theorem 7.6].

3.9 Type I groups

We have not (and will not) given the definition of a type I group. However,we can give a characterization using the following theorem [Fol95, Theorem7.6]:

Theorem 3.9.1 If G is a second countable locally compact group then thefollowing are equivalent:

(a) The group G is of type I.

(b) The Fell topology on G is T0.

(c) The map G −→ Prim(G) is injective. ut

We point out that the groups of primary concern to us in this book arefortunately of type I:

Theorem 3.9.2 If G is an algebraic group over a local field F then G(F ) isof type I. ut

Proof. For the archimedean case see [Dix77, §13.11.12]. For the nonar-chimedean case see [Sli84]. ut

One can find a proof of the following in [Clo07, Appendix]:

Theorem 3.9.3 If G is a reductive group over a global field F then G(AF )is of type I. ut

It should be pointed out that if G is not reductive then G(AF ) need not beof type I [Moo65].

For π ∈ G let Vπ be the space of (a particular realization of) π. The basicrefinement of (3.12) is the following (see [Clo07, Theorem 3.3]):

Theorem 3.9.4 Let (ρ, V ) be a unitary representation of G. Then there ex-ists a Borel measurable multiplicity function

G −→ 1, 2, . . . ,∞π 7−→ m(π)

and a positive measure µ on G such that

(ρ, V ) =

∫ ⊕G

(⊕m(π)

Vπ

)dµ(π).


The decomposition is unique up to changing the measures and multiplicityfunctions on sets of measure zero. ut

One application of this theorem is that it allows us to define the discretespectrum of a representation V . Since µ is a measure on a standard Borelspace Lebesgue’s decomposition theorem implies that there is a unique de-composition

µ = µpt + µcont

where µpt is zero outside a countable set and µcont is zero on every countableset. The discrete spectrum of V is then

Vdisc :=

∫ ⊕G

m(π)⊕Vπ

µpt(π) (3.14)

and the continuous spectrum is

Vcont :=

∫ ⊕G

m(π)⊕Vπ

µcont(π). (3.15)

We note that since the measure in (3.14) is supported in a countable set thisdirect integral is really a direct sum.

Of course in practice it would be better to have an explicit description ofthe decomposition of Theorem 3.9.4 for naturally occurring representationsV . Langlands’ profound and foundational work on Eisenstein series providesuch a decomposition in our primary case of interest, namely when V =L2([G]). We will return to this point in §10.4 below.

3.10 Why affine groups?

In the preceding discussion we have always assumed that G is an affine alge-braic group. There are two possible avenues for generalization. First, we couldlook at covering groups of G(AF ) that are not algebraic. There is a great dealof beautiful theory that has developed around this topic and it is starting tobecome systematic. However, due to the authors’ lack of experience with thetheory and additional complications that come up we have decided not totreat this topic.

If we forgo covering groups we could still generalize from affine groups toalgebraic groups that are not necessarily affine. To get some idea of how thisenlarges the set of groups in question we recall that a general algebraic groupG over a number field F (not necessarily affine) has a unique linear algebraicsubgroup Gaff E G such that GA := G/Gaff is an abelian variety by a famoustheorem of Chevalley [Con02].

3.10 Why affine groups? 69

Now consider an abelian variety A. In [Con12b] one finds a proof of a resultof Weil stating that A(AF ) can still be defined as a topological space. In fact itis compact [Con12b, Theorem 4.4]. Moreover A(F ) ≤ A(AF ) is still a discretesubgroup. However, it fails to be closed as soon as it is infinite [Con12b,Theorem 4.4] because A is proper over F . Thus the quotient A(F )\A(AF )is not even Hausdorff. Thus A(AF ) and the adelic quotient A(F )\A(AF ) arevery different objects than in the affine case, and we do not know whether ornot it makes sense to pursue automorphic forms in this setting.

Exercises

3.1. In the situation of the proof of Theorem 3.1.1 prove that if ϕ ∈ Cc(X)then ‖π(g)ϕ− ϕ‖∞ → 0 as g → 1.

3.2. Prove that compact (Hausdorff) topological groups are unimodular.

3.3. Show that for a locally compact topological group G the following areequivalent:

dr(hg) = δ(h)drg,

d`(gh) = δG(h)−1d`g,

dr(g−1) = δG(g)−1drg,

d`(g−1) = δG(g)d`g.

Show in addition that

drg = δG(g)d`g.


3.5. Let G be a reductive algebraic group over a global field F , let P ≤ G bea parabolic subgroup, and let K ≤ G(AF ) be a maximal compact subgroupsuch that the Iwasawa decomposition

G(AF ) = P (AF )K

holds (see Theorem 2.7.1). Show that the Haar measures on G(AF ), P (AF ),and K can be normalized so that

dg = d`pdk.

3.6. Show that if Ω ⊂ G(AF ) is a compact set, then there is a finite set ofplaces S of F depending on Ω such that the projection of Ω to G(ASF ) iscontained in a maximal compact open subgroup KS ≤ G(ASF ). Deduce thatthe products (3.9) are left and right Haar measures on G(AF ), respectively.


3.7. Let F be a nonarchimedean local field. Let dx be a Haar measure onF , normalized so that dx(OF ) = 1, and let dx× be a Haar measure on F×,normalized so dx×(O×F ) = 1. Show that

1

1− q−1dx(O×F ) = dx×(O×F )

where q is the order of the residue field of OF .

3.8. Let F be a nonarchimedean local field and let dg be the Haar measureon GL2(F ) such that dg(GL2(OF )) = 1. For n ≥ 0 compute

dg(g ∈ gl2(OF ) ∩GL2(F ) : |det g| = q−n).

3.9. Suppose that G is a connected reductive group over a global field F andthat AG 6= 1. Prove that G(F )\G(AF ) has infinite volume with respect tothe measure induced by a Haar measure on G(AF ).

3.10. Suppose that G is a connected reductive group over a global field F .Show that [G] has finite volume using reduction theory (Theorem 2.7.2).

3.11. Prove that

d (xij) =∧i,jdxij|detxij |

is a (right and left) Haar measure on

GLn(R) := (xij) ∈ gln(R) : det(xij) ∈ R×

if R is either a local field or the adeles of a global field and | · | is the normon the local field or the adelic norm, respectively.

3.12. Let F be a global field and let R be an F -algebra. For (xij) ∈ GL2(R)let

ω =dx11 ∧ dx21 ∧ dx12 ∧ dx22

x11x22 − x21x12

viewed as a left-invariant top-dimensional differential form on G. For a nonar-chimedean place v of F , compute

d|ω|v(GL2(OFv )).

Prove that ∏v

d|ω|v(GL2(OFv ))

does not converge, but that∏v

d|ω|v(GL2(OFv ))

ζF (1)

does converge.

3.10 Why affine groups? 71

3.13. Prove that representation of a finite group G on a finite-dimensional(complex) vector space is completely reducible, i.e. it decomposes into a finitedirect sum of irreducible representations.

3.14. Let B ≤ GL2 be the Borel subgroup of upper triangular matrices.Prove that the natural representation

B(C) −→ GL2(C)

is not completely reducible; that is, it does not decompose into a direct sumof irreducible subrepresentations.

3.15. Let G be a locally compact group. Prove that the convolution producton Cc(G) is associative.

3.16. If π : G → GL(V ) is a unitary representation of G then show thatπ(f) ∈ End(V ) for all f ∈ L1(G).

3.17. Let G be a locally compact group and let f, g ∈ Cc(G). Prove that forany representation π of G one has π(f ∗ g) = π(f) ∗ π(g).

Chapter 4

Archimedean Representation Theory

I have often pondered over theroles of knowledge or experience,on the one hand, and imaginationor intuition, on the other, in theprocess of discovery. I believethat there is a certainfundamental conflict between thetwo, and knowledge, byadvocating caution, tends toinhibit the flight of imagination.Therefore, a certain naivete,unburdened by conventionalwisdom, can sometimes be apositive asset.

attributed to Harish-Chandra byLanglands

Abstract In this chapter we introduce the basic players in representation the-ory on real Lie groups. In particular we define admissibility, (g,K)-modules,infinitesimal characters, and end the chapter with a brief discussion of theLanglands decomposition.

4.1 The passage between analysis and algebra

For the moment let F be a number field and let G be an affine algebraicgroup over F . Consider a Hilbert space representation V of

G(F∞) =∏v|∞

G(Fv).

73

74 4 Archimedean Representation Theory

Since we are interested in automorphic representations, the example oneshould keep in mind, given Definition 3.3, is L2([G]). The fact that V isoften not a space of smooth functions is often inconvenient. It is also notconsonant with the classical theory of automorphic forms (to be discussed inChapter 6), which defines automorphic forms as space of smooth, even realanalytic functions satisfying specific differential equations.

The goal of this chapter is to explain how one passes from the abstractrepresentation V to a space of smooth, even real analytic functions thatessentially determines the original space. An added benefit of taking on thisprocess is that it allows us to pass back and forth from representations ofG(F∞), which involve analysis, (g,K)-modules, which can be studied viaalgebraic means. The §4.2 through §4.4 establish crucial vocabulary for theremainder of the book. At the end of §4.4 we will use this vocabulary todescribe the remaining sections of the chapter, which could be omitted on afirst reading and then referred back to as needed.

Before continuing we note that much of [Bum97, §2.4] is reproduced herewithout essential change. The presentation in loc. cit. seemed nearly perfectso we did not feel the need to modify it. We have of course added materialthat frames these basic results in a form suitable for our purposes later inthe book.

There is a substantial literature dealing solely with archimedean repre-sentation theory, and we have passed over much of it in silence. We cannot,however, omit explicit mention of Harish-Chandra. Anyone studying reduc-tive Lie groups must at some point walk the roads laid by him. His originalpapers are still the only reference for certain results. We mention that Knapp[Kna02] and Wallach [Wal88, Wal92] have written lucid introductions to thetheory in complementary styles.

4.2 Smooth vectors

In this section G denotes an algebraic group over the archimedean field F .Recall that there exists an exponential map

exp : g −→ G(F ).

In the case where G = GLn, the Lie algebra gln is the collection of n × nmatrices. The exponential is simply the matrix exponential in this case:

exp(X) :=

∞∑j=0

Xj

j!.

In general, if one has a faithful representation G → GLn, then this inducesan inclusion g→ gln, and the exponential on g is just obtained by restriction.

4.2 Smooth vectors 75

Let (π, V ) be a Hilbert representation of G(F ). Given ϕ ∈ V we write

π(X)ϕ =d

dtπ(exp(tX))ϕ|t=0

=

(limh→0

π(exp(t+ h)X)ϕ− π(exp(tX))ϕ

h

)|t=0

if the limit exists. We will sometimes simply write Xϕ for π(X)ϕ. We saythat a vector ϕ ∈ V is C1 if for all X ∈ g, the derivative Xϕ is defined. Wedefine Ck inductively by stipulating that ϕ ∈ V is Ck if ϕ is Ck−1 and Xϕis Ck−1 for all X ∈ g. A vector ϕ ∈ V is C∞ if it is Ck for all k ≥ 1.

Definition 4.1. A vector ϕ ∈ V is said to be smooth if ϕ is C∞. Thesubspace of smooth vectors is denoted by Vsm ≤ V .

Lemma 4.2.1 The space Vsm is invariant under G(F ).

Proof. Let g ∈ G(F ) and let X ∈ g. Then

X(π(g)ϕ) = limt→0

1

t(π(exp(tX)g)ϕ− π(g)ϕ)

= π(g) limt→0

1

t(π(exp(tAd(g−1)X)ϕ− ϕ)

where Ad(g−1)X = g−1Xg. The limit exists if ϕ is C1. This implies thatπ(g)ϕ is C1. One shows that if ϕ is Ck then π(g)ϕ is Ck for all k by induction.

ut

Lemma 4.2.2 Let (π, V ) be a Hilbert space representation of G(F ). Thenthe action of g defined above is a Lie algebra representation.

Proof. Recall that C∞(G(F )) is a representation of the Lie algebra g wherethe action is given by sending X ∈ g to dX, that is, differentiation in thedirection of X. The strategy of the proof is to reduce to this case.

Let ϕ0 ∈ V . We claim that one has a g-equivariant map

I := Iϕ0: Vsm −→ C∞(G(F ))

ϕ 7−→ (g 7→ 〈π(g)ϕ,ϕ0〉) .

To prove that I is an intertwining map it suffices to verify that

(dX I)ϕ(g) = (I X)ϕ(g)

(see [God15, §25-26] for more details, especially [God15, Theorem 39]). Forthis we compute:

d

dt(I(ϕ))(g exp(tX))

∣∣∣t=0

=d

dt〈π(g)π(exp(tX))ϕ,ϕ0〉

∣∣∣t=0


= 〈π(g)Xϕ,ϕ0〉= (I X)(ϕ)(g).

Since we are assuming that ϕ is smooth, we see that this function is smoothas well.

In order to verify that Vsm is a representation of g, we must check that

X(Y ϕ)− Y (Xϕ) = [X,Y ]ϕ

for all X,Y ∈ g and all ϕ ∈ Vsm. By duality, it suffices to prove that thisidentity holds after pairing with ϕ0 for all ϕ0 ∈ V , in other words, that

Iϕ0(X(Y ϕ))− Iϕ0(Y (Xϕ)) = Iϕ0([X,Y ]ϕ)

for all ϕ, ϕ0. This is a consequence of the fact that C∞(G(F )) is a represen-tation of g as recalled at the beginning of the proof. ut

Thus Vsm affords a representation of G(F ) and of g. Note that so far wedon’t even know if Vsm is nonzero; it ought to be large in order for thisnotion to be useful. Fortunately, it is indeed large. To make this precise, iff ∈ C∞c (G(F )) then define

π(f)ϕ =

∫G(F )

f(g)π(g)ϕdrg

where drg is a right Haar measure. Let d`g = δG(F )(g)−1drg. It is a left Haarmeasure.

Proposition 4.2.3 If f ∈ C∞c (G(F )) and ϕ ∈ V then π(f)ϕ ∈ Vsm. More-over the space Vsm is dense in V .

Proof. For X ∈ g let

fX(g) =d

dtf(exp(−tX)g)

∣∣∣t=0

.

Then fX ∈ C∞c (G(F )) and∫G(F )

fX(g)

δG(F )(g)π(g)ϕdrg =

d

dt

∫G(F )

f(exp(−tX)g)π(g)ϕd`g∣∣∣t=0

=d

dt

∫G(F )

f(g)π(exp(tX)g)ϕd`g∣∣∣t=0

=d

dtπ(exp(tX))π(fδ−1

G(F ))ϕ∣∣∣t=0

= Xπ(fδ−1G(F ))ϕ.

Since f is arbitrary, this implies that π(f)ϕ ∈ C1. By induction we see thatπ(f)ϕ ∈ Vsm.

4.3 Restriction to compact subgroups 77

For the second claim let ε > 0. The action map

G(F )× V −→ V

(g, ϕ) 7−→ π(g)ϕ

is continuous. This implies that for all ε > 0 there exists a neighborhoodU ⊆ G(F ) of the identity such that |π(g)ϕ− ϕ| < ε for all g ∈ U . Choose anonnegative function f ∈ C∞c (G(F )) supported in U such that∫

G(F )

f(g)drg = 1.

Then

‖π(f)ϕ− ϕ‖2 =

∥∥∥∥∥∫G(F )

f(g)(π(g)ϕ− ϕ)drg

∥∥∥∥∥2

≤∫G(F )

f(g)‖π(g)ϕ− ϕ‖2drg ≤ ε

which implies that π(f)ϕ is as close to ϕ as we wish. Hence Vsm is dense. ut

The subspace of Vsm spanned by vectors of the form π(f)ϕ for some f ∈C∞c (G(F )) is known as the Garding subspace. Note that what we actuallyproved was that the Garding subspace is contained in Vsm and is dense in V .In fact, one has the following [DM78]:

Theorem 4.2.4 (Dixmier-Malliavin theorem) The Garding subspace isequal to Vsm.

Another result from the same paper, known as the Dixmier-Malliavin lemma,also comes up often in practice:

Theorem 4.2.5 (Dixmier-Malliavin lemma) Every function f ∈ C∞c (G(F ))can be written as a finite sum of convolutions:

f =∑i

hi1 ∗ hi2

where hij ∈ C∞c (G(F )). ut

4.3 Restriction to compact subgroups

The representation theory of compact groups is much simpler than the rep-resentation theory of noncompact groups. For example, any irreducible rep-resentation of a compact group is unitarizable and finite-dimensional (seeTheorem 4.3.3 below). A profitable strategy in representation theory is to


analyze the restriction of a given representation to subgroups. We will applythis strategy to reductive groups over the reals in the following section, butin this section we start by recalling what we require from the representationtheory of compact groups.

Let G be a locally compact (Hausdorff) topological group and let K ≤ Gbe a compact subgroup.

Lemma 4.3.1 Let π be a representation of G on a Hilbert space V . Thereexists a Hermitian inner product ( , ) : V × V → C which gives the sametopology as the given pairing on V but with respect to which π|K is unitary,i.e.

(π(k)ϕ1, π(k)ϕ2) = (ϕ1, ϕ2)

for all k ∈ K and ϕ1, ϕ2 ∈ V .

In particular, taking G(F∞) to be compact we see that any representationof a compact Lie group is unitarizable.

Proof. Let 〈 , 〉 denote the original Hilbert space pairing and ‖·‖2 the originalnorm. Define

(ϕ1, ϕ2) =

∫K

〈π(k)ϕ1, π(k)ϕ2〉dk.

By construction it is K-invariant so we need only check the claim about thetopology. The maps K → C given by

k 7−→ 〈π(k)ϕ, π(k)ϕ〉

where ϕ 6= 0 form a family of non-vanishing continuous functions. Thus theoperators π(k) : k ∈ K are a bounded family of operators which implies bythe uniform boundedness principle that their operator norm is bounded. Inparticular there is some nonzero constantC such that ‖π(k)ϕ‖2 < C‖ϕ‖2. Wecan likewise find a similar bound for π(k−1) and so C−1‖ϕ‖2 ≤ ‖π(k)ϕ‖2 ≤C‖ϕ‖2. From this we find that

‖ϕ‖22,new :=

∫K

〈π(k)ϕ, π(k)ϕ〉dk

satisfiesC−2meas(K)‖ϕ‖22 < ‖ϕ‖22,new < C2meas(K)‖ϕ‖22.

This implies the result. ut

We now prepare to state the Peter-Weyl theorem, which says that all ofthe representation theory of a compact Lie group K is contained in L2(K).

Definition 4.2. Let (π, V ) be a continuous representation of G on a Hermi-tian vector space V . A matrix coefficient of π is a function of the form

m : G −→ C

4.3 Restriction to compact subgroups 79

g 7−→ (π(g)ϕ1, ϕ2)

for some ϕ1, ϕ2 ∈ V .

The following proposition implies that irreducible representations of K canbe recovered from their matrix coefficients:

Proposition 4.3.2 Suppose K is a compact Lie group and (π1, V1) and(π2, V2) are two representations of K with π2 unitary. If there exist matrixcoefficients m1,m2 for π1, π2 respectively that are not orthogonal in L2(K)then there exists a non-trivial intertwining operator I : V1 → V2.

Proof. Write ( , )i for the Hermitian pairing on Vi. Let x1, y1 ∈ V1 andx2, y2 ∈ V2 be such that∫

K

(π1(k)x1, y1)1(π2(k)x2, y2)2dk 6= 0.

Let

I(ϕ) =

∫K

(π1(k)ϕ, y1)1π2(k−1)y2dk.

We claim that I gives an intertwining map

I : V1 −→ V2.

Indeed,

π2(g) I(ϕ) =

∫K

(π1(k)ϕ, y1)1π2(gk−1)y2dk.

If we change variables k 7→ kg this becomes I π1(g)ϕ and so I is an inter-twining operator. We now verify that it is nonzero. One has

(I(x1), x2)2 =

(∫K

(π1(k)x1, y1)1π2(k−1)y2dk, x2

)2

=

∫K

(π1(k)x1, y1)1(π2(k−1)y2, x2)2dk

=

∫K

(π1(k)x1, y1)1(π2(k)x2, y2)2dk (V2 is unitary)

6= 0.

Thus I is nonzero. ut

The following theorem tells us that all of the representation theory of acompact Lie group K is contained in L2(K):

Theorem 4.3.3 (Peter-Weyl Theorem) Let K be a compact Lie group.

(a) The matrix coefficients of finite dimensional unitary representations of Kare dense in C(K) and Lp(K) for all 1 ≤ p ≤ ∞.


(b) Any irreducible unitary representation of K is finite dimensional.(c) If (π, V ) is a unitary representation of K, then V decomposes into a

Hilbert space direct sum of irreducible unitary subrepresentations. ut

If we assume that K admits a faithful representation then the proof is notdifficult (see Exercise 4.2). This case suffices for the purposes of this book.However, the usual proof that K admits a faithful representation uses thePeter-Weyl theorem [Kna02, §I.5]

4.4 (g,K)-modules

Let G be an affine algebraic group G over the archimedean field F and letK ≤ G(F ) be a maximal compact subgroup. Our goals in this section areto introduce the notion of admissibility, define (g,K)-modules, and attacha (g,K)-module to each admissible representation. With these concepts inhand, at the end of the section we will describe the content of the remainderof the chapter.

Let (π, V ) be a representation of K. For each equivalence class of irre-ducible representation σ of K we write

V (σ) = ϕ ∈ V : 〈π(k)ϕ : k ∈ K〉 ∼= σ

where the brackets denote the C-span. This is the σ-isotypic subspace. Avector in V (resp. a subspace) of V is said to have K-type σ if it is anelement (resp. a subspace) of V (σ).

Definition 4.3. A vector ϕ ∈ V is K-finite if ϕ ∈ V (σ) for some irreduciblerepresentation σ of K. The algebraic direct sum

Vfin := ⊕σ∈KV (σ)

is the space of K-finite vectors in V .

Here, as in §3.8, K is the unitary dual of K. This can be identified with theset of all equivalence classes of irreducible representations of K because allrepresentations of K are unitarizable. We will later show in Proposition 4.4.2that if V is a Hilbert space, then Vfin is dense in V , and hence V is the Hilbertspace direct sum of the V (σ).

Definition 4.4. A representation V of G(F ) is admissible if for each σ ∈ Kthe dimension of V (σ) is finite.

The notion of admissibility isolates a subcategory of the category of repre-sentations of G that is closed under many natural operators in representationtheory (for example parabolic induction, see §4.9). This category properlycontains the subcategory of unitary representations [Wal88, Theorem 3.4.10]:

4.4 (g,K)-modules 81

Theorem 4.4.1 (Harish-Chandra) If G is reductive then unitary repre-sentations of G(F ) are admissible. ut

Remark 4.1. Unitary representations of nonreductive groups are not in gen-eral admissible. A precise discussion of this phenomenon together with exam-ples that are relevant to classical automorphic forms is contained in [BS98,§3.2].

The basic tool used to analyze admissible representations is the notion ofa (g,K)-module:

Definition 4.5. A (g,K)-module is a vector space V with a representationof π of g and K which satisfy the following:

(a) The space V is a countable algebraic direct sum V = ⊕iVi with each Via finite dimensional K-invariant vector space.

(b) For X ∈ k and ϕ ∈ V we have:

π(X)ϕ = Xϕ =d

dtπ(exp(tX))ϕ

∣∣∣t=0

= limh→0

1

h(π(exp(hX))ϕ− ϕ).

In particular, the limit on the right exists.(c) For k ∈ K and X ∈ g we have π(k)π(X)π(k−1)ϕ = π(Ad(k)X)ϕ.

We say that the (g,K)-module is admissible if V (σ) is finite dimensional

for all σ ∈ K.

A morphism of (g,K)-modules is simply a vector space morphism equivariantwith respect to the action of g and K, and an isomorphism is a morphismthat induces an isomorphism of the underlying vector spaces. One importantdifference between (g,K) modules and Hilbert space representations is thatwe impose no topology on the former. In particular, a submodule of a (g,K)-module V is just a vector subspace of V fixed by the actions of g and K. Sincewe have not required V to be topological, we do not require the subspace tobe closed. A (g,K)-module V is irreducible if it admits no subspaces fixedunder the action of g and K.

The following proposition states that an admissible representation of G(F )yields a (g,K)-module:

Proposition 4.4.2 Let (π, V ) be a representation of G(F ) on a Hilbert spaceV . Let Vfin ≤ V be the space of K-finite vectors. Then Vfin ∩ Vsm is dense inV and invariant under the action of g. If V is admissible, then Vfin ≤ Vsm.

Thus the space of K-finite vectors in an admissible Hilbert space representa-tion of G(F ) is in a natural manner an admissible (g,K)-module. We remarkthat every admissible (g,K)-module is canonically the space of K-finite vec-tors of an admissible representation of G(F ) on a smooth Frechet space ofmoderate growth by a theorem of Casselman and Wallach [Cas89] [Wal92,Chapter 11].

Before giving the proof we state a lemma which we leave as an exercise:


Lemma 4.4.3 Let k = Lie(K). The following are equivalent:

(a) The vector ϕ ∈ V is K-finite.(b) The space 〈π(k)ϕ|k ∈ K〉 is finite dimensional.

If ϕ is smooth, then this is equivalent to

(c) The space 〈π(x)ϕ|x ∈ k〉 is finite dimensional. ut

Thus K-finiteness can be detected using the Lie algebra of K. Assuming thislemma we give the proof of Proposition 4.4.2:

Proof of Proposition 4.4.2: We assume without loss of generality that π|K isunitary. Write

V0 := Vsm ∩ Vfin.

We first prove that V0 is dense in V . Let U be a neighborhood of 1 in G(F )and let ε > 0. Suppose that f is a nonnegative smooth function on G(F )with support in KU such that∫

G(F )

f(g)drg = 1 and

∫G(F )−U

f(g)drg < ε. (4.1)

By making U and ε sufficiently small we can make π(f)ϕ as close as we liketo ϕ for all ϕ in V by an analogue of the argument in the proof of Proposition4.2.3.

It therefore suffices to show that for arbitrary U and ε > 0 we can choose anf satisfying (4.1) such that π(f)ϕ is K-finite. To construct such an f , let letU1 ⊂ G(F ) and W ⊂ K be neighborhoods of 1 such that WU1 ⊂ U , and let f1

be a nonnegative smooth function supported in U1 such that∫G(F )

f1(g)drg =

1. By the Peter-Weyl theorem, there exists a matrix coefficient f0 of a finitedimensional representation of K that is nonnegative such that

∫Kf0(k)dk = 1

and∫K−W f0(k)dk < ε. Let

f(g) :=

∫K

f0(k)f1(k−1g)dk.

Clearly, f has support contained in KU1 ⊂ KU and∫G(F )

f(g)drg = 1.

Moreover, since WU1 ⊂ U , if k ∈ K is such that there exists g ∈ G(F ) − Uwith f1(k−1g) 6= 0 then k 6∈W . Indeed, otherwise we would have

g = k(k−1g) ∈WU1 ⊂ U.

Therefore∫G(F )−U

|f(g)|drg ≤∫G(F )−U

∫K

|f0(k)||f1(k−1g)|dkdrg

=

∫G(F )−U

∫K−W

|f0(k)||f1(k−1g)|dkdrg

4.4 (g,K)-modules 83

≤∫K−W

|f0(k)|∫G(F )

|f1(k−1g)|drgdk

=

∫K−W

f0(k)dk < ε.

Thus f satisfies (4.1). Here we have used the fact that δG(F )(k) = 1 becauseK is compact.

We now show that π(f)ϕ is K-finite. Let ρ be a finite dimensional unitaryrepresentation of which f0 is a matrix coefficient. Thus f0(k) = (ρ(k)ξ, ζ) forsome ξ, ζ in the space of ρ. Then if k1 ∈ K:

f(k−11 g) =

∫K

f0(k)f1(k−1k−11 g)dk

=

∫K

(ρ(k)ξ, ζ)f1(k−1k−11 g)dk

=

∫K

(ρ(k−11 )ρ(k)ξ, ζ)f1(k−1g)dk

=

∫K

(ρ(k)ξ, ρ(k1)ζ)f1(k−1g)dk.

Therefore the linear span of the functions f(k−11 g) is contained in the linear

span of the functions g 7→∫K

(ρ(k)ξ, ζ)f1(k−1g)dk for varying ξ, ζ in thespace of ρ, and this space is finite dimensional. Thus the space spanned bythe vectors π(k1)π(f)ϕ =

∫G(F )

f(g)π(k1g)ϕdg =∫G(F )

f(k−11 g)π(g)ϕdg as

k1 varies over K is finite dimensional, so π(f)ϕ ∈ Vfin. Moreover, π(f)ϕ issmooth for any vector ϕ by Proposition 4.2.3. It follows that V0 is dense inVsm, which is dense in V .

Next we prove that Vfin ≤ Vsm if V is admissible. First observe that V0

is K-invariant since Vsm is by Lemma 4.4.3. Let σ be an irreducible unitaryrepresentation of K. Then V0(σ) ≤ V (σ). Since Vfin is an algebraic directsum of the V (σ) it suffices to show that V0(σ) = V (σ).

Since V (σ) is finite-dimensional by admissibility, V0(σ) admits a well-defined orthogonal complement in V (σ) (this is the only part of the proofwhere admissibility is used). If ϕ is in this orthogonal complement then ϕis orthogonal to all of V0, because it is orthogonal to V (τ) for every τ 6= σ.Therefore ϕ = 0, since V0 is dense. This establishes that V0(σ) = V (σ), andhence V0 = Vfin ≤ Vsm.

Finally, must show that Vfin is invariant under g. Let ϕ ∈ Vfin, let W bethe span of ϕ under K, and let

W1 := 〈Y ϕ | Y ∈ g and ϕ ∈W 〉,

which is clearly finite dimensional. We claim that W1 is fixed by K.


Indeed, if X ∈ k = Lie(K), and Y ϕ ∈ W1, then X(Y ϕ) = [X,Y ]ϕ +Y (Xϕ), which is an element in W1. Therefore the elements of W1 are K-finite by Lemma 4.4.3, and hence Y ϕ is K-finite for all Y ∈ g. 2

Two admissible representations are infinitesimally equivalent if theirunderlying (g,K)-modules are isomorphic. Certainly equivalent representa-tions are infinitesimally equivalent, but the converse is not true in general[Bum97, Exercise 2.6.1]. However, the situation is better if we assume thatboth representations are unitary [HC53, Theorem 8]:

Theorem 4.4.4 If G is reductive then two infinitesimally equivalent unitaryrepresentations of G(F ) are unitarily equivalent. ut

Now that we have introduced the notions of admissibility and the (g,K)-module of an admissible representation we can begin to make use of them.One manifestation of the power of admissibility is that allows us to under-stand the infinite-dimensional space V in terms of finite dimensional sub-spaces. An example of this is given in §4.5.

There is a useful invariant attached to any irreducible (g,K)-module calleda infinitesimal character; its construction will be recalled in §4.6. We then givean example of how (g,K)-modules can be computed and classified explicitlyin the simplest nontrivial case in §4.7.

The real power of admissibility is that one can classify admissible repre-

sentations. Thus the description of the unitary dual G(F ) is reduced to thequestion of which admissible representations are unitary. This is still verydifficult, and at the present time the unitary dual has not been computed forgeneral reductive groups.

The classification itself (which has several forms) is known as the Lang-lands classification. It can be viewed as establishing the local Langlands cor-respondence for archimedean fields. Historically this was the first case of thelocal Langlands conjecture proven. We will review the Langlands classifica-tion in §4.9 after recalling some prerequisites involving tempered and discreteseries representations in §4.8.

4.5 Hecke algebras with K-types

Let σ be an irreducible representation of K. For k ∈ K let

eσ(k) := dim(σ)−1tr(σ)(k−1) (4.2)

This is a smooth function on K. For any Hilbert representation (π, V ) ofG(F ) the operator π(eσ) is defined:

4.5 Hecke algebras with K-types 85

π(eσ) : V −→ V (σ)

ϕ 7−→∫K

eσ(k)π(k)ϕdk(4.3)

These operators are idempotents:

π(eσ)π(eσ′) =

Id if σ ∼= σ′

0 otherwise.(4.4)

One also can define the left and right convolution of eσ with any elementf ∈ C∞c (G(F )):

f ∗ eσ(g) :=

∫K

f(gk−1)eσ(k)dk (4.5)

eσ ∗ f :=

∫K

eσ(k−1)f(kg)dk. (4.6)

These are again elements of C∞c (G(F )).

For any finite subset Ξ of the unitary dual K of K we define

eΞ :=∑σ∈Ξ

eσ

andV (Ξ) := ⊕σ∈ΞV (σ).

(algebraic direct sum).

Definition 4.5.1 The Hecke algebra of K-type Ξ is

C∞c (G(F ), Ξ) := eΞ ∗ C∞c (G(F )) ∗ eΞ . (4.7)

We observe that V (Ξ) is naturally a representation of C∞c (G(F ), Ξ). Weinvite the reader to compare this with the nonarchimedean setting discussedin §5.3. The algebra C∞c (G(F ), Ξ) is often called the spherical Hecke algebraof type Ξ, but we will avoid this terminology because it is usually reservedfor the special case where Ξ is one-element set consisting of the the trivialrepresentation.

If (π, V ) is admissible, then V (Ξ) is finite-dimensional for all Ξ. This ob-servation is quite useful in practice as it allows us to reduce certain questionsabout the infinite-dimensional representation V of G(F ) to the family offinite-dimensional representations of the algebras C∞c (G(F )). We give someexamples in the remainder of this section.

First we state a very general version of Schur’s lemma. Let V be a vectorspace and let Λ ⊆ End(V ) be a subset of its set of endomorphisms. We saythat Λ acts irreducibly if for any subspace W of V such that ΛW ≤ W


either W = 0 or W = V . For a proof of the following version of Schur’slemma see [Wal88, §0.5].

Lemma 4.1. Assume that V is a vector space of countable dimension andthat Λ ⊆ End(V ) acts irreducibly. If T ∈ End(V ) commutes with every ele-ment of Λ then T is a scalar multiple of the identity operator. ut

We also require the following version of the Jacobson density theorem:

Theorem 4.5.2 Let V be a vector space of countable dimension and let A <End(V ) be a C-subalgebra (possibly without identity) that acts irreducibly.Then for any ϕ1, . . . , ϕn, ϕ

′1, . . . , ϕ

′n ∈ V with ϕ1, . . . , ϕn linearly independent

there is an a ∈ A such that aϕi = ϕ′i for all i. ut

We warn the reader that the Jacobson density theorem is usually statedfor modules over rings with identity. For a version that avoids the idenityassumption we refer to [Jac64, §II.2].

Proposition 4.5.3 A Hilbert space representation (π, V ) of G(F ) is irre-ducible if and only if V (Ξ) is an irreducible C∞c (G(F ), Ξ)-module for allΞ.

Proof. Suppose V is reducible, that is, V = V1 ⊕ V2 as a representation ofG(F ), where V1 and V2 are closed subrepresentations. Then

V (Ξ) = V1(Ξ)⊕ V2(Ξ)

as C∞c (G(F ), Ξ)-modules for all finite subsets Ξ ⊂ K. By Proposition 4.4.2V1(Ξ) and V2(Ξ) are nonzero for Ξ large enough.

Conversely, suppose V is irreducible, and suppose that

V (Ξ) = V1 ⊕ V2

as C∞c (G(F ), Ξ)-modules for some Ξ ⊂ K. Here V1 and V2 are two subspacesof V (Ξ). If V1 6= 0 then

V1 = C∞c (G(F ), Ξ)V1 = eΞ ∗ C∞c (G(F )) ∗ eΞV1 = eΞ ∗ C∞c (G(F ))V1 (4.8)

where we have used Theorem 4.5.2 for the first equality. It is clear thateΞ ∗ C∞c (G(F ))V1 is a subspace of V (Ξ). If it is a proper subspace then theinverse image of eΞ ∗ C∞c (G(F ))V1 under

eΞ : V 7→ V (Ξ)

is a proper closed G(F )-invariant subspace subspace of V , contradicting ourirreducibility assumption. Thus eΞ ∗ C∞c (G(F ))V1 = V (Ξ), and we deducefrom this and (4.8) that V1 = V (Ξ) and hence V2 = 0. ut

Combining (4.5.3) and Lemma 4.1 we obtain the following corollary:

4.6 Infinitesimal characters 87

Corollary 4.5.4 If (π, V ) is an irreducible admissible Hilbert space rep-resentation of G(F ) then T ∈ End(Vfin) commuting with the action of

C∞c (G(F ), Ξ) for all finite subsets Ξ ⊂ K is a scalar multiple of the identityoperator. ut

The following proposition will be used in the proof of Proposition 16.2.3.It is an immediate consequence of Theorem 4.5.2 and Corollary 4.5.4:

Proposition 4.5.5 Let (π, V ) be an admissible Hilbert space representation

of G(F ) and let Ξ ⊂ K be finite. Let ϕ1, . . . , ϕn, ϕ′1, . . . , ϕ

′n ∈ ⊕σ∈ΞV (σ)

with ϕ1, . . . , ϕn linearly independent. Then there exists an f ∈ C∞c (G(F ), Ξ)such that π(f)ϕi = ϕ′i for all i. ut

4.6 Infinitesimal characters

Let g be a Lie algebra over an archimedean field F . We let

gC := g⊗R C.

We use this odd notation as a reminder that if F is complex then gC 6∼= g,whereas gC := g⊗C C is isomorphic to g in this case.

We denote the universal enveloping algebra of gC by U(g). This is a uni-tal associative algebra. Any associative algebra becomes a Lie algebra withbracket [X,Y ] = XY − Y X, and thus we can regard U(g) as a Lie algebra.The universal enveloping algebra is equipped with an injective Lie algebramorphism

g⊗R C −→ U(g)

by the Poincare-Birkoff-Witt theorem. This morphism is univeral in thatgiven any Lie algebra homomorphism from gC → A to a unital associativealgebra A (equipped with the Lie algebra structure mentioned above) thereis a unique morphism U(g)→ A such that the following diagram commutes:

gC //

!!

U(g)

A

We denote by Z(g) the center of U(g).We now present the explicit description of Z(g) (for reductive g) afforded

by the Harish-Chandra isomorphism. Assume that g is reductive and let t ≤ gbe a Cartan subalgebra. For example, we could choose a connected reductivealgebraic group G over F with Lie algebra g and a maximal torus T ≤ Gwith Lie algebra t. We then obtain a root system


Φ := Φ(G×Spec(R) Spec(C), T ×Spec(R) Spec(C))

as in §1.8. Let ∆ ⊂ Φ be a base and let Φ+ ⊂ Φ be the associated set ofpositive roots. We define the Harish-Chandra homorphism

γ : Z(g) −→ U(t) −→ Z(t)

X 7−→ X − ρ(X)1

where the first map is the projection, and

ρ := 12

∑α∈Φ+

α. (4.9)

Remark 4.2. This is our first time explicitly mentioning ρ, the ubiquitoushalf-sum of positive roots. It is incorporated into the definition to make itcompatible with parabolic induction (see §4.9).

Let W := W (G,T )(C) denote the Weyl group of T in G; it acts on tC andhence U(t). For the proof of the following theorem of Harish-Chandra, see[Wal88, Theorem 3.2.3]:

Theorem 4.6.1 The linear map γ is an algebra homomorphism, is indepen-dent of the choice of base ∆, has image in U(t)W , and induces an isomor-phism

γ : Z(g) −→ U(t)W .

ut

The theorem allows us, in particular, to explicitly describe the characters ofZ(g) as we now explain. Suppose we are given λ ∈ (tC)∨, the C-linear dual oftC. The linear map λ : tC → C is tautologically a Lie algebra map because tC

is commutative. By the universal property of universal enveloping algebrasthis extends uniquely to an algebra morphism λ : U(t) −→ C. We let

χλ := λ|U(t)W γ : Z(g) −→ C.

This is an algebra morphism (i.e. a character) from Z(g) to C.

Proposition 4.6.2 Every algebra morphism from Z(g) to C is of the formχλ for some λ ∈ (tC)∨. Two λ define the same χλ if and only if they are inthe same orbit under W .

Proof. Since tC is a commutative Lie algebra (i.e. its bracket is identicallyzero) U(t) is just the symmetric algebra Sym(tC). This is canonically isomor-phic to the polynomial algebra C[(tC)∨]. Thus we obtain a bijection

Spec(C[(tC)∨],C)−→Hom(U(t),C).

4.7 Classification of (g,K)-modules for GL2R 89

The set on the left can be identified with the maximal ideals of (tC)∨, which,by the Hilbert Nullstellensatz, are in natural bijection with elements of (tC)∨.Taking W invariants we obtain the proposition. ut

A g-module V is said to admit an infinitesimal character if Z(g) acts viaa character on V . By Theorem 4.6.1 and Proposition 4.6.2 such a charactercan be identified with an element of (tC)∨.

The following is an immediate corollary of Lemma 4.1:

Corollary 4.6.3 Let V be an irreducible admissible (g,K)-module and letΛ : V → V be a C-linear map commuting with the actions of g and K. ThenΛ is multiplication by a scalar. utIt follows immediately from the corollary that any irreducible (g,K)-moduleadmits an infinitesimal character. This is a very useful invariant that is sur-prisingly closely tied to arithmetic.

4.7 Classification of (g,K)-modules for GL2R

In this section we state the classification of admissible irreducible (g,K)-modules when

g := gl2 and K := O2(R).

We refer to [Bum97, §2.5] for proofs and many more details.Define

H :=

(1 00 −1

), X :=

(0 10 0

), Y :=

(0 01 0

), Z :=

(1 00 1

). (4.10)

These elements form a basis for g. If we set

∆ = (1/4)(H2 + 2XY + 2Y X)

then Z(g) = 〈∆,Z〉 (see Exercise 4.8). Finally we set

wθ :=

(cos θ − sin θsin θ cos θ

).

For every (s, µ) ∈ C2 consider the character

χ : Z(g) −→ CZ 7−→ µ

∆ 7−→ s(1− s).

Then for each ε ∈ 0, 1 one has a (g,K)-module

(π, V ) = (πs,µ,ε, Vs,µ,ε)


with infinitesimal character χ constructed as follows. The space is

V =⊕

`≡ε (mod 2)

Cv`.

The action is given by

(a) π(wθ)v` = ei`θv` and π(

1 00 −1

)v` = v−`

(b)Xv` =(s+ `

2

)v`+2,

(c) Y v` =(s− `

2

)v`−2,

(d)∆v` = s(1− s)v`,(e) Zv` = µv`.

This is an admissible (g,K)-module. It is irreducible unless s = k2 where

k is an integer congruent to ε modulo 2. If k ≥ 1 then π k2 ,µ,ε

has has a

unique irreducible infinite dimensional subrepresentation denoted by πk. Theset of K-types of πk consist of the v` with |`| ≥ k. The quotient by thissubrepresentation is a finite dimensional representation, and all irreduciblefinite dimensional representations are obtained in this manner.

If πs,µ,ε is irreducible it is known as an irreducible principal seriesrepresentation. If it is reducible, πk is known as a discrete series represen-tation if k 6= 1 and a limit of discrete series if k = 1. This terminologyshould make more sense after the next two sections. It is clear (upon con-sidering K-types, for example) that if πk1

∼= πk2 then k1 = k2, and that anirreducible representation cannot be both in the principal series and in thediscrete series. Moreover, it is easy to see that if

πs1,µ1,ε1∼= πs2,µ2,ε2

then µ1 = µ2, ε1 = ε2, and either s1 = s2 or s1 = 1− s2.The following is the classification result mentioned in the title of this

section:

Theorem 4.7.1 Any infinite dimensional irreducible admissible (g,K)-moduleis isomorphic to πs,µ,ε for some s, µ, ε or πk for some k ≥ 1. ut

4.8 Matrix coefficients

Let G be a reductive group over the local field F . For the moment we do notassume that F is archimedean. Let (π, V ) be a representation of G(F ). InDefinition 4.2 above we defined notion of a matrix coefficient. In this sectionwe explain how matrix coefficients can be used to isolate important classesof representations of G(F ).

We have already seen in §2.6 that the center of G often causes analyticproblems for somewhat trivial reasons. One often circumvents this difficulty

4.8 Matrix coefficients 91

by defining classes of representations based on their restriction to appropriatesubgroups. In analogy with (2.10) we let

G(F )1 :=⋂

χ∈X∗(G)

ker (| · | χ : G(F )→ R>0) . (4.11)

Definition 4.6. An irreducible representation (π, V ) of G(F ) is in the dis-crete series or square integrable (resp. is tempered) if its matrix coef-ficients lie in L2(G(F )) (resp. L2+ε(G(F )) for any ε > 0). It is essentiallysquare integrable (resp. essentially tempered) if the matrix coefficientsof π|G(F )1 lie in L2(G(F )1) (resp. L2+ε(G(F )1)).

We concentrate on square integrable and tempered representations in thefollowing discussion, but similar results hold for essentially square integrableand essentially tempered representations.

Discrete series representations can be realized as subrepresentations ofL2(G(F )), and hence are always unitarizable (see [Dix77, Chapter 14] fordetails). Additionally, tempered representations are infinitesimally equivalentto unitary representations in the archimedean case [Kna02, Theorem 8.53]and equivalent to unitary representations in the nonarchimedean case [Sil79,Corollary 4.5.13]. We note that in many references, including [Sil79, §4.5],one finds a more technical definition of temperedness. Once one knows thattempered representations are unitary, one can prove that the more technicaldefinition (which we have not and will not give) is equivalent to the easilystated one above using the arguments of [CHH88].

Now that we have defined these classes of representations, we ought tocomment briefly on their significance. We concentrate on the notion of tem-peredness. If π is an admissible representation of G(F ) let Bπ be an orthonor-mal basis of the space of π. For f ∈ C∞c (G(F )) the sum

Θπ(f) =∑ϕ∈Bπ

〈π(f)ϕ,ϕ〉

converges absolutely (this is obvious in the nonarchimedean case, see §8.5, andin the archimedean case it is proven in [Kna02, Theorem 10.2]). Technicallyspeaking we have not defined admissibility in the nonarchimedean case; it isdefined in §5.3. The linear functional

Θπ : C∞c (G(F )) −→ C

is known as the character of π.In the context of Fourier theory on R, tempered distributions are precisely

those which admit a Fourier transform; in other words, they are analyticallyfairly well-behaved. In the archimedean case tempered representations areprecisely those whose characters define a tempered distribution on a certainSchwartz space containing C∞c (G(F )) [Kna02, Theorem 12.23] [Sil79, §4.5].


There is also a version of the Plancherel theorem for G(F ) relating spaces offunctions on G(F ) and tempered representations of G(F ). It takes the form

f(g) =

∫G(F )

Θπ(g)dµPl(π)

where µPl(π) is the Plancherel measure on the unitary dual. Though thisis called the Plancherel theorem, it is really more like a Fourier inversiontheorem. In the archimedean case this is [Wal92, Theorem 13.4.1]. In thenonarchimedean case this is [Wal03].

Finally the condition of temperedness is also a representation theoreticencapsulation of how well-behaved automorphic L-functions are. This will bediscussed in the context of the Ramanujan conjecture in Conjecture 10.6.4.

4.9 The Langlands classification

The Langlands classification gives a description of representations of reduc-tive groups over archimedean local fields in terms of tempered representationsof Levi subgroups. We state it in this section. There is a refinement wherethe tempered representations are replaced by discrete series representationsand limits of discrete series representations. We will touch on this in §10.5 inthe case G = GLn. For the general case we refer to [Kna86, §XIV.17].

Let G be a reductive group over the archimedean local field F . Fix a min-imal parabolic subgroup P0 and call a parabolic subgroup P ≤ G standardif it contains P0. Let P ≤ G be a standard parabolic subgroup. Since F hascharacteristic zero, it admits a Levi decomposition as in Theorem 1.5.4

P = MN

where M is a Levi subgroup of P and N is it unipotent radical. We choose amaximal compact subgroup K ≤ G(F ) such that the Iwasawa decompositionG(F ) = P (F )K holds (see Appendix A).

We let AM be the identity component in the real topology of the largestR-split torus T in the center of ResF/RM and we let

M(F )1 :=⋂

χ∈X∗(M)

ker (| · | χ : M(F ) −→ R>0) ,

where the reader recalls the notion of X∗(M)F in §1.7. Then AMM(F )1 =M(F ) and the product is direct. The Langlands decomposition of P (F )is

P (F ) = N(F )M(F )1AM .

4.9 The Langlands classification 93

If we let aM := Hom(X∗(M)F ,R) then aM can be identified with LieAM .We then have a map

HM : M(F ) −→ aM (4.12)

defined by〈HM (g), χ〉 = log |χ(g)|

for χ ∈ X∗(M)F and we can identify M(F )1 := kerHM ∩M(F ). We nowextend HM to HP by extending the characters in X∗(M)F to P by declaringthem to be trivial on N . We then obtain HP : P (F ) −→ aM ; this providesan isomorphism X∗(M)F →X∗(P )F .

For each λ ∈ a∗MC := Hom(aM ,C) we therefore obtain a character

P (F ) −→ C×

p 7−→ e〈HP (p),λ〉. (4.13)

Now consider the set of roots of a∗M can be identified with the vector spaceunderlying the associated root system Φ(G,T ). We let ρ denote half the sumof the positive roots defined by the parabolic subgroup P (see §1.9). For eachrepresentation (σ, V ) ofM(F )1 one can form the representation (σ, λ) of P (F )by extending σ trivially to P (F ) and twisting by the character 〈HP (·), λ〉.We can then form the (normalized) induced representation

I(σ, λ)

of G(F ). A dense subspace of the space of I(σ, λ) consists of continuousfunctions ϕ : G(F ) −→ V such that

ϕ(nmg) = e〈HP (m),λ+ρ〉σ(m)ϕ(g). (4.14)

This space is equipped with an inner product 〈ϕ1, ϕ2〉 =∫Kϕ1(k)ϕ2(k)dk.

The whole space is the completion with respect to this inner product. Finallythe action of G(F ) is given by

I(σ, λ)(g)ϕ(x) := ϕ(xg).

The ρ is incorporated so that if σ ⊗ 〈HP (·), λ〉 is unitary then so is I(σ, λ)[Kna02, §VII.1]. This also explains the parenthetic “normalized” mentionedabove.

The following is known as the subquotient theorem (see [Kna02, The-orem 7.24]):

Theorem 4.9.1 If σ is unitary and tempered and λ lies in the positive Weylchamber then I(σ, λ) admits a unique irreducible quotient J(σ, λ). ut

The quotient J(σ, λ) is known as the Langlands quotient and σ, λ areknown as Langlands data.


We can now give the Langlands classification [Kna02, Theorem 8.54]:

Theorem 4.9.2 Every irreducible admissible representation of G(F ) is iso-morphic to some J(σ, λ). Moreover, if we insist that the parabolic sub-group defining J(σ, λ) is standard, fix a Levi decomposition of each standardparabolic subgroup, and stipulate that λ is in the positive Weyl chamber thenevery irreducible representation of G(F ) is isomorphic to a J(σ, λ) that isunique up to replacing σ by another representation of M(F )1 equivalent toσ. ut

There is also a version of the Langlands classification in the nonar-chimedean setting, for this we refer to §8.4.

Exercises

4.1. Let T be a torus over R and let K ≤ T (R) be a maximal compactsubgroup. Show that there exists an isomorphism between the C-vector spaceof K-finite functions in C∞c (T (R)) and C∞c (AT )×C C(K). Here by C(K) wemean the free vector space on the isomorphism classes of irreducible unitaryrepresentations of K.

4.2. Let K be a compact Lie group that admits a faithful representationK → GLn(C). We can then view K as a subgroup of gln(C) which can beviewed as a real vector space. Call a function on K polynomial if it is therestriction to K of a polynomial on this real vector space. Show that everypolynomial function on K is the matrix coefficients of a finite dimensionalrepresentations ofK, and deduce part (a) of the Peter-Weyl theorem from thisand the Stone-Weierstrass theorem (Theorem 4.3.3). Using part (a) deduceparts (b) and (c) as well.


4.4. Let (π, V ) be an irreducible unitary representation of G(F ) where G isa reductive group over the local field F and V is a Hilbert space. Show thatthere is a quasi-character ωπ : ZG(F ) → C× such that π(z) acts via ωπ(z)on V for all z ∈ ZG(F ). The quasi-character ωπ is called the central quasi-character of π. Show that if χ : G(F )→ C× is a quasi-character, then thecentral quasi-character of π ⊗ χ is χ|ZG(F )ωπ, where ZG is the center of G.

4.5. Give an example of a reductive group G over R and an irreducible ad-missible representation of G(R) that is not unitary.

4.6. Let g be a semisimple Lie algebra over an archimedean field F and letgC := g ⊗R C. Let B be the Killing form on g and denote also by B itsextension to gC. It is nondegenerate. Let Xi denote a basis for gC anddenote by Xi the dual basis with respect to B. The element

4.9 The Langlands classification 95

∆ :=∑i

XiXi ∈ U(g)

is known as the Casimir element. Prove that ∆ is independent of the choiceof basis and that ∆ ∈ Z(g).

4.7. Prove that the image of eσ is contained in V (σ) as asserted in (4.3).

4.8. Prove that if g := gl2 then Z(g) = 〈H,∆〉 in the notation of (4.7).

4.9. Prove that the space Vs,µ,ε of §4.7 is indeed a (g,K)-module with thegiven action.

4.10. In the notation of §4.7 prove that if πk1∼= πk2

then k1 = k2, and thatif

πs1,µ1,ε1∼= πs2,µ2,ε2

then µ1 = µ2, ε1 = ε2, and either s1 = s2 or s1 = 1− s2.

4.11. Prove that all finite dimensional (gl2,O2(R))-modules are quotients(resp. subrepresentations) of some Vs,µ,ε for some s, µ, ε, and identify theVs,µ,ε of which they are quotients (resp. subrepresentations).

Chapter 5

Representations of Totally DisconnectedGroups

I will tell you a false proof. Butlike every fairy tale, it has akernel of truth.

D. Kazhdan

Abstract In this chapter our goal is to develop enough of the representa-tion theory of locally compact totally disconnected groups (or td groups forshort) to state a refined definition of an automorphic representation in thenext chapter. At the end of the chapter we prove Flath’s theorem. This funda-mental result implies that automorphic representations can be factored intolocal representations indexed by the places of the global field over which therepresentation is defined.

5.1 Totally disconnected groups

We start with the following definition, following [Car79]:

Definition 5.1. A topological group G is td or of td-type if every neigh-borhood of the identity contains a compact open subgroup.

Here and the rest of this chapter, the letters td stand for totally disconnected.Our basic examples are given by the next lemma, which follows readily

from the definition of the topology on the points of an algebraic group from§2.2:

Lemma 5.1.1 Let G be an algebraic group over a local nonarchimedean fieldF . Then G(F ) is of td-type. Similarly, if F is a global field and S is a finiteset of places of F including the archimedean places if F is a number field,then G(ASF ) is a td group. ut

97

98 5 Representations of Totally Disconnected Groups

A td group is Hausdorff and locally compact, so the theory of §3.1 isapplicable. A td-type group is also totally disconnected, which explains theterminology (see Exercise 5.1). Thus the topology on a td group is verydifferent from that of a connected Lie group. In fact, in stark contrast to thedefining property of td-type group, a connected Lie group satisfies has nosmall subgroups in a sense made precise in the following lemma:

Lemma 5.1.2 Let G be a connected Lie group. Then there is a neighborhoodU of the identity so that for all g ∈ U − 1 there is an integer N dependingon g such that gN 6∈ U .

Proof. It is a standard result that there are neighborhoods V0 of 0 in Lie(G)and U1 of 1 in G such that the exponential map exp : V0 → U1 is a diffeomor-phism. Let U = exp

(12V0

). Then if g = exp( 1

2v) ∈ U − 1 for some v ∈ V0

one has

gn =

n times︷︸︸︷exp

(1

2v

). . . exp

(1

2v

).

Choosing n large enough that n2 v 6∈ V0 we deduce the result. ut

This proof illustrates the difference between nonarchimedean and archimedeanmetrics, see Exercise 5.8.

Our aim in this chapter is to use the special topological properties of tdgroups to refine our understanding of their representation theory. We dealwith the basic theory in this chapter, including the local Hecke algebra (see§5.2), the definition of admissible in this context (see §5.3, and Flath’s the-orem on decomposition of admissible representations (Theorem 5.7.1). Thereader will notice that in this section we essentially make no use of the factthat the groups in question arise as the points of reductive groups in nonar-chimedean local fields. The (deeper) part of the theory that requires thisstructure is relegated to Chapter 8.

Unless otherwise specified for the remainder of this chapter we let G be alocally compact td group.

5.2 Smooth functions in td groups

The first step in understanding the representation theory of a td-group is todefine what we mean by a smooth function:

Definition 5.2. A function f : G → C is smooth if it is locally constant.The complex vector subspace of C(G) (resp. Cc(G)) consisting of smoothfunctions is denoted by C∞(G) (resp.C∞c (G)).

It turns out that C∞c (G) is preserved under convolution, defined as in§3.3. Thus C∞c (G) is an algebra under convolution. It is known as the Hecke

5.2 Smooth functions in td groups 99

algebra of G. If K ≤ G is a compact open subgroup, then we let

C∞c (G //K) ≤ C∞c (G)

denote the subalgebra of functions that are right and left K-invariant.

Lemma 5.2.1 Any element f ∈ C∞c (G) is in C∞c (G //K) for some compactopen subgroup K. If f ∈ C∞c (G //K), then f is a finite C-linear combinationof elements of the form

1KγK

for γ ∈ G. Here 1X denotes the characteristic function of a set X.

Proof. By local constancy of f for every γ in the support supp(f) we canchose a compact open subgroup K(γ) ≤ G so that f is constant on γK(γ).Then ⋃

γ

γK(γ)

is an open cover of supp(f), so it admits a finite subcover⋃ni=1 γiK(γi) since

f has compact support. Let Kr :=⋂ni=1K(γi); it is a finite intersection of

compact open subgroups so it is itself a compact open subgroup. We seethat f is right Kr-invariant. Similarly we can find a compact open subgroupK` ≤ G such that f is left K`-invariant, and letting K := K` ∩ Kr we seethat f ∈ C∞c (G //K).

Assume f ∈ C∞c (G //K). The union⋃γ∈supp(f)

KγK

is an open cover of supp(f), which therefore admits a finite subcover. Thelast claim in the lemma follows. ut

Note that C∞c (G //K) ≤ C∞c (G //K ′) if K ≥ K ′. Thus, in view of thelemma, to explicitly describe the convolution operation on C∞c (G) it sufficesto describe

1KαK ∗1KβKfor a fixed compact open subgroup K ≤ G and α, β ∈ G. We can write

KαK = qiKαi and KβK = qjβjK

where both sums are finite and αi, βj ∈ G. We then have

1KαK ∗1KβK =∑i,j

1KαiβjK . (5.1)

Example 5.1. All compact open subgroups of GLn(A∞Q ) are of the form


KS

∏p 6∈S

GLn(Zp)

for S a finite set of finite primes and KS is a compact open subgroup ofGLn(QS). The subgroup GLn(Z) =

∏p GLn(Zp) ≤ GLn(A∞Q ) is a maximal

compact open subgroup, and all maximal compact open subgroups are conju-gate to this maximal compact open subgroup [Ser06, Chapter IV, Appendix1]. Examples of nonmaximal compact open subgroups are given by the kernelof the reduction map

GLn(Z) −→ GLn(Z/m)

for integers m.

Let G be an algebraic group over F and assume that we have chosena model for G over OF which we denote by the same letter by abuse ofnotation. If γ ∈ G(A∞F ), then γ ∈ G(OSF ) for some finite set S of places. IfK∞ ≤ G(A∞F ) is a compact open subgroup, then upon enlarging S we can

assume that KS = G(OSF ). For such a choice of S we have

1K∞γK∞ = 1KSγSKS ⊗1KS

for some finite set of nonarchimedean places S. This reduces the study ofnonarchimedean Hecke algebras to the study of the local Hecke algebras

C∞c (G(Fv))

as v varies over nonarchimedean places v of F . This idea will be formalizedin a very useful manner in §5.7 below.

5.3 Smooth and admissible representations

Definition 5.3. A representation (π, V ) of G on a complex vector space Vis smooth if the stabilizer of any vector in V is open in G.

Equivalently, V is smooth if and only if

V =⋃K≤G

V K

where the superscript denotes the subspace of fixed vectors and where theunion is over all compact open subgroups K ≤ G.

In this definition we do not assume that the representation V is continuous,or for that matter even give a topology on V . In fact, V is smooth if and onlyif the representation is continuous if with respect to the given td topology onG and the discrete topology on V . We note that the smooth representations of

5.3 Smooth and admissible representations 101

a given td group form a category where morphisms are simply G-equivariantC-linear maps. The category is even abelian.

It is sometimes useful to rephrase the condition of smoothness in terms ofthe associated representation of the algebra C∞c (G). For this we recall thata module M for an algebra A is nondegenerate if every element of M canbe written as a finite sum

a1m1 + · · ·+ anmn

for some (ai,mi) ∈ A ×M . Of course, this is trivially true if A contains anidentity, but we will be applying this concept when A = C∞c (G), which doesnot have an identity. However, it has approximate identities in the followingsense. For each compact open subgroup K ≤ G let

eK :=1

meas(K)1K .

Then eK is the identity element of the algebra

C∞c (G //K) = eK ∗ C∞c (G) ∗ eK .

If V is a smooth representation of G, then it is naturally a C∞c (G)-moduleand C∞c (G //K) preserves

V K = eKV.

This observation is in fact the key to the proof of the following lemma, whichwe leave as an exercise:

Lemma 5.3.1 There is an equivalence of categories between nondegenerateC∞c (G)-modules and smooth representations of G. ut

Using this equivalence we prove the following irreducibility criterion:

Proposition 5.3.2 A smooth G-module V is irreducible if and only if V K

is an irreducible C∞c (G //K)-module for all compact open subgroups K ≤ G.

Proof. Suppose V is reducible, that is, V = V1 ⊕ V2 as C∞c (G)-modules,where V1 and V2 are nonzero. Then

V K = V K1 ⊕ V K2

as C∞c (G //K)-modules for all compact open subgroups K ≤ G, and forsufficiently small K the spaces V K1 and V K2 are nonzero by smoothness.Conversely, suppose V is irreducible, and suppose that

V K = V1 ⊕ V2

as C∞c (G //K)-modules for some compact open subgroup K ≤ G. Here V1

and V2 are two subspaces of V K (necessarily fixed by K). If V1 6= 0 then byirreducibility of V one has


V1 = C∞c (G //K)V1 = eK ∗ C∞c (G) ∗ eKV1 = eK ∗ C∞c (G)V1 = eKV = V K .

Thus V2 = 0. ut

We now come to the most important definition of the chapter:

Definition 5.4. A representation V of G is admissible if it is smooth andV K is finite dimensional for every compact open subgroup K ≤ G. A C∞c (G)-module V is admissible if it is nondegenerate and eKV is finite dimensionalfor all compact open subgroups K ≤ G.

It is immediate that a representation V of G is admissible if and only if itsassociated C∞c (G)-module is admissible.

Jacquet and Langlands introduced this definition in their classic work[JL70], though some indication of the definition in the real case appearedin work of Harish-Chandra [HC53]. The importance of the definition is thatit isolates exactly the correct category of representations to study, and at thesame time eliminates extraneous topological assumptions (e.g. the presenceof a Hilbert or pre-Hilbert space structure on V ). One indication that thisis the correct category is the following theorem, which is proven as part ofTheorem 8.3.5 below:

Theorem 5.1. Let F be a nonarchimedean field. Assume that G is the F -points of a reductive algebraic group over F . Then an irreducible smoothrepresentation of G is admissible.

Another indication that admissible representations are the correct categoryto study is the fact that unitary representations of G give rise to admissiblerepresentations. To explain this we require some preparation. Recall thatin the archimedean setting of §4.4 we passed from Hilbert representationsto their subspace of K-finite vectors (K a maximal compact subgroup ofthe relevant group) in order to define a notion of admissibility for them. Werequire a similar process here, but it is slightly simpler. Given a Hilbert spacerepresentation V of G we let

Vsm :=⋃K

V K ≤ V

where the union is over all compact open subgroups K ≤ G. Then Vsm ≤V is evidently a smooth subrepresentation of G. We say that the Hilbertrepresentation V is admissible if Vsm is admissible. We note that in thiscase Vsm is a pre-Hilbert representation.

Lemma 5.3.3 Let V be a Hilbert representation of G. The subspace Vsm ≤ Vis dense and G-invariant. Moreover, if V and W are Hilbert representationsand Vsm →Wsm is a G-intertwining morphism then it extends uniquely to aG-intertwining continuous morphism V → W . If the morphism Vsm → Wsm

is an isomorphism, then so is V →W .

5.4 Contragredients 103

We leave the proof as an exercise (see Exercise 5.11). Thus, by the lemma,we can detect isomorphisms of Hilbert representations using the underlyingrepresentations in the smooth category. We also have the following analogueof Theorem 4.4.4:

Proposition 5.3.4 If V1, V2 are unitary representations of G and theirspaces of smooth vectors V1sm and V2sm are equivalent then V1 and V2 areunitarily equivalent.

We give the proof of Proposition 5.3.4 in §5.4. Finally, we have the followingnonarchimedean analogue of Theorem 4.4.1:

Theorem 5.2 (Harish-Chandra, Bernstein). Let F be a nonarchimedeanfield and let G be the F -points of a reductive group over F . Then all irreducibleunitary representations of G are admissible. ut

The proof was reduced to a conjectural estimate on the dimensions of theV K in [HC70b] (which invoked earlier work of Godement, see loc. cit.). Thisestimate was proven in [Ber74].

Taken together, Lemma 5.3.3, Proposition 5.3.4 and Theorem 5.2 tell usethat the study of unitary irreducible representations of G that are the F -points of reductive groups over F is equivalent to the study of preunitaryadmissible representations of this type of group.

5.4 Contragredients

In this section we use the proof of Proposition 5.3.4 as a convenient excusefor introducing the useful concepts of the contragredient representation of agiven smooth representation of a td group G.

Let (π, V ) of be a smooth representation of G. Then there is an action ofG on the space of C-linear functionals Hom(V,C) on V given by

G×Hom(V,C) −→ Hom(V,C) (5.2)

(g, λ) 7−→ λ π(g−1).

The smooth dual V ∨ ⊆ Hom(V,C) is the subspace of smooth linearfunctionals, this is precisely the set of λ that are fixed by some compactopen subgroup K ≤ G. The action (5.2) preserves V ∨ and affords a smoothrepresentation (π∨, V ∨) of G.

Definition 5.5. The contragredient representation (π∨, V ∨) is the rep-resentation of G on the smooth dual V ∨ given above.

We note that ∨ is a contravariant functor from the category of admissiblerepresentations of G to itself.


We now define a variant of the contragredient representation. For anyC-vector space V let

V := C⊗(λ 7→λ),C V.

This has the same underlying additive group as V , but with scalar multipli-cation defined by (λ, v) 7→ λv (the bar denoting complex conjugation). Therepresentation

(π∗, V ∗) := (π∨, V ∨)

is known as the Hermitian contragredient. The map ∗ again defines acontravariant functor from the category of admissible representations of G toitself. If V is admissible then there are natural isomorphisms

V −→ (V ∨)∨ and V −→ (V ∗)∗

(see Exercise 5.13).The following lemma, which is [Lau96, Lemma D.6.3], is easily seen to

imply Proposition 5.3.4:

Lemma 5.4.1 Let (π, V ) be an admissible representation of G. There is acanonical bijection between the set of G-invariant, definite, Hermitian scalarproducts on V and the set of isomorphisms

ι : V −→ V ∗

of smooth representations such that ι∗ = ι. If (π, V ) is irreducible any twoG-invariant, definite (resp. positive definite) Hermitian scalar products on Vare R× (resp. R>0) multiples of each other.

Here we use the canonical isomorphism (V ∗)∗ = V to regard ι∗ as a mapfrom V to itself.

Proof. The last claim follows from Schur’s lemma (in the form of Exercise 5.5)and the first claim. Suppose that we have a G-invariant, definite, Hermitianinner product (·, ·) on V (C-linear in the first variable and C-antilinear in thesecond). We then obtain a G-equivariant morphism

V −→ V ∗

ϕ0 7−→ (ϕ 7−→ (ϕ,ϕ0)). (5.3)

It is clearly injective. Now if K ≤ G is a compact open subgroup, thendimC((V ∗)K) = dimC((V ∨)K) = dimC(V K). Thus for all compact open sub-groups K ≤ G we have that (5.3) induces an isomorphism.

V K−→(V ∗)K .

Hence by the smoothness of V and V ∗ the injection (5.3) is an isomorphism.Since (ϕ1, ϕ2) = (ϕ2, ϕ1) we deduce that ι = ι∗. Conversely if ι : V → V ∗

5.5 The unramified Hecke algebra 105

is an isomorphism of admissible representations satisfying ι = ι∗, we canassociate to ι the G-invariant Hermitian inner product

(ϕ1, ϕ2) := ι(ϕ2)(ϕ1).

ut

5.5 The unramified Hecke algebra

Let G be a reductive group over F , where F is a nonarchimedean local field.Recall from §2.4 that G is unramified if G is quasi-split and split over anunramified extension of F . In this case there is a unique G(F )-conjugacyclass of hyperspecial subgroups K (see §2.4).

Definition 5.6. If G is unramified and K ≤ G(F ) is a hyperspecial sub-group, then

C∞c (G(F ) //K)

is known as the spherical Hecke algebra.

Here C∞c (G(F ) //K) is the subalgebra of functions invariant on the left andright under K.

The following is arguably the most important fact about this algebra:

Theorem 5.5.1 The spherical Hecke algebra C∞c (G(F ) //K) is commuta-tive.

One way to prove this is via the Satake isomorphism (see Theorem 7.2.1).In special cases this can also be proven using a trick due to Gelfand (seeExercise 5.14). Let us describe the algebra in more detail in a special case:

Example 5.2. Let G = GLn, viewed as a group over Qp. A hyperspecial sub-group is GLn(Zp). The spherical Hecke algebra in this case is

C∞c (GLn(Qp) // GLn(Zp)).

Letλ = (λ1, . . . , λn).

Let T be the maximal torus of diagonal matrices in GLn. Then λ defines acocharacter λ : Gm → T given on points by

λ(x) =

(xλ1

. . .

xλn

).

Consider


1GLn(Zp)λ(p)GLn(Zp) : λ1 ≥ . . . ,≥ λn

The Smith normal form for matrices over Qp from the theory of elementarydivisors implies that the set above is a C-vector space basis for the sphericalHecke algebra.

Definition 5.7. Assume that G is unramified with hyperspecial subgroupK ≤ G(F ). An irreducible admissible representation (π, V ) of G(F ) is un-ramified, or spherical if V K 6= 0.

The following consequence of Theorem 5.5.1 is often referred to as unique-ness of the spherical vector, although it is more correct to say that it estab-lishes the uniqueness of the spherical line:

Corollary 5.5.2 Assume that G is unramified and that (π, V ) is an irre-ducible admissible unramified representation of G(F ). Then dimC V

K = 1.

Proof. Since V is irreducible, V K , if nonzero, must be an irreducible repre-sentation of the commutative algebra C∞c (G(F ) //K) by Proposition 5.3.2.

ut

5.6 Restricted tensor products of modules

In the representation theory of compact or algebraic groups one very quicklyreduces the representation theory of products of groups to representation the-ory of the individual factors, thus simplifying problems significantly. We nowexplain how one can do this in the context of automorphic representations.The problem is that G(AF ) is a restricted direct product

G(AF ) ∼=′∏v

G(Fv),

not a direct product, so we should not expect representations of G(AF ) todecompose into a direct external product of representations. What one doesinstead is define a notion of restricted direct products of representations. Wemake this precise in the current section, and then prove in Theorem 5.7.1 thatautomorphic representations to indeed factor into restricted direct productsindexed by the places in F .

We start by defining a restricted direct product of vector spaces. Let Ξ bea countable set, let Ξ0 ⊂ Ξ be a finite subset, let

Wv : v ∈ Ξ

be a family of C-vector spaces and for each v ∈ Ξ−Ξ0 let w0v ∈Wv−0. Forall sets

5.6 Restricted tensor products of modules 107

Ξ0 ⊆ S ⊆ Ξ

of finite cardinality set WS :=∏v∈SWv. If S ⊆ S′ there is a map

WS −→WS′ (5.4)

⊗v∈Sϕv 7−→ ⊗v∈Sϕv ⊗ (⊗v∈S′−Sϕ0v) .

Consider the vector space

W := ⊗′Wv := lim−→S

WS

where the transition maps are given by (5.4). This is the restricted tensorproduct of the Wv with respect to the ϕ0v. Thus W is the set of sequences

(ϕv)v∈Ξ ∈ ⊗vWv

such that wv = ϕ0v for all but finitely many v ∈ Ξ. We note that if we aregiven for each v ∈ Ξ a C-linear map

Bv : Wv −→Wv

such that Bv(w0v) = w0v for all but finitely many v ∈ Ξ then this gives amap

B = ⊗vBv : W −→W

⊗ϕv 7−→ ⊗Bv(ϕv).

We now define a restricted directed product of algebras. Suppose we aregiven C-algebras (not necessarily with unit) Av : v ∈ Ξ and idempotentsa0v ∈ Av for all v ∈ Ξ −Ξ0. If S ⊆ S′ there is a map

AS −→ AS′ (5.5)

⊗v∈Sav 7−→ ⊗v∈Sav ⊗ (⊗v∈S′−Sa0v) .

Consider the algebraA := ⊗′v∈ΞAv := lim−→

S

AS

where the transition maps are given by (5.5). This is the restricted tensorproduct of the Av with respect to the a0v. Finally, if Wv is an Av-modulefor all v ∈ Ξ such that a0vϕ0v = ϕ0v for almost all v, then ⊗′vWv is anA-module. The isomorphism class of W as an A-module in general dependson the choice of ϕ0v. However, if we replace the ϕ0v by nonzero scalarmultiples we obtain isomorphic A-modules.

An easy example of this construction is the ring of polynomials in infinitelymany variables. It can be given the structure of a restricted tensor productof algebras


C[X1, X2, . . .] = ⊗′iC[Xi]

where we take a0i to be the identity in C[Xi].Let G be a reductive group over the global field F and let ∞ be the set

of infinite places. Let Ξ0 be a finite set of places of F including the infiniteplaces. Enlarging Ξ0 if necessary we assume that if v 6∈ Ξ0 then GFv isunramified. If v 6∈ Ξ0 we letKv ≤ G(Fv) be a choice of hyperspecial subgroup.The fundamental example of the constructions above for our purposes is theisomorphism

C∞c (G(A∞F )) ∼= ⊗′v-∞C∞c (G(Fv))

The idempotents are

eKv :=1

meas(Kv)1Kv (5.6)

where Kv ≤ G(Fv) is a (choice of) hyperspecial subgroup for all v outside afinite set of places of F including∞. Here the measure in question is the Haarmeasure used in the definition of the convolution product on C∞c (G(Fv)).

We really can take any choice of hyperspecial compact open subgroup here(see Exercise 5.15), so the isomorphism above is not canonical. Of course,since all choices lead to isomorphic algebras, the representation theory willbe essentially the same (see Exercise 5.16).

5.7 Flath’s theorem

Flath’s theorem implies that every automorphic representation can be fac-tored into components indexed by the places of the global field over whichthe representation is defined. This fact is fundamental to the study of auto-morphic representations. In this section we state and prove Flath’s result.

Let G be a reductive group over a global field F . A C∞c (G(A∞F ))-moduleW is factorizable if we can write

W ∼= ⊗′vWv (5.7)

where the restricted direct product is with respect to elements φ0v ∈WKvv , dim(WKv

v ) = 1, and the isomorphism (5.7) intertwines the action ofC∞c (G(A∞F )) with the action of ⊗′v-∞C

∞c (G(Fv)), the restricted direct prod-

uct with respect to the idempotents eKv as in (5.6) above.In view of the assumption that dim(WKv

v ) = 1 the module ⊗′vWv onlydepends on the choice of the φ0v up to isomorphism. Note that in this caseW is admissible and irreducible if and only if the Wv are for all v.

5.7 Flath’s theorem 109

Theorem 5.7.1 (Flath) Every admissible irreducible representation W ofC∞c (G(A∞F )) is factorizable.

We start with the following easy case of Theorem 5.7.1:

Theorem 5.7.2 Let G1, G2 be td groups and let G = G1 ×G2.

(a) If Vi is an admissible irreducible representation of Gi for i = 1, 2 thenV1 ⊗ V2 is an admissible irreducible representation of G.

(b) If V is an admissible irreducible representation of G then there existsadmissible irreducible representations Vi of Gi for i = 1, 2 such thatV ∼= V1 ⊗ V2. Moreover the isomorphism classes of the Vi are uniquelydetermined by V .

Proof. We first prove (a). It is easy to see that V1⊗V2 is admissible. We mustcheck that it is irreducible. By the irreducibility criterion in Proposition 5.3.2for every compact open subgroup K1 ×K2 ≤ G1 ×G2 the finite-dimensionalC∞c (Gi //Ki)-modules V Kii are irreducible for i ∈ 1, 2. Since the V Kii are

finite dimensional it it is easy to see that V K11 ⊗V K2

2 is irreducible as a moduleunder

C∞c (G1 //K1)⊗ C∞c (G2 //K2).

But

C∞c (G1 ×G2 //K1 ×K2) = C∞c (G1 //K1)⊗ C∞c (G2 //K2)

(V1 ⊗ V2)K1×K2 = V K11 ⊗ V K2

2

so V1 ⊗ V2 is irreducible by Proposition 5.3.2.Conversely, let V be an admissible representation of G. Choose K = K1×

K2 such that V K 6= 0 (this is possible by smoothness). Then since V K is finitedimensional there exist finite-dimensional irreducible C∞c (Gi //Ki)-modulesVi(Ki) and an isomorphism of C∞c (G //K) modules V K→V1(K1)⊗ V2(K2).Varying K, we obtain a decomposition

V ∼= V1 ⊗ V2

as C∞c (G) ∼= C∞c (G1 ×G2)-modules, where

V1 := lim−→K1

V1(K1) and V2 := lim−→K2

V2(K2).

The Vi are evidently admissible, and they are irreducible by Proposition 5.3.2.ut

We now prove Flath’s theorem:

Proof of Theorem 5.7.1: For each finite set S of places of F containing theinfinite places and the set of places v where GFv is ramified let


KS :=∏v 6inS

Kv ≤ G(ASF )

be a fixed compact open subgroup with Kv hyperspecial for every v.Choose an isomorphism

C∞c (G(A∞F )) ∼= ⊗′v-∞C∞c (G(Fv)) (5.8)

where the restricted direct product is constructed using the idempotents eKvfor Kv ≤ G(Fv) with v 6∈ S. Use (5.8) to identify these two algebras for theremainder of the proof. We then have a well-defined subalgebra

AS := C∞c (G(F∞S ))⊗ eKS ≤ C∞c (G(A∞F )),

where eKS = ⊗v/∈SeKv is 1meas(KS)

1KS .

By Corollary 5.5.2 and Theorem 5.7.2, as a representation of AS we havean isomorphism

WKS ∼= ⊗v∈S−∞Wv ⊗WS

where WS is a one-dimensional C-vector space on which eKS acts trivially.Hence, by admissibility,

W =⋃S

WKS ∼= lim−→S

⊗v∈S−∞Wv ⊗WS (5.9)

with respect to the obvious transition maps (compare §5.6). On the otherhand, (5.8) induces an identification

C∞c (G(A∞F )) =⋃S

AS = lim−→S

AS (5.10)

where the direct limit is taken with respect to the obvious transition maps(again, compare §5.6), and it is clear that (5.9) is equivariant with respect to(5.10). 2

Exercises

Throughout these exercises G is a group of td type and (π, V ) is a represen-tation of G.

5.1. Prove that a group of td-type is Hausdorff, locally compact, and totallydisconnected.

5.2. DefineVsm :=

⋃K

V K

5.7 Flath’s theorem 111

where the union is over all compact open subgroups K ≤ G. Prove that Vsm

is preserved by G and is a smooth subrepresentation of G. The subspaceVsm ≤ V is the space of smooth vectors in V (in the current setting of tdgroups).

5.3. Suppose that V is a Hilbert space representation. Assume that G issecond countable. Define Vsm as in the previous exercise. Show that Vsm

admits a countable algebraic (also known as Hamel) basis. Show that if Vis infinite dimensional then V does not admit a countable algebraic basis.Conclude that if V is infinite-dimensional then Vsm ( V .

5.4. Suppose that V is admissible and preunitary, that is, there is a non-degenerate inner product on V that is preserved by G. Prove that everyG-invariant subspace V1 ≤ V admits a complement, that is, a G-invariantsubspace V2 ≤ V such that V1 ⊕ V2 = V .

5.5. Assume that G is second countable and V is smooth and irreducible.Show that any linear map Λ : V → V commuting with the action of G is ascalar.

5.6. Assume that G is second countable and V is smooth and irreducible. Letϕ1, . . . , ϕn, ϕ′1, . . . , ϕ

′n be two sets of elements of V with ϕ1, . . . , ϕn linearly

independent. Then there is an f ∈ C∞c (G(F )) such that

π(f)ϕi = ϕ′i

for all 1 ≤ i ≤ n.

5.7. Assume that G is second countable and V is smooth and irreducible. LetZG be the center of G. Show that there is a quasi-character ωπ : ZG → C×such that π(z) acts via ωπ(z) on V for all z ∈ ZG. The quasi-character ωπis called the central quasi-character of π. Show that if χ : G → C× is aquasi-character, then the central quasi-character of π ⊗ χ is χ|ZGωπ.

5.8. Let p be a prime number in Z. Prove that there are no constants c1, c2 ∈R>0 such that

c1| · |∞ ≤ | · |p ≤ c2| · |∞,

where | · |∞ denotes the usual archimedean norm on Q and | · |p denotes thep-adic norm.

5.9. Prove the equality (5.1).



5.12. Show that for any admissible representation V of a td group G


Hom(V K ,C) = (V ∨)K

for all compact open subgroups K ≤ G. Conclude that the contragredientand Hermitian contragredient of an admissible representation of a td groupis again admissible.

5.13. Prove that if V is an admissible representation of a td group then thereare canonical isomorphisms V → (V ∨)∨ and V → (V ∗)∗.

5.14. . Let F be a nonarchimedean local field. Consider the spherical Heckealgebra C∞c (GLn(F ) // GLn(OF )). For f ∈ C∞c (GLn(F )) let f†(g) := f(gt).Show that for every f1, f2 ∈ C∞c ((GLn(F ))

(f1 ∗ f2)†(g) = (f†2 ∗ f†1 )(g).

Show, on the other hand, that f† = f for f ∈ C∞c (GLn(F ) // GLn(OF )).Conclude that C∞c (GLn(F ) // GLn(OF )) is commutative.

5.15. Let F be a global field. For all v outside a finite set of places includingthe infinite places choose hyperspecial subgroups Kv ≤ G(Fv). Show that

G(A∞F ) ∼=′∏

v-∞

G(Fv)

where the restricted tensor product is taken with respect to the Kv. Note thatwe allow for infinitely many of the Kv’s (perhaps all of them) to be differentfrom those obtained by choosing a model of G as in Proposition 2.3.1.

5.16. Let F be a global field. For all v outside a finite set of places includingthe infinite places choose hyperspecial subgroups K1v,K2v ≤ G(Fv). Fori = 1, 2, let Ai be ⊗′v-∞C

∞c (G(Fv)) where the restricted direct product is

taken with respect to eKiv (see (5.6)). Construct an equivalence of categoriesbetween admissible A1-modules and admissible A2-modules.

Chapter 6

Automorphic Forms

Discovery is the privilege of thechild: the child who has no fearof being once again wrong, oflooking like an idiot, of not beingserious, of not doing things likeeveryone else.

Alexander Grothendieck

Abstract In this chapter we give a new definition of automorphic represen-tations. There are two reasons for introducing the new definition. The first isthat it is easier to relate the new definition to spaces of automorphic formson locally symmetric spaces (in the number field case). The second is thatthe new definition allows us to move from the category of Hilbert space rep-resentations to the category of admissible representations. In practice it isoften easier to work in the latter category.

6.1 Smooth functions

Throughout this chapter we let G be a reductive group over a global fieldF . Our goal in this chapter is to give a new definition of an automorphicrepresentation and explain its relation to the old definition. There are tworeasons for introducing a new definition. The first is that it is easier to relatethe new definition to spaces of automorphic forms on locally symmetric spaces(in the number field case). The second is that the new definition allows us tomove from the category of Hilbert space representations to the category ofadmissible representations.

As preparation for this in the current section we comment on the relationbetween AG\G(AF ) and G(AF )1 and how one defines smooth functions ineach context. Recall that in §3.3 we defined an automorphic representation

113

114 6 Automorphic Forms

of G(AF )1 to be an irreducible unitary representation π of G(AF )1 isomor-phic to a subquotient of L2([G]). Since we have assumed G is reductive in thischapter we can identify [G] = G(F )AG\G(AF ) via the canonical isomorphismG(AF )1−→AG\G(AF ). We can then define an automorphic representa-tion of AG\G(AF ) to be an irreducible unitary representation of AG\G(AF )isomorphic to a subquotient of L2([G]). In the current chapter, we refer tothese concepts as automorphic representations in the L2 sense.

Both conventions have advantages and disadvantages and it is sometimesuseful to switch back and forth between them. For example, there is an ob-vious isomorphism

AG\G(AF ) ∼= AG\G(F∞)×G(A∞F )

Analogously test functions on AG\G(AF ) factor in an obvious way.On the other hand, the groups G(AF )1 seem to be useful when working

with subgroups (which is perhaps why Arthur uses them in his trace formula[Art78, Art80]), are more natural when working over function fields, andadmit a definition whose natural generalization to the nonreductive settingis well-behaved.

It is useful to say explicitly what we mean by a smooth function fromeither optic. We define

C∞(G(AF )) : = C∞(G(F∞))⊗ C∞(G(A∞F )), and (6.1)

C∞c (AG\G(AF )) : = C∞c (AG\G(F∞))⊗ C∞c (G(A∞F )). (6.2)

Then the compactly supported smooth functions are simply C∞c (G(AF )) =C∞c (G(F∞))⊗ C∞c (G(A∞F )).

By definition, a smooth function on G(AF )1 is the restriction of a smoothfunction on G(AF )

C∞(G(AF )1) := f |G(AF )1 : f ∈ C∞(G(AF )). (6.3)

The canonical isomorphism G(AF )1→AG\G(AF ) induces an isomorphism

C∞c (AG\G(AF )) −→ C∞c (G(AF )1)

via pullback of functions (see Exercise 6.9).

6.2 Classical automorphic forms

For this section F is a number field, K∞ ≤ G(F∞) is a maximal compactsubgroup, and K∞ ≤ G(A∞F ) is a compact open subgroup. Recall the home-omorphism

6.2 Classical automorphic forms 115

G(F )\G(AF )/K∞−→∐i

Γi(K∞)\G(F∞) (6.4)

of (2.14). By Theorem 2.6.1 there are finitely many indices in this disjointunion. The important point for the current discussion is not the precise defi-nition of the Γi(K

∞), but just that they are discrete arithmetic subgroups ofG(F∞) such that the quotient Γi(K

∞)\G(F∞) has finite volume (see Theo-rem 2.6.2). Clearly automorphic representations are related to functions onthe left side. In this section and the following we make this relationship pre-cise. We work only at infinity in the current section and then pass to theadelic setting in the next.

Let

ι′ : G −→ GLn

be a closed embedding and let ι : G→ GL2n be the embedding defined by

ι(g) :=

(ι′(g)

ι′−t(g)

). (6.5)

We then define the norm

‖g‖ := ‖g‖ι =∏v|∞

supp1≤i,j≤2n|gij |v.

Let g := LieGF∞ and let K∞ ≤ G(F∞) be a maximal compact subgroup.Let U(g) be the universal enveloping algebra of gC and Z(g) its center (see§4.6).

Definition 6.1. A function

ϕ : G(F∞) −→ C

is of moderate growth or slowly increasing if there are constants c, r ∈R>0 such that

|ϕ(g)| ≤ c‖g‖r.

It is of uniform moderate growth if there exists an r ∈ R>0 such that forall X ∈ U(g) there exists a cX ∈ R>0 such that

|ϕ(g)| ≤ cX‖g‖r.

It turns out that different choices of ι lead to norms that are equivalent ina sense made precise by Exercise 6.1. In particular, the notion of moderategrowth and uniform moderate growth is independent of the choice of ι.

For the definition of automorphic forms we will require the notion of K∞-finite functions and Z(g)-finite functions. The notion of K∞-finiteness is givenin Definition 4.3. The definition of Z(g)-finite is similar.


Definition 6.2. Let V be a Z(g)-module. A vector ϕ ∈ V is Z(g)-finite ifZ(g)ϕ is finite-dimensional, or, equivalently, if there is an ideal J ≤ Z(g) offinite codimension that annihilates ϕ.

We now finally come to the notion of an automorphic form:

Definition 6.3. Let Γ ⊆ G(F ) be an arithmetic subgroup. A smooth func-tion ϕ : G(F∞) → C of moderate growth is an automorphic form on Γif it is left Γ -invariant, K∞-finite, and Z(g)-finite. We denote the space ofautomorphic forms on Γ by A(Γ ).

Often it is convenient to isolate the subspace that is annihilated by aparticular ideal J ≤ Z(g). It is denoted A(Γ, J). This space decomposes stillfurther:

A(Γ, J) := ⊕σ∈KA(Γ, J, σ)

where A(Γ, J, σ) := A(Γ, J)(σ) in the notation of §4.4. We record the follow-ing important result of Harish-Chandra [HC68]:

Theorem 6.2.1 (Harish-Chandra) For each ideal J ≤ Z(g) of finite codi-mension the space A(Γ, J) is an admissible (g,K∞)-module. In particular,A(Γ, J, σ) is finite-dimensional for each σ. ut

One might ask about the relationship betweenA(Γ ) and L2(ΓAG\G(F∞)).There is no obvious relationship between them except under more stringentassumptions. For example, Eisenstein series (which will be discussed in Chap-ter 10) provide examples of elements of A(Γ ) need not even be square inte-grable. It turns out to be more convenient to explain the relationship betweenthe so-called cuspidal subspaces of A(G) and L2(ΓAG\G(F∞)). We will deferthe discussion of this connection to the following section, in which we discussautomorphic forms adelically.

6.3 Adelic automorphic forms over number fields

Let ι : G→ SL2n be the embedding of (6.5). In this section we assume thatF is a number field. For a place v of F let

‖g‖v := ‖g‖ι,v = supp1≤i,j≤2n|gij |v (6.6)

and for a set of places S of F (finite or infinite) let

‖g‖S :=∏v∈S‖g‖v.

If S is the set of all places of F then we omit it from notation.As in the archimedean setting we have a definition of a function of mod-

erate growth:

6.3 Adelic automorphic forms over number fields 117


ϕ : G(AF ) −→ C

is of moderate growth or slowly increasing if there are constants c, r ∈R>0 such that

|ϕ(g)| ≤ c‖g‖r.

A function ϕ is of uniform moderate growth if there exists an r ∈ R>0

such that for all X ∈ U(g) there exists a cX ∈ R>0 such that

|Xϕ(g)| ≤ cX‖g‖r.

It is often convenient to phase the notion of moderate growth in terms ofSiegel sets:

Lemma 6.3.1 A functionϕ : [G] −→ C

is of moderate growth if and only if it is smooth and for all compact subsetsΩ ⊂ [G] and t ∈ R>0 there are constants c, r ∈ R>0 (depending on Ω and t)such that one has

|ϕ(sy)| ≤ cα(s)r

for all s ∈ AT0(t) and α ∈ ∆. It is of uniform moderate growth if and onlyif it is smooth and there exists an r ∈ R>0 such that for all X ∈ U(g) thereexists a cX ∈ R≥0 such that

|Xϕ(g)| ≤ cXα(s)r

ut

Here we are using notation from our discussion of Siegel sets and reductiontheory in §2.7. For a proof of this lemma we refer to [Bor07, §5.4].

By definition, a (g,K∞)×G(A∞F )-module (π, V ) is a complex vector spaceV that is both a (g,K∞) and G(A∞F )-module (we denote the two actionmaps by π) such that the two actions commute. It is admissible if for eachK∞ ≤ G(A∞F ) the space of fixed vectors V K∞ is admissible as a (g,K∞)-module. A morphism of (g,K∞)×G(A∞F ) is a complex linear map commutingwith the actions of (g,K∞) and G(A∞F ) (no continuity condition is assumed.We define an irreducible (g,K∞)×G(A∞F )-module to be one that admits noinvariant submodule. There is no topology on V , so we do not assume thatthe submodule is closed.

Let Kmax ≤ G(AF ) be a maximal compact subgroup; thus Kmax =K∞ ×K∞max where K∞ ≤ G(F∞) and K∞max ≤ G(A∞F ) are maximal compactsubgroups. As before, we say that a function ϕ : G(AF ) → C is K-finite ifthe span of translates

x 7→ ϕ(xk) : k ∈ Kmax


is finite dimensional. An alternate way of phrasing this condition is givenin Exercise 6.3. The reason that we do not include the subscript max atthe infinite component is that it is rare to consider nonmaximal compactsubgroups in this setting, although it is very natural to consider nonmaximalcompact open subgroups at the finite places. The reason stems from the factthat there are no “small subgroups” in the archimedean setting, compareLemma 5.1.2.

Definition 6.5. A smooth function

ϕ : G(AF ) −→ C

of moderate growth is an automorphic form onG if it is leftG(F )-invariant,Kmax-finite, and Z(g)-finite. The C-vector space of automorphic forms isdenoted by A or A(G).

The space A is equipped with an action of (g,K∞) and an action of G(A∞F )by right translations:

(g,K∞)×G(A∞F )×A −→ A(X, k, g, ϕ) 7−→ (x 7→ Xϕ(xkg)) . (6.7)

It is important to note that the action of G(A∞F ) does not extend to an actionof G(AF ). Indeed, let ϕ ∈ A and let g ∈ G(F∞). Then x 7→ ϕ(xg) is smooth,but it is only finite under gK∞g

−1, not K∞.If σ is an irreducible representation of K∞ we denote by A(σ) the subspace

of vectors with K∞-type σ, and if J ≤ Z(g) is an ideal the subspace of vectorsannihilated by J is denoted by A(J) (resp. A(J, σ)). The action of G(A∞F )preserves these spaces, and the action of (g,K∞) preserves A(J) (but ofcourse not A(J, σ) in general).

The following is the adelic analogue of Theorem 6.2.1:

Theorem 6.1. Let J ≤ Z(g) be an ideal of finite codimension. Then A(J)is an admissible (g,K∞)×G(A∞F )-module.

Proof. We are to show that A(J, σ)K∞

is finite dimensional for each K∞-typeσ and each compact open subgroup K∞ ≤ G(A∞F ). We start by recalling thedescription of the adelic quotient [G]/K∞ given by (2.14). Letting t1, . . . , thdenote a set of representatives for the finite set G(F )\G(A∞F )/K∞ one has ahomeomorphism

h∐i=1

Γi(K∞)\G(F∞) −→ G(F )\G(AF )/K∞

given on the ith component by Γi(K∞)g∞ 7→ G(F )g∞ti where Γi(K

∞) =G(F ) ∩ tiK∞t−1

i (see (2.14)). By pullback we obtain an isomorphism of C-vector spaces

6.5 The cuspidal subspace 119

A(J)K∞−→ ⊕i=1 A(Γi(K

∞), J)

φ 7−→ (xi 7→ ϕ(xiti)) (6.8)

which intertwines the natural action of (g,K∞) on both sides. We can nowdeduce the theorem by applying Theorem 4.4.1. ut

In view of the previous theorem the following definition is reasonable:

Definition 6.6. An automorphic representation of G(AF ) is an irre-ducible admissible (g,K∞)×G(A∞F ) isomorphic to a subquotient of A.

6.4 Adelic automorphic forms over function fields

The refined definition of an automorphic form in the function field case isfairly straightforward:


ϕ : G(F )\G(AF ) −→ C

invariant on the right under a compact open subgroup of G(AF ) is an auto-morphic form on G if the C-vector space spanned by the functions

x 7→ ϕ(xg) : g ∈ G(AF )

is an admissible representation of G(AF ).

As in the number field case, we denote by A or A(G) the C-vector space ofautomorphic forms. It is naturally equipped with an action of G(AF ).

Definition 6.8. An automorphic representation of G(AF ) is an irre-ducible admissible representation of G(AF ) that is isomorphic to a subquo-tient of A.

In analogy with the number field case, two automorphic representationsof G(AF ) are equivalent if they are equivalent in the category of admissiblerepresentations of G(AF ).

6.5 The cuspidal subspace

We now isolate what is arguably the most important subspace of A:

Definition 6.9. An automorphic form ϕ ∈ A is said to be cuspidal if forevery proper parabolic subgroup P < G with unipotent radical N one has

120 6 Automorphic Forms∫[N ]

ϕ(ng)dn = 0

for all g ∈ G(AF ).

The subspace of cuspidal forms will be denoted Acusp.Analogously, the cuspidal subspace

L2cusp([G]) ⊆ L2([G]).

is the closure of the space of continuous functions ϕ such that for all properparabolic subgroups P < G with unipotent radical N one has∫

[N ]

ϕ(ng)dn = 0.

It is clear that this is a G(AF )-subrepresentation of L2([G]).

Definition 6.10. A cuspidal automorphic representation of G(AF ) isan automorphic representation equivalent to a subquotient of Acusp.

Let us say that an irreducible subquotient of L2cusp([G]) is a cuspidal

automorphic representation in the L2- sense. We will show in Corollary9.1.2 below that one has a discrete decomposition

L2cusp([G]) =

⊕π

L2cusp(π)

where L2cusp(π) is the π-isotypic subspace of L2

cusp([G]) and the sum is over allisomorphism classes of cuspidal automorphic representations π of AG\G(AF ).In view of this fact every irreducible subquotient of L2

cusp([G]) is in fact asubrepresentation.

One justification for our claim that Acusp is the most important subspaceof A is that any element of A can be obtained from elements of Acusp([M ]) asM runs over the Levi subgroups of parabolic subgroups of G. We will discussthe L2-version of this fact in §10.4 below.

Though the relationship between automorphic representations in the senseof Definition 6.6 and automorphic representations in the L2-sense is compli-cated, they are very closely linked in the cuspidal case. To explain the rela-tionship, first note that if ϕ is in the space of a cuspidal automorphic repre-sentation π of G(AF ), then by Schur’s lemma (Lemma 4.1) the subgroup AGacts via a scalar. Let us refine this a little. Let aG := Hom(X∗(G)F ,R) andlet

HG : G(F ) −→ aG (6.9)

be defined via 〈HG(g), λ〉 = log |λ(g)| for λ ∈ X∗(G)F (compare §4.9). Thereis a unique λ ∈ a∗GC such that the representation


π ⊗ e〈HG(·),λ〉

is trivial on AG. In the case where G = GLn, we can be more explicit: thereis a unique s ∈ C such that π ⊗ |det |s is trivial on AG. In any case, there isnot much loss in generality in assuming that an automorphic representationof G(AF ) is trivial on AG. Let AAGcusp ≤ Acusp be the subspace invariant underAG.

Theorem 6.5.1 The space of cuspidal automorphic forms AAGcusp is a densesubspace of L2

cusp([G]).

We defer the proof of Theorem 6.5.1 to §9.8. We also have the followingtheorem:

Theorem 6.5.2 Any automorphic representation in the L2 sense defines,via passage to K-finite vectors, an automorphic representation. Two auto-morphic representations in the L2 sense are unitarily equivalent if and onlyif the associated automorphic representations are equivalent.

Proof. Suppose we are given an automorphic representation in the L2-sense.Passage to K-finite vectors yields an admissible representation of G(AF )in the function field case or (g,K∞) × G(A∞F ) in the number field case byTheorem 4.4.1 and Theorem 5.2. If automorphic representations in the L2-sense have equivalent underlying automorphic representations, then they areunitarily equivalent by Theorem 4.4.4, Lemma 5.3.3 and Proposition 5.3.4.

ut

In view of Theorem 6.5.2, for many purposes we can work with eitherthe definition of automorphic representation in terms of admissible repre-sentations presented in this chapter, or automorphic representations in theL2-sense, whichever is convenient for the situation at hand. Both are oftenconvenient for different reasons. The L2 definition is convenient for abstractharmonic analysis, but the definition in terms of admissible representationshas the advantage that one can immediately reduce questions to finite di-mensional settings, either by passing to K∞-types or vectors fixed undera compact open subgroup. We will also explain in the next few sectionshow spaces of automorphic forms, though not representations of G(AF ), are(g,K∞) × G(A∞F )-modules and hence give rise to automorphic representa-tions.

We also warn the reader that later in this book we will often not be specificabout whether we are viewing an automorphic representation in the L2 senseor in the sense of an admissible representation of G(AF ) (in the functionfield case) or (g,K∞)×G(A∞F ) (in the number field case). Indeed, in practiceone often has to switch back and forth between the two perspectives. Theparticular point of view must be deduced from the context.


6.6 From modular forms to automorphic forms

For readers familiar with modular forms we make the connection betweenmodular forms and automorphic forms precise in this section.

Let Γ ≤ GL2(Z) be a congruence subgroup. For example, we could set Γequal to

Γ0(N) :=(

a bc d

)∈ GL2(Z) : N |c

. (6.10)

We recall the definition of a classical modular form on Γ :

Definition 6.11. Let k ∈ Z>0 and let H be the complex upper half plane.The space of weight k modular forms for Γ is the space Mk(Γ ) of holo-morphic functions f : H→ C satisfying the following conditions:

(a) f(az+bcz+d

)= (cz + d)kf(z) for all

(a bc d

)∈ Γ ∩ SL2(Z) and all z ∈ H,

(b) f extends holomorphically to the cusps.

If f additionally vanishes at the cusps we say that f is a cusp form. Thespace of weight k cusp forms is denoted Sk(Γ ).

There are many books on modular forms that the reader can consult for moredetails. We mention [DS05, GG12, Kob93, Lan95, Miy06, Ser73, Shi94]. Arecent extension of the theory to larger classes of functions that arise naturallyin applications is explained in [BFOR17].

We remark that if k is even then Mk(Γ ) can be identified with a subspaceof the space of holomorphic differential forms H0(Γ ∩SL2(Z)\H, Ω⊗k/2). Thisexplains the moniker “modular form.” This is explained in great detail in aslightly more general setting in [GG12, Chapter 6].

We now relate the space Mk(Γ ) to a space of automorphic forms. The ideais to pull back an automorphic form on H along the quotient map

AGL2\GL2(R) −→ AGL2

\GL2(R)/O2(R) = H . (6.11)

In this settingAGL2

:= ( r r ) : r > 0 .

We setj(g, z) = det(g)−1/2(cz + d)

for

g =

(a bc d

)∈ GL2(R)+ := g ∈ GL2(R) : det g > 0.

This is an example of an automorphy factor (see [GG12, §6.3] for details).Set

ϕf (g) = j(g, i)−kf(gi) : GL2(R)+ −→ C

6.6 From modular forms to automorphic forms 123

where g acts on i by fractional linear transformations. This will give us anautomorphic form, but to specify the type we require further notation. Letσk be the induction of the representation(

cos θ − sin θsin θ cos θ

)7−→ e2πikθ (6.12)

of SO2 to O2 and let

∆ =1

4(H2 + 2XY + 2Y X)

be the Casimir element of glC2 (see §4.7). It and the element Z = ( 11 )

generate Z(gl2).We leave the following proposition as an exercise:

Proposition 6.6.1 For each integer k ≥ 1 the C-linear map f 7→ ϕf inducesan isomorphism

Sk(Γ ) −→ Acusp

(Γ, 〈∆− 1

4(k2 − 1), Z〉, σk

)f 7−→ ϕf .

ut

The isomorphism of Proposition 6.6.1 can also be phrased adelically. Let

K0(N) := Γ0(N) =(

a bc d

)∈ GL2(Z) : N |c

where the hat denotes profinite completion. Composing the isomorphism ofProposition 6.6.1 with (6.8) we obtain an isomorphism

Sk(Γ0(N))−→Acusp

(〈∆− 1

4(k2 − 1)〉, σk

)K0(N)

One might ask if all cuspidal automorphic forms on GL2 arise from ele-ments of Sk(Γ ). In fact we have missed many of them. The missing auto-morphic representations correspond to Maass forms. From the automorphicperspective it is very simple to define these objects; they are the automorphicrepresentations of GL2(AQ) whose associated (gl2,O2(R)) module is in theprincipal series in the sense of §4.7. Defining them classically involves at leastas much work as required to formalize the notion of an element of Sk(Γ ).We refer the reader to [Bum97, §3.2] for more details. The ease with whichwe can talk about Maass forms and holomorphic cusp forms on equal footingillustrates the utility of the language of automorphic representation theory.


6.7 Ramanujan’s ∆-function

We would be remiss not to give an example of a modular form from theclassical perspective, and to an expert it should not be a surprise that wechoose Ramanujan’s ∆-function

∆(z) = e2πiz∞∏i=1

(1− e2πinz)24, z ∈ H.

This ∆ should not be confused with the Casimir operator from the previoussection.

One has the following basic theorem:

Theorem 6.7.1 One has

S12(SL2(Z)) = C∆(z).

The proof of this fact can be found in any of the standard references [DS05,Lan95, Miy06, Kob93, Ser73, Shi94].

Define τ(n) ∈ C to be the unique integers such that

∞∑n=1

τ(n)e2πinz = ∆(z).

In other words the τ(n) could be viewed as the Fourier coefficients on ∆(z).Ramanujan conjectures that

∞∑n=1

τ(n)

nt=∏p

1

1− τ(p)p−t + p11−2t(6.13)

for a sufficiently large real number t [Ram00b, (101)]. Remarkably this isexactly his notation. He also conjectures that

|τ(p)| ≤ 2p11/2 (6.14)

[Ram00b, (104)]. For the uninitiated, we pause and explain how fantasti-cally prescient these two assertions are. In (6.13) Ramanujan has writtendown the L-function of the automorphic representation attached to ∆(z)over 20 years prior to Hecke’s general theory of L-functions for modularforms [Hec37b, Hec37a] (Ramanujan’s assertion had been proven some timeearlier by Mordell [Mor]). The assertion (6.14) was proven by Deligne as aconsequence of his proof of the Weil conjectures [Del73a], however, the nat-ural generalization of (6.14) to automorphic representations of GLn remainsopen even in the case n = 2; it will be discussed in Conjecture 10.6.4 later.

6.7 Ramanujan’s ∆-function 125

Exercises

6.1. Let ι′1 : G → GLn1, ι′2 : G → GLn2

be a pair of faithful representationsof an algebraic group G over a global field F and let S be a set of places ofF (finite or infinite). Let ‖g‖ι1,S , ‖g‖ι2,S be the corresponding norms. Provethat there are constants c1, c2 ∈ R and r ∈ R>1 such that

c1‖g‖1/rι1,S≤ ‖g‖ι2,S ≤ c2‖g2‖rι1,S .

Deduce that the notion of moderate growth and uniform moderate growth isindependent of the choice of faithful representation ι′ : G→ GLn.

6.2. Let F be an archimedean local field, G an affine algebraic group overF and let K ≤ G(F ) be a compact subgroup. For every irreducible finite-dimensional representation σ of K let χσ be the character of σ and let d(σ)be its degree. For any K-module (π, V ) let

V −→ V

ϕ 7−→∫K

π(k)ϕχσ(k)dk

d(σ).

Show that this linear map is an idempotent that projects V onto the σ-isotypic subspace V (σ).

6.3. Let F be a global field and let K = K∞K∞ ≤ G(AF ) be a maximal

compact subgroup. Show that a continuous function ϕ : [G]→ C is K-finiteif and only if for all x∞ ∈ G(A∞F ), x∞ 7→ ϕ(x∞g

∞) is K∞-finite and thereis a compact open subgroup K ′∞ ≤ G(A∞F ) such that ϕ(xk) = ϕ(x) for allx ∈ G(AF ) and k ∈ K ′∞.

6.4. Let F be a local field and let G be a connected reductive group over F .Let K1,K2 ≤ G(F ) be maximal compact subgroups. If F is nonarchimedean,show that K1 ∩ K2 is a compact open subgroup of G(F ) and deduce thata function on G(F ) is K1-finite if and only if it is K2-finite. Construct anexample to show that if F is archimedean then a function that is K1-finiteneed not be K2-finite.

6.5. Let P0 ≤ G be a minimal parabolic subgroup, and call a parabolicsubgroup P standard if it contains P0. Prove that a function ϕ ∈ L2([G]) iscuspidal if and only if for all standard maximal parabolic subgroups P ≤ Gwith unipotent radical N one has∫

[N ]

ϕ(ng)dn = 0

for almost every g ∈ G(AF ).


6.6. Using the classification of §4.7 identify the underlying g,K)-module ofAcusp(Γ, 〈∆− 1

4 (k2 − 1), Z〉, σk).

6.7. Prove Proposition 6.7.

6.8. Prove Ramanujan’s assertion that (6.13) is valid assuming Theorem6.7.1.

6.9. Prove that the canonical isomorphism G(AF )1→AG\G(AF ) induces anisomorphism

C∞c (AG\G(AF )) −→ C∞c (G(AF )1)

via pullback of functions.

Chapter 7

Unramified Representations

Demazure nous indique que,derriere cette terminologie[epinglage], il y a l’image dupapillon (que lui a fournieGrothendieck): le corps est untore maximal T, les ailes sontdeux sous-groupes de Borelopposees par rapport a T, ondeploie le papillon en etalant lesailes, puis on fixe des elementsdans les groupes additifs (desepingles)...

M. Demazure

Abstract In this chapter we describe the classification of unramified repre-sentations of connected reductive groups over nonarchimedean local fields.Along the way we discuss the Satake isomorphism and the Langlands dualgroup.

7.1 Unramified representations

By Flath’s theorem, if π is an automorphic representation of G(AF ), then

π ∼= ⊗′vπv

where for almost every v the representation πv is unramified in the sense thatit contains a (unique) vector fixed under a hyperspecial subgroup K ≤ G(Fv).In this section we discuss the classification of unramified representations. Itturns out that they can be explicitly parametrized in terms of conjugacyclasses in the dual group of G (see Theorem 7.2.1 and Theorem 7.5.1). This

127

128 7 Unramified Representations

fundamental fact will be used in §7.7 to state a version of the Langlandsfunctoriality conjecture.

In this section we let G be a reductive group over a non-archimedean localfield F . Our purpose here is to study unramified representations. Recall thatG is unramified if G is quasi-split and split over an unramified extension ofF . In this case there exists a hyperspecial subgroup K ≤ G(F ).

Definition 7.1. An irreducible representation (π, V ) of G(F ) is called un-ramified if G is unramified and V K 6= 0.

Notice that we do not assume that V is admissible or even smooth. In fact, anunramified representation is automatically smooth and admissible (see Ex-ercise 7.2). It is important to realize also that when we speak of unramifiedrepresentations we should really speak of K-unramified representations. Hy-perspecial subgroups of G are not necessarily G(F )-conjugate, they are onlyconjugate under Gad(F ) (see [Tit79, §2.5]). Here Gad := G/ZG is the adjointquotient of G. We will however follow tradition and suppress the dependenceon K (but see §12.5).

Let K ≤ G(F ) be a hyperspecial subgroup. Then the subalgebra

C∞c (G(F ) //K) ≤ C∞c (G(F ))

is known as the unramified Hecke algebra of G(F ) (with respect to K).Let f ∈ C∞c (G(F ) //K) and let π be unramified. Then π(f) acts via a scalaron V K . It is sensible to denote the scalar by trπ(f) (see also §8.5). The map

C∞c (G(F ) //K) −→ Cf 7−→ trπ(f)

is called the Hecke character of π.

Proposition 7.1.1 An unramified representation π is determined up to iso-morphism by its Hecke character.

Proof. Say that a representation (π, V ) of G(F ) is generated by a subspaceW ≤ V if V = π(G)W .

There is an equivalence of categories

representations of G(F ) generated by V K−→C∞c (G(F ) //K)-modulesV 7−→ V K .

Here we have given the objects of the categories on both sides; morphismsare simply intertwining maps.

Any irreducible representation is generated by any nonzero subspace, soan unramified representation is generated by V K . We deduce the proposition.

ut

7.2 Satake isomorphism 129

7.2 Satake isomorphism

If we did not know anything about C∞c (G(F ) //K), then we could hardlyregard Proposition 7.1.1 as useful. However, it turns out that C∞c (G(F ) //K)has a simple description:

Theorem 7.2.1 (Satake) Assume that G is split. There is an isomorphismof algebras

S : C∞c (G(F ) //K) −→ C[T ]W (G,T )(C)

where G is the complex connected reductive algebraic group with root datumdual to that of G and T ≤ G is a maximal torus. ut

To gain intuition for the Satake isomorphism, let us consider the specialcase of GLn. The Hecke algebra C∞c (GLn(F ) // GLn(OF )), as a C-module,has a basis given by

1λ := 1

GLn(OF )

$λ11

. . .

$λnn

GLn(OF )

with λ := (λ1, . . . , λn) ∈ Zn, λi ≥ λi+1 for all 1 ≤ i ≤ n − 1 (this followsfrom the theory of elementary divisors). As a C-algebra it is generated by 1λ

with λ = (1r) := (1, . . . , 1, 0, . . . , 0) (r ones and n − r zeros) for 1 ≤ r ≤ nand λ = ((−1)n) := (−1, . . . ,−1).

On the generating set above the Satake isomorphism is given by

S(1(1r)) = qr(n−r)/2tr (∧rCn) (7.1)

where Cn is the standard representation of GLn(C), ∧rCn is the rth exteriorpower of the standard representation, and tr (∧rCn) is the trace of a diagonalmatrix in GLn(C) acting on the given representation (see [Gro98, (3.14)]).

It is instructive to indicate various ways of rephrasing this basic result. Bythe Chevalley restriction theorem, the inclusion T → G induces an isomor-phism

Spec(C[T ]W (G,T )) = TW (G,T ) −→ G/conj = Spec(C[G]G

)(7.2)

where the quotient on the right is the scheme theoretic quotient of G bythe action of conjugation (for more on scheme-theoretic quotients over fields

[MFK94] is the canonical reference). In particular G/conj(C) is just the set ofclosed conjugacy classes in G(C) . The closed orbits are precisely the orbitsof semisimple elements; a nice reference for this is [Ste65, Corollary 6.13].Thus we have a sequence of isomorphisms


Hom(C∞c (G(F ) //K),C)(S−1)∗−→ Hom(C[T ]W (G,T ),C) −→ Gss(C)/conj

(7.3)

where the first map is pullback along the inverse of the Satake isomorphism.Here homomorphism means homomorphism of C-algebras, and Gss ⊂ G is theclosed subscheme consisting of semisimple elements. In Proposition 7.1.1 wesaw that unramified representations were in bijection with Hecke characters,which are precisely elements of Hom(C∞c (G(F ) //K),C). We have thereforeproven the following corollary of the Satake isomorphism:

Corollary 7.2.2 Assume that G is split. The composite isomorphism (7.3)

induces a bijection between semi-simple conjugacy classes in G(C) and iso-morphism classes of irreducible unramified representations of G(F ). ut

From the point of view of automorphic representation theory the fact thatthe Satake isomorphism is only valid for split groups is problematic. Indeed,suppose for the moment that G is a reductive group over a global field F .Then for all but finitely many places v of F the group GFv is unramified byProposition 2.4.3 and hence in particular is quasi-split. However, the groupGFv can be nonsplit for infinitely many v (see Exercise 7.3).

Langlands was able to extend the Satake isomorphism to the quasi-splitcase. Historically this was important because it gave crucial hints as to thestructure of the Langlands dual group, which he introduced at the same time[Lan] (see also [Lan70]). We will define the Langlands dual group in the nextsection and use it to extend the Satake isomorphism in §7.5.

7.3 The Langlands dual group

For the moment, let G be a connected reductive group over a global or localfield F and let T ≤ G be a maximal torus. To these data we associated in§1.8 a root datum Ψ(G,T ) = (X∗(T ), X∗(T ), Φ, Φ∨). We remind the readerthat X∗(T ) and X∗(T ) are the character groups, Φ ⊂ X∗(T ) is the set ofroots of T in g and and Φ∨ ⊂ X∗(T ) is the set of coroots. We also remindthe reader that the root datum characterizes GF sep where F sep is a separableclosure of F . To ease on notation, we let

GalF := Gal(F sep/F ).

If G is split, then we set

LG := G(C)×GalF

where G is the complex dual group defined as in §1.8; it is the connectedreductive group over C with root datum (X∗(T ), X∗(T ), Φ∨, Φ).

7.3 The Langlands dual group 131

If G is not split then the L-group has a more complicated definition involv-ing a Galois action on G that records the fact that G is a nonsplit group overF . To define it we must construct an action of GalF on G(C). It is obviousthat the root datum should be involved in this process. Indeed, given a Galoisaction on GF sep we obtain one on Ψ(GF sep , TF sep) and hence tautologically

we obtain one on Ψ(G, T ). This is not enough, as we really need a Galois

action on G(C). It would suffice to produce a splitting of the surjective map

Aut(G(C)) −→ Aut(Ψ(G, T ))

described in Proposition 7.3.2 below. What we actually do is to produce arefinement Ψ(G, B, T ) of Ψ(G, T ) called a based root datum and a surjectivemap

Aut(G(C)) −→ Aut(Ψ(G, B, T )).

We then provide an explicit description of splittings of this map via somethingknown as a pinning, or epilange, of G. This then suffices to construct ourdesired action of Gal(F sep/F ) on G(C).

We now begin this process. We assume until otherwise specified that G isa split, connected reductive group over a separably closed field k. Let ∆ ⊂ Φbe a base for Φ (see (1.15)). The set ∆∨ ⊂ Φ∨ of coroots dual to ∆ forms abase of Φ∨. With this in mind, a tuple

(X,Y,∆,∆∨) (7.4)

is called a based root datum if there is a root datum (X,Y, Φ, Φ∨) suchthat ∆ ⊆ Φ is a maximal set of simple roots (with respect to some set ofpositive roots) and ∆∨ is the dual set. We note that ∆ spans Φ and ∆∨

spans Φ∨ as Z-modules, so there is no ambiguity in the notation (7.4). Thereis an obvious notion of isomorphism of root data; it is simply a pair of linearisomorphisms on the first two factors preserving the pairing and the sets ofsimple roots.

We let Ψ(G,B, T ) := (X∗(T ), X∗(T ), ∆,∆∨) be a choice of based rootdatum, and Ψ(G,T ) the root datum it defines. The reason for the B in thisnotation is the following lemma:

Lemma 7.3.1 The choice of a set of simple roots ∆ ⊆ Φ is equivalent to thechoice of a Borel subgroup B ≤ G containing T .

We have already mentioned a more general result in §1.9, but we did notprove it there. We pause here to sketch the proof. For each α ∈ Φ there is aunique subgroup Nα ≤ G normalized by T such that LieNα = gα. It is calledthe root group attached to α. For example if G = GLn and α = eij in thenotation of (1.9) then the corresponding root group is In +Reij .

Proof of Lemma 7.3.1: We just explain the bijection, leaving the details tothe standard references mentioned at the beginning of Chapter 1. Given a


choice of a set of simple roots ∆ the group

B = 〈T, Nαα∈∆〉 (7.5)

is a Borel subgroup. Conversely, given a Borel subgroup the set of roots of Tin LieB form a set of positive roots which provides us with a set of simpleroots. 2

To better describe Ψ(G,B, T ) we recall the notion of a pinning:

Definition 7.2. A pinning of G is a tuple

(B, T, Xαα∈∆)

where T is a maximal torus and B is a Borel subgroup containing it, ∆ isthe set of simple roots attached to B, and Xα ∈ gα − 0 for all α.

We let Aut(B, T, Xαα∈∆) be the group of automorphisms of G thatpreserve B and T and the set Xαα∈∆. The theorem of Chevalley and De-mazure (Theorem 1.8.3) provides lifts of automorphisms of root data of (G,T )to automorphisms of the pair (G,T ); using it we obtain an isomorphism

Aut(B, T, Xαα∈∆)−→Aut(Ψ(G,B, T )) (7.6)

where we use ∆ to define Ψ(G,B, T ).

Proposition 7.3.2 There is a split exact sequence

1 −→ Inn(G(k))a−→ Aut(G)

b−→ Aut(Ψ(G,B, T )) −→ 1. (7.7)

The splittings are in bijection with pinnings (B′, T ′, Xαα∈∆) up to conju-gation by T ′(k).

Proof. The arrow a is the obvious injection. To describe b, note that forany automorphism φ ∈ Aut(G(k)) there is a g ∈ G(k) such that φ Ad(g)preserves B and T since all Borel subgroups and tori are conjugate underG(k). Thus φ Ad(g) restricts to define an automorphism of the root datumΨ(G,B, T ). It already preserves B, and by replacing g by tg for some t ∈ T (k)we can assume that

φ Ad(g)

preserves the simple roots. Thus we obtain the map b : Aut(G(k))/ Inn(G(k))→Aut(Ψ(G,B, T )). It is surjective by Theorem 1.8.3. Moreover, by loc. cit. givenan automorphism φ′ ∈ Aut(Ψ(G,T )) there is an automorphism φ ∈ Aut(G)such that φ(T ) = T and b(φ) = φ′. If φ in addition preserves the simple rootsthen it follows that the lift is uniquely determined (see [Con14, §1.5] for moredetails). This proves that Im(a) = ker(b).

Since Aut((B, T, Xαα∈∆)) is defined to be a subset of G, we see that apinning defines a section of Aut(G(k)) −→ Aut(Ψ(G,B, T )).

7.3 The Langlands dual group 133

On the other hand, if g ∈ G(k), then the conjugate of this section may bethought of as Aut(gBg−1, gTg−1, gXαg

−1α∈∆), which is also a pinning ofGk. ut

With this preparation complete we can now define the Langlands dualgroup. Assume that G is a connected reductive group over a local or globalfield F . Choose a Borel subgroup B ≤ GF sep and a maximal torus T ≤ B ≤GF sep . We obtain a based root datum

Ψ(G,B, T ) := Ψ(GF sep , B, T ).

Using the exact sequence of Proposition 7.3.2 the homomorphism GalF →Aut(GF sep) giving G its F structure yields an action of GalF on the basedroot datum. Now

(X∗(T ), X∗(T ), ∆∨, ∆)

is again a based root datum with associated root datum Ψ(G, T ), so we have

a Borel subgroup B ≤ G such that

(X∗(T ), X∗(T ), ∆∨, ∆) = Ψ(G, B, T ). (7.8)

We therefore obtain an action of GalF on Ψ(G, B, T ).

Via a choice of pinning of G we obtain a section of the map

Aut(G)→ Aut(Ψ(G, B, T ))

and hence a map GalF −→ Aut(G). We define the Langlands dual groupof G or the L-group of G to be the semidirect product

LG := G(C) o GalF

with respect to this action. Notice that the action is canonical up to conju-gation by G(C) by Proposition 7.3.2.

A morphism of L-groupsLH −→ LG

is simply a homomorphism commuting with the projections to GalF such thatits restriction to the neutral components is induced by a map of algebraicgroups H −→ G. Morphisms are also sometimes called L-maps.

Assume for the moment that F is a global field. For every place v uponchoosing an embedding F sep → F sep

v we obtain an embedding

GalFv −→ GalF ,

unique up to conjugacy. This induces a morphism

LGFv −→ LGF (7.9)


which allows us to relate local and global L-groups.Occasionally one works with modifications of the L-group. For example,

we can replace GalF by its quotient Gal(E/F ) where E/F is a finite degreeGalois extension such that GE splits. In some applications one wants tomodify or enlarge GalF (see [Mok15]). In general we can replace GalF byany topological group Γ admitting a continuous homomorphism Γ → GalFthat surjects onto Gal(E/F ) for any field E such that GE is split; all of theconstructions above have obvious analogues in this level of generality.

It is useful to give some examples of L-groups in this nonsplit situation.First let G = GLn over a field F and let E/F be a field extension (of finitedegree). One has

LResE/FGLn ∼= GLn(C)σ:E→F sep o GalF (7.10)

where GalF acts via permuting the factors. For a slightly more complicatedexample let M/F be a quadratic extension and let σ ∈ Gal(M/F ). For anF -algebra R, consider the quasi-split unitary group

U(R) :=

g ∈ GLn(M ⊗F R) :

(1...

1

)σ(g)−t

(1...

1

)= g

.

One has

LU ∼= GLn(C) o GalF (7.11)

where GalF acts via its quotient Gal(M/F ), which acts on GLn(C) via theisomorphism g 7→ g−t. We mention that there is an L-map

LU −→ LResM/FGLn

g o σ 7−→ (g, g−t) o σ.

7.4 Parabolic subgroups of L-groups

LetG be a connected reductive group over a local or global field F . In Chapter12 we will require the notion of a parabolic subgroup of an L-group. It ismost natural to discuss this immediately after discussing the L-group, butthe reader should feel free to skip this section and then refer back to it asneeded.

We follow the discussion in [Bor79]. A parabolic subgroup of LG is a

closed subgroup Q such that Q ∩ G(C) is the complex points of a parabolic

subgroup of G and the restriction of the canonical map LG → GalF to Q issurjective. Clearly

LB := B(C) o GalF (7.12)

7.4 Parabolic subgroups of L-groups 135

is a parabolic subgroup, where B is the group of (7.8). A parabolic subgroupof LG is standard if it contains LB. As in the case of algebraic groups,every parabolic subgroup of LG is conjugate under G(C) to one and only onestandard parabolic subgroup.

We now isolate a subset of relevant parabolic subgroups of LG. This willallow us to define a dual parabolic subgroup LP to any proper parabolicsubgroup P ≤ G. We will also explain how this duality behaves with respectto Levi subgroups.

For every field extension F sep ≥ k ≥ F we let P(k) denote the set ofparabolic subgroups of Gk. Let (P/G)(k) denote the set of G(F sep)-conjugacyclasses of parabolic subgroups in G(F sep) that are fixed by Gal(F sep/k).There is a natural map

P(k) −→ (P/G)(k)

which is surjective if G is quasi-split.Similarly, let LP denote the set of parabolic subgroups of LG and let

LP/LG denote the set of LG-conjugacy classes of parabolic subgroups of LG.The bijection ∆↔ ∆∨ yields a bijection

(P/G)(k)←→ LP/LG.

Note that we view (P/G)(k) above as a group scheme opposed to the factthat LP/LG is a group.

We say that a parabolic subgroup of LG is relevant if its LG-conjugacyclass is in the image of the composite

P(k) −→ (P/G)(k) −→ LP/LG. (7.13)

If P ∈ P(k) we denote by LP the standard parabolic subgroup in the classthat is the image of P under the map (7.13).

Let Q be a parabolic subgroup of LG. The unipotent radical N of Q isnormal in Q and will also be called the unipotent radical of Q. Then Q is thesemidirect product of N by the normalizer in Q of any Levi subgroup M ofQ. Those normalizers will be called the Levi subgroups of Q.

For given parabolic subgroup P ≤ G, a uniqueG(F sep)-conjugate gPF sepg−1

of PF sep contains BF sep . Since the parabolic subgroups of GF sep containingB are in bijection with subsets of ∆ as explained in Theorem 1.9.2 we cantherefore associate a subset J(P ) ⊆ ∆ to P . Given a Levi subgroup M ≤ P ,there is a g′ ∈ G(F sep) such that g′Mg′−1 has based root datum

(X∗(T ), X∗(T ), J(P ), J(P )∨)

and hence a Levi subgroup of LP with based root datum

(X∗(T ), X∗(T ), J(P )∨, J(P )).


The semidirect product of this Levi subgroup of LP with Gal(F sep/F ) isthen a Levi subgroup of LP denoted by LM . A Levi subgroup of a parabolicsubgroup LP of LG is relevant if LP is.

7.5 The Satake isomorphism for quasi-split groups

Let G be an unramified connected reductive group over a local field F andlet K ≤ G(F ) be a hyperspecial subgroup. We wish to describe Langlands’extension of the Satake isomorphism to this setting. Since G is unramifiedwe can take the L-group to be

G(C) o FrZ

where Fr ∈ Gal(F sep/F ) is a choice of Frobenius element. In the split case,the Satake transform could be phrased in terms of the quotient

G(C)/conj.

In the nonsplit case, the appropriate quotient is

Go Fr/ ∼

where the action is given on C-algebras R by

G(R)× G(R) o Fr −→ G(R) o σ

(g, (ho Fr)) 7−→ ghFr(g)−1 o Fr.

This action can also be described as the action of G on itself via Fr-conjugacy,a notion that plays a key role in the twisted trace formula (see §??? below).

We now describe the analogue of the Chevalley restriction theorem (7.2)

in this setting. Given a maximal torus T ≤ G, its dual T is a maximal torusof G, and we can consider its Weyl group W (G, T ). Now

W (G,T )(F ) ≤ Aut(Ψ(G,T )) = Aut(Ψ(G, B, T ))

so we can consider W (G,T )(F ) as a subgroup of W (G, T ) which is a quo-

tient of the normalizer NG(T )(C). Let FN ≤ NG(T )(C) denote the inverse

image of W (G,T )(F ) under the quotient map NG(T )(C) → W (G, T )(C).The analogue of the Chevalley restriction theorem in this setting is

T o Fr/FN −→ Go Fr/ ∼ (7.14)

(see [Bor79, Lemma 6.5]). With this isomorphism in mind the following the-orem is natural:

7.5 The Satake isomorphism for quasi-split groups 137

Theorem 7.5.1 (Langlands and Satake) There is an isomorphism

S : C∞c (G(F ) //K) −→ C[T o Fr]FN .

ut

As before, we can rephrase this as an isomorphism

Hom(C∞c (G(F ) //K),C)→ (T o Fr/FN)(C) −→ (Go Fr)Fr-ss(C)/ ∼(7.15)

where (G × Fr)Fr-ss is the subscheme of elements that are Fr-semisimple; a

concise way to phrase this condition is that the orbit under the action of Gis closed. Thus we obtain the following analogue of Corollary 7.2.2:

Corollary 7.5.2 Assume that G is unramified over F . The composite iso-morphism (7.15) induces a bijection

(Go Fr)Fr-ss(C)/ ∼

←→

isomorphism classesof irreducible unramifiedrepresentations of G(F )

.

ut

By way of terminology the Fr-semisimple conjugacy class attached toan isomorphism class of unramified representations is called its Langlandsclass. In the split case, the eigenvalues of a representative of the conjugacyclass are called its Satake parameters.

We will not prove either Theorem 7.2.1 or Theorem 7.5.1, though we willsay more about the definition of the map S in a moment. The standardreferences are [Car79, Theorem 4.1], [Bor79, §7], [Sat63], which we follow.There is a categorified version of the Satake correspondence due to Mirkovicand K. Vilonen that is known as the geometric Satake correspondence, see[MV07]. This result is an important tool in the geometric Langlands program.Moreover it has now been extended to mixed characteristic by work of Zhuand Bhatt-Scholze.

Let B ≤ G be a Borel subgroup containing a maximal torus T of G and letK be a hyperspecial subgroup. We let N denote the unipotent radical of B.We say that K is in good position with respect to (B, T ) if the Iwasawadecomposition

G(F ) = KB(F )

holds andB(F ) ∩K = (K ∩ T (F ))(K ∩N(F )).


We also can and do assume that KT := K ∩ T (F ) is a maximal compactsubgroup of T . In fact, we given B and T we can always choose K so thatthese assumptions hold (see Theorem 2.7.1).

Using the Iwasawa decomposition G(F ) = KB(F ) = KT (F )N(F ) wehave a decomposition of measures

dg = dkdrb = δB(t)dkdtdn (7.16)

(see Proposition 3.2.1); we always assume that dk(K) = 1. We define a C-linear map

C∞c (G(F ) //K) −→ C∞c (T (F )/KT )

f 7−→ fB

where

fB(t) := δB(t)1/2

∫N(F )

f(tn)dn. (7.17)

The function fB is known as the constant term of f along B.The following lemma is implied by Lemma 8.6.2 and Proposition 8.7.2

below:

Lemma 7.5.3 The map f 7→ fB is an algebra homomorphism and has imagein C∞c (T (F )/KT )W (G,T )(F ). ut

Let Ts ≤ T be the greatest F -split torus in T . Thus we can write T = TsTawhere Ta is anisotropic and Ts ∩ Ta is finite. Let Ks = K ∩ Ts(F ); it is amaximal compact subgroup of Ts(F ). We note that

W (G,T )(F ) = W (G,Ts)(F ) = W (G,Ts)(Fsep).

Lemma 7.5.4 There is a W (G,T )(F ) ≤ W (G,T )(F sep) = W (G, T )(C)-equivariant isomorphism

C∞c (T (F )/KT )W (G,T )(F )−→C[Ts]W (G,T )(F ).

Proof. We note that Ta(F ) is compact, so via restriction we obtain an iso-morphism

C∞c (T (F )/KT )W (G,T )(F )−→C∞c (Ts(F )/Ks)W (G,T )(F ).

We note that there is an isomorphism of C-algebras

C∞c (Ts(F )/Ks) −→ Hom(X∗(Ts)F ,C) = C[X∗(Ts)F ]

1tKs 7−→ (χ 7→ ordv(χ(t))).

7.5 The Satake isomorphism for quasi-split groups 139

It is not hard to see that this isomorphism is W (G,T )(F )-equivariant. Sinceone has an identification

C[X∗(Ts)F ] = C[X∗(Ts)C] = C[Ts],

we deduce the lemma. ut

Lemma 7.5.5 The inclusion Ts → T induces a bijection

(T o Fr)(C)/ ∼−→ Ts(C)/W (G,T )(F )

where on the left the quotient is modulo the conjugation action of FN .

Proof. Let ν : T → Ts be the map induced by the inclusion Ts → T viaduality.

As mentioned after [Bor79, Lemma 6.4] every element of FN is of the form

ws where w ∈W (G,T )(F ) and u ∈ T (C). Moreover

(wu)−1(to Fr)wu = u−1Fr(u)w−1tw o Fr.

Since Ts is the fixed points of Fr on T , we see that we obtain a map

ν : (T o Fr)(C) −→ Ts(C)

(to Fr) 7−→ ν(t)

that is equivariant with respect to the action of FN (which acts via its quo-tient W (G,Ts)(F ) on the right hand side). It is clearly surjective. To prove

injectivity, assume that t, t′ ∈ T (C) and

ν(to Fr) = w−1ν(to Fr)w

for some w ∈ W (G,T )(F ). Then ν(t) = ν(w−1t′w) since w is fixed by Fr.Thus t = xw−1t′w for some x ∈ ker(ν). By Hilbert’s theorem 90 all x ∈ ker(ν)are of the form u−1Fr(u), and we deduce that toFr = (wu)−1(t′oFr)wu. ut

Combining lemmas 7.5.3, 7.5.4 and the map of coordinate rings inducedby Lemma 7.5.5 we see that we have constructed the Satake morphism

S : C∞c (G(F ) //K) −→ C∞c (T (F )/KT )W (G,T )(F ) −→ C[T o Fr]FN , (7.18)

though we have not proved that it is injective or surjective. The first map inthis factorization of the Satake isomorphism is a special case of a parabolicdescent map. In general, any construction that relates objects on the groupG to objects on Levi subgroups of its parabolic subgroups (in this case theLevi subgroup T of the Borel subgroup B) is known as parabolic descent. Wewill use this idea in the next section to give a more explicit parametrizationof unramified representations.


7.6 The Principal series

We now explain how to explicitly realize unramified representations. Let Gbe an unramified connected reductive group with Borel subgroup B ≤ Gand maximal torus T ≤ B. We assume that B and T are chosen so that theIwasawa decomposition

G(F ) = B(F )K

holds and that KT := T (F ) ∩ K is a maximal compact open subgroup ofT (F ); it is always possible to arrange this by Lemma A.5.2. Let N ≤ B bethe unipotent radical of B.

We recall from Proposition 3.5.1 that the modular character

δB := δB(F ) : B(F )→ C (7.19)

isδB(b) := |det(Ad(b))|.

The last ingredient we need to define unramified principal series representa-tions is unramified quasi-characters.

Definition 7.3. A quasi-character χ : T (F ) → C× is unramified if it istrivial on KT .

Let us explicitly describe unramified quasi-characters. As in the previoussection, we let Ts ≤ T be the maximal split torus and choose an anisotropictorus Ta ≤ T such that TaTs = T and Ta ∩ Ts is finite. We remind thereader that X∗(T )F is the abelian group of homomorphisms T → Gm; thisis in general a proper subgroup of the group X∗(T )F sep of homomorphismsTF sep → GmF sep . In fact

X∗(T )F = X∗(Ts)F .

In analogy with (4.12) we define a map

HT : T (F ) −→ aT := Hom(X∗(T )F ,R)

via

e〈HT (t),χ〉 = |χ(t)|

for χ ∈ X∗(T ). For each

λ ∈ a∗TC := X∗(T )F ⊗ C

we then obtain a quasi-character

t 7−→ e〈HT (t),λ〉.

7.6 The Principal series 141

It is clearly unramified. Moreover, it depends only on the image of λ underthe map

X∗(T )F ⊗ C −→ X∗(T )F ⊗ C× = Ts(C)

induced by the map x 7→ e−x (for the last equality see Exercise 7.5). Thus

for every element of Ts(C) we obtain an unramified quasi-character of T (F ).Taking T = G in the following lemma we see that all unramified quasi-characters arise in this manner:

Lemma 7.6.1 One has natural FN -equivariant bijections

Hom(T (F )/KT ,C×)←− Ts(C) −→ (T (C) o Fr)FN

(t 7→ e〈HT (t),λ〉)←− [ e−λ

provided that we let FN act via its quotient W (G,T )(F ) ≤W (G,T )(F sep) =

W (G, T )(C) on the left two groups.

Proof. For the first isomorphism we start by noting that the inclusion Ts → Tinduces a W (G,T )(F )-equivariant isomorphism

Ts(F )/Ks−→T (F )/KT ,

where Ks := K ∩ Ts(F ) is a maximal compact subgroup of Ts(F ). Thus weare immediately reduced to the case where T is split.

We henceforth assume Ts = T . The map

Ts(C) −→ Hom(Ts(F )/Ks,C×)

e−λ 7−→(t 7→ e〈HT (t),λ〉

)is W (G,T )(F )-equivariant, so we need only check it is an isomorphism. Bychoosing an isomorphism Ta ∼= Gdm we are reduced to observing that

C× −→ Hom(F×/O×F ,C×)

e−λ 7−→ (t 7→ qλ log |t|)

is a bijection (here we represent an element of C× as e−λ for some λ ∈ C). Thiscompletes the proof of the first bijection. The second bijection has alreadybeen proven in Lemma 7.5.5. ut

For λ ∈ a∗TC we define the representation

I(λ) := IndGB(t 7→ e〈HT (t),λ〉)

to be the (smooth) representation of G(F ) on the space of functions

142 7 Unramified Representationsϕ ∈ C∞(G(F )) :

ϕ(tng) = δB(t)1/2e〈HT (t),λ〉ϕ(g)for all (t, n, g) ∈ T (F )×N(F )×G(F )

.

Here G(F ) acts via right translation:

I(λ)(g)ϕ(x) := ϕ(xg). (7.20)

This is an example of an induced representation, see §8.2 for the general

construction. The factor of δ1/2B is present so that if

λ ∈ ia∗T

then I(λ) is pre-unitarizable; this assertion is part of the following specialcase of Proposition 8.2.2 later:

Proposition 7.6.2 The representations I(λ) are admissible and satisfy

I(λ)∨ ∼= I(−λ).

If λ ∈ ia∗T then I(λ) is pre-unitarizable.

We note that the condition λ ∈ ia∗T is equivalent to the statement that thequasi-character t 7→ e〈HT (t),λ〉 is unitary (i.e. is a character).

Definition 7.4. An unramified principal series representation of G(F )is a representation isomorphic to I(λ) for some λ ∈ a∗TC

This definition is traditional, but it is a little misleading. Indeed, unrami-fied principal series representations need not be irreducible, but we do havethe following lemma:

Lemma 7.6.3 There is a unique line in I(λ) fixed by K.

Proof. A function in I(λ) is uniquely determined by its restriction to K. ut

If I(λ) is reducible, the situation is slightly more complicated, but notinsufferably so. To describe what happens, let us start by remarking thatW (G,T )(F ) acts on X∗(T )F , hence a∗TC by duality. It therefore makes senseto talk of I(w(λ)) for λ ∈ a∗TC and w ∈W (G,T )(F ). We say that λ ∈ a∗TC isregular if w(λ) = λ for w ∈W (G,T )(F ) implies that w is the identity. Thisis equivalent to λ lying in an open Weyl chamber in a∗TC. With this action ofW (G,T )(F ) in mind we are ready to state a result known as the unramifiedsubquotient theorem:

Theorem 7.6.4 Let λ, λ′ ∈ a∗TC. Then one has the following:

(a) The representations I(λ) and I(λ′) are isomorphic if and only if λ = w(λ′)for some w ∈W (G,T )(F ).

(b) Every unramified representation is isomorphic to a subrepresentation ofan I(λ), and every I(λ) admits a unique unramified subquotient.

7.7 Weak global L-packets 143

(c) Assume that λ ∈ iaT . Under this assumption the representation I(λ) isirreducible if and only if λ is regular.

ut

For the proof of the theorem one can refer to [Car79, §3-§4]. These results aredue to Casselman, and one can also find proofs of them in his unpublishednotes [Cas]. For λ ∈ a∗TC we let J(λ) be the unique unramified subquotientof I(λ). The reason for the apparent asymmetry in item (b) is that J(λ)is not necessarily an irreducible subrepresentation of I(λ), merely a sub-quotient. It is however an irreducible subrepresentation of I(w(λ)) for somew ∈W (G,T )(F ).

By the theorem, every unramified representation is isomorphic to a J(λ)where λ is unique up to the action of W (G,T )(F ). This is consonant withCorollary 7.5.2, the parametrization of unramified representations affordedby the Satake isomorphism, and it is useful to explicitly relate Corollary7.5.2 and Theorem 7.6.4:

Proposition 7.6.5 Let f ∈ C∞c (G(F ) //K). For λ ∈ a∗TC one has

tr J(λ)(f) = S(f)(e−λ).

Proof. One hastr I(λ)(f) = tr e〈HT (·),λ〉(fB)

by Proposition 8.6.1, and by unwinding the definition of the Satake isomor-phism one obtains

tr e〈HT (·),λ〉(fB) = S(f)(e−λ).

Thus to complete the proof it suffices to show that tr I(λ)(f) = tr J(λ)(f).Invoking Theorem 7.6.4 we see that upon replacing λ by wλ for some w ∈W (G,T )(F ) we can assume that J(λ) is a subrepresentation of I(λ).

There is a unique line Cϕ0 in the space of I(λ) fixed by K by Lemma7.6.3. It follows that I(λ)(f) acts via the scalar tr I(λ)(f) on Cϕ0. On theother hand there is an equivariant map

J(λ) −→ I(λ). (7.21)

Since J(λ), being unramified, has a unique spherical line this line its imagemust be Cϕ0 under (7.21). ut

7.7 Weak global L-packets

Let G be a reductive group over a global field F .

Definition 7.5. We say that two irreducible admissible representations π1,π2 of G(AF ) are weakly globally L-indistinguishable if πS1

∼= πS2 for


some finite set of places S of F . An equivalence class of L-indistinguishableadmissible representations is called an weak global L-packet. If an adelicL-packet contains an automorphic representation, we say that it is a weakautomorphic L-packet.

Here when we say an admissible representation of G(AF ) we mean an ad-missible (g,K∞)×G(A∞F )-module in the archimedean case and an admissibleG(AF )-module in the function field case. Later on we will define a notion oflocal L-packets that complements this definition. The definition of local L-packets will be used to give a refinement of the notion of a global L-packet;this is why the adjective weak is used above.

For the remainder of this section, we will omit the word global to save ink.It may seem odd that we include merely admissible, and not automorphic,representations in a weak automorphic L-packet, but this turns out to beconvenient for technical reasons. In practice one describes the weak L-packetof an automorphic representation, and then asks which representations in theweak global L-packet are automorphic.

To describe weak L-packets in more detail we set some notational con-ventions. If S is a finite set of places of F including the infinite places suchthat GFv is unramified for v 6∈ S we say that G is unramified outside ofS. Likewise if π is an admissible representation of G(AF ) then we say π isunramified outside of S if πv is unramified for v 6∈ S.

The Satake isomorphism provides a convenient way of thinking about weakL-packets. Assume that G is unramified outside of S. Notice that if π is anadmissible representation of G(AF ) unramified outside of S then it defines acharacter

trπS : C∞c (G(ASF ) //KS) −→ C

where KS ≤ G(ASF ) is a hyperspecial subgroup, i.e. Kv ≤ G(Fv) is hyper-special for all v 6∈ S. By the Satake isomorphism, giving this character isequivalent to giving, for each v 6∈ S, an element cv ∈ LG o Frv/ ∼, whereFrv is a choice of Frobenius element at v and the quotient is with respect tothe conjugation action of LG; the element cv will be Frv-semisimple. Let

Πw(G) : = weak L-packets of G(AF )Πw,aut(G) : = weak automorphic L-packets of G(AF )

(7.22)

and

c(G) := (cv) ∈∏v 6∈S

LG o Frv/ ∼: cv is Frv-semisimple for all v 6∈ S/ ∼

where we say that∏v 6∈S cv ∼

∏v 6∈S c

′v if and only if cv = c′v for almost every

v 6∈ S (i.e. all v outside of a finite set). There is no need to be specific aboutthe set of places S in the notation here because we can always enlarge S by afinite set. Any weak L-packet has a representative that is unramified outsideof S, so we have a map


c : Πw(G) −→ c(G) (7.23)

defined in the obvious manner. We denote by c(Π) the image of an L-packetΠunder this map. We refer to it as the Langlands class or Satake parameterof Π. If G(Fv) has a unique G(Fv)-conjugacy class of hyperspecial subgroupsfor almost every v then by the Satake isomorphism the map c is bijective.In general hyperspecial subgroups are only conjugate under Gad(Fv) [Tit79,§2.5] so the map can have infinite fibers.

We are now in a position to discuss a naıve form of the Langlands functo-riality conjecture. Let

r : LH −→ LG

be an L-map. This gives rise to a map

c(H) −→ c(G).

We can now state a weak form of the Langlands functoriality conjecture:

Conjecture 7.7.1 (Langlands) Given a weak global L-packet Π of auto-morphic representations of H(AF ) there is a global L-packet r(Π) of auto-morphic representations of G(AF ) such that c(r(Π)) = r(c(Π)).

One can also phrase this as the existence of a top arrow making the followingdiagram commute:

Πw,aut(H)

c

∃? // Πw,aut(G)

c

c(H)

r // c(G)

The fact that the class c(Π) is only defined up to a finite set of placesis an irritant, but it is not as horrible as it first appears. To explain, wewill require some results described in more detail in §10.6. Let n1, . . . , ndbe a collection of positive integers with

∑di=1 ni = n. Let P = MN ≤

GLn be the parabolic subgroup of type (n1, . . . , nd) with its standard Levidecomposition (see Example 1.10). For each i let πi be a cuspidal automorphicrepresentation of AGLni

\GLni(AF ) and let λ ∈ a∗M . We can then form theinduced representation

I(π1, . . . , πnd , λ)

as in §10.3. In general it may have several irreducible subquotients, all ofwhich turn out to be automorphic. However it always has a canonical sub-quotient that will be described in more detail in §10.6. Automorphic repre-sentations equivalent to this canonical subquotient are known as isobaricrepresentations. In particular, if I(π1, . . . , πnd , λ) is irreducible then it isisobaric; this holds in particular if n1 = n which is to say that the represen-tation is cuspidal.

We now record the following foundational fact from [JS81b, JS81a]:


Theorem 7.7.2 (Jacquet and Shalika) Every weak automorphic L-packetin Π(GLn) contains a unique isobaric element. ut

This tells us that, at least for GLn, admissible representations that areautomorphic are very special; they are determined by their local factors atany cofinite set of places of F (see Exercise 7.8).

On the other hand, if we fix a finite set of places S of F and choose an ana-lytically well-behaved irreducible admissible representation πS of AG\G(FS),then πS is almost always the local factor at S of an automorphic represen-tation. To make this precise, recall the notion of a tempered representationfrom Definition 4.6 and the Fell topology on the unitary dual from §3.8. Fora proof of the following theorem see [Clo07, §3.3] and the references therein:

Theorem 7.7.3 (Burger, Li and Sarnak) Suppose that πS is a unitaryirreducible tempered representation of the semisimple group G over F . Thenthere is an automorphic representation π′ of G(AF ) such that π′S in the clo-sure of the point πS is in the unitary dual of G(FS). ut

This theorem can be interpreted as saying that almost every irreducibletempered representation of AG\G(FS) is the local factor of an automorphicrepresentation.

Exercises

7.1. Let S2 act on C[t±1 , t±2 ] by letting the nontrivial element act by t1 7→ t2.

Define a linear map

S ′ : C∞c (GL2(F ) // GL2(OF )) −→ C[t±1 , t±2 ]S2

by setting, for each n ∈ Z, k ∈ Z≥0,

S ′b k2 c∑i=0

1(i+n,k−i+n)

= (t1t2)nSymk(t1, t2).

Show that this is an algebra isomorphism and deduce that S = S ′.

7.2. Let G be a td-group and let K ≤ G be a compact open subgroup. Arepresentation (π, V ) of G is said to be generated by a subspace W ≤ V ifπ(G)W = V . Prove that there is an equivalence of categories

representations of G generated by V K−→C∞c (G //K)-modulesV 7−→ V K .

In particular, deduce that every representation of G generated by V K issmooth and admissible.


7.3. Let E/F be a quadratic extension of number fields with Galois auto-morphism σ. For any F -algebra R, let

U(R) :=g ∈ GL2(E ⊗F R) : ( 1

1 )σ(g)−t ( 11 ) = g

.

This is a unitary group in two variables over F . Prove that U is quasi-splitover F , UFv is split at every place v of F where E/F is split, and UFv isnonsplit at every place v of F where E/F is nonsplit.

7.4. Prove the isomorphisms (7.10) and (7.11).

7.5. Let T be a torus over C. Prove that the set valued functors on thecategory of C algebras are all naturally equivalent: For C-algebra R,

R 7−→ Ts(R)

R 7−→ X∗(T )⊗C R,R 7−→ Hom(X∗(T ), R).

7.6. Let F be a local field and let T be a torus over F . If T (F ) is compact,then T is anisotropic.

7.7. Let G be an affine algebraic group over a local field F and let V be atopological C-vector space (possibly of infinite dimension). Prove that a map

G(F )→ V

is continuous with respect to the natural topology on G(F ) and the giventopology on V if and only if it is locally constant. Conclude that the mapG(F ) → V is continuous if and only if it is continuous when we give V thediscrete topology.

7.8. Let π be a cuspidal automorphic representation of AGLn\GLn(AF ) forsome global field F . Show that if S is a finite set of places of F and π′S 6∼= πSis an admissible representation of GLn(FS) then

π′S ⊗ πS

is never an isobaric automorphic representation of AGLn\GLn(AF ).

Chapter 8

Nonarchimedean Representation Theory

Let’s assume we know nothing,which is a reasonableapproximation.

D. Kazhdan

Abstract In this chapter we explain how general admissible representationsare built up out of supercuspidal representations via the process of parabolicinduction.

8.1 Introduction

Let G be a reductive group over a nonarchimedean local field F , let P ≤ Gbe a parabolic subgroup, and let P = MN be a Levi decomposition withM ≤ P a Levi subgroup and N the unipotent radical of P . This notationwill be in force throughout the chapter.

We recall from Proposition 3.5.1 that the modular quasicharacter

δP := δP (F ) : P (F ) −→ R× (8.1)

isδP (p) := |det(Adp(p) : p(F )→ p(F ))|.

Let (σ, V ) be a smooth irreducible representation of M(F ). Using the mod-ular character we define the parabolically induced representation (orsimply induced representation) I(σ) := IndGP (σ) to be the (smooth) rep-resentation of G(F ) on the space of functions

IndGP (V ) :=

locally constant ϕ : G(F ) −→ V : ϕ(mng) = δP (m)1/2σ(m)ϕ(g)

149

150 8 Nonarchimedean Representation Theory

for all (m,n, g) ∈M(F )×N(F )×G(F ). Note that a function ϕ : G(F )→ Vis locally constant if and only if it is continuous with respect to the naturaltopology on G(F ) and the usual topology (or even the discrete topology) onV . The group G(F ) acts via right translation:

I(σ)(g)ϕ(x) = ϕ(xg). (8.2)

The factor of δ1/2P is present so that if σ is unitarizable then I(σ) is also

unitarizable (see Proposition 8.2.2). We note that this procedure yields afunctor

IndGP : RepsmM(F ) −→ RepsmG(F ), (8.3)

from smooth representations of M(F ) to smooth representations of G(F )(in these categories objects are smooth representations and morphisms areequivariant maps, see §5.3). This functor is called parabolic induction.

In the previous chapter we gave the classification of unramified representa-tions of G(F ). In this chapter we delve deeper into the representation theoryof G(F ).

If V is a smooth representation of G(F ) then its contragredient V ∨ isdefined as in §5.4. A function of the form

m(g) := mϕ,ϕ∨(g) := 〈π(g)ϕ,ϕ∨〉 (8.4)

for (ϕ,ϕ∨) ∈ V ×V ∨ is called a matrix coefficient of π (for the relationshipbetween this notion and that of §4.8 see Exercise 8.1).

Definition 8.1. A supercuspidal (resp. quasicuspidal) representation ofG(F ) is an admissible (resp. smooth) representation all of whose matrix co-efficients are compactly supported modulo the center of G(F ).

The main result of this chapter is a theorem of Jacquet which states thatevery irreducible admissible representation is obtained as a subquotient ofthe parabolic induction of a supercuspidal representation for an appropri-ately chosen parabolic subgroup (see Corollary 8.3.5 for the precise state-ment). One can profitably compare this result with the Langlands classifica-tion from §4.9 and the classification of automorphic representations explainedin Chapter 10. These results can all be viewed as manifestations of whatHarish-Chandra called the “philosophy of cusp forms” [HC70a]. Interpretedbroadly, this is a slogan for the statement that all irreducible representationsof reductive groups can all be obtained via parabolic induction from cuspi-dal representations of parabolic subgroups. Of course, the correct notion ofcuspidal representation varies based on the context.

We will not give a full proof of Theorem 8.3.5, but we will develop muchof the machinery used in the proof because it is useful in many contexts. Wewill also discuss the notion of a trace of an admissible representation, and

8.2 Parabolic induction 151

in addition develop some of the basic properties of supercuspidal representa-tions, including the fact that they admit coefficients (see Proposition 8.5.3),a fact that we will later use in our treatment of simple trace formulae.

The canonical reference for the topics covered in this chapter is the unpub-lished manuscripts of Casselman [Cas] and [Ber]. The paper [Car79] containssketches of proofs and refers to [Cas] for the details. We have sometimes fol-lowed the exposition in [BH06] and [Mur], which fill in many details omittedin the previous two sources.

8.2 Parabolic induction

Let us now prove some basic facts about parabolic induction. Let (σ, V ) bea smooth representation of M(F ).

We start with the following observation:

Lemma 8.2.1 If K ≤ G(F ) is a compact open subgroup and ϕ ∈ IndGP (V )K

thenϕ(g) ∈ VM(F )∩gKg−1

.

Proof. Under the assumptions in the lemma if k ∈ g−1M(F )g ∩K we have

ϕ(g) = ϕ(gk) = ϕ(gkg−1g) = σ(gkg−1)ϕ(g).

ut

Proposition 8.2.2 One has the following:

(a) If σ is admissible then I(σ) is admissible.(b) If σ is unitary then I(σ) is unitarizable.

Proof. Assume that (σ, V ) is admissible. Let K ≤ G(F ) be a compact opensubgroup. We are to show that IndGP (V )K is finite dimensional. An elementϕ ∈ IndGP (V ) is determined by its value on any set of representatives X forthe double cosets in

P (F )\G(F )/K.

This set of double cosets is finite by the Iwasawa decomposition, and henceX is finite. On the other hand, for any x ∈ X one has ϕ(x) ∈ VM(F )∩xKx−1

by Lemma 8.2.1, and this space is finite dimensional by admissibility. Thisproves (a).

For (b), let (·, ·)V be an M(F )-invariant inner product on V . Choose amaximal compact open subgroup Kmax ≤ G(F ) so that the Iwasawa decom-position G(F ) = P (F )Kmax holds (see Theorem 2.7.1), and normalize theHaar measures dg on G(F ), dk on Kmax, and d`p on P (F ) so that dg = d`pdk(see Exercise 3.5).

For ϕ1, ϕ2 ∈ IndGP (V ) we define


(ϕ1, ϕ2) :=

∫K

(ϕ1(k), ϕ2(k))V dk.

Since every element of IndGP (V ) is determined by its restriction to K this isa positive definite inner product. We must check that it is G(F )-invariant.

Choose a nonnegative f ∈ C∞(G(F )) such that∫P (F )

f(px)drp = 1

for all x ∈ G(F ) (see Exercise 8.3). For x ∈ G(F ) one has∫Kmax

(ϕ1(kx), ϕ2(kx))V dk

=

∫Kmax

(ϕ1(kx), ϕ2(kx))V

∫P (F )

f(pkx)drpdk

=

∫P (F )×Kmax

(ϕ1(kx), ϕ2(kx))V f(pkx)δ−1P (p)d`pdk (see Exercise 3.3)

=

∫P (F )×Kmax

(ϕ1(pkx), ϕ2(pkx))V f(pkx)d`pdk

=

∫G(F )

(ϕ1(gx), ϕ2(gx))V f(gx)dg

=

∫G(F )

(ϕ1(g), ϕ2(g))V f(g)dg.

This expression is independent of x, so we deduce the invariance of our innerproduct. ut

Proposition 8.2.3 If σ is smooth then I(σ∨) ∼= I(σ)∨.

Proof. Fix a maximal compact open subgroup Kmax ≤ G(F ) so that theIwasawa decomposition G(F ) = P (F )Kmax holds (see Theorem 2.7.1).

Let ϕ ∈ IndGP (V ) and ϕ∨ ∈ IndGP (V ∨). We then have a pairing

(ϕ,ϕ∨) :=

∫Kmax

〈ϕ(k), ϕ∨(k)〉dk (8.5)

where the pairing on the right is the pairing between V and V ∨. Arguing asin the proof of part (b) of Proposition 8.2.2 we deduce that this pairing isG(F )-invariant.

Since ϕ∨ is smooth the linear form 〈·, ϕ∨〉 on V is smooth and we thereforeobtain a C-linear map

IndGP (V ∨) −→ IndGP (V )∨ (8.6)

ϕ∨ 7−→ (·, ϕ∨).

8.2 Parabolic induction 153

It is an intertwining map because (8.5) is G(F )-equivariant. We must showthat it is bijective. By smoothness it suffices to show that it induces anisomorphism

IndGP (V ∨)K −→ Ind(V )∨K

for all compact open subgroups K ≤ G(F ).To do this we first describe a basis of Ind(V )K . We let X be a set of

representatives for the double cosets in P (F )\G(F )/K. Since P (F )\G(F ) is

compact X is a finite set. For each x ∈ X let Bx be a basis of V P (F )∩gKg−1

.By Lemma 8.2.1, ϕ(g) ∈ VM(F )∩xKx−1

. For each w ∈ Bx let ϕx,w be thefunction supported on P (F )xK such that

ϕx,w(mnxk) = σ(m)w

for (m,n, k) ∈M(F )×N(F )×K. Then

ϕx,w : x ∈ X,w ∈ Bx (8.7)

is a C-basis of I(σ)K . Now for each x let B∨x be the basis of V ∨M(F )∩xKx−1

dual to Bx. We define ϕx,w∨ by replacing V with V ∨ in the constructionabove, and we obtain a basis

ϕx,w∨ : x ∈ X,w∨ ∈ B∨x (8.8)

for IndGP (V ∨)K . It is clear that (8.7) and (8.8) are a basis and dual basis withrespect to the pairing (8.5). ut

It is often useful to introduce a variant of the parabolic induction functor.We recall briefly the Harish-Chandra map (4.12) discussed in §4.9. We definea map

HM : M(F ) −→ aM := Hom(X∗(M)F ,R) (8.9)

via〈HM (m), λ〉 = log |λ(m)|

for λ ∈ X∗(M)F . For each λ ∈ a∗MC we then obtain a quasi-character

m 7−→ e〈HM (m),λ〉.

It is clearly unramified. We then define

I(σ, λ) := I(σ ⊗ e〈HM (·),λ〉). (8.10)

The point of introducing this extra notation is that it makes clear that theinduced representation I(σ, λ) is part of a continuous family, indexed by a∗MC,of representations. In practice this can be very useful. We employed this


construction in the special case where P is a Borel subgroup and σ is trivialin our construction of the unramified principal series in §7.6.

8.3 Jacquet modules

We now define left-adjoints to the functors IndGP . Let (π, V ) be a smoothrepresentation of G(F ). Set

V (N) := 〈π(n)ϕ− ϕ : ϕ ∈ V, n ∈ N(F )〉 and VN := V/V (N). (8.11)

The space VN is refereed to as the space of coinvariants. We note that sinceM(F ) normalizes N(F ) it normalizes V (N) and hence M(F ) acts on VN . Wecan therefore define

N : Repsm(G(F )) −→ Repsm(M(F )) (8.12)

(π, V ) 7−→ (πN := π|M(F ) ⊗ δ1/2P , VN ).

The representation (πN , VN ) is referred to as the Jacquet module of (π, V )(with respect to N).

A version of Frobenius reciprocity is valid for this functor:

Proposition 8.3.1 Restriction is left adjoint to induction; in other wordsfor smooth representations V of G(F ) and W of M(F ) one has a bijection

HomG(F )(V, IndGP (W ))−→HomM(F )(VN ,W )

that is functorial in V and W .

Proof. There is an M(F )-equivariant map

ev1 : IndGP (W ) −→W

ϕ 7−→ ϕ(1)

where 1 ∈ G(F ) is the identity; it is clearly surjective. Thus we have a M(F )-equivariant map

ev1 (·) : HomG(F )(V, IndGP (W )) −→ HomM(F )(VN ,W )

given by composition with ev1. Here we are using the fact that N acts triv-ially on IndGP (W ) and thus any G(F )-equivariant map V → IndGP (W ) fac-tors through VN . To construct the inverse, suppose we are given an M(F )-intertwining map Φ : VN →W . We define

Φ : V −→ IndGP (W )

ϕ 7−→ (g 7→ Φ(g.ϕ)).

8.3 Jacquet modules 155

The map Φ 7→ Φ is inverse to ev1 (·). ut

One would hope that the functor N preserves admissibility, and this isindeed the case:

Theorem 8.3.2 (Jacquet) The functor N takes admissible representationsto admissible representations. ut

Jacquet also proved the following elegant characterization of quasicuspidalrepresentations using these functors:

Theorem 8.3.3 (Jacquet) A smooth irreducible representation (π, V ) ofG(F ) is quasicuspidal if and only if VN = 0 for all parabolic subgroups P ≤ G.

ut

It is because of these theorems that VN called the Jacquet module. Strictlyspeaking Jacquet wrote up his results only for the case of GLn [Jac71], butthe proof carries over to the general case and was worked out in [Cas].

We will not prove theorems 8.3.2 or 8.3.3, referring the reader to [Cas,§3-4]. In the remainder of this section we use theorems 8.3.2 and 8.3.3 toprove a subrepresentation theorem for admissible representations over nonar-chimedean local fields (see Theorem 8.3.5).

Before proving Theorem 8.3.5 we prove a warm-up result. Recall that asupercuspidal representation is an admissible quasicuspidal representation.

Proposition 8.3.4 A quasicuspidal irreducible representation is supercusp-idal.

Proof. Let (π, V ) be a quasicuspidal irreducible representation of G(F ). Fixa nonzero ϕ ∈ V . For all compact open subgroups K ≤ G(F ) one has

V K = π(eK)V = 〈π(eK)π(g)ϕ : g ∈ G(F )〉,

where

eK :=1

meas(K)1K

is the idempotent attached to K. We must show V K is finite dimensional.Since π admits a central quasi-character (see Exercise 5.7) if V K is not finitedimensional then there exists gn : n ∈ Z>0 ⊆ G(F ), all inequivalent moduloZG(F ), such that π(eK)π(gn)ϕ are linearly independent. Let W ⊆ V K be aC-vector subspace space such that

V K = W ⊕ 〈π(eK)π(gn)ϕ : gn ∈ G(F ), n ∈ Z>0〉.

As V = V K ⊕ kerπ(eK), we can define ϕ∨ ∈ Hom(V,C) such that

〈π(eK)π(gn)ϕ,ϕ∨〉 = n


for all n and ϕ∨|W⊕kerπ(eK) = 0. Then ϕ∨ is fixed by K, hence is smooth, andhence is an element of V ∨. On the other hand, by construction the support ofthe matrix coefficient 〈π(g)ϕ,ϕ∨〉 is not compact modulo the center ZG(F ).This implies the proposition. ut

Combined with Proposition 8.3.4, Theorem 8.3.2 and 8.3.3 allow us todeduce the following concrete manifestation of the philosophy of cusp forms:

Theorem 8.3.5 If (π, V ) is a smooth irreducible representation of G(F )then one has the following:

(a) There exists a parabolic subgroup P = MN ≤ G, a supercuspidal repre-sentation (σ,W ) of M(F ) and a nonzero intertwining map V → IndGP (W ).

(b) The representation π is admissible.

Proof. The first assertion implies the second, as induction preserves admis-sibility by Proposition 8.2.2. For the first, we proceed by induction on thedimension of G (as an F -algebraic group, say). If G has dimension 1 then itis a torus, and so the result is trivial.

Assume that for all proper parabolic subgroups P there is no nonzerointertwining map V → IndGP (W ) where (σ,W ) is a smooth irreducible rep-resentation of M(F ). Applying Frobenius reciprocity (Proposition 8.3.1) wesee that

HomM(F )(VN ,W ) = 0

for all parabolic subgroups P = MN and all smooth representations σ ofM(F ), which implies π is supercuspidal by Theorem 8.3.3.

Now assume that there is a proper parabolic subgroup P < G, a Levisubgroup M ≤ P , a smooth representation (σ,W ) of M(F ), and a nonzero(hence injective) intertwining map V → IndGP (W ). By Frobenius reciprocity,there is a nonzero intertwining map

VN −→W

of representations of M(F ) so we can apply our inductive hypothesis to de-duce that there is a parabolic subgroup Q ≤M with Levi subgroup MQ anda supercuspidal representation (ρ, U) of MQ(F ) and a nonzero intertwiningmap

VN −→W −→ IndMQ (U).

Applying Frobenius reciprocity again we obtain a nonzero intertwining map

V −→ IndGP IndMQ (U) ∼= IndGQN (U)

(see Exercise 8.4 for the last isomorphism). ut

8.4 The Bernstein-Zelevinsky classification 157

8.4 The Bernstein-Zelevinsky classification

Theorem 8.3.5 provides the first step towards a classification of all irreducibleadmissible representations of G(F ) in terms of supercuspidal representations.The remaining task is to understand isomorphisms between subquotients ofinduced representations.

It is useful to start by stepping backwards and recalling the Langlands clas-sification in the nonarchimedean setting. This classifies irreducible admissiblerepresentations in terms of tempered representations and is the analogue inthe current setting of Theorem 4.9.2.

Fix a minimal parabolic subgroup P0 ≤ G and call a parabolic subgroupstandard if it contains P0. Let M(F )1 := kerHM where HM is defined asin (8.9). Let (σ, V ) be an irreducible admissible representation of M(F )1.We extend it trivially to M(F ). Given λ ∈ a∗M we can form the inducedrepresentation I(σ, λ) as in (8.10).

The following is the analogue of Theorem 4.9.1 in this setting:

Theorem 8.4.1 If σ is unitary and tempered and λ is in the positive Weylchamber then I(σ, λ) admits a unique irreducible quotient J(σ, λ). ut

The representation J(σ, λ) is known as the Langlands quotient of I(σ, λ).

Theorem 8.4.2 Every irreducible admissible representation of G(F ) is iso-morphic to some J(σ, λ). Moreover if we insist that the parabolic sub-group defining J(σ, λ) is standard, fix a Levi decomposition of each standardparabolic subgroup, and stipulate that λ is in the positive Weyl chamber thenevery irreducible representation of G(F ) is isomorphic to a J(σ, λ) that isunique up to replacing σ by another representation of M(F )1 equivalent toσ. ut

For the proofs of these theorems we refer to [BW00, §XI.2]. See also [Kon03].In view of these theorems to classify the admissible irreducible representa-

tions of G(F ) it suffices to classify the irreducible tempered representationsof G(F ). This work is largely complete in the case when G was a classicalgroups, see [Mg02, MgT02, Jan14] for example. The situation for general lin-ear groups is arguably the simplest and we will discuss it in this section. Theresults are due to Bernstein and Zelevinsky [BZ77, Zel80], and the theory isknown as the Bernstein-Zelevinsky classification.

We start with the classification of representations that are essentiallysquare integrable. Let n > 1, let a|n, and let σ be an irreducible super-cuspidal representation of GLn/a(F ). The external tensor product σa := σ⊗a

can then be viewed as a (supercuspidal) representation of GLan/a, which weview as a Levi subgroup of the standard parabolic subgroup of GLn of type(n/a, . . . , n/a) (see Example 1.10 for our conventions regarding standardparabolic subgroups).

There is then a unique irreducible subquotient J(σa, λa) of I(σa, λa) where


λa :=

(a− 1

2,a− 3

2, · · · ,−a− 3

2,−a− 1

2

)∈ X∗(M)F ⊗ C.

This subquotient is essentially square-integrable. Conversely, all irreducibleessentially square integrable representations arise in this manner:

Theorem 8.4.3 (Bernstein) Every irreducible (admissible) essentially square-integrable representation π of GLn(F ) is isomorphic to J(σa, λa) for a pair(σ, a) where a|n and σ is an irreducible supercuspidal representation ofGLn/a(F ). The parameter a is uniquely determined by π, and σ is deter-mined by π up to isomorphism. ut

The only reference for the proof appears to be [Ber, Proposition 42]; thetheorem is stated in [Zel80, §9.1]. There is a sketch of the proof in [JS83,§1.2].

To proceed we need the notion of linked representations. Following [BZ77]one calls any set of supercuspidal representations of GLr(F ) for some r ofthe form

σ ⊗ e〈HGLr (·),1〉, σ ⊗ e〈HGLr (·),2〉, · · · , σ ⊗ e〈HGLr (·),d〉

for some integer d a segment. Two segments are linked if neither is includedin the other and their union is a segment.

To each J(σa, λa) we associate the segment

σ ⊗ e〈HM (·), a−12 〉, . . . , σ⊗〈HM (·),− a−1

2 〉.

We say that

J(σa, λa) and J(σ′a′, λa′) (8.13)

are linked if their associated segments are linked.Now assume

∑ji=1 ni = n and ai|ni for all i. If P is the standard parabolic

of type (n1, . . . , nj) then for any collection of square-integrable representa-tions J(σaii , λai) of GLni(F ) as above we can form the induced representation

π = Ind(J(σa11 , λa1

)⊗ · · · ⊗ J(σajj , λaj ), 0). (8.14)

In the case where J(σaii , λai) is linked with J(σajj , λaj ) if and only if i = j

we say that none of the J(σaii , λai) are linked.The following theorem is [Zel80, Theorem 9.7(a)]:

Theorem 8.4.4 If none of the J(σaii , λai) are linked then the representationπ in (8.14) is irreducible. ut

A representation π is irreducible and generic in the sense of §11.2 if andonly if it is isomorphic to a π as in (8.14) above where none of the J(σaii , λai)are linked [Zel80, Theorem 9.7].

8.5 Traces, characters, coefficients 159

Theorem 8.4.5 An irreducible representation of GLn(F ) is tempered if andonly if it is of the form (8.14) where all of the J(σaii , λai) are square integrable.The J(σaii , λai) are uniquely determined up to reordering indices.

To clarify the assumptions in the theorem, we note that the J(σaii , λai) arealways essentially square integrable, so the assumption of square integrabilityin the theorem amounts to assuming that the central character of J(σaii , λai)is unitary.

Proof. In [JS83, §1.2] one finds an argument that proves that an irreduciblepre-unitary tempered representation is generic. In fact any tempered rep-resentation is pre-unitary (see Definition 4.6), so any irreducible temperedrepresentation is generic. Now since the J(σaii , λai) are all square integrablenone of them are linked by Exercise 8.15. Thus we can then use [Zel80, The-orem 9.7(b)] to conclude the assertion of the theorem. ut

Combining the Langlands classification of Theorem 8.4.2 and the subse-quent description of tempered representations given by Theorem 8.4.5 we cangive an analogue of the Langlands classification in terms of square-integrablerepresentations. We defer a discussion of this to §10.5.

8.5 Traces, characters, coefficients

Let π be an admissible representation of G(F ). Then for all f ∈ C∞c (G(F ))one has an operator

π(f) : V −→ V.

There is a compact open subgroup K ≤ G(F ) such that f ∈ C∞c (G(F ) //K),and hence π(f) induces an operator

π(f) : W −→W

for any finite dimensional subspace V K ≤ W ≤ V . We define the trace ofπ(f) by

trπ(f) := trπ(f)|W (8.15)

for any such W . The notion of the trace of a representation will play a crucialrole in the trace formula in later chapters.

In this nonarchimedean setting, a distribution on G(F ) is simply a com-plex linear functional on C∞c (G(F )). Thus the trace map

trπ : C∞c (G(F )) −→ C

is a distribution on G(F ), called the character of π. Of course this distribu-tion depends on a choice of Haar measure. Let Greg ≤ G denote the (open)


subscheme consisting of regular semisimple elements. The notion of a regularsemisimple element will be discussed in more detail in §17.1 below. For themoment we point out that for fields E/F

Greg(E) := γ ∈ G(E) : Cγ ≤ GE is a maximal torus. (8.16)

The following is a fundamental and deep result:

Theorem 8.5.1 (Harish-Chandra) The distribution trπ is represented bya locally constant function Θπ with support in Greg(F ). ut

(see [HC99] for the proof and [Kot05] for a detailed exposition).In other words there is a locally constant function Θπ on Greg(F ) such

that

tr π(f) =

∫G(F )

Θπ(g)f(g)dg

for all f ∈ C∞c (G(F )). This result tells us that we can almost regard trπ asa function.

The following is a version of linear independence of characters adapted tothis setting:

Proposition 8.5.2 (Linear independence of characters) If π1, . . . , πn isa finite set of admissible irreducible representations such that πi ∼= πj impliesi = j, then the distributions trπi are linearly independent.

Proof. We use admissibility to reduce the assertion to a finite-dimensionalsetting. Fix a compact open subgroup K ≤ G(F ) such that V Ki 6= 0 for all i.This implies that V Ki is a finite family of finite dimensional C-vector spaceswith an action of C∞c (G(F ) //K). They are all simple, that is irreducible, forthis action. Moreover, they are pairwise nonisomorphic as C∞c (G(F ) //K)-modules by an analogue of the argument in the proof of Proposition 7.1.1.Let

A := Im

(C∞c (G(F ) //K) −→

∏i

EndC(V Ki )

).

Then A is a finite-dimensional C-algebra and the V Ki are a finite fam-ily of finite-dimensional nonisomorphic simple A-modules. Hence the tracestrπi|C∞c (G(F ) //K) are linearly independent. As a reference for this last state-ment we give [GW09, Lemma 4.1.18]. A simpler argument that depends onthe fact that C∞c (G(F ) //K) has a unit is given as Exercise 8.9. ut

Thus traces can be used to distinguish between a finite set of representa-tions. In particular, if π1, . . . , πn is a finite set of pairwise nonisomorphic ir-reducible representations then linear independence of characters implies thatwe can find an f ∈ C∞c (G(F )) such that

trπi(f) = 0 if and only if i 6= 1.

8.5 Traces, characters, coefficients 161

For a refinement of this result at the level of operators see Exercise 8.10.One can ask for more. Let π be an admissible irreducible representation. A

coefficient of π is a smooth function fπ ∈ C∞c (G(F )) such that trπ(fπ) 6= 0and trπ′(fπ) = 0 for π′ 6∼= π. Thus if a coefficient for π exists, we can use itto isolate π among any set of irreducible admissible representations, finite ornot. For general π, it is not necessarily true that such functions f exist. To putthis in perspective, it is useful to recall the Heizenberg uncertainty principle,namely that the Fourier transform of a compactly supported smooth functionon a real vector space cannot again be compactly supported. Thus if wereplaced G(F ) with the nonreductive group R there would be no way toconstruct coefficients. However, we are not in this setting, and in certaincircumstances we can construct coefficients:

Proposition 8.5.3 Assume that ZG(F ) is compact, and let (π, V ) be anirreducible supercuspidal representation of G(F ). Then for all f ∈ C∞c (G(F ))there exists a unique fπ ∈ C∞c (G(F )) such that

π(fπ) = π(f) and π′(fπ) = 0 if π′ 6∼= π.

As an immediate consequence, we see that coefficients exist for supercuspi-dals. The proof we give depends on our original definition of supercuspidalrepresentations, that is, they are admissible representations whose matrixcoefficients are compactly supported modulo the center.

To prove Proposition 8.5.3 we collect some observations on the space ofendomorphisms of a representation. For any smooth representation (π, V ) ofG(F ) the C-vector space

End(V ) := HomC(V, V )

is naturally a G(F ) × G(F )-module, where one copy of G acts via precom-position and the other via postcomposition. The action is given explicitly by(g1, g2).A = π(g1) A π(g−1

2 ). We let

Endsm(V ) ≤ End(V )

denote the subspace consisting of smooth endomorphisms, that is, en-domorphisms that are left and right invariant by a compact open subgroupK ≤ G. This is a smooth representation of G(F )×G(F ).

The usual isomorphism

V ⊗HomC(V,C) −→ End(V )

given on pure tensors by

ϕ⊗ ϕ∨ 7−→ (ϕ0 7−→ 〈ϕ0, ϕ∨〉ϕ)


is G(F ) × G(F )-equivariant and upon restriction induces an isomorphismV ⊗V ∨ −→ Endsm(V ) (here all of the tensor products are over C). Thus theaction of G(F ) × G(F ) on Endsm(V ) can be reasonably denoted by π ⊗ π∨(for more details on product representations see Theorem 5.7.2 above). Thisis a special case of an exterior tensor product; unfortunately, it seems thatthe notation ⊗ is used more commonly for this than . If (π, V ) is admissible,then Endsm(V ) is also admissible (see Exercise 8.13).

One also has an intertwining map

β : (π ⊗ π∨,Endsm(V )) −→ (ρ, C∞(G(F )))

given byβ(A)(g) = tr(π(g) A).

Here ρ acts via ρ(g1, g2)(f)(h) = f(g−11 hg2). We note that for each A ∈

Endsm(V ) the function β(A) is a sum of matrix coefficients. Indeed, if wechoose a compact open subgroup K ≤ G(F ) such that A is fixed on the leftand right under K, choose a basis B of V K and a dual basis B∨ of V ∨K then

β(A)(g) =∑ϕ∈B〈π(g) Aϕ,ϕ∨〉. (8.17)

Proof of Proposition 8.5.3: Since ZG(F ) is compact by assumption and π issupercuspidal, we have β(Endsm(V )) ≤ C∞c (G(F )). Since there are ϕ ∈ Vand ϕ∨ ∈ V ∨ such that

〈ϕ,ϕ∨〉 = 〈π(1)ϕ,ϕ∨〉 6= 0,

we have that β is not identically zero. Thus since the exterior tensor productπ ⊗ π∨ is irreducible β is an embedding.

Consider

β′ : (ρ, C∞c (G(F ))) −→ (π ⊗ π∨,Endsm(V )) (8.18)

f 7−→ π(f).

Then β′ β is an endomorphism of the irreducible representation Endsm(V )of G(F )×G(F ). Hence β′ β is scalar by Schur’s lemma (see Exercise 5.5),say β′ β = λId for some λ ∈ C. We will now show that λ 6= 0 and that wecan take

fπ := λ−1β β′(f).

First,π(β β′(f)) = β′ β β′(f) = λβ′(f) = λπ(f).

To show that λ 6= 0, note that we can find f ∈ C∞c (G(F )) such that π(f) 6=0 (take, for example, f to be the characteristic function of a sufficiently

8.6 Parabolic descent of representations 163

small compact open subgroup). Thus β′(f) = π(f) 6= 0, and since β is anembedding, we deduce that β β′(f) 6= 0.

Second, let (π′, V ′) be a smooth irreducible representation of G(F ), andlet ϕ′ ∈ V ′ be a non-zero vector. Let

γ : (ρ|G(F )×1, C∞c (G(F ))) −→ (π′, V ′) (8.19)

f 7−→ π′(f)ϕ′.

As a representation of G(F ) one has that Endsm(V )|G(F )×1 is isomorphic to adirect sum of copies of π, and thus the same is true of γ(β(Endsm(V )). Thusγ(β(Endsm(V ))) = 0 unless π′ ∼= π (since whenever the former is nonzero weobtain an intertwining operator between π′ and π). It follows that π′(fπ) = 0if π′ 6∼= π. This completes the proof of the proposition. 2

8.6 Parabolic descent of representations

Parabolic descent is a term for the process by which one passes from objects(say, representations) on a group to objects on a Levi subgroup of a parabolicsubgroup. In this section we give two examples of this phenomenon. First weshow that the character of a parabolically induced representation is easilyrelated to the character of the inducing representation (Proposition 8.6.1).

Let P ≤ G be a parabolic subgroup with Levi decomposition P (F ) =M(F )N(F ) and let K ≤ G(F ) be a maximal compact subgroup.

Definition 8.2. A maximal compact subgroup K ≤ G(F ) is said to be ingood position with respect to (P,M) if G(F ) = P (F )K and P (F )∩K =(M(F ) ∩K)(N(F ) ∩K).

Given a parabolic subgroup P we can always find a maximal compactsubgroup in good position with respect to P . We can even assume that

KM := M(F ) ∩K

is a maximal compact open subgroup (see Theorem A.4.2); we will also as-sume this in what follows.

We assume throughout this section that Haar measures are chosen so that

dg = δP (m)dkdmdn (8.20)

where dk(K) = dn(N(F ) ∩K) = 1 (see Exercise 8.2).The constant term of f ∈ C∞c (G(F )) along P is the function

fP (m) := δ1/2P (m)

∫N(F )

f(mn)dn. (8.21)


This is equal to (7.17) in the special case considered in §7.5.Let T ≤M be a maximal split torus, and let

C∞c (G(F );K) := f ∈ C∞c (G(F )) : f(k−1xk) = f(x) for all k ∈ K.

We note that C∞c (G(F );K) is precisely the image of the map

C∞c (G(F )) −→ C∞c (G(F );K)

f 7−→ fK (8.22)

where fK(x) :=∫Kf(k−1xk)dk. The space C∞c (G(F );K) ≤ C∞c (G(F )) is a

subalgebra under convolution, and one trivially has

trπ(f) = trπ(fK). (8.23)

Proposition 8.6.1 The constant term gives a map

C∞c (G(F );K) −→ C∞c (M(F );KM )

f 7→ fP .

If (σ, V ) is an admissible representation of M(F ), then

tr I(σ)(f) = trσ(fP ).

Remark 8.1. The archimedean analogue of this proposition is valid as well.One reference is [Kna02, (10.23)].

We follow the proof of [Lau96, Lemma 7.5.7] in the following:

Proof. Since K is compact, δP (k) = 1 for all k ∈ M(F ) ∩K. It follows thatfP ∈ C∞c (M(F );KM ) as claimed.

We now prove the equality of traces. Consider the space IndGP (V )|K oflocally constant functions

ϕ : K −→ V

such that ϕ(mnk) = σ(m)ϕ(k) for (m,n, k) ∈ KM × (N(F )∩K)×K. Thereis a representation I(σ|K) of K on this space given by

I(σ|K)(k)ϕ(g) = ϕ(gk). (8.24)

By the Iwasawa decomposition restriction of functions to K induces an iso-morphism

IndGP (V ) −→ IndGP (V )|K (8.25)

that intertwines ResG(F )K (I(σ)) with I(σ|K). Since this is an isomorphism,

I(σ)(f) induces an endomorphism of IndGP (V )|K , and tr I(σ)(f) is equal tothe trace of this endomorphism on IndGP (V ). Define

8.6 Parabolic descent of representations 165

Φk,k′(m) := δ1/2P (m)

∫N(F )

f(k−1mnk′)dn

for k, k′ ∈ K and m ∈M(F ). Then if ϕ ∈ IndGP (V ) one has

I(σ)(f)ϕ(k) =

∫G(F )

f(g)ϕ(kg)dg

=

∫G(F )

f(k−1g)ϕ(g)dg

=

∫M(F )

∫N(F )

∫K

f(k−1mnk′)ϕ(mnk′)dk′dndm

=

∫M(F )

∫N(F )

∫K

f(k−1mnk′)δP (m)1/2σ(m)ϕ(k′)dk′dndm

=

∫K

σ(Φk,k′)ϕ(k′)dk′.

It follows thattr I(σ)(f)

is the trace of the endomorphism ϕ 7→ (k 7→∫Kσ(Φk,k′)ϕ(k′)dk′). This en-

domorphism has trace ∫K

trσ(Φk,k)dk.

On the other hand

fP =

∫K

Φk,kdk.

The lemma follows. ut

As an addendum, we prove the following lemma:

Lemma 8.6.2 The constant term provides an algebra homomorphism

C∞c (G(F ) //K) −→ C∞c (M(F ) //KM ).

We used this lemma in the special case where G is unramified and P = Bin §7.5:

Proof. Let f1, f2 ∈ C∞c (G(F ) //K). One has

(f1 ∗ f2)P (m) = δP (m)1/2

∫N(F )

(f1 ∗ f2)(mn)dn

= δP (m)1/2

∫N(F )

(∫G(F )

f1(mng−1)f2(g)dg

)dn.

Using Exercise 8.2, we rewrite this as


δ1/2P (m)

∫N(F )

(∫K×M(F )×N(F )

δP (g′)f1(gn(k′m′n′)−1)f2(k′m′n′)dk′dm′dn′

)dn

= δ1/2P (mm′)

∫N(F )

(∫K×M(F )×N(F )

f1(mm′−1nn′−1)f2(m′n′)dk′dm′dn′

)dn

= δ1/2P (mm′)

∫M(F )

(∫N(F )

f1(mm′−1n)dn

∫N(F )

f2(m′n′)dn′

)dm′

= fP1 ∗ fP2 (m).

This proves our assertion that the constant term is an algebra morphism. ut

8.7 Parabolic descent of orbital integrals

In our discussion of the Satake isomorphism in §7.5 we claimed that theconstant term map has image in Weyl invariant functions on the maximaltorus. We prove this in the current section and use it as an opportunity todiscuss descent of orbital integrals.

We keep the notation of the previous section; thus P = MN is a parabolicsubgroup of G, K ≤ G(F ) is a maximal compact subgroup in good positionwith respect to P , and KM := M(F ) ∩K is maximal in K.

Let γ ∈ G(F ) and let Cγ be the centralizer of γ:

Cγ(R) = g ∈ G(R) : g−1γg = γ. (8.26)

Earlier in Definition 1.12 we only defined semisimple elements for perfectfields. If F is not perfect then we say that γ ∈ G(F ) is semisimple if theimage of γ in G(F ) is semisimple. If γ ∈ G(F ) is semisimple then the orbitO(γ) ⊂ G of γ is closed [Ste65, Corollary 6.13, Proposition 6.14]. For asemisimple element γ the neutral component of the stabilizer Cγ is reductive.For more details (including proofs and references) we refer to §17.1.

Assume that γ ∈ G(F ) is semisimple. Then Cγ(F ) is unimodular, andupon choosing a Haar measure dgγ on Cγ(F ) we can form a right G(F )-

invariant Radon measure dgdgγ

on the quotient Cγ(F )\G(F ) (see Theorem

3.2.2). We then define the orbital integral

Oγ(f) :=

∫Cγ(F )\G(F )

f(g−1γg)dg

dgγ. (8.27)

Note that this is not the same as the local version of the orbital integralappearing in (18.11), it differs in that in (18.11) we use Cγ instead of Cγ .This is why we there is a superscript in the notation. The quotient map

8.7 Parabolic descent of orbital integrals 167

Cγ(F )\G(F )→ Cγ(F )\G(F )

is proper as is the map

Cγ(F )\G(F ) −→ G(F )

Cγ(F )g 7−→ g−1γg

by the argument in the proof of Theorem 17.4.1. It follows as in the proof ofTheorem 17.4.1 that the integral (8.27) is absolutely convergent.

Let H ≤ G be a connected subgroup. For F -algebras R and h ∈ H(R) itis convenient to define

DH\G(h) := det(I −Ad(h)−1 : h(F )\g(F )→ h(F )\g(F )

)(8.28)

where (as usual) h := LieH and g := LieG and where I is the identity map.There is a dual relationship between orbital integrals and characters that

can be made precise in many ways; one of the most profound is the absolutetrace formula (see §18.4). The following proposition is consonant with thisduality:

Proposition 8.7.1 Assume that γ ∈ G(F ) is closed, γ ∈ M(F ), and Cγ ≤M . If DM\G(γ) 6= 0 then one has

Oγ(f) = |DM\G(γ)|−1/2Oγ(fP ).

Here we normalize the Haar measure dg on G(F ) as in (8.20) and we takethe Haar measures on Cγ(F ) to be equal on either side of the equation.If we replace the assumption that Cγ ≤ M with the stronger assumptionthat Cγ ≤ M and define Oγ(f) with Cγ(F ) replaced by Cγ(F ) then theproposition remains valid, with an identical proof.

The following proof is taken from [Lau96, Proposition 4.3.11]:

Proof. Upon taking a change of variables n 7→ mnm−1 we see that the nor-malization of the Haar measure in (8.20) is equivalent to

d(mnk) = dmdndk.

Thus one has

Oγ(f) =

∫Cγ(F )\M(F )

∫N(F )

∫K

f(k−1n−1m−1γmnk)dkdndm

dmγ.

For each m ∈ M(F ) there is a morphism of affine F -schemes N → N givenon points in an F -algebra R by

N(R) −→ N(R)

n 7−→ (m−1γm)−1n−1(m−1γm)n.(8.29)


We claim that this morphism is an isomorphism. The image is the orbit of theidentity under an action of the unipotent group N , and is therefore closed inN by [Mil17, Theorem 17.64]. On the other hand the morphism is injectiveat the level of points. Indeed if γn−1γn = γn′−1γn′ for n, n′ ∈ N(R) thenn′n−1 ∈ Cγ(R) which implies n′n−1 = I by our assumption that Cγ ≤ M(see [Hum95, §2.2, Theorem]). The same is true if we replace γ by m−1γmand injectivity follows. In particular the stabilizer of the identity element istrivial. We concluded using Proposition 17.1.2 that (8.29) is an isomorphismonto its image, a closed subscheme ofN . By considering dimensions we deducethat (8.29) is an isomorphism.

The Jacobian depends only on γ and is equal to

J(γ) = |det (I −Ad(γ) : n(F )→ n(F ))| .

We note that|DM\G(γ)| = δP (γ)J(γ)2.

Therefore we can change variables and deduce that the integral above is∫Mγ (F )\M(F )

∫N(F )

∫K

f(k−1m−1γmnk)dkdn

J(γ)

dm

dmγ

and deduce the proposition. ut

This is the first time we have used the usual change of variables formulain the nonarchimedean setting. A general discussion of integration on mani-folds over nonarchimedean fields, including the change of variables formula,is in [Igu00, Chapter 7]. We refer especially to [Igu00, Proposition 7.4.1] and[Igu00, §7.6].

We now prove the following proposition, used in §7.5:

Proposition 8.7.2 Assume that G is quasi-split with Borel subgroup B, thatM ≤ B is a maximal torus and that T ≤ M is its maximal split subtorus.The constant term induces a homomorphism

C∞c (G(F ) //K) −→ C∞c (T (F )/KT )W (T,G)(F ).

In the case at level M , Proposition 8.7.2 is a refinement of Lemma 8.6.2.To prove the proposition we require the following proposition:

Proposition 8.7.3 With assumptions as in Proposition 8.7.2 for any γ ∈M(F )

|DM\G(γ)| 6= 0

if and only if Cγ = M . The set

γ ∈M(F ) : Cγ = M (8.30)

is dense in M(F ) in the natural topology.


Proof. Clearly M ≤ Cγ because M is connected and commutative. SinceI −Ad(γ) acts by zero on the subspace

m\gγ ≤ m\g

we deduce that |DM\G(γ)| 6= 0 if and only if m = gγ , which is to say thatCγ = M . Here m = LieM and gγ = LieCγ .

To deduce that the set of elements of M(F ) where Cγ = M is densein M(F ) in the natural topology we remark that the set of points where anonzero polynomial on a (nonarchimedean) analytic manifold does not vanishis necessarily dense. ut

Remark 8.2. There are surprisingly few textbook treatments of analytic man-ifolds in the nonarchimedean setting; one is [Ser06] and some additional topicsare in [Igu00]. Perhaps the reason is that more sophisticated treatments in-volving rigid analytic spaces, Berkovich spaces, etc. are often needed andhave drawn more attention.

We now prove Proposition 8.7.2:

Proof of Proposition 8.7.2: Every element of W (T,G)(F ) has a representativew ∈ NG(T )(F ), the normalizer of T in G(F ) (see above [Spr09, 16.1.3]). Thusit suffices to show that for f ∈ C∞c (G(F ) //K) one has

fB(w−1tw) = fB(t)

for t ∈ T (F ). We will in fact show the stronger statement that

fB(w−1γw) = fB(γ) (8.31)

for all γ ∈M(F ).The set of γ ∈M(F ) such that Cγ = M is dense by Proposition 8.7.3, and

for such a γ one has |DM\G(γ)| 6= 0. Let dm be the Haar measure on M(F )used in (8.20). We then have

Oγ(f) = |DM\G(γ)|−1/2Oγ(fB)

= |DM\G(γ)|−1/2fB(γ)

by Proposition 8.7.1. Here the last equality follows since M(F ) is commuta-tive. It is clear that Ow−1γw(f) = Oγ(f), so at least for γ ∈M(F ) such that

Cγ = M we deduce (8.31). This set is dense in M(F ) by Proposition 8.7.3.

It is clear that fB is continuous, so we deduce (8.31) for all γ ∈M(F ). 2


Exercises

In all of these exercises G is a connected reductive group over a local field Fand P ≤ G is a parabolic subgroup with a fixed Levi subgroup M ≤ P andunipotent radical N .

8.1. Let (π, V ) be a unitary representation of G(F ) on a Hilbert space V .Let Vsm ≤ V be the subspace of smooth vectors (see Exercise 5.2). Constructan isomorphism Vsm

∼= (Vsm)∨. Using this isomorphism, show that a matrixcoefficient in the sense of (8.4) is a matrix coefficient in the sense of §4.8.

8.2. Assume that K ≤ G(F ) is a maximal compact subgroup in good positionwith respect to (P,M). Show that we can normalize the Haar measures dg,dk, dm, dn on G(F ), K, M(F ), and N(F ), respectively, so that

dg(kmn) = δP (m)dkdmdn

where dk(K) = dn(N(F ) ∩K) = 1.

8.3. Show that we can choose a function f ∈ C∞(G(F )) such that∫P (F )

f(px)drp = 1

for all x ∈ G(F ).

8.4. Let Q ≤M be a parabolic subgroup. There is a natural transformationof functors

IndGP IndMQ∼= IndGQN

The analogous statement for Jacquet modules is true as well.

8.5. The functor IndGP is exact (that is, it preserves exact sequences) andadditive (that is, takes direct sums to direct sums).

8.6. The functor π 7→ πN is exact (sends exact sequences to exact sequences)and additive (sends direct products to direct products).

8.7. Let (π, V ) be a smooth representation of G(F ). An element ϕ ∈ Vis in V (N) if and only if

∫KN

π(n)ϕdn = 0 for some compact subgroup

KN ≤ N(F ).

8.8. Prove that Greg, defined as in (8.16), is an open subscheme of G.

8.9. Assume that A is a finite-dimensional (not necessarily commutatives)C-algebra with unit and that V1, . . . , Vn are finite-dimensional pairwise non-isomorphic simple nonzero A-modules. Prove that the functions

A −→ C


a 7−→ tr(a : Vi → Vi)

are linearly independent over C.

8.10. Let (π1, V1), . . . , (πn, Vn) be a finite set of irreducible admissible rep-resentations of G(F ) such that πi ∼= πj if and only if i = j. Let K ≤ G(F )be a compact open subgroup such that V K1 6= 0. Show that there exists anf ∈ C∞c (G(F ) //K) such that π1(f)|V K1 is the identity and πi(f) = 0 if1 < i ≤ n.

8.11. Without using the Bernstein-Zelevinsky classification, prove that anunramified irreducible admissible representation of GLn(F ) is tempered ifand only if its Satake parameters have complex norm 1.

8.12. Suppose that π is a supercuspidal representation of G(F ). Show thatthere exists a function fπ ∈ C∞c (G(F )) such that trπ(fπ) = 1 and iftrπ′(fπ) 6= 0 for some irreducible admissible representation π′ of G(F ) thenπ ∼= π′ ⊗ χ for some character χ : ZG(F )→ C×.

8.13. Let (π, V ) be an admissible representation of a td group G. Prove thatthe space of smooth endomorphisms Endsm(V ) is an admissible representa-tion of G×G.

8.14. Prove (8.23).

8.15. In the notation of §8.4, whow that if J(σa, λa) and J(σ′a′, λa′) are both

square integrable (not just essentially square integrable) then they are notlinked.

Chapter 9

The Cuspidal Spectrum

God exists since mathematics isconsistent, and the Devil existssince we cannot prove it.

Andre Weil

Abstract The cuspidal spectrum of L2([G]) decomposes discretely into aHilbert space direct sum with finite multiplicities. We give a complete proofof this fact in this chapter.

9.1 Introduction

Let G be a reductive group over a global field F . For x ∈ G(AF )1, one hasthe operator

R(x) : L2([G]) −→ L2([G]) (9.1)

ϕ 7→ (g 7→ ϕ(gx)) .

This defines the regular representation of G(AF )1) on L2([G]). For a smoothversion, one considers the integral operator

R(f) : L2([G]) −→ L2([G]) (9.2)

ϕ 7→

(g 7→

∫G(AF )1

f(x)ϕ(gx)dx

),

where f ∈ C∞c (G(AF )1). Later in this book we will be interested in variousnotions of the trace for operators of this type. There is a problem, however,in that in general R(f) does not have a well-defined trace. Instead, one re-stricts the operator to the cuspidal subspace of Definition 6.9 and proves

173

174 9 The Cuspidal Spectrum

that this restriction is of trace class. One is then left with understanding thecomplement of the cuspidal subspace; this is possible thanks to the theory ofEisenstein series which will be discussed in Chapter 10.

Before stating the main theorems of this chapter we pause and explainwhat one means by a trace in this infinite-dimensional setting.

Definition 9.1. Let V be a Hilbert space. A linear operator A : V → V isHilbert-Schmidt if there is an orthonormal basis (ϕi)

∞i=1 of V such that

∞∑i=1

‖Aϕi‖22 <∞.

The operator A : V → V is of trace class if there is an orthonormal basis(ϕi)

∞i=1 of V such that

∞∑i=0

‖Aϕi‖2 <∞.

If A is Hilbert-Schmidt we let

‖A‖HS =

∞∑i=1

‖Aϕi‖22 and ‖A‖tr =

∞∑i=1

‖Aϕi‖2. (9.3)

This is known as the Hilbert-Schmidt norm (resp. trace norm) of A. IfA is of trace class we define its trace to be

trA :=

∞∑i=1

(Aϕi, ϕi).

The Hilbert-Schmidt norm, trace norm, and trace, are independent of thechoice of basis ϕ (Exercise 9.1).

For the reader’s convenience we recall Definition 6.9:

Definition 9.2. A function ϕ ∈ L2([G]) is cuspidal if for every parabolicsubgroup P ≤ G with unipotent radical NP one has∫

[NP ]

ϕ(ug)du = 0

for almost every g ∈ G(AF )1.

Our aim in this chapter is to prove the following fundamental theorem:

Theorem 9.1.1 (Gelfand and Piatetski-Shapiro) The subspace

L2cusp([G]) ≤ L2([G])

is closed and Rcusp(f) is of trace class for all f ∈ C∞c (G(AF )1).


Here Rcusp(f) is the restriction of R(f) to the cuspidal subspace. As a fairlystraightforward consequence we will prove the following corollary in §9.3:

Corollary 9.1.2 The subspace L2cusp([G]) decomposes into a Hilbert space

direct sum of irreducible representations of G(AF )1, each occurring with finitemultiplicity.

In other words we can write

L2cusp([G]) = ⊕πL2

cusp(π), (9.4)

where the sum is over isomorphism classes of cuspidal automorphic repre-sentations of AG\G(AF ) and the multiplicity of π in L2

cusp(π) is finite. Itdoes not matter if we take this to mean automorphic representations in theL2-sense or with the refined definition of Chapter 6; see §9.8 below.

Let

L2disc([G]) ≤ L2([G]) (9.5)

denote the discrete spectrum. It is defined as in (3.14). It is the Hilbert spacedirect sum of all irreducible subrepresentations of L2([G]). An automorphicrepresentation of G(AF )1 is discrete if it can be realized as subrepresentationof L2

disc([G]). This is equivalent to it being a subrepresentation of L2([G])and not merely a subquotient. Thus cuspidal representations are discrete byCorollary 9.1.2, but the converse is false. A far more difficult theorem thanCorollary 9.1.2 is the following:

Theorem 9.1.3 Assume F is a number field. The restriction of R(f) to thefull discrete spectrum of L2([G]) is of trace class. ut

Muller [Mul89] first proved the theorem in the special case where f is finiteunder a maximal compact subgroup K∞ ≤ G(F∞). Ji [Ji98] and Muller[Mul98] then independently removed this assumption.

For the proof of theorem 9.1.1 we will follow [God66] and fill in somedetails in his presentation using results from [Bor97]. These two referencesproceed classically whereas we will proceed adelically. This leads to somesimplifications (mostly notational) because one can work with rational pointsof groups instead of integral points of groups. We remark that the readershould feel free to skip the remainder of this chapter and refer back to it asnecessary; for the most part one can get by just knowing the statement ofTheorem 9.1.1 and Corollary 9.1.2.

9.2 The cuspidal subspace

We begin with the easiest part of Theorem 9.1.1:


Lemma 9.2.1 The space L2cusp([G]) is closed in L2([G]).

A direct proof of the lemma is possible; see [Bor97, Proposition 8.2]. In-stead we will prove a stronger result which will give us an opportunity todiscuss Poincare series. To keep straight the techniques used in the proof wedo not assume that G is reductive or even connected; any affine algebraicgroup over a global field will do. Thus we recall the adelic quotient

[G] = G(F )\G(AF )1

as in (2.12). Let H ≤ G be a subgroup and let

H(AF ) := H(AF ) ∩G(AF )1.

Assume that H(F )\H(AF ) is compact and let χ : H(AF ) → C× be a char-

acter trivial on H(F ). The fact that H(F )\H(AF ) is compact implies that

H(AF ) is unimodular (see Exercise 9.6); we let dh be a choice of Haar measureon it.

For continuous functions ϕ : [G]→ C we let

Pχ(ϕ) :=

∫H(F )\H(AF )

ϕ(h)χ(h)dh. (9.6)

This is an example of a period integral. It is absolutely convergent sinceH(F )\H(AF ) is compact.

We consider the subspace

V :=ϕ ∈ L2([G]) : Pχ(R(g)ϕ) = 0 for almost every g ∈ G(AF )1

,

where R is defined as (9.1). We will show that V is closed:

Proposition 9.2.2 The subspace V ≤ L2([G]) is closed.

The proof of Proposition 9.2.2 will be given at the end of this section. As acorollary of proposition 9.2.2 it is easy to obtain Lemma 9.2.1:

Proof of Lemma 9.2.1: For parabolic subgroups P ≤ G with unipotent radicalNP let VP be the space V of Proposition 9.2.2 in the special case whereH = NP and χ is the trivial character. Then

L2cusp([G]) =

⋂P

VP .

Thus L2cusp([G]), being the intersection of closed subspaces, is closed. 2

For f ∈ C∞c (G(AF )1) we consider the kernel function

Kf (x, y) =∑

γ∈G(F )

f(x−1γy). (9.7)


These functions will be revisited in §16.1 below; there we will prove that theyare absolutely convergent and define smooth functions on G(AF )1. We definethe Poincare series

Pef (g) := Pef,H,χ(g) :=

∫H(F )\H(AF )

Kf (h, g)χ(h)dh. (9.8)

Since H(F )\H(AF ) is compact the integral converges absolutely. If G is con-nected and reductive, H = N is the unipotent radical of a Borel subgroupand χ is a generic character of N(AF ) in the sense of §11.2 below, then thiscoincides with the Poincare series studied in many places, including [Fri87]and [Ste87].

Lemma 9.2.3 One has Pef ∈ L2([G]).

Proof. One has that

Pef (g)Pef (g) =

∫H(F )\H(AF )

∑γ1∈G(F )

f(h1γ1g)χ(h1)dh1

(9.9)

×

∫H(F )\H(AF )

∑γ2∈G(F )

f(h2γ2g)χ(h2)dh2

.

Choose a fundamental domain F for H(F )\H(AF ). Then

f(g) :=

∫Ff(hg)χ(h)dh ∈ C∞c (G(AF )1). (9.10)

Thus from (9.9) and (9.10), we obtain that

∫[G]

Pef (g)Pef (g)dg =

∫[G]

∑γ1∈G(F )

f(γ1g)∑

γ2∈G(F )

f(γ2g)

dg.

Since∑γ∈G(F ) f(γg) is a compactly supported continuous function on [G],

this integral converges. ut

In the proof of the next lemma we will encounter for the first time thetrivial, but useful, technique known as unfolding. We formalize this in thefollowing lemma, the proof of which is an application Theorem 3.2.2:

Lemma 9.2.4 Suppose that G is a Hausdorff, locally compact, second count-able topological group with right Haar measure dg. If f ∈ L1(G) and Γ ≤ Gis a discrete subgroup such that the modular character of G is trivial on Γthen

178 9 The Cuspidal Spectrum∫Γ\G

∑γ∈Γ

f(γg)dg =

∫G

f(g)dg.

ut

Here in the lemma we have denoted also by dg the right G-invariant measureon Γ\G induced by dg. Use this technique, we prove the following lemma.

Lemma 9.2.5 One has∫[G]

Pef (g)ϕ(g)dg = Pχ(R(f)ϕ).

Proof. Let ϕ ∈ L2([G]). Then by Lemma 9.2.3 the integral∫[G]

Pef (g)ϕ(g)dg

is absolutely convergent. It is equal to∫[G]

∫H(F )\H(AF )

Kf (h, g)χ(h)dhϕ(g)dg.

Manipulating formally, we have∫[G]

∫H(F )\H(AF )

Kf (h, g)χ(h)dhϕ(g)dg

=

∫H(F )\H(AF )

(∫[G]

Kf (h, g)ϕ(g)dg

)χ(h)dh

=

∫H(F )\H(AF )

(∫G(AF )1

f(g)ϕ(hg)dg

)χ(h)dh.

Here in the last equality we have unfolded as in Lemma 9.2.4 to deduce that∫[G]

Kf (h, g)ϕ(g)dg =

∫G(AF )1

f(g)ϕ(hg)dg. (9.11)

The integral on the right converges absolutely and defines a continuous func-tion of h. Combining this with the compactness of H(F )\H(AF ) and domi-nated convergence we deduce that our formal manipulations above are justi-fied. ut

Given our preparation, the proof of Proposition 9.2.2 is now straightfor-ward:

Proof of Proposition 9.2.2: Each f ∈ C∞c (G(AF )1) gives rise to a linear form(·,Pef ) on L2([G]). It is continuous by Lemma 9.2.3. In view of Lemma 9.2.5

9.3 Deduction of the discreteness of the spectrum 179

the space V is the intersection over all f ∈ C∞c (G(AF )1) of the kernels ofthese linear forms. 2

9.3 Deduction of the discreteness of the spectrum

We now explain how to deduce the discreteness of the spectrum in Corollary9.1.2 from the assertion that certain operators are of trace class in Theorem9.1.1. As in the previous section, we do not yet assume that G is reductive.

A Dirac sequence on G(AF )1 is a sequence of nonnegative (real valued)functions fn ∈ C∞c (G(AF )1) indexed by n ∈ Z>0 such that

(a) For any open neighborhood U 3 1 in G(AF )1 with compact closure thesupport of fn is contained in U for n large enough.

(b) One has fn(x−1) = fn(x).(c) One has ∫

G(AF )1

fn(x)dx = 1

for all n.

It is not difficult to see that a Dirac sequence on G(AF )1 exists (Exercise9.7). They are known as Dirac sequences because for representations (R, V )of G(AF )1 and ϕ ∈ V one has R(fn)ϕ→ ϕ as n→∞ (Exercise 9.8). We alsonote that if fn is a Dirac sequence and (R, V ) is a unitary representation ofG(AF )1 then R(fn) is self-adjoint for every n by condition (b).

We recall that an operator T on an infinite dimensional Hilbert space Vis compact if and only if it can be written as

T =

∞∑n=1

λn〈ϕn, ·〉ϕ′n (9.12)

operator where ϕnn>0, ϕ′nn>0 are orthonormal sequences of vectors inV and λnn>0 ⊂ C is a sequence of numbers such that limn→∞ λn = 0. IfT is self-adjoint, then we can take ϕn = ϕ′n, and in particular in this case Thas an orthonormal basis of eigenvectors. A nice reference is [Zhu93, §1.3].

Lemma 9.3.1 Let (R, V ) be a unitary representation of G(AF )1. Assumethat there is a Dirac sequence fn on G(AF )1 such that R(fn) is compactfor all n. Then (R, V ) decomposes into a Hilbert space direct sum of irre-ducible closed G(AF )1-invariant representations of G(AF )1, each with finitemultiplicities.

Proof. The following is an adaptation of the proof of [Bor97, Lemma 16.1]:We first show that a closed G(AF )1-invariant nonzero subspace W ≤ V

contains a closed G(AF )1-invariant subspace that is minimal among nonzero


closed G(AF )1-invariant subspaces. Let W be a closed G(AF )1-invariant sub-space. Clearly there exists a j such that R(fj)|W 6= 0. Let λ be a nonzeroeigenvalue of R(fj) in W and let Eλ 6= 0 be the corresponding finite dimen-sional eigenspace in W . Let M be a minimal element among the nonzerointersections of Eλ with the closed G-invariant subspaces of W ; such an el-ement exists because Eλ is finite-dimensional. Let ϕ ∈ M \ 0 and let Pbe the smallest closed G(AF )1-invariant subspace containing ϕ. By construc-tion, P ∩ Eλ = M . We claim that P is a minimal closed G(AF )1-invariantsubspace of W . Let Q be a closed G(AF )1-invariant subspace of P and Q⊥

be its orthogonal complement in P . Then P = Q⊕Q⊥. The spaces P , Q, Q⊥

and Eλ are invariant under R(fj), hence M = P ∩Eλ = (Q∩Eλ)⊕(Q⊥∩Eλ).Therefore, either M = Q ∩ Eλ with P = Q and Q⊥ = 0 or M = Q⊥ ∩ Eλwith P = Q⊥ and Q = 0.

Next, we prove that (R, V ) has a discrete decomposition into closed ir-reducible G(AF )1-invariant subspaces. Let S be the set consisting of thenonzero closed G(AF )1-invariant subspaces admitting a discrete decomposi-tion. Then S is not empty by the prior claim. This set is partially orderedby the relation X ≤ Y if the space of X is contained in that of Y . Let Wbe the space given by a maximal element. If W 6= V , then we could add toW a minimal closed nonzero G(AF )1-invariant subspace in the orthogonalcomplement of W , which exists by the previous claim. This contradictionimplies that W = V .

Finally, we will show that the multiplicities are finite. Let π be an irre-ducible unitary representation occurring in V and let mπ be its multiplicity.Let j be such that π(fj) is not zero on the speace of π, let λ be a nonzeroeigenvalue of π(fj) and mλ be its multiplicity. Then the dimension of theeigenspace of R(fj) in V with eigenvalue λ is finite and is at least mλmπ.Therefore mπ is finite. ut

In view of Lemma 9.2.1 and Lemma 9.3.1, to prove Theorem 9.1.1 it sufficesto show that Rcusp(f) is compact for all f ∈ C∞c (G(AF )1). The followinglemma gives us a criteria for compacity:

Lemma 9.3.2 Let f ∈ C∞c (G(AF )1). Let V ≤ L2([G]) be a closed G(AF )1-invariant subspace. Assume one has an estimate

‖R(f)ϕ‖∞ f ‖ϕ‖2

for all ϕ ∈ V . Then the operator R(f)|V is of trace class, hence Hilbert-Schmidt, hence compact.

The assertions that trace class implies Hilbert-Schmidt implies compact areleft as Exercise 9.3. In the lemma and below ‖·‖∞ denotes the uniform norm.

Proof. This follows from the argument of the proof of [Bor97, Theorem 9.5]which we will recall here: For given x ∈ G(AF )1, the assumption implies that

9.3 Deduction of the discreteness of the spectrum 181

the map ϕ 7→ R(f)ϕ(x) is a continuous linear form on V . Hence by the Rieszrepresentation theorem there exists an element KR(f),x ∈ V such that

R(f)ϕ(x) =

∫[G]

KR(f),x(y)ϕ(y)dy

for all ϕ ∈ V . Taking ϕ = KR(f),x we obtain

‖KR(f),x‖22 ≤ ‖R(f)KR(f),x‖∞ f ‖KR(f),x‖2

so it follows that ‖KR(f),x‖2 ≤ C for some constant C > 0. Write

KR(f)(x, y) := KR(f),x(y)

for x, y ∈ [G]. Then∫[G]

∫[G]

|KR(f)(x, y)|2dxdy =

∫[G]

(∫[G]

KR(f),x(y)KR(f),x(y)dy

)dx

≤ C2

∫[G]

dx

is finite and

R(f)ϕ(x) =

∫[G]

KR(f)(x, y)ϕ(y)dy

for ϕ ∈ V . Therefore the operator R(f) on V is represented by an L2-kernel.This implies that it is Hilbert-Schmidt (see Exercise 9.4).

It follows from the Dixmier Malliavin lemma (Theorem 4.2.5) that f is afinite linear combination of convolutions f1 ∗ f2 with f1, f2 ∈ C∞c (G(AF )1)(Exercise 9.9). Therefore R(f) is a finite linear combination of convolutionsof two Hilbert-Schmidt operators and hence is of trace class (see Exercise9.2). ut

In view of lemmas 9.3.1 and 9.3.2, to prove Theorem 9.1.1 it suffices toprove that for f ∈ C∞c (G(AF )1) one has an estimate

‖R(f)ϕ‖∞ f ‖ϕ‖2 (9.13)

for ϕ ∈ L2cusp([G]). This estimate is really the heart of the proof. In the func-

tion field case we will be able to deduce something much stronger, and in thenumber field case we derive it using the argument of [God66]. It is only inthis part of the proof where we use the assumption that G is reductive. Wehave been careful in isolating when this assumption is used for two reasons.First, it is helpful to see at what point the argument passes from abstractrepresentation theory to something that uses the specific structure of theadelic quotient in the reductive case. Second, there ought to be a notion ofa cuspidal representation on more general groups than those that are reduc-


tive, but we do not know the natural level of generality. We feel that thisis an interesting research problem. One example that has been worked outreasonably thoroughly is the case where G is the Jacobi group, a semi-directproduct of SL2 and the three-dimensional Heisenberg group [BS98, §4.2]. Thework in [Sli84] is probably relevant here as well.

9.4 The function field case

Assume for this section that F is a function field and that G is reductive.The asymptotic behavior of cuspidal functions is much simpler in this casethan their behavior in the number field case.

Theorem 9.4.1 (Harder) Let ϕ ∈ L2cusp([G]) be a function that is invari-

ant under the compact open subgroup K ′ ≤ G(AF )1. Then ϕ(x) = 0 for xoutside a compact subset of [G] that depends only on K ′.

Before proving this theorem let us indicate how it can be used to proveTheorem 9.1.1 in the function field case.

Proof of Theorem 9.1.1 (for function field): Let f ∈ C∞c (G(AF )1). As ex-plained in more detail in §16.1 the operator

R(f) : L2([G]) −→ L2([G])

is represented by the kernel

Kf (x, y) :=∑

γ∈G(F )

f(x−1γy).

It is not hard to see that this function is continuous as a function of (x, y) ∈[G]× [G] (for more details see §16.1). Thus for any compact subset Ω ⊆ [G]the restriction of Kf (x, y) to Ω ×Ω is in L2(Ω ×Ω), and hence is a Hilbert-Schmidt operator.

Now there is a compact open subgroup K ≤ G(AF )1 such that f ∈C∞c (G(AF )1 //K). Thus for any ϕ ∈ L2

cusp([Q]) such that R(f)ϕ 6= 0 one

has ϕ ∈ L2([G])K . Using Theorem 9.4.1 choose a compact subset Ω ⊂ [G]such that any element of L2

cusp([G])K is supported in Ω. In other words,

R(f)|L2cusp([G]) = R(f)|L2(Ω)∩L2

cusp([G]).

Now R(f)|L2(Ω) is Hilbert-Schmidt, so its restriction to the closed subspaceL2(Ω) ∩ L2

cusp([G]) of L2(Ω) is also Hilbert-Schmidt (see Exercise 9.5).Thus we have proven that R(f) : L2

cusp([G]) → L2cusp([G]) is Hilbert-

Schmidt. On the other hand, every f ∈ C∞c (G(AF )1) is a finite linear com-

bination of convolutions: f =∑ki=1 hi1 ∗ hi2 where hij ∈ C∞c (G(AF )1) (see

9.4 The function field case 183

Exercise 9.9). We deduce that

R(f) =

k∑i=1

R(hi1) ∗R(hi2).

This says that R(f) is a finite sum of convolutions of two Hilbert-Schmidtoperators, and hence is of trace class (Exercise 9.2). 2

We now begin the proof of Theorem 9.4.1. We follow Harder’s originalproof [Har74]. Let P = MN be a minimal parabolic subgroup of G and letH be a maximal torus torus of Gder contained in M . Thus T := ZGT0 isa maximal torus of G. Fix Ω ⊆ M(AF )N(AF ), t ∈ R>0 and a maximalcompact subgroup K ≥ K ′ and let S(t) be the corresponding Siegel set asin (2.16). We assume that Ω and K are chosen so that for t sufficiently smallone has

AGG(F )S(t) = G(AF );

this is possible by Theorem 2.7.2.Let Φ+ be the set of positive roots of the split maximal torus T ≤ M

acting on N . For each simple root α ∈ ∆, let

Φ+α ⊂ Φ+

be the set of roots of T in G of the form λ =∑α′ mα′α

′ with mα > 0.Moreover let

n(Φ+α ) := ⊕λ∈Φ+

αnλ. (9.14)

It is clear that this is an ideal in n. Let N(Φ+α ) / N be the closed connected

subgroup ofN with Lie algebra n(Φ+α ). It is the unipotent radical of a maximal

parabolic subgroup of G containing P .

Lemma 9.4.2 If t > 0 is chosen sufficiently large, then for any h ∈ S(t)one has

N(Φ+α )(F )(N(Φ+

α )(AF ) ∩ hK ′h−1) = N(Φ+α )(AF ).

Proof. Temporarily denote by p the residual characteristic of F . Observe thatfor any open subset ω ⊂ AF there is a constant cω such that if a ∈ A×F satisfies|a| ≥ pcω then F + aω = AF . Write

N(Φ+α ) =

∏λ∈Φ+

α

Nλ

where τλ : Ga−→Nλ is a one-parameter subgroup with LieNλ = nλ. ThenNλ(AF )∩hK ′h−1 ⊂ Nλ(AF ) is a compact open subgroup for any h ∈ G(AF ).Thus by our observation one has Nλ(F )(Nλ(AF ) ∩ hK ′h−1) = Nλ(AF ) forh ∈ S(t) if t is sufficiently large.


We now wish to complete the proof using an inductive argument. Forthis purpose choose an ordering of the set of positive roots Φ+ that gives atotal ordering on Φ+

α with respect to which the simple roots are positive. Forλ′ ∈ Φ+

α , let

N(α, λ′) :=∏

λ∈Φ+α :λ≥λ′

Nλ.

We have that N(Φ+α ) :=

⋃λ∈Φ+

αN(α, λ), i.e. N(Φ+

α ) is an increasing chain ofproducts of the Nλ. We can now apply induction to deduce the lemma. ut

Proof of Theorem 9.4.1: Let ϕ ∈ L2cusp([G])K

′. Choose t0 > 0 and a Siegel

set S(t0) := ΩAT0(t0)K so that AGG(F )S(t0) = G(AF ).

Choose t > t0 so that the conclusion of Lemma 9.4.2 holds. Then forh ∈ S(t) the function

N(Φ+α )(AF ) −→ C

n 7−→ ϕ(nh)

is left invariant under N(Φ+α )(F ) and right invariant under N(Φ+

α )(AF ) ∩hK ′h−1. Thus by Lemma 9.4.2 it is constant. On the other hand N(Φ+

α ) isthe unipotent radical of a proper parabolic subgroup of G and we obtain

0 =

∫[N(Φ+

α )]

ϕ(nh)dn = measdn([N(Φ+α )])ϕ(h)

so ϕ(h) = 0. Thus ϕ vanishes on S(t) for sufficiently large t. The theoremfollows. 2

9.5 Estimates for the Fourier transform of the kernel

We now assume that G is reductive. Our aim is prove (9.13). To motivatethe approach, let’s consider the case G = GL2 over Q. To make things asconcrete as possible, we assume that

ϕ : [G]→ C

is a smooth cuspidal function invariant under O2(R)GL2(Z); in this case wecan identify ϕ with a function

ϕ : SL2(Z)\H −→ C

(compare (2.14), (6.11)). We can choose a fundamental domain for SL2(Z)contained in the Siegel set

9.5 Estimates for the Fourier transform of the kernel 185z ∈ H : Im(z) >

1

2, 0 ≤ Re(z) ≤ 1

(9.15)

by Exercise 2.15. Thus it suffices to bound ϕ in this region. Let B ≤ GL2 bethe Borel subgroup of upper triangular matrices and NB ≤ B the unipotentradical of B. Then ϕ is invariant under NB(Z) on the left, which is to saythat it is invariant under z 7→ z + 1. Thus it can be expanded in a Fourierseries

ϕ(z) =

∞∑n=1

ane2πinz.

The key point here is that cuspidality implies that only positive n contribute.Since ϕ is continuous the an are bounded and this allows one to deduce (9.13)in this case. In the discussion above we have used Fourier analysis on

Z\R ∼= NB(Z)\NB(R)

to establish (9.13). The general case follows this basic outline, using Fourieranalysis on the unipotent radical of a minimal parabolic subgroup.

We now begin our discussion of the general case, so we let G be a reductiveconnected group over a number field F . By applying Weil restriction of scalarswe can and do assume F = Q. Let P be a fixed minimal parabolic subgroupof G with Levi decomposition P = MN (N being the unipotent radical). LetT ≤ M be a maximal split torus and let T0 ≤ Gder a split torus such thatT0ZG = T . The exponential map induces an isomorphism of affine schemesover Q:

exp : n −→ N.

Here as usual n denotes the Lie algebra of N and we are viewing n as anaffine space over F , isomorphic to GdimF n

a . Note that we are not claimingthat the exponential map is an isomorphism with respect to any sort ofgroup structure.

Any element ϕ ∈ L2([G]) is left-invariant under G(Q). Thus for f ∈C∞c (G(AQ)1) we can write

R(f)ϕ(x) =

∫G(AQ)1

f(x−1y)ϕ(y)dy (9.16)

=

∫N(Q)\G(AQ)1

∑γ∈N(Q)

f(x−1γy)dyϕ(m)dm

=

∫N(Q)\G(AQ)1

Kf,P (x, y)ϕ(y)dy

where for x, y ∈ G(AQ) we set

Kf,P (x, y) =∑

γ∈N(Q)

f(x−1γy) =∑

η∈nP (Q)

f(x−1 exp(η)y). (9.17)


We view nP as an affine abelian algebraic group over Q with addition as thelaw of composition. In particular,

[nP ] := nP (Q)\nP (AQ)

is an abelian group. We note that for fixed x, y ∈ G(AQ) the function

η 7→ f(x−1 exp(η)y)

is smooth on nP (AQ). Let ψ : Q\AQ → C× be a nontrivial character and let

〈 , 〉 : nP × nP −→ Ga

be a perfect pairing (over Q). Then by Poisson summation one has

Kf,P (x, y) =∑

η∈nP (Q)

∫nP (AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn (9.18)

for an appropriate choice of (additive) Haar measure nP (AQ). For a primeron Poisson summation over global fields see Appendix B.

Using the Iwasawa decomposition (2.15) we write a general element ofG(AQ)1 as

y = uymysyky

with (uy,my, sy, ky) ∈ N(AQ) ×M(AQ)1 × AT0(t) ×K for some t > 0. We

shall denoteS(t) = ΩN .ΩM .AT0(t).K

a fixed Siegel domain in G(AQ). Here ΩN and ΩM are fixed compact subsetsof N(AQ) and M(AQ)1, respectively.

Lemma 9.5.1 Assume x ∈ S(t). Then s−1x x lies in a compact subset of

G(AQ) depending only on S(t).

Proof. If x ∈ S(t) we have

x ∈ ΩNΩMsxK = ΩNsxΩMK.

for some compact subsets ΩN ⊂ N(AQ), ΩM ⊂ M(AQ)1. Here we are usingthe fact that sx commutes with M(AQ).

As x varies over S(t) one has that s−1x ΩNsx lies in a compact subset of

N(AQ) (see Exercise 9.10). Thus x ∈ sxΩG for some compact subset ΩG ofG(AQ)1. ut

Lemma 9.5.2 Fix a compact subset ΩN ⊂ N(AQ). Assume that x ∈ S(t)and y ∈ ΩNM(AQ)1K. If

f(x−1γy) 6= 0

9.5 Estimates for the Fourier transform of the kernel 187

for some γ ∈ N(F ) then x−1sx and s−1x y lie in compact subsets of G(AQ)1

depending only on the support of f and on ΩN . Moreover my lies in a compactsubset of M(AQ)1 that depends only on the support of f and on ΩN .

Proof. Since x ∈ S(t) one has by Lemma 9.5.1 that x ∈ sxΩG for somecompact subset ΩG of G(AQ)1 depending only on the Siegel set S(t). We cantake ΩG to be invariant under g 7→ g−1 and then it follows that x−1 ∈ ΩGs−1

x .Now if f(x−1γy) 6= 0 then x−1γy lies in a compact subset ΩG of G(AQ)1

depending only on f . By our earlier considerations we see that upon enlargingΩG if necessary

s−1x γy = s−1

x γuysxs−1x mysyky ∈ ΩG (9.19)

and hence (again enlarging ΩG if necessary) s−1x γuysxmys

−1x sy ∈ ΩG.

We deduce that my lies in a compact subset ΩM ≤M(AQ)1 and s−1x sy lies

in a compact subset of AT0(t) depending only on f and ΩN . Using Lemma9.5.1 we deduce that s−1

x y lies in a compact subset of G(AQ)1 depending onlyon f,ΩN . ut

Let Φ be the set of roots of T in G. The choice of simple roots ∆ yields aset of positive roots Φ+. Choose a norm

‖·‖n

on the the real vector space n(R).

Proposition 9.5.3 Assume that y ∈ ΩNM(AQ)KM for some fixed compactsubset ΩN ⊂ N(AQ). Let B > 0. There is a compact subset Ω∞n ⊂ n(A∞Q )such that for any λ ∈ Φ+ and x ∈ S(t) one has∫

n(AQ)

f(x−1 exp(n)y)ψ (〈η, n〉) dn

B,ΩN ‖η‖−Bn 1Ω∞n(η)δP (sx)supλ(sx)−B ,

where the supremum is taken over among λ such that

〈η, ·〉|n(λ) 6= 0.

Proof. One has∫n(AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn

=

∫n(AQ)

f(x−1sx exp(Ad(s−1x )n)s−1

x y)ψ(〈η, n〉)dn

= δP (sx)

∫n(AQ)

f(x−1sx exp(n)s−1x y)ψ(〈η,Ad(sx)n〉)dn.


Using Lemma 9.5.2 and integration by parts at the infinite place we concludethat for all B > 0∫

n(AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn δP (sx)‖Ad∨(s−1x )η‖−Bn 1Ω∞n

(η).

The proposition follows. ut

For each simple root α ∈ ∆, let

Φ+α ⊂ Φ+

be the set of roots of T in G of the form λ =∑α′ mα′α

′ with mα > 0.Moreover let

n(Φ+α ) := ⊕λ∈Φ+

αnλ. (9.20)

It is clear that this is an ideal in n. Let N(Φ+α ) / N be the closed connected

subgroup ofN with Lie algebra n(Φ+α ). It is the unipotent radical of a maximal

parabolic subgroup of G containing P .

Lemma 9.5.4 Let α ∈ ∆, and assume that 〈η, ·〉 is trivial on n(Φα). Then

y 7−→∫n(AQ)


is left invariant under multiplication by N(Φ+α ).

Proof. Let n ∈ n(AQ) and u ∈ N(Φ+α ). One has

exp(n)u = exp(n+ w(n, u))

for some w(n, u) ∈ n(AQ) by the Baker-Campbell-Hausdorff formula. As men-tioned above, n(Φ+

α ) is an ideal in n. Thus since w(n, u) is given in terms ofcommutators of n and u one has w(n, u) ∈ n(Φ+

α ). Thus∫n(AQ)

f(x−1 exp(n)uy)ψ(〈η, n〉)dn

=

∫n(AQ)

f(x−1 exp(n+ w(n, u))y)ψ(〈η, n〉)dn

=

∫n(AQ)

f(x−1 exp(n)y)ψ(〈η, n− w(n, u)〉)d(n− w(n, u))

=

∫n(AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn.

ut

9.6 The basic estimate 189

9.6 The basic estimate

The main result of this section is the following proposition:

Proposition 9.6.1 Let ϕ ∈ L2cusp([G]). Then for any f ∈ C∞c (G(AQ)1) and

any B ∈ R>0 one has

|Rcusp(f)ϕ(x)| f,B maxα∈∆(α(sx))−B‖ϕ‖2

for x ∈ S(t).

Combined with Lemma 9.3.1 and Lemma 9.3.2 this completes the proofof Theorem 9.1.1.

Proof. Using (9.16) and (9.18) we obtain

R(f)ϕ(x) =

∫N(Q)\G(A)1

Kf,P (x, y)ϕ(y)dy (9.21)

=

∫N(Q)\G(AQ)1

∑η∈n(Q)

∫n(AQ)


ϕ(y)dy.

To prove the proposition it suffices to provide a bound

∫N(Q)\G(AQ)1

∑η∈n(Q)

∫n(AQ)

f(ux−1 exp(n)y)ψ(〈η, n〉)dn

ϕ(y)dy

f,B δP (sx)α(sx)−B‖ϕ‖2

for each α ∈ ∆.Now if η ∈ n(Q) has the property that 〈η, ·〉 is identically zero on n(Φ+

α )then

y 7−→∫n(AQ)


is left-invariant under N(Φ+α ) by Lemma 9.5.4. Thus for such an η

measdu([N(Φ+α )])

∫N(Q)\G(AQ)1

(∫n(AQ)


)ϕ(y)dy

=

∫[N(Φ+

α )]

∫N(Q)\G(AQ)1

(∫n(AQ)

f(x−1 exp(n)uy)ψ(〈η, n〉)dn

)ϕ(y)dydu

=

∫N(Q)\G(AQ)1

(∫n(AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn∫

[N(Φ+α )]

ϕ(u−1y)du

)dy

= 0


by cuspidality of ϕ. Here we are using the fact that [N(Φ+α )] is compact

to justifying manipulating the integrals. Thus in (9.21) one can omit thesummands indexed by those η for which 〈η, ·〉 is identically zero on n(Φ+

α ).Recall that we have assumed x ∈ S(t). Thus for an η such that 〈η, ·〉 is not

identically zero on n(Φ+α ) we can apply Proposition 9.5.3 to see that there is

a compact subset Ω∞n ⊂ n(A∞Q ) such that for any B > 0 one has∫n(AQ)

f(x−1 exp(n)y)ψ(〈η, n〉)dn ‖η‖−Bn 1Ω∞n(η)δP (sx)α(sx)−B .

Here we have replaced the supremum of Proposition 9.5.3 by α because theother possible weights can only make the bound smaller.

Note that∑η∈n(Q)‖η‖−Bn 1Ω∞n (η) is bounded for B sufficiently large. Thus

for B sufficiently large one can sum over η and apply Lemma 9.5.2 to obtaina bound

R(f)ϕ(x) δP (sx)α(sx)−B∫sxΩG

ϕ(y)dy

where ΩG ⊂ G(AQ)1 is a dy measurable compact set. By the Cauchy-Schwartz inequality this is bounded by

δP (sx)α(sx)−Bmeasdy (ΩNsxΩG)1/2

(∫ΩNsxΩG

|ϕ(y)|2dy)1/2

. (9.22)

One has (∫ΩNsxΩG

|ϕ(y)|2dy)1/2

‖ϕ‖2 (9.23)

andmeasdy (ΩNsxΩG) = measdy

(s−1x ΩNsxΩG

) δ(sx)−1.

Thus (9.22) is bounded by a constant times

δ1/2P (sx)α−B(sx)‖ϕ‖2

as required. ut

9.7 Rapidly decreasing functions

The estimate of Proposition 9.6.1 for cuspidal functions motivates the follow-ing definition:


ϕ : [G] −→ C

9.8 Cuspidal automorphic forms 191

is rapidly decreasing if it is smooth and for all compact subsets Ω ⊂ [G],and t, B ∈ R>0 there is a constant C ∈ R>0 (depending on t, Ω and B) suchthat one has

|ϕ(sy)| ≤ Cα(s)−B

for all s ∈ AT0(t), y ∈ Ω, and α ∈ ∆.

Here we are using notation from our discussion of Siegel sets and reductiontheory in §2.7.

Using this terminology we see that Theorem 9.4.1 and Proposition 9.6.1imply the following theorem:

Theorem 9.7.1 If f ∈ C∞c (G(AF )1) and ϕ ∈ L2cusp([G]) then R(f)ϕ is

rapidly decreasing. ut

Corollary 9.7.2 Any smooth ϕ ∈ L2cusp([G]) is rapidly decreasing.

Here when we say that ϕ is smooth we mean that there is a compact opensubgroup K ≤ G(A∞F ) such that ϕ ∈ L2

cusp([G])K , and ϕ lies in the set of

vectors in L2cusp([G])K that are smooth with respect to the action of G(F∞)

in the sense of §4.2.

Proof. It suffices to show that such a ϕ can be written as a finite sum

ϕ =∑i

R(fi)ϕi

for some fi ∈ C∞c (G(AF )1) and ϕi ∈ L2cusp([G]). This follows from Theorem

4.2.4. ut

9.8 Cuspidal automorphic forms

We now wish to prove Theorem 6.5.1, which is the statement that AAGcusp is adense subspace of L2

cusp([G]).To prove this we first state the following result:

Theorem 9.8.1 An element of AAGcusp is compactly supported in the functionfield case and rapidly decreasing in the number field case. ut

We refer to [MW95, §1.2.12] for the proof. It is similar to the proofs ofTheorem 9.4.1 and Proposition 9.6.1.

Let K = K∞K∞ ≤ G(AF ) be a maximal compact subgroup. We also

require the following lemma:

Lemma 9.8.2 Let (π, V ) be an irreducible unitary representation of AG\G(AF )on a Hilbert space V . The subspace Vfin of K-finite vectors is dense. For everyϕ ∈ Vfin there is an f ∈ C∞c (AG\G(AF )) such that π(f)ϕ = ϕ.


Proof. If F is a function field, then the density of Vfin in V follows fromLemma 5.3.3. In this case every vector ϕ ∈ Vfin is fixed by a compact opensubgroup K ′ ≤ K, so π(1K′)ϕ = ϕ.

If F is a number field then we can write Vfin = V∞ ⊗ V∞ where V∞ is anirreducible admissible (g,K∞)-module and V∞ is an admissible irreduciblerepresentation of G(A∞F ). The density assertion is implied by Proposition4.4.2 and Lemma 5.3.3. The fact that every element in Vfin is of the formπ(f)ϕ for some ϕ ∈ V is implied by the corresponding statements for V∞and V∞. For V∞ one proceeds as in the function field case. For V∞ it is ageneral fact (mentioned already in §4.2) that every element of V∞ is of theform π∞(f)ϕ for some f ∈ C∞c (AG\G(F∞) and ϕ ∈ V∞ [DM78]. ut

Proof of Theorem 6.5.1: By Theorem 9.8.1 an element of AAGcusp is bounded.

Since [G] has finite measure we deduce that AAGcusp ≤ L2cusp([G]).

We now prove density. For a cuspidal automorphic representation π ofAG\G(AF ) let L2

cusp(π) be the π-isotypic subspace of L2cusp([G]). In view of

Corollary 9.1.2 it suffices to show that AAGcusp ∩ L2(π) is dense in π.Let K = K∞K

∞ be a maximal compact subgroup of G(AF ). We claimthat for each cuspidal automorphic representation π of AG\G(AF ) (in the L2

sense) the space of K-finite vectors in L2cusp(π) is exactly

AAGcusp ∩ L2cusp(π).

The space of K-finite vectors Vfin in L2cusp(π) is an admissible G(AF )-

representation in the function field case and an admissible (g,K∞)×G(A∞F )-module in the number field case. Moreover, in the number field case this(g,K∞)×G(A∞F )-module admits an infinitesimal character by Schur’s lemma(Lemma 4.1). Invoking Lemma 9.8.2, Theorem 9.4.1 and Proposition 9.6.1we deduce the theorem. 2

Exercises

9.1. Prove that the Hilbert-Schmidt norm of a Hilbert-Schmidt operator isindependent of the choice of basis and the trace norm and trace of a trace-class operator is independent of the choice of basis.

9.2. Prove that if A and B are Hilbert-Schmidt operators then A B is atrace class operator.

9.3. Prove that a trace class operator is Hilbert-Schmidt and a Hilbert-Schmidt operator is compact.

9.8 Cuspidal automorphic forms 193

9.4. Let (Y, µ) be a σ-finite measure space. Then L2(Y, dµ) is a separableHilbert space. Let K(x, y) ∈ L2(Y × Y, dµ× dµ). Prove that the operator

L2(Y, dµ) −→ L2(Y, dµ)

ϕ 7−→(x 7→

∫Y

K(x, y)ϕ(y)dµ(y)

)is Hilbert-Schmidt.

9.5. Let V be a Hilbert space and let A : V → V be a Hilbert-Schmidtoperator. Prove that the restriction of A to a closed subspace W ≤ V is alsoa Hilbert-Schmidt operator.

9.6. Let G be a connected reductive group over a global field F and letH ≤ G be a subgroup such that H(F )\H(AF ) ∩ G(AF )1 is compact. Provethat H(AF ) ∩G(AF )1 is unimodular.

9.7. Prove that a Dirac sequence on G(AF )1 exists.

9.8. If fn ∈ C∞c (G(AF )1) is a Dirac sequence and (R, V ) is a representationof G(AF )1 and ϕ ∈ V then

R(fn)ϕ −→ ϕ

as n→∞.

9.9. Using Theorem 4.2.5 prove that every f ∈ C∞c (G(AF )1) can be writtenas

f =

r∑i=1

f1r ∗ f2r (9.24)

for some integer r and some fij ∈ C∞c (G(AF )1).

9.10. With notation as in Lemma 9.5.1 prove that when x varies over S(t)one has that s−1

x ΩNsx lies in a compact subset of N(AF ).

9.11. For n ≥ 1,let J :=(

0 In−In 0

). For Q-algebras R let

Sp2n(R) = g ∈ GL2n(R) : Jg−tJ−1 = g.

Let Hn be the set of n× n symmetric complex matrices Z = X + iY whoseimaginary part Y is positive definite (this is called Siegel’s upper-half spaceof degree n). The group Sp2n(R) acts on Hn via

(A BC D ) · Z = (AZ +B)(CZ +D)−1.

Here A,B,C,D are n× n matrices.


(a) Show that Sp2n(R) acts transitively on Hn.(b) Let B ⊂ Sp2n be the Borel subgroup of upper triangular matrices. Com-

pute the image of the associated Siegel domain S(t) under

Sp2n(AQ) −→ Sp2n(R) −→ Hn

for n = 1 and n = 2.

Chapter 10

Eisenstein Series

Langlands’ calculation of theconstant terms of Eisensteinseries was one of the mostinfluential computations of the20th century.

E. Lapid

Abstract In this chapter we survey the theory of Eisenstein series. Our maingoal is to state Langlands’ decomposition of L2([G]).

10.1 Induced representations

In §8.1 we first mentioned the philosophy of cusp forms, a slogan coinedby Harish-Chandra [HC70a] that states that the irreducible representationsof a connected reductive group ought to all be obtained as subquotients ofcuspidal representations induced from Levi subgroups of parabolic subgroups.This philosophy is in part based on the foundational results of Langlands[Lan76] which we survey in this chapter. We largely follow [Art05, §7] in ourexposition. It is beyond the scope of this book to provide proofs. Indeed, wholebooks have been devoted to solely to the theorems we state here [MW95].

We begin by setting notation for the induction of automorphic represen-tations from Levi subgroups of G to G itself. Intertwining operators relatingsuch representations on different (but isomorphic) Levi subgroups are dis-cussed in §10.2. Eisenstein series, defined in §10.3 allow one to prove thatthese induced representations are again automorphic, and in §10.4 we useEisenstein series to give Langlands’ decomposition of L2([G]). In the case ofGLn all of this theory simplifies significantly. In particular, one can use it todefine an analogue of direct sums in the category of automorphic represen-tations of general linear groups. This is discussed in §10.5 and 10.6. We end

195

196 10 Eisenstein Series

the chapter with a statement, in §10.7 of Moeglin and Waldspurger’s foun-dational result which precisely describes the discrete spectrum of L2([GLn])in terms of cuspidal representations of smaller general linear groups.

Let G be a connected reductive group over a global field F with minimalparabolic subgroup P0 ≤ G. As usual, parabolic subgroups containing P0 tobe standard. We fix Levi decompositions P0 = M0N0. There is then a uniqueLevi decomposition

P = MN

for each standard parabolic subgroup P such that M contains M0; we alwaysuse this Levi decomposition. In this global setting we pick a maximal compactsubgroup K ≤ G(AF ) in good position with respect to P0 (and hence all P )and normalize the Haar measures dg on G(AF ), dk on K and d`p on P (AF )so that

dg = d`pdk

(see Exercise 3.5). Let aM := Hom(X∗(M)F ,R). We have a map

HM : M(AF ) −→ aM (10.1)

defined by〈HM (m), λ〉 = log |λ(m)|

for λ ∈ X∗(M)F . Using the Iwasawa decompositionG(AF ) = N(AF )M(AF )Kwe define a morphism

HP : G(AF ) −→ aM

nmk 7−→ HM (m)

for (n,m, k) ∈ N(AF )×M(AF )×K.Let (σ, V ) be an AM\M(AF )-subrepresentation of the discrete series

L2disc([M ]) ≤ L2([M ]). The precise definition of the discrete series is given

in (3.14); it is the largest subspace of L2([M ]) that decomposes as a Hilbertspace direct sum of irreducible representations. We denote by Rdisc the re-striction of the regular representation R of L2([M ]) to this subspace.

For λ ∈ a∗MC we then form a global induced representation

(I(σ, λ), IndGP (V )).

Here IndGP (V ) is the space of measurable functions

ϕ : N(AF )M(F )AM\G(AF ) −→ C

such that for all x ∈ G(AF ) the function

m 7→ ϕ(mx)

lies in V ≤ L2disc([M ]) and such that

10.2 Intertwining operators 197

‖ϕ‖2 =

∫K

∫AMM(F )\M(AF )

|ϕ(mk)|2dmdk <∞.

Thus IndGP (V ) is a Hilbert space. The action is given by

I(σ, λ)(g)ϕ(x) = ϕ(xg)e〈HP (xg),λ+ρP 〉e−〈HP (x),λ+ρP 〉

where ρP ∈ a∗M is half the sum of the positive roots of a maximal split torusof G contained in P ; of course we use P to denote the notion of positivitydefined by P (see §1.9).

Suppose now that σ is irreducible, so it is an automorphic representationof AM\M(AF ). Since

e〈HM (m),λ〉 =∏v

e〈HM (mv),λ〉

for m ∈M(AF ), we have a factorization

I(σ, λ) ∼=∏v

I(σv, λ)

where the local factors I(σv, λ) are defined as in §4.9 in the archimedeansetting and §8.2 in the nonarchimedean setting. There is a slight differencebetween our conventions in the global setting and in the local setting in termsof the spaces on which I(σ, λ) acts. In the global setting we are normalizingthings so that the space on which I(σ, λ) acts is independent of λ. In the localsetting we incorporated λ into the definition of the functions themselves. Thetwo definitions only differ by the character e−〈HP (xv),λ+ρP 〉 at a place v.

10.2 Intertwining operators

Let T0 ≤M0 be a maximal split torus and let

W0 := W (T0, G)(F ) = W (T0, G)(F )

be the Weyl group of T0 in G. For a pair of standard parabolic subgroupsP, P ′ we let

W (aP , aP ′)

be the set of linear isomorphisms from aP onto aP ′ obtained by restrictingelements in the Weyl group W0. If W (aP , aP ′) is nonempty then P and P ′

are said to be associate. Let

P = MN and P ′ = M ′N ′


be the unique Levi decompositions such that M and M ′ contain M0.

Lemma 10.2.1 If P and P ′ are associate then M = M ′. Moreover, w pre-serves M = M ′, where w ∈W (aP , aP ′).

In view of the lemma, it is therefore natural to expect a relationship be-tween representations of G induced from P and those induced from P ′.

This relationship is obtained by introducing intertwining operators

M(w, λ) : IndGP (V ) −→ IndGP ′(V )

defined by

M(w, λ)ϕ(x) :=

∫ϕ(w−1nx)e〈HP (w−1nx),λ+ρP 〉e〈HP ′ (x),−wλ+ρP ′ 〉dn, (10.2)

where the integral is over N ′(AF )∩ wN(AF )w−1\N ′(AF ). Here w ∈ G(F ) isany representative for w. At the moment this is just a formal definition. Tomake it rigorous let K ≤ G(AF ) be a maximal compact subgroup. We firstrestrict to the dense subspace

IndGP (V )0 ≤ IndGP (V )

that consists of K-finite functions in the function field case and automorphicforms in the sense of Definition 6.5 in the number field case.

When restricted to IndGP (V )0 the integral M(w, λ) converges for λ suffi-ciently large in a sense we now make precise. Let Φ denote the set of rootsof ZM in G and let ∆P be a base for the set of positive roots defined by P(see §1.9). Set

(a∗PC)+ := λ ∈ a∗MC : λ(α) > 0 for all α ∈ ∆P .

The following proposition is stated in [Art05, Lemma 7.1]; a discussion iscontained in [MW95, §II.1.6].

Proposition 10.2.2 (Langlands) If Re(λ) ∈ ρP + (a∗PC)+ then M(w, λ)converges absolutely. ut

It turns out that the operators M(w, λ) admit meromorphic continuations[MW95, §IV.1.8, §IV.1.10] [Art05, Theorem 7.2].

Theorem 10.2.3 The operators M(w, λ) admit meromorphic continuationsto a∗PC that are holomorphic and unitary on iaP . If P, P ′, P ′′ are standardparabolic subgroups then the operators satisfy the functional equation

M(w1w2, λ) = M(w1, w2λ) M(w2, λ)

for w1 ∈W (aP ′ , aP ′′) and w2 ∈W (aP , aP ′) . ut

10.3 Eisenstein series 199

The subspace iaP ⊆ a∗PC plays a distinguished role, because the represen-tations I(σ, λ) are unitary for λ ∈ iaP (see Exercise 10.5).

10.3 Eisenstein series

Now I(σ, λ) is a representation of G(AF ) on a space of functions on G(AF ),but the functions in IndGP (V ) are not left invariant under G(F ). The functionsin its space are left invariant under P (F ), however. We therefore do the mostnaıve thing possible to make them invariant under G(F ), namely average.Given

ϕ ∈ IndGP (L2disc([M ]))

we let

E(x, ϕ, λ) :=∑

δ∈P (F )\G(F )

ϕ(δx)e〈HP (δx),λ+ρP 〉. (10.3)

This is an Eisenstein series. It provides an intertwining map from I(σ, λ)to the regular action of G(AF ) on functions on G(AF ):

E(xg, ϕ, λ) = E(x, I(σ, λ)(g)ϕ, λ) (10.4)

(see Exercise 10.4) provided that the series E(x, ϕ, λ) converges absolutely.The problem is that E(x, ϕ, λ) only converges for λ “sufficiently large”

[Art05, Lemma 7.1]:

Proposition 10.3.1 (Langlands) If Re(λ) ∈ ρP + (a∗PC)+ then E(x, ϕ, λ)converges absolutely. ut

On the other hand the representations I(σ, λ) are unitary when λ ∈ iaP ,which is never in the plane of absolute convergence of the Eisenstein series.Instead, one has to analytically continue the Eisenstein series as a complex an-alytic function of λ to the line λ ∈ iaM in order to construct pieces of L2([G]).In the case G = GL2 this was accomplished by Selberg [Sel56, Sel63]. Thoughimportant, it offered only limited insight into the difficulties presented by thegeneral case.

Langlands was the first to appreciate the difficulties presented by the gen-eral case, and was able to overcome them [Lan76]. It is hard to overestimatethe impact of Langlands’ work on Eisenstein series to automorphic represen-tation theory (and indeed much of mathematics, especially number theory).This work, for example, led to the discovery of Langlands reciprocity as enun-ciated in Langlands letter to Weil [Lan, Lan70], led to Langlands-Shahiditheory of automorphic L-functions [Lan71, Sha10], and also is required inputinto any general treatment of a trace formula, either in the Arthur-Selbergsense [Art78, Art80], or in Jacquet’s sense [Jac05a, JLR99].


If P and P ′ are associate standard parabolics, then the intertwining op-erators of the previous section give a relationship between representationsinduced from P to representations induced from P ′. This suggests a rela-tionship between the associated Eisenstein series. The precise relationship iscontained in the following theorem of Langlands (see [Art05, Theorem 7.2]and [MW95, §IV.1.10])

Theorem 10.3.2 (Langlands) The Eisenstein series E(g, ϕ, λ) admits ameromorphic continuation to a∗PC. It satisfies the functional equation

E(g,M(w, λ)ϕ,wλ) = E(g, ϕ, λ)

for w ∈W (aP , aP ′). ut

Sometimes in applications one needs more precise information about thesize of Eisenstein series and the intertwining operators. For an example werefer to [GL06] for an important example where this is crutial. This paperrefers to the equally important paper [M00], which proves that Eisensteinseries the Eisenstein series E(x, ϕ, λ) is a quotient of functions of finite orderunder suitable assumptions on ϕ.

10.4 Decomposition of the spectrum

Assume for this section that F is a number field. It turns out that there is afinite direct sum decomposition

L2([G]) = ⊕PL2P([G]) (10.5)

where the direct sum is over all association classes P of parabolic subgroups.Consider the Hilbert space of families F of measurable functions

FP : iaP −→ IndGP (L2disc([M ]))

indexed by P ∈ P and satisfying the Weyl invariance condition

FP ′(wλ) = M(w, λ)FP (λ)

for w ∈W (aP , aP ′). The relevant inner product is

(F1,F2) =∑P∈P

n−1P

∫iaP

(F1P (λ), F2P (λ))dλ,

where F1P ∈ F1 and F2P ∈ F2 for P ∈ P. Here

nP :=∑P ′∈P

|W (aP , aP ′)|. (10.6)

10.5 Local preparation for isobaric representations 201

Given such a family of functions we can define∑P∈P

n−1P

∫iaP

E(x, FP (λ), λ)dλ ∈ L2([G])

at least when FP is compactly supported as a function of λ with image ina finite dimensional subspace of IndGP (L2

disc([M ])) consisting of automorphicforms in the sense of Definition 6.5 in the number field case and K-finite inthe function field case. Here, as usual, K ≤ G(AF ) is a maximal compactsubgroup. The closure of the subspace of L2([G]) spanned by functions ofthis type is denoted

L2P([G]).

Theorem 10.4.1 (Langlands) One has

L2([G]) = ⊕PL2P([G])

where the sum is over association classes P of parabolic subgroups of G. ut

This is [Art05, Theorem 7.2].

10.5 Local preparation for isobaric representations

In this section we collect some local preliminaries for our discussion of isobaricrepresentations in §10.6. Thus for this section we assume that F is a localfield.

We have already discussed in §4.9 and §8.4 how to classify admissible rep-resentations of G(F ) for reductive F -groups G in terms of tempered repre-sentations. When G = GLn and F is nonarchimedian, we have also discussedhow this can be refined to a classification of the tempered representationsin terms of the square-integrable representations. We now explain a usefulrefinement of these results.

Assume G = GLn. Let n =∑ki=1 ni be a partition of n and for each i let

πi be an irreducible essentially square-integrable representation of GLni(F ).Let

GLni(F )1 := ker(HGLni: GLni(F ) −→ aGLni

) (10.7)

where HGLniis defined as in (8.9). Thus

GLni(F ) := GLni(F )1 ×AGLni,

where


AGLni=

R>0Ini if F is archimedean and

$ZIni if F is nonarchimedean.

Here in the nonarchimedean case $ is a uniformizer for F .For λi ∈ C, let π1

i be the unique irreducible representation of GLni(F )and such that

πi(ag) = |a|λiπ1i (g) (10.8)

for a ∈ AGLni.

After changing the order of the partition we can assume that

Re(λ1) ≥ · · · ≥ Re(λk).

Let P be the standard parabolic attached to the partition n1, . . . , nk withstandard Levi subgroup M . Let λ = (λ1, . . . , λk). Then we can form theinduced representation

I

(k∏i=1

π1i , λ

).

Theorem 10.5.1 The induced representation I(∏k

i=1 π1i , λ)

has a unique

irreducible quotient J(∏k

i=1 π1i , λ)

. If

J

(k∏i=1

π1i , λ

)∼= J

k′∏i=1

π1′i, λ′

with Re(λ1) ≥ · · · ≥ Re(λk) and Re(λ′1) ≥ · · · ≥ Re(λ′k′) then k = k′ andthere is a permutation τ of 1, . . . , k such that π′i

∼= πτ(i) (which impliesλ′i = λτ(i)).

Proof. For the nonarchimedean case, this is consequence of the Bernstein-Zelevinsky classification discussed in §8.4. For the archimedean case a dis-cussion with references is given after [Kna94, Theorem 1 and 4]. ut

Let π1· · ·πk be the irreducible quotient of the theorem. In this notationwe do not assume that the πi are arranged so that Re(λ1) ≥ · · · ≥ Re(λk),so by definition

ki=1πi∼= ki=1πτ(i)

for any permutation τ of 1, . . . , k. Because of the theorem, this notation isunambiguous.

Theorem 10.5.2 Any irreducible admissible representation of GLn(F ) isisomorphic to ki=1πi for some set of essentially square-integrable πi.

10.6 Isobaric representations 203

Proof. Again, this is consequence of the Bernstein-Zelevinsky classificationin §8.4 in the nonarchimedean case and the discussion following [Kna94, The-orem 1 and 4] in the archimedean case. ut

Given Theorem 10.5.1 and Theorem 10.5.2, we can define an irreducibleadmissible representation ki=1πi of GLn(F ) for any tuple π1, . . . , πk of ir-

reducible admissible representations of GLni(F ) with∑ki=1 ni = n. In par-

ticular, we can remove the assumption that the πi are essentially square-integrable. For each πi we write πi = kij=1π

′ij and then set

ki=1πi = ki=1

(kij=1π

′ij

). (10.9)

The symbol behaves something like a formal direct sum on the categoryof admissible representations, but we emphasize that ki=1πi is always irre-ducible.

It is useful to explicitly point out the relationship between the classificationof unramified representations in terms of the Satake correspondence and theisobaric sum. For this let Tn ≤ GLn be the maximal torus of diagonal matricesand let Bn ≤ GLn be the Borel subgroup of upper triangular matrices.

Theorem 10.5.3 Let χ1, . . . χn : F× → C× be n unramified quasicharactersand let

χ : Tn(F ) −→ C×

(tij) 7−→n∏i=1

χi(tii).

Then the isobaric sum ni=1χi is the unique unramified subquotient of theinduced representation IndGLn

Bn(χ).

Proof. We know that there is a unique unramified subquotient of IndGLnBn

(χ)by Theorem 7.6.4. On the other hand the isobaric sum ni=1χi is unramifiedby [Mat13, Corollary 1.2]. ut

10.6 Isobaric representations

Assume now that F is a number field. So far we have explained how the entirespectrum of L2([G]) can be described in terms of automorphic representationsof Levi subgroups of parabolic subgroups of G. We again restrict to the caseG = GLn and discuss refinements of this result.

As a first step in this section we discuss an operation on automorphicrepresentations of GLn that plays the role in automorphic representationtheory of the direct sum in ordinary representation theory. In fact, under the


conjectural Langlands correspondence (to be discussed in Chapter 12) thisoperation should be induced by taking direct sums of L-parameters.

Let

n =

k∑i=1

ni (10.10)

and for each i let σi is a cuspidal automorphic representation ofAGLni\GLni(AF ).

Let P be a standard parabolic subgroup of GLn (i.e. a parabolic subgroupcontaining the Borel subgroup of upper triangular matrices) with Levi sub-

group isomorphic to∏ki=1 GLni . Let σ =

∏ki=1 σi. We form the induced

representation

I(σ, λ) (10.11)

as in (10.1). For the proof of the following theorem see the appendix to [BJ79](see also [Lan79a]):

Theorem 10.6.1 (Langlands) Any irreducible subquotient of I(σ, λ) is anautomorphic representation, and every automorphic representation is of thisform. ut

This theorem leaves open the question of whether the automorphic rep-resentation π of GLn(AF ) can arise in two different manners. In fact, it canarise in essentially only one manner. More precisely, assume that

k∑i=1

ni =

k′∑j=1

n′j = n

and σi (resp. σ′j) are cuspidal automorphic representations ofAGLni\GLni(AF )

(resp. AGLn′j\GLn′j (AF )) for 1 ≤ i ≤ k and 1 ≤ j ≤ k′. We let

σ =

k∏i=1

σi and σ′ =

k′∏j=1

σ′j .

We view these as representations of standard parabolics P and P ′ attachedto

k∑i=1

ni and

k′∑j=1

n′j ,

respectively. The following theorem is [JS81a, Theorem 4.4]:

Theorem 10.6.2 (Jacquet and Shalika) Let S be a finite set of places ofF including the infinite places such that I(σ, λ) and I(σ′, λ′) are unramified

10.6 Isobaric representations 205

outside of S. If the irreducible spherical subquotients of I(σ, λ)v and I(σ′, λ′)vare isomorphic for v 6∈ S then there is a permutation τ of the set of k elementssuch that

σi ∼= σ′τ(i) and λi = λ′τ(i).

ut

This immediately implies the following corollary:

Corollary 10.6.3 If π is an irreducible subquotient of I(σ, λ) and I(σ′, λ′)then the conclusion of Theorem 10.6.2 holds ut

Given the Bernstein-Zelevinsky classification described in §8.4, it would beaesthetically pleasing if σv were tempered for each v. Indeed, if this where thecase, then upon rearranging the σi we could assume that λ is in the positiveWeyl chamber. There would then be a canonical irreducible subquotient ofI(σ, λ), namely ∏

v

J(σv, λ)

with the local factors J(σv, λ) defined as in Theorem 4.9.1 and Theorem 8.4.1.An important open conjecture is that cuspidal representations always havetempered local factors:

Conjecture 10.6.4 (The Ramanujan conjecture) If π is a cuspidal au-tomorphic representation of AGLn\GLn(AF ) then πv is tempered for all v.

A discussion of this conjecture and what is known towards it is contained in[Sar05].

Since the Ramanujan conjecture remains unproven to find a canonicalirreducible subquotient of I(σ, λ) we must proceed differently. The first stepis the following refinement of Theorem 10.6.1:

Theorem 10.6.5 Assume

Re(λ1) ≥ · · · ≥ Re(λk).

The induced representation I(σ, λ) has a unique irreducible quotient J (σ, λ).If

J (σ, λ) ∼= J (σ′, λ′)

with Re(λ1) ≥ · · · ≥ Re(λk) and Re(λ′1) ≥ · · · ≥ Re(λ′k′) then k = k′

and there is a permutation τ of 1, . . . , k such that σ′i∼= στ(i) and hence

λ′i = λτ(i).

Proof. The uniqueness of the irreducible quotient follows from the corre-sponding local uniqueness assertion in Theorem 10.5.1 provided that thesequotients can be taken to be spherical outside of finitely many places. This isimplied by Theorem 10.5.3 The second assertion of the theorem then followsfrom Theorem 10.6.2. ut


This is the global analogue of the local result Theorem 10.5.1. To makethe analogy clearer, it is convenient to adjust our notation slightly. Let πi becuspidal automorphic representations of GLni(AF ) for 1 ≤ i ≤ k. Then thereis a unique λ ∈ a∗M such that

⊗ki=1πi∼= σ ⊗ e〈HM (·),λ〉

where σ is a cuspidal automorphic representation of AM\M(AF ). We denoteby

π1 · · · πk (10.12)

the representation J (σ, λ) of the theorem. Here by definition (10.12) is inde-pendent of the ordering of the factors. This convention is justified by Theorem10.6.5.

Definition 10.1. An automorphic representation of GLn(AF ) is isobaric ifit is of the form ki=1πi for some k-tuple of cusp forms π1, . . . , πk.

Not all automorphic representations are isobaric. In the notation of Theorem10.6.5, if I(σ, λ) is reducible, then any subquotient that is not J(σ, λ) providesan example. A concrete example is given in Exercise 10.9 below.

In view of Theorem 10.6.5 we can extend the definition of to arbitrarytuples of isobaric automorphic representations in a well-defined manner justas in (10.9). The operation on the set of isomorphism classes of isobaricrepresentations behaves something like a direct product, but just as in thelocal case isobaric sums are always irreducible representations. We close thissection with the following local-global compatibility result:

Lemma 10.6.6 For a k-tuple π1, . . . , πk where πi is a cuspidal automorphicrepresentation of GLni(AF ) one has

ki=1πi∼=∏v

ki=1πiv

where ki=1πiv is defined as in (10.9).

Proof. Since ki=1πi is defined to be the unique irreducible quotient of anappropriate representation, and the right hand side of the isomorphism pro-vides such a quotient by the local definition of an isobaric representation thelemma follows. ut

10.7 A theorem of Moeglin and Waldspurger

For this section we assume that F is a number field. Let G be a reductivegroup over F . The discrete spectrum L2

disc([G]) of L2([G]) is the largest

10.7 A theorem of Moeglin and Waldspurger 207

closed subspace that decomposes as a direct sum of irreducible representa-tions (see §3.7 for more information). The spectral decomposition of L2([G])proven by Langlands leaves open the important question of how to describethe discrete spectrum of L2([M ]) for Levi subgroups M of G. Some informa-tion on this question is obtained in the course of Langlands’ proof of Theorem10.4.1, but nothing definitive.

In [Jac71] Jacquet gave a precise conjectural description of L2disc([GLn])

which was later proven by Moeglin and Waldspurger in [MgW89]. We statethis result in this section. Since the Levi subgroups of GLn are products ofgeneral linear groups, by induction this gives a complete spectral decompo-sition of L2([GLn]).

Fix an integer n. Call a parabolic subgroup P ≤ GLn standard if it con-tains the Borel subgroup of upper triangular matrices. For each factorizationn = md there is a unique standard parabolic subgroup P with Levi sub-group M isomorphic to GL×md . Suppose we are given a cuspidal automorphicrepresentation π of AGLd\GLd(AF ). We can then form the tensor productrepresentation π⊗m of AM\M(AF ). Let

L2(π⊗m) ≤ L2([M ])

be the π⊗m-isotypic subspace. We can then consider a form

ϕ ∈ IndGP (L2(π⊗m))

and the corresponding Eisenstein series

E(x, ϕ, λ).

Jacquet constructed an irreducible subrepresentation (σ,m) of L2([GLn]) bytaking an iterated residue of E(x, ϕ, λ). If we pick an appropriate isomorphisma∗M∼= Rm then the iterated residue is at certain point. The representation

(σ,m) is called a Speh representation, since Speh constructed the localanalogues of these representations (see [Spe83]). To get a feel for these repre-sentations, it is useful to note that if v is a nonarchimedean place of F whereσ is unramified then

L(s, (σ,m)v) =

m∏i=1

L(s− i+ (m+ 1)/2, σv).

Here L(s, (σ,m)v) is the L-function of (σ,m)v (see §11.8). Jacquet then con-jectured the following theorem, which was proven in [MgW89]:

Theorem 10.7.1 (Moeglin and Waldspurger) Any irreducible subrepre-sentation of L2([GLn]) is isomorphic to a Speh representation (σ,m) for aunique factorization md = n and a unique cuspidal automorphic representa-tion σ of AGLd\GLd(AF ). ut


Thus the theorem gives a complete description of L2disc([GLn]). This the-

orem forms the basis of much of what we know about the discrete spectrumfor other groups. For example, it together with the Jacquet-Langlands cor-respondence is used to describe the discrete spectrum of inner forms of GLnin [Bad08]. As a consequence of the theory of twisted endoscopy, Arthur[Art13] and Mok [Mok15] have given a description of the discrete spectrumof quasi-split classical groups in terms of the parametrization of Moeglin andWaldspurger. This is explained in §13.8 below.

Exercises

10.1. Let G = GL2 and let P be the Borel subgroup of upper triangularmatrices with Levi subgroup M , the diagonal matrices. Let

χ : [M ] −→ C×

( a1a2

) 7−→ χ1(a1)χ2(a2)

be a character. Let Φ ∈ S(A2F ) (the usual Schwartz space). Let

ϕΦ,χ,λ(g) := χ1(det g)|det g|λ∫A×F

Φ((0, x)g)χ1χ−12 (x)|x|2λdx×.

Show that ϕΦ,χ,0(g) ∈ IndGP (χ) (with our conventions on the induced actionfrom §10.1) and that

E(g, ϕΦ,χ,0, λ) =∑

γ∈P (F )\G(F )

ϕΦ,λ+1/2(γg).

In particular show that this series converges absolutely for Re(λ) largeenough.

10.2. Using Poisson summation, prove that E(g, ϕ, λ) admits a meromorphiccontinuation to the plane and satisfies the functional equation

E(g, ϕΦ,χ,0, λ) = E(g−t, ϕΦ,χ−1,0,−λ).

10.3. Prove that E(g, ϕΦ,χ,0, λ) is holomorphic unless χ1 = χ2, in whichcase it has at worst simple poles at s = ± 1

2 . Prove that the residue ofE(g, ϕΦ,χ,0, λ) is independent of g, and hence can be thought of as a vec-tor in the trivial representation of SL2(AF ).

10.4. Prove the intertwining relation (10.4).

10.5. Prove that I(σ, λ) is unitary for λ ∈ iaM .

10.7 A theorem of Moeglin and Waldspurger 209

10.6. Prove theorems 10.5.1 and 10.5.2 using the results on the Bernstein-Zelevinsky classification stated in §8.4.

For the remaining problems, let triv denote the trivial representation ofGL1(AF ). Let

π := I(triv, ( 12 ,−

12 ));

it is a representation of GL2(AF ).

10.7. Prove that every irreducible subquotient of I(triv, ( 12 ,−

12 )) is automor-

phic and that the (unique) isobaric subquotient is the trivial representationof GL2(AF ).

10.8. Prove that πv has a composition series of length 2 (the unique irre-ducible subrepresentation σv is known as the Steinberg representation).

10.9. Prove that the irreducible subquotients of π are in bijection with finitesets of places of F : the set S of places corresponds to the representation(∏

v|S σv

)⊗ πS

Chapter 11

Rankin-Selberg L-functions

Artin and Hecke were together atGottingen, but neither realizedthe intimate connection betweenthe two different types ofL-functions they wereconstructing. The moral of thestory is to talk with yourcolleagues.

L. Saper

Abstract In this chapter we sketch the theory of generic representations andRankin-Selberg L-functions.

11.1 Paths to the construction of automorphicL-functions

Let F be a global field and let π and π′ be a pair of cuspidal automorphicrepresentations of AGLn\GLn(AF ) and AGLm\GLm(AF ), respectively. Animportant analytic invariant of this pair is the Rankin-Selberg L-function

L(s, π × π′).

This is a meromorphic function on the complex plane satisfying a functionalequation whose poles can be explicitly described in terms of π and π′. TheseL-functions play a crucial role in automorphic representation theory. On theone hand they are used in the statement of the local Langlands correspon-dence for GLn (and hence, implicitly, for the global Langlands correspon-dence). This will be discussed in §12.5.

In this chapter we describe one method by which they can be defined,namely the Rankin-Selberg method. This requires the introduction of the no-

211

212 11 Rankin-Selberg L-functions

tion of a generic representation. It turns out that all cuspidal representationsof GLn(AF ) are generic, but cuspidal representations of other groups need notbe generic. A much more detailed treatment of the Rankin-Selberg methodis contained in [Cog07], which was our primary reference in the preparationof this chapter.

There are other approaches to defining these L-functions. One is via theso-called Langlands-Shahidi method. This method also uses the notion ofa generic representation. The idea is that Rankin-Selberg L-functions (andsome more general L-functions) occur in the constant term of certain Eisen-stein series. This approach was historically important because it suggestedboth the general definition of a Langlands L-function, discussed in §12.7,and the general formulation of Langlands functoriality. We do not discussthis construction. It is discussed at length in the book [Sha10].

In the special case where π′ is the trivial representation there is an alter-nate approach to defining these L-functions due to Godement and Jacquet[GJ72]. It is a direct generalization of Tate’s construction of the L-functionsof Hecke characters in his famous thesis [Tat67], which in turn is an adelicformulation of the classical work of Hecke.

11.2 Generic characters

Let G be a quasi-split group over a global or local field F , let B ≤ G be aBorel subgroup, let T ≤ B a maximal torus and let N ≤ B the unipotentradical of B. We choose a pinning

(B, T, Xαα∈∆)

as in Definition 7.2. Let E/F be a Galois extension over which B and Tsplit. Then since T (E) is preserved by Gal(E/F ) the root spaces Xα arepermuted by Gal(E/F ) and we can choose a set of representatives β forthe Gal(E/F ) orbits; let O(β) denote the orbit of β. For E-algebras R theexponential map yields a group homomorphism

exp :∏

α∈O(β)

Ga(R) −→ GE(R)

(xα) 7−→∏

α∈O(β)

exp(xαXα).

Its image is a unipotent subgroup whose Lie algebra is spanned by Xαα∈O(β).This unipotent subgroup is in fact stable under Gal(E/F ), and hence is theE points of a unipotent subgroup N(β) ≤ N that is isomorphic to ResE/FGa.Varying β, we see that we have a subgroup

11.3 Generic representations 213∏β

N(β) ≤ N,

where the product is over all Gal(E/F )-orbits of simple roots α. We notethat inclusion induces an isomorphism∏

β

N(β)−→N/Nder.

Definition 11.1. If F is a local field, a character ψ : N(F ) → C× is calledgeneric if ψ|N(β)(F ) is nontrivial for each Gal(E/F )-orbit β. If F is global acharacter ψ : N(AF )→ C× trivial on N(F ) is called generic if ψv is genericfor all places v.

Example 11.1. Take G = GLn and let Bn ≤ GLn be the Borel subgroup ofupper triangular matrices. Moreover let Nn ≤ Bn be the unipotent radical.If F is local any character of Nn(F ) is of the form

Nn(F ) −→ C×1 x12 ∗ ∗ ∗

1 x23 ∗ ∗. . .

. . . ∗1 x(n−1)n

1

7−→ ψ(m1x12 + · · ·+mn−1x(n−1)n)(11.1)

for some m1, . . . ,mn−1 ∈ F , where ψ : F → C× is a nontrivial character. Thecharacter is generic if and only if all of the mi are nonzero. If F is global anycharacter of Nn(AF ) trivial on Nn(F ) is of the same form, but now the mi

must be in F and ψ must be trivial on F . Again, the character is generic ifand only if

∏n−1i=1 mi 6= 0. If all mi = 1 we call this the standard character

attached to ψ.

11.3 Generic representations

Let G be a quasi-split reductive group over a local field F and let ψ : N(F )→C× be a character. Let (π, V ) be an admissible representation of G(F ).

Definition 11.2. Assume F is nonarchimedean. A ψ-Whittaker func-tional on V is a continuous linear functional λ : V −→ C such that

λ(π(n)ϕ) = ψ(n)λ(ϕ)

for ϕ ∈ V and n ∈ N(F ).

In this nonarchimedean setting, continuous means locally constant.


If F is archimedean, we assume that V is a Hilbert space and even that πis unitary. Then for every X ∈ U(g) we can define a seminorm ‖ · ‖X on Vsm

via‖ϕ‖X := ‖π(X)ϕ‖2.

We can give Vsm a locally convex topology via these seminorms, and it thenmakes sense to speak of continuous linear functionals. It is this notion ofcontinuity we use in the following definition:

Definition 11.3. Assume F is archimedean. A ψ-Whittaker functionalon V is a continuous linear functional λ : V −→ C such that

λ(π(n)ϕ) = ψ(n)λ(ϕ)

for ϕ ∈ Vsm and n ∈ N(F ).

Assume, as above, that in the archimedean setting (π, V ) is unitary. As-sume in addition that ψ is generic.

Definition 11.4. An irreducible admissible representation (π, V ) of G(F ) isψ-generic if it has a nonzero ψ-Whittaker functional on V .

An irreducible admissible representation (π, V ) is simply said to be generic ifthere exists a Borel subgroup B ≤ G with unipotent radical N and a genericcharacter ψ : N(F )→ C× such that (π, V ) is ψ-generic.

The following theorem of Shalika [Sha74] has turned out to be extremelyimportant:

Theorem 11.3.1 (Shalika) The space of ψ-Whittaker functionals on an ir-reducible admissible representation of G(F ) is at most 1-dimensional. ut

We now define the related notion of a Whittaker function. The space ofWhittaker functions on G(F ) is the space of smooth functions

W(ψ) := smooth W : G(F )→ C satisfying W (ng) = ψ(n)W (g). (11.2)

This space admits an action of G(F ):

G(F )×W(ψ) −→W(ψ)

(g,W ) 7−→ (x 7→W (xg)).

Let (π, V ) be an irreducible admissible representation π of G(F ). A Whit-taker model of π is a nontrivial G(F )-intertwining map

Vsm −→W(ψ)

(in the nonarchimedean case, V = Vsm by assumption). As a corollary ofTheorem 11.3.1 we have the following (see Exercise 11.2):

11.3 Generic representations 215

Corollary 11.3.2 A Whittaker model of π exists if and only if π is generic.If π is generic, then it admits a unique Whittaker model. ut

Motivated by Corollary 11.3.2 if (π, V ) is generic we let

W(π, ψ) (11.3)

be the image of any intertwining map Vsm → W(ψ). We write the mapexplicitly as

Vsm −→W(π, ψ)

ϕ 7−→Wϕψ .

By Schur’s lemma this map is well-defined up to multiplication by a nonzerocomplex number.

Now assume that F is global, let π be a cuspidal automorphic represen-tation of G(AF ) realized in a subspace V ≤ L2

cusp([G]), and let ϕ ∈ Vsm. Wethen define the global ψ-Whittaker function

Wϕψ (g) :=

∫[N ]

ϕ(ng)ψ(n)dn. (11.4)

We say that (π, V ) is globally ψ-generic if

Wϕψ (g) 6= 0

for some ϕ ∈ V and g ∈ G(AF )1. We remark that a priori this notion dependson the realization of π in L2

cusp([G]) if it does not occur with multiplicity 1.It is not hard to check that if π is globally ψ-generic and π ∼= ⊗′vπv then eachπv is ψv-generic (Exercise 11.3).

The notion of a generic representation is only useful if we have an interest-ing supply of generic representations. The following theorem implies that allcuspidal automorphic representations of GLn(AF ) are generic, and moreoverthey admit an expansion in terms of Whittaker functionals:

Theorem 11.3.3 Let ϕ ∈ L2cusp([GLn]) be a smooth vector in the space of a

cuspidal automorphic representation π of AGLn\GLn(AF ). If ψ : Nn(AF )→C× is a generic character then one has

ϕ(g) =∑

γ∈Nn−1(F )\GLn−1(F )

Wϕψ (( γ 1 ) g) . (11.5)

ut

The expression for ϕ in (11.5) is called its Whittaker expansion. It is ageneralization (but not the only generalization) of the well-known Fourierexpansion of a modular form.

We now prove this theorem in the special case n = 2. Consider the function


x 7→ ϕ (( 1 x1 ) g) . (11.6)

This is a continuous function on the compact abelian group F\AF , and henceadmits a Fourier expansion. If we fix a nontrivial character ψ : F\AF → C×,then all other characters are of the form ψα(x) := ψ(αx) for α ∈ F (seeLemma B.1.2). Thus the Fourier expansion of the function (11.6) is∑

α∈F

∫F\AF

ϕ (( 1 x1 ) g)ψ(αyx)dy. (11.7)

Since ϕ is cuspidal, the α = 0 term vanishes identically. Taking a change ofvariables x 7→ α−1x for each α ∈ F× and using the left GL2(F )-invariance ofϕ we see that (11.6) is equal to∑

α∈F×Wϕψ (( α 1 ) ( 1 x

1 ) g) . (11.8)

Setting x = 0 we obtain Theorem 11.3.3 when n = 2. We will not proveTheorem 11.3.3 because we cannot improve on the exposition in [Cog07,§1.1]. The basic idea is to proceed as in the n = 2 case, using abelian Fourieranalysis on certain abelian subgroup of Nn(AF ). However, since Nn(AF ) isno longer abelian for n > 3 one has to combine this with a clever inductiveargument using the fact that ϕ is cuspidal.

Using Theorem 11.3.3, we obtain the following related theorem of Shalika[Sha74, Theorem 5.5]:

Theorem 11.3.4 (Multiplicity one) An irreducible admissible represen-tation of GLn(AF ) occurs with multiplicity at most one in L2

cusp([GLn]).

We note that Theorem 11.3.4 is false for essentially every reductive groupthat is not a general linear group.

Proof. Let (π1, V1), (π2, V2) be two realizations of a given cuspidal automor-phic representation (π, V ). Choose equivariant maps Li : V → Vi. We thenobtain Whittaker functionals

λi : Vsm −→ C

ϕ 7−→WLi(ϕ)ψ (In),

where In is the identity element. They are nonzero by Theorem 11.3.3. ByTheorem 11.3.1 one therefore has λ1 = cλ2 for some c ∈ C×. Thus

L1(ϕ)(g) =∑


WL1(ϕ)ψ (( γ 1 ) g)

= c∑


WL2(ϕ)ψ (( γ 1 ) g)

11.4 Formulae for Whittaker functions 217

= L2(ϕ)(g).

This implies in particular that V1 and V2 have nonzero intersection, and henceare equal. ut

11.4 Formulae for Whittaker functions

Let (π, V ) be a globally generic cuspidal representation of the quasi-splitgroup G, realized on a closed subspace V ≤ L2([G]).

Using Flath’s theorem we can choose an isomorphism

Vsm−→ ⊗′v Vv −→ ⊗vW(πv, ψv). (11.9)

Let ϕ ∈ Vsm be a pure tensor, i.e. a vector that maps to a vector of the form⊗′vϕv under the first isomorphism. Then, by local uniqueness of Whittakermodels upon normalizing the composite isomorphism Vsm by multiplying bya suitable nonzero complex number we have

Wϕψ (g) =

∏v

Wϕvψv

(g). (11.10)

Thus to compute the global Whittaker function Wϕψ (g) it suffices to com-

pute the local Whittaker functions Wϕvψv

. When πv and ψv are unramifiedthis can be accomplished using famous Casselman-Shalika formula [CS80].

For the remainder of this section let F be a nonarchimedean local field andlet ψ : F → C× be a nontrivial unramified character. The Casselman-Shalikaformula is valid for any unramified reductive group over F , but to simplifyour discussion we will assume that G is a split group. Let B ≤ G be theBorel subgroup with split maximal torus T ≤ B. We let K ≤ G(F ) be ahyperspecial subgroup in good position with respect to (B, T ). We note thatby the Iwasawa decomposition one has

G(F ) =∐

µ∈X∗(T )

N(F )µ($)K (11.11)

where N ≤ B is the unipotent radical of B.Using the notation of §7.6, for λ ∈ a∗TC we can then form the induced

representation I(λ), where the induction is with respect to the Borel subgroupB. Its unique unramified subquotient is denoted by J(λ). Any unramifiedrepresentation is equivalent to such a J(λ). The choice of T and B, by duality,

give rise to a maximal torus and Borel subgroup T ≤ B ≤ G as explainedin §7.3. This gives rise to a notion of dominant weight of T in B, say µ ∈X∗(T ) = X∗(T ). By Cartan-Weyl theory, there is a unique isomorphism class

V (µ) of irreducible representation of G attached to µ. Let χµ be its character.


Theorem 11.4.1 (Casselman and Shalika) Any function W in the one-dimensional space W(J(λ), ψ)K satisfies W (I) ∈ C×, I is the identity ele-ment. If µ is a dominant weight then

W (µ($))

W (I)= δ

1/2B (µ($))χµ(e−λ).

If µ is not a dominant weight then W (µ($)) = 0. ut

Here one recalls that eλ is discussed in Lemma 7.6.1. This is proven in [CS80];to translate the formula in loc. cit. to the formula above one uses the Weylcharacter formula.

It is instructive to write this formula down more explicitly when G = GLn.In this case it is due to Shintani [Shi76]. We assume that B = Bn is the Borelsubgroup of upper triangular matrices and T = Tn ≤ Bn is the maximaltorus of diagonal matrices. In this case weights can be identified with tuples(k1, . . . , kn) ∈ Zn as in (1.9). The cocharacter µ attached to such a tuple isrecorded in (1.9). The dominant weights are those with

k1 ≥ · · · ≥ kn.

The associated irreducible representation of GLn is Sk1,...,kn(Vst), where Vst

is the standard representation and Sk1,...,kn is the Schur functor attached tothe partition k1, . . . , kn of

∑ni=1 kn. Let χk1,...,kn be its character.

Corollary 11.4.2 (Shintani) If G = GLn and W ∈ W(J(λ), ψ)GLn(OF ) isthe unique vector satisfying W (In) = 1 then

W

($k1

. . .

$kn

)=

δ

1/2Bn

(µ($))χk1,...,kn(e−λ) if k1 ≥ · · · ≥ kn0 otherwise.

ut

11.5 Local Rankin-Selberg L-functions

Let F be a local field and let ψ : F → C× be a nontrivial character. Our goalin this section is to define local Rankin-Selberg L-functions for generic repre-sentations, at least when F is nonarchimedean. The fundamental idea here isthat these L-functions are the smallest rational functions in q−s that cancelthe poles of a family of zeta functions. The definition of this family was orig-inally given in [JPSS83], and the archimedean case was treated definitivelyin [Jac09]. We refer to these papers for proofs of the unproved statements wemake below.

Let π, π′ be irreducible admissible generic representations of GLn(F ) andGLm(F ), respectively. Let ψ : F → C× be an additive character. Use it to

11.5 Local Rankin-Selberg L-functions 219

define a standard generic character of Nn(F ) and Nm(F ) as in Example 11.1.We denote these characters again by ψ. Let

W ∈ W(π, ψ), W ′ ∈ W(π′, ψ).

If m < n let

Ψ(s;W,W ′) :=

∫W(hIn−m

)W ′(h)|det(h)|s−(n−m)/2dh,

Ψ(s;W,W ′) :=

∫ ∫W(hx In−m−1

1

)dxW ′(h)|deth|s−(n−m)/2dh,

(11.12)

where the top integral is over Nm(F )\GLm(F ), the outer integral on thebottom is over Nm(F )\GLm(F ), and the inner integral on the bottom is over(n−m− 1)×m matrices with entries in F .

If m = n, then for each Φ ∈ S(Fn) (the Schwartz space) let

Ψ(s;W,W ′, Φ) :=

∫Nn(F )\GLn(F )

W (g)W ′(g)Φ(eng)|det g|sdg. (11.13)

Here en ∈ Fn is the elementary vector with 0’s in the first n− 1 entries and1 in the last entry.

Proposition 11.5.1 Assume that F is nonarchimedean.

(a) The integrals (11.12) and (11.13) converge for Re(s) sufficiently large.For π and π′ unitary they converge absolutely for Re(s) ≥ 1. For π and π′

tempered, they converge absolutely for Re(s) > 0.(b) Each integral is a rational function of q−s.(c) The C-linear span of the integrals in C[qs, q−s] as W,W ′ (and when m = n

the Schwartz function Φ) is a principal ideal I(π, π′). ut

In this nonarchimedean case, one proves that there is a unique polynomialPπ,π′ ∈ C[x] satisfying Pπ,π′(0) = 1 such that Pπ,π′(q

−s)−1 is a generator ofI(π, π′). One sets

L(s, π × π′) := Pπ,π′(q−s)−1. (11.14)

In the archimedean case one actually defines L(s, π × π′) using the localLanglands correspondence, which was established very early on in the theoryby Langlands himself. We will explain this in more detail in §12.3 below.Right now we note the following important bound:

Theorem 11.5.2 If π and π′ are unitary then L(s, π × π′) is holomorphicand nonzero for Re(s) ≥ 1. ut

See [BR94a, §2] for a proof. Technically speaking they only state that theL-function is holomorphic at this point, but the argument actually provesthat it is nonzero as well.


To discuss the archimedean case we say that a holomorphic function f(s)of s ∈ C is bounded in vertical strips if it is bounded in

Vσ1,σ2:= s : σ1 ≤ Re(s) ≤ σ2

for all σ1 < σ2 ∈ R. A meromorphic function f(s) is bounded in verticalstrips if for each σ1 < σ2 there is a polynomial Pσ1,σ2

∈ C[x] such thatPσ1,σ2

(s)f(s) is holomorphic and bounded in vertical strips in the originalsense.

Theorem 11.5.3 Assume F is archimedean. The local Rankin-Selberg inte-grals (11.12) and (11.13) converge absolutely for Re(s) sufficiently large andadmit meromorphic continuations to functions of s that are bounded in ver-tical strips. ut

Let

wm,n :=

(Im

wn−m

), wn :=

(1

. ..

1

)∈ GLn(Z)

and for functions f on GLn(F ) or GLn(AF ) let ρ(wm,n)f(x) := f(xwm,n)

and f(g) := f(wng−t).

Theorem 11.5.4 There is a meromorphic function γ(s, π× π′, ψ) ∈ C(q−s)in the nonarchimedean case such that if m < n we have

Ψ(1− s, ρ(wn,m)W , W ′) = ω′(−1)n−1γ(s, π × π′, ψ)Ψ(s,W,W ′)

and if m = n we have

Ψ(1− s, W , W ′, Φ) = ω′(−1)n−1γ(s, π × π′, ψ)Ψ(s;W,W ′, Φ)

for all W ∈ W(π, ψ), W ′ ∈ W(π′, ψ). Here ω′ is the central quasi-characterof π′.

We warn the reader that the functional equations in [Jac09] are different. The

root of this is that the definition of the contragredient W of W in loc. cit. isdifferent.

We need one other factor, the local ε-factor:

ε(s, π × π′, ψ) :=γ(s, π × π′, ψ)L(s, π × π′)

L(1− s, π∨ × π′∨). (11.15)

Proposition 11.5.5 In the nonarchimedean case ε(s, π × π′) is a functionof the form cq−fs for some real number f and c ∈ C×.

The ε-factor is a useful arithmetic invariant of the pair (π, π′). Assumethat π′ is trivial. In this case the ideal $f is called the conductor of π. One

11.6 The unramified Rankin-Selberg L-functions 221

writes f := f(π), and calls it the exponent of the conductor, or sometimesjust the conductor. Let.

K1($f(π)) :=

g ∈ GLn(OF ) : g ≡

∗∗...∗

0 . . . 0 1

(mod $f(π))

.

Then one has the following (see [Mat13]):

Theorem 11.5.6 The number f(π) is the smallest nonnegative integer suchthat the space V of π satisfies

V K1($f(π)) 6= 0.

Moreover, dimC VK1($f(π)) = 1. ut

11.6 The unramified Rankin-Selberg L-functions

In this section we assume that F is nonarchimedean and that π and π′ areirreducible unramified representations of GLn(F ) and GLm(F ), respectively.Assume that ψ : F → C× is unramified and let W ∈ W(π, ψ) and W ′ ∈W(π′, ψ) be the unique spherical vectors satisfying W (In) = W ′(Im) = 1.We assume that π ∼= J(λ) and π′ ∼= J(λ′), where λ ∈ a∗TnC and λ′ ∈ a∗TmC.

Our aim in this section is to prove the following identity:

Theorem 11.6.1 For Re(s) sufficiently large if n > m one has

Ψ(s,W,W ′, ψ) = det(Imn − q−se−λ ⊗ e−λ

′)−1

.

If m = n one has

Ψ(s,W,W ′,1OnF ) = det(In2 − q−se−λ ⊗ e−λ

′)−1

.

Implicit in the definition of the local Rankin-Selberg integrals is a choice ofmeasure on GLm(F ) and Nm(F ) so that we can form a quotient measure onNm(F )\GLm(F ). In order for the equalities above to be valid we must choosethe unique Haar measures that assign volume 1 to GLm(OF ) and Nm(OF ),respectively.

Proof. For eachm ∈ Z≥1 let T+(m) be the set of the tuples µ = (k1, · · · , km) ∈Zm with k1 ≥ · · · ≥ km ≥ 0. This can be identified with a subset of the dom-inant weights of Tm or a cocharacter of Tm. If n > m let


T+(m) → T+(n)

be the injection given by extending the tuple k1, . . . , km by zeros.Assume for the moment that n > m. Then by Corollary 11.4.2 we have

Ψ(s;W,W ′)

=∑

µ∈T+(m)

W

(µ($)

In−m

)W ′(µ($))|det(µ($))|s−(n−m)/2δ−1

Bm(µ($))

=∑

µ∈T+(m)

χk1,...,km(e−λ′)χk1,...,km,0,...,0(e−λ)q−|µ|s

= det(Imn − q−se−λ′⊗ e−λ)−1,

where we let |µ| = k1+· · ·+km and, as before, χk1,...,km,0,...,0 denotes the char-acter associated to Sk1,...,km,0,...,0. This last identity is known as the Cauchyidentity [Bum13, Chapter 38]. Similarly, when n = m we have

Ψ(s;W,W ′,1OnF )

=∑

µ∈T+(n)

W(µ($)

)W ′(µ($))1OnF (enµ($)) |det(µ($))|sδ−1

Bn(µ($))

=∑

µ∈T+(n)

χk1,...,kn(e−λ)χk1,...,kn(e−λ′)q−|µ|s

= det(In2 − q−se−λ ⊗ e−λ′)−1.

ut

The calculation above implies that det(Imn−q−se−λ⊗e−λ′) divides L(s, π×

π′)−1 (as a polynomial in q−s)). More is true (see [JPSS83]):

Theorem 11.6.2 One has L(s, π × π′) = det(Imn − q−se−λ ⊗ e−λ′)−1. ut

11.7 Global Rankin-Selberg L-functions

Let π and π′ be cuspidal automorphic representations of AGLn\GLn(AF ) andAGLm\GLm(AF ), respectively.

We have previously defined the local Rankin-Selberg L-functions L(s, πv×π′v). The global Rankin-Selberg L-function is the product

L(s, π × π′) :=∏v

L(s, πv × π′v). (11.16)

If ψ : F\AF → C× is a nontrivial character we also define

11.7 Global Rankin-Selberg L-functions 223

ε(s, π × π′) :=∏v

ε(s, πv × π′v, ψv).

The reason that ψ is not encoded in to the left hand side of this equation isthat the right hand side is in fact independent of the choice of ψ.

The following theorem collects the basic facts about Rankin-Selberg L-functions:

Theorem 11.7.1 The Rankin-Selberg L-function admits a meromorphic con-tinuation to the plane, holomorphic except for possible simple poles at s = 0, 1.There are poles at s = 0, 1 if and only if m = n and π ∼= π′∨. One has afunctional equation

L(s, π × π′) = ε(s, π, π′)L(1− s, π∨ × π′∨). (11.17)

One extremely important consequence of this theorem is the strong mul-tiplicity one theorem:

Theorem 11.7.2 (Strong multiplicity one) Let π and π′ be cuspidal au-tomorphic representations of GLn(AF ) and let S be a finite set of places ofF . If πS ∼= π′S then π ∼= π′.

We leave the proof as Exercise 11.6. It is a consequence of Proposition 11.5.1,Theorem 11.5.2 and Theorem 11.7.1. Despite its name Theorem 11.7.2 doesnot imply the Theorem 11.3.4 above. For a beautiful generalization of thestrong multiplicity one theorem we refer the reader to the work of DinakarRamakrishnan on automorphic analogues of the Chebatarev density theorem.More generally, one has the following (see Exercise 11.7).

Corollary 11.7.3 Let π and π′ be isobaric automorphic representations ofGLn(AF ) and let S be a finite set of places of F . If πS ∼= π′S then π ∼= π′.

Let ϕ be in the space of π and ϕ′ in the space in π′. We outline the basicidea behind this theorem in the case m = n − 1. We will not justify any ofthe convergence statements made in our outline. For full details we refer thereader to [JPSS83].

In this case we take

I(s;ϕ,ϕ′) :=

∫GLn−1(F )\GLn−1(AF )

ϕ ( h 1 )ϕ′(h)|deth|s−1/2dh. (11.18)

Let ϕ(g) := ϕ(g−t) and ϕ′(g) := ϕ(g−t). Taking a change of variables h 7→h−t we see that

I(s;ϕ,ϕ′) = I(1− s, ϕ, ϕ′). (11.19)

This simple change of variables is what ultimately powers the proof of theo-rem 11.7.1.

Define


Ψ(s;Wϕψ ,W

ϕ′

ψ) =

∫Nn−1(AF )\GLn−1(AF )

Wϕψ

(( h 1 )

)Wϕ′

ψ(h)|deth|s−1/2dh.

This is a global Rankin-Selberg integral.

Theorem 11.7.4 For Re(s) sufficiently large I(s;ϕ,ϕ′) = Ψ(s,Wϕψ ,W

ϕ′

ψ).

In particular, the functional equation (11.19) immediately implies an anal-

ogous one for the global Rankin-Selberg integral Ψ(s,Wϕψ ,W

ϕ′

ψ).

Proof. Replacing ϕ by its Whittaker expansion we have

I(s;ϕ,ϕ′) =


ϕ ( h 1 )ϕ′(h)|deth|s−1/2dh

=


∑γ

Wϕψ

(( γ 1 ) ( h 1 )

)ϕ′(h)|deth|s−1/2dh,

where the inner sum is over γ ∈ Nn−1(F )\GLn−1(F ). Since ϕ′(h) is automor-phic on GLn−1(AF ) and |det(γ)| = 1 for γ ∈ GLn−1(F ), we may interchangethe order of summation and integration and obtain

I(s;ϕ,ϕ′) =

∫Nn−1(F )\GLn−1(AF )

Wϕψ

(( h 1 )

)ϕ′(h)|deth|s−1/2dh.

This integral is absolutely convergent for Re(s) 0 which justifies the inter-change.

Let us first integrate over [Nn−1]. View Nn−1 → Nn as matrices of theform ( u 1 ). Then for u ∈ Nn−1(AF ) one has Wϕ

ψ (ug) = ψ(u)Wϕψ (g).

I(s;ϕ,ϕ′)

=


∫[Nn−1]

Wϕψ

(( u 1 ) ( h 1 )

)ϕ′(uh)du|deth|s−1/2dh

=


Wϕψ

(h

1

) ∫[Nn−1]

ψ(u))ϕ′(uh)du|deth|s−1/2dh

=


Wϕψ

(h

1

)Wϕ′

ψ(h)|deth|s−1/2dh

= Ψ(s;Wϕψ ,W

ϕ′

ψ).

ut

In the case m = n− 1 above Theorem 11.7.1 can be deduced from theorems11.5.4 and 11.7.4; we leave this as Exercise 11.8.

11.9 Converse theory 225

11.8 The nongeneric case

In the theory above crucial use was made of the fact that cuspidal auto-morphic representations are globally generic. Indeed, our definition of localL-functions was given in terms of the Whittaker model.

We now explain how to handle the general case. Assume first that F isa local field. From §10.6 we know that every pair of admissible irreduciblerepresentation π of GLn(F ) and π′ of GLm(F ) can be written as an isobaricsum

π ∼= ki=1πi and π′ ∼= k′

j=1π′i

where the πi and π′i are essentially square integrable and hence generic [JS83,§1.2].

We then define, for nontrivial characters ψ : F → C×, that

L(s, π × π′) =k∏i=1

k′∏j=1

L(s, πi × π′j),

γ(s, π × π′, ψ) =

k∏i=1

k′∏j=1

γ(s, πi × π′j , ψ),

ε(s, π × π′, ψ) =

k∏i=1

k′∏j=1

ε(s, πi × π′j , ψ).

Of course, one has to check that this is consistent with our earlier definitions.In other words, we must know that if π and π′ are generic, then this proceduregives the same result as if we used our original definition of the local factors.This can be done.

We then adopt the analogous convention in the global case and obtainRankin-Selberg integrals for isobaric automorphic representations. The ana-lytic properties of these L-functions can be read off from the correspondingones for cuspidal representations (see Theorem 11.7.1).

11.9 Converse theory

Let π be an admissible irreducible representation of GLn(AF ). If π is a cusp-idal automorphic representation then for all cuspidal automorphic represen-tations π′ of GLm(AF ) we have seen in Theorem 11.7.1 that the L-functionL(s, π×π′) admits a functional equation and a meromorphic continuation tothe plane that is bounded in vertical strips. Converse theory asks if automor-phic representations can be characterized by the analytic properties of theirL-functions. It has been used to establish cases of Langlands functoriality[CKPSS04]. In addition, it is philosophically important because it says that


an L-function that satisfies reasonable assumptions has to come from an au-tomorphic representation, solidifying the tight connection between these twoobjects.

Theorem 11.9.1 Let π be an irreducible admissible representation of GLn(AF ).Assume that for all cuspidal automorphic representations π′ of GLm with1 ≤ m ≤ max(n− 2, 1) the L-functions L(s, π × π′) and L(s, π∨ × π′∨) haveanalytic continuations to the plane that are holomorphic and bounded in ver-tical strips of finite width. Assume moreover that for all π′ as above

L(s, π × π′) = ε(s, π × π′)L(1− s, π∨ × π′∨).

Then π is a cuspidal automorphic representation. ut

Here to define ε(s, π × π′) we fix a nontrivial character ψ : F\AF → C× andformally define

ε(s, π × π′) :=∏v

ε(s, πv × π′v, ψv). (11.20)

We refer to [CPS99] for a proof of Theorem 11.9.1 and for variants.

Exercises

11.1. Let B1, B2 ≤ GLn be Borel subgroups defined over a nonarchimedeanlocal field F . For 1 ≤ i ≤ 2 let Ni be the unipotent radical of Bi and letψi be a generic character of Ni(F ). Prove that an irreducible admissiblerepresentation π of GLn(F ) is ψ1-generic if and only if it is ψ2-generic.

11.2. Prove Corollary 11.3.2.

11.3. Let G be a split reductive group over a global field F and assume thatπ is a cuspidal globally ψ-generic representation of V . If π ∼= ⊗′vπv prove thateach πv is ψv-generic.

11.4. Prove Theorem 11.4.2 when n = 2.

11.5. Assume that J(λ) is irreducible unitary unramified generic represen-tation of GLn(F ) where λ ∈ a∗TnC. Let ψ : F → C× be unramified, let

W ∈ W(π, ψ) and W ′ ∈ W(π∨, ψ) be the unique spherical Whittaker func-tionals satisfying W (In) = W ′(In) = 1. Using the identity

det(I2 − q−seλ ⊗ e−λ)Ψ(s,W,W ′,1OnF ) = 1

and Proposition 11.5.1 prove that every eigenvalue of eλ lies in (q−1/2, q1/2).

11.9 Converse theory 227

11.6. If π and π′ are cuspidal automorphic representations of GLn(AF ) suchthat πS ∼= π′S for some finite set of places S of F then π ∼= π′.

11.7. If π and π′ are isobaric automorphic representations of GLn(AF ) suchthat πS ∼= π′S for some finite set of places S of F then π ∼= π′.

11.8. Prove Theorem 11.7.1 in the case m = n−1 using Theorem 11.7.4 andthe local functional equations of Theorem 11.5.4.

Chapter 12

Langlands Functoriality

The local Langlands conjecture isone of those hydra-likeconjectures which seems to growas it gets proved.

M. Harris and R. Taylor

Abstract Langlands functoriality has been a driving force in modern numbertheory, representation theory, and beyond since it was first posed in a famousletter of Langlands to Weil. We survey progress toward the local Langlandsconjecture and state a rough form of the global Langlands functoriality con-jecture in this chapter.

12.1 The Weil group

In this chapter we state the Langlands reciprocity and functoriality conjec-tures. Though the conjectures are mostly open, extremely important progresshas been made.

We begin with reciprocity. Let G be a reductive group which is unramifiedover a nonarchimedean local field F (recall that by our conventions reductivegroups are connected). By Corollary 7.5.2 one has a bijection

(Go Fr)Fr-ss(C)/ ∼

←→

isomorphism classesof irreducible unramifiedrepresentations of G(F )

.

This is the archetypical example of a Langlands reciprocity law or Langlandscorrespondence . Roughly speaking, Langlands reciprocity relates represen-tations of generalized Galois groups with image in LG and representations ofthe points of G in locally compact rings. One refers informally to the Galois

229

230 12 Langlands Functoriality

side of the correspondence and the automorphic side of the correspondence.This terminology is standard, but a bit loose, because the Galois side reallyconsists of representations of generalized Galois groups and the representa-tions on the other side of the correspondence are only automorphic in theglobal setting. It is not difficult to see how the bijection above fits into thisframework but we will postpone a discussion to §12.5.

We now describe more precisely the objects involved on the Galois sideof the correspondence. We start by indicating why working with traditionalGalois groups is not sufficient. Let F be a global or local field, let F sep be aseparable closure of F and let

GalF := Gal(F sep/F )

be the absolute Galois group of F . It is endowed with the profinite topology.A continuous homomorphism GalF → GLn(C) necessarily has finite im-

age. On the other hand, there are many continuous homomorphisms GalF →GLn(Q`) with infinite image.

Example 12.1. If E is an elliptic curve over Q without CM, then the Tatemodule of E gives a representation with image isomorphic to GL2(Z`) foralmost all primes ` [Ser72].

Example 12.2. A more elementary example is given by the cyclotomic char-acter, which is the character

χ` : GalQ −→ Z×`

defined as follows. If σ ∈ GalQ and ζn is a primitive `nth root of unity thenσ(ζn) = ζ

aσ,nn for some aσ,n ∈ (Z/`n)×. We then define

χ`(σ) = lim←−n

aσ,n.

One can check that if p 6= ` then χ`(Frp) = p.

It would be nice to view these examples as complex-valued representationsof a single group. At least in the special case where we are only consideringrepresentations into GL1(C) Weil was able to accomplish this by constructinga modification of the Galois group. We now discuss this modification followingthe exposition of [Tat79].

Definition 12.1. Let F be a local or global field. Then a Weil group for Fis a tuple (WF , φ, ArtE) where WF is a topological group,

φ : WF −→ GalF

is a continuous homomorphism with dense image, and, for each finite exten-sion E/F ,

12.1 The Weil group 231

WE := φ−1(GalE),

and

ArtE : CE −→W abE (12.1)

is an isomorphism, where

CE :=

E× if F is local,

E×\A×E if F is global.

Here the superscript ab denotes its maximal abelian Hausdorff quotient, thatis, the quotient by the closure of the commutator subgroup. Moreover, thesedata are required to satisfy the following assumptions:

(a) For each finite extension E/F the composite

CEArtE−−−−→ W ab

E

induced by φ−−−−−−−−→ GalabE

is the reciprocity map of class field theory.(b) Let w ∈WF and σ = φ(w) ∈ GalF . For each E the following diagram

CEArtE //

σ

W abE

CEσ

ArtEσ// W ab

Eσ

commutes. Here the right arrow is induced by conjugation by w.(c) For E′ ⊆ E the following diagram

CE′ArtE′ //

inclusion

W abE′

transfer

CE

ArtE

// W abE

commutes.(d) Finally, the map

WF → lim←−WE/F

is an isomorphism, where WE/F := WF /WE , the bar denoting closure.

For the proof of the following theorem we refer to [AT09]:

Theorem 12.1.1 A Weil group exists and it is unique up to isomorphism.ut


In many cases the Weil group can be described explicitly:

Example 12.3. Let F be a local field with residue field k and for all finiteextensions E/F let kE be the residue field of E. Put k =

⋃E kE . Let q be

the cardinality of k. The group WF is the dense subgroup of GalF generatedby the σ ∈ GalF such that on k, σ acts as x 7→ xq

n

for some n ∈ Z. Theautomorphism ArtE(a) acts as x 7→ x|a| on k.

Example 12.4. In the global function field case, the Weil group is defined asin the previous example, but one replaces the residue field above with theconstant field.

Example 12.5. If F = C then WF is just C×, φ is the trivial map and ArtFis the identity map.

Example 12.6. If F = R then WF is C× ∪ jC× where j2 = −1 and jcj−1 = cfor c ∈ C×. In this case φ takes C× to 1 and jC× to the nontrivial elementof Gal(C/R).

For number fields one doesn’t have a nice intrinsic description like in theabove examples.

Almost by definition of the Weil group, one obtains the following corre-spondence:

Theorem 12.1.2 (The Langlands correspondence for GL1) There is abijection between isomorphism classes of irreducible automorphic representa-tions of GL1(AF ) and continuous representations χ : WF → GL1(C).

Proof. An irreducible automorphic representation of GL1(AF ) can be iden-tified with a character of F×\A×F , which is isomorphic to W ab

F by definitionof the Weil group. ut

12.2 The Weil-Deligne group and L-parameters

For better or worse the Weil group is still not big enough. The correct en-largement of WF when F is a number field should be the as yet hypotheticalLanglands group. On the other hand, the correct enlargement of WF when Fis a local field is known, so in this section we restrict to this case. We followthe treatment in [GR10].

The Weil-Deligne group of a local field F is

W ′F :=

WF if F is archimedean,

WF × SL2(C) if F is nonarchimedean.(12.2)

We also denote by φ the map

12.2 The Weil-Deligne group and L-parameters 233

φ : W ′F → GalF (12.3)

that is just φ in the archimedean case and it the composite of the naturalquotient map W ′F →WF with φ in the nonarchimedean case.

Let G be a group scheme over C whose neutral component is a reductivegroup. If F is archimedean, then a representation of W ′F into G(C) is simplya homomorphism ρ : WF → G(C).

Assume until stated otherwise that F is nonarchimedean. We now prepareto define a representation of W ′F . Let k be the residue field of F , k a choiceof algebraic closure, and Galk := Gal(k/k). The action of GalF preserves thering of integers and the (unique) prime ideal of the ring of integers of anyfinite extension field E/F contained in F . There is thus an exact sequence

1 −→ IF −→ GalF −→ Galk −→ 1

where IF is the inertia subgroup, which can be defined as the kernel of themap GalF → Galk.

The Weil group WF is a subgroup of GalF and contains IF . We thus havean exact sequence

1 −→ IF −→WF −→ Galk.

The last map is not surjective. Its image is isomorphic to Z, whereas Galk isisomorphic to Z, the profinite completion of Z. We deduce that

WF∼= IF o 〈Fr〉

where Fr is a Frobenius element.

Definition 12.2. Let G be a group scheme over C whose neutral componentis a reductive group. If F is nonarchimedean a representation of W ′F intoG(C) is a homomorphism

ρ : W ′F −→ G(C)

such that ρ is trivial on an open subgroup of IF , ρ(Fr) is semisimple, andρ|SL2(C) is induced by a morphism of algebraic groups SL2 → G.

By a representation of W ′F we mean a representation of W ′F into GLn(C)for some integer n. There are various equivalent alternate definitions of arepresentation of the Weil-Deligne group in the literature, see [GR10]. Wealso note that in the nonarchimedean case what we call a representation moreproperly ought to be called a Frobenius semisimple representation.

Recall that the L-group LG of a reductive group G over F admits a canon-ical map LG→ GalF given by taking the quotient by the neutral component.This map occurs in the following definition:

Definition 12.3. An L-parameter is a representation ρ of W ′F into LG suchthat the composite


W ′Fρ−→ LG −→ GalF

is the map φ of (12.3). Two L-parameters are equivalent if they are conju-

gate by an element of G(C).

We note that given an L-parameter ρ : W ′F → LH and an L-map

LH −→ LG

the composite homomorphism W ′F → LG is an L-parameter.When G is split, the L-group is a direct product:

LG := G(C)×GalF .

In particular in this case an L-parameter ρ : W ′F → LG induces a represen-

tation into G(C)

ρ : W ′F −→ LG −→ G(C)

and every representation of W ′F into G(C) arises in this manner. Thus L-

parameters may be identified with representations into G(C) in this setting,

and the notion of equivalence of L-parameters corresponds to G(C)-conjugacyof the associated representations.

We now define L-functions, ε-factors, and γ-factors. We start by givingourselves a representation ρ : W ′F −→ GL(V ). Recall that the isomorphismclasses of irreducible representations of SL2 are given by the symmetric powerrepresentations Symn of the standard representation. We can therefore de-compose V as a representation of W ′F = WF × SL2(C):

V =

∞⊕n=0

Vn ⊗ Symn.

Here Vn is a representation of WF (that is zero for all but finitely many n).We then define

L(s, ρ) :=

∞∏n=0

det(

1− ρ(Fr)q−(s+n/2)|V IFn)−1

. (12.4)

Assume that E/F is a field extension of either archimedean or nonar-chimedean fields. If ρ : W ′E → GL(V ) is a representation then it is not hardto see that if we define the induction

IndFE(ρ) := IndW ′FW ′E

(ρ) : W ′F −→ GL(V ⊕[W ′F :W ′E ]) (12.5)

in the usual manner then it is a representation of W ′F into GL(V ⊕[W ′F :W ′E ]

)in the sense of Definition 12.2. One checks that when F is nonarchimedean

L(s, ρ) = L(s, IndFE(ρ)

). (12.6)

12.2 The Weil-Deligne group and L-parameters 235

One says that L-functions are invariant under induction. Moreover, if ρi :W ′F → GL(Vi), i ∈ 1, 2, are a pair of representations then the direct sum,defined in the usual manner, is a representation into GL(V1⊕V2) in the senseof Definition 12.2. Again when F is nonarchimedean one checks that

L(s, ρ1 ⊕ ρ2) = L(s, ρ1)L(s, ρ2). (12.7)

One says that L-functions are additive.We now assume until otherwise stated that F is archimedean. Part of

the data of the Weil group is an isomorphism ArtF : F×→W abF . Thus we

can identify quasi-characters of WF and quasi-characters of F×. Define thequasi-character

χs,m : F× −→ C×

z 7−→ |z|s zm

(zz)m/2.

(12.8)

Here if F is complex the bar denotes complex conjugation. If F is real thebar is the identity map and we assume m ∈ 0, 1. In particular χ0,1 is thesign character. We leave the proof of the following lemma as Exercise 12.7:

Lemma 12.2.1 Every quasi-character of F× is of the form χs,m for a uniques ∈ C and m ∈ Z where we assume m ∈ 0, 1 if F is real. ut

Unless otherwise specified, when we discuss the character χs,m we will alwaysassume that s ∈ C, that m ∈ Z and if F is real that m ∈ 0, 1.

Let

ΓF (s) :=

π−

s2Γ(s2

)if F = R

2(2π)−sΓ (s) if F = C.(12.9)

Here Γ (s) is the usual Γ -function. We then define

L(s, χs′,m) = ΓF

(s+ s′ + |m|

[F :R]

). (12.10)

Now WC is just C× by (12.5) and WR contains WC as a normal subgroupof index 2. Using this fact the following proposition is not hard to prove (seeExercise 12.8):

Proposition 12.2.2 Any representation of WF for archimedean F is com-pletely reducible. If F = C then irreducible representations are 1-dimensional,and if F = R then irreducible representations are 1 or 2 dimensional. Theirreducible two-dimensional representations of WR are the induced represen-tations

IndRC(χs,m)

where m 6= 0. For m 6= 0 one has


IndRC(χs,m) ∼= IndRC(χs′,m′) (12.11)

if and only if s = s′ and m = ±m′. ut

Motivated by (12.6) we define

L(s, IndRC(χs′,m)

):= L(s, χs′,m).

By Proposition 12.2.2 a general representation ρ : WF −→ GL(V ) is a di-rect sum of its irreducible subrepresentations. All these subrepresentationsare either 1 or 2 dimensional and for such representations we already havea definition of an L-function. Motivated by (12.7) we then define the L-function of such a representation to be the product of the L-functions of itsirreducible subrepresentations. This completes the definition of L(s, ρ) in thearchimedean case.

Now we allow F to be either archimedean or nonarchimedean and discuss εfactors. If ψ : F → C× is an additive character, d×x is a Haar measure on F×,dx is a Haar measure on F and χ is a quasi-character, i.e., a homomorphism

χ : WF −→ GL1(C),

then we define ε(s, χ, ψ) := ε(s, χ, ψ, dx) to be the nonzero complex numbersuch that∫

F×f(x)χ−1(x)|x|1−sd×xL(1− s, χ−1)

= ε(s, χ, ψ)

∫F×

f(x)χ(x)|x|sd×xL(s, χ)

, (12.12)

where f(y) :=∫Ff(x)ψ(xy)dx is the Fourier transform of f . In certain cases

ε(s, χ, ψ) can be explicitly computed. When F is nonarchimedean and ab-solutely unramified, χ and ψ are unramified, and dx is normalized so thatdx(OF ) = 1 then ε(s, χ, ψ) = 1.

Langlands and Deligne defined in general an ε-factor ε(s, ρ, ψ) [Del73b,Del75]. The ε-factor encodes important arithmetic information about therepresentation ρ, including its Artin conductor [Tat79]. Finally we set

γ(s, ρ, ψ) :=ε(s, ρ, ψ)L(1− s, ρ∨)

L(s, ρ). (12.13)

Now given an L-parameter ρ : W ′F → LG and a representation

r : LG −→ GL(V )

we obtain a representation r ρ : W ′F → GL(V ). We then define

12.3 The archimedean Langlands correspondence 237

L(s, ρ, r) : = L(s, r ρ),

ε(s, ρ, r, ψ) : = ε(s, r ρ, ψ),

γ(s, ρ, r, ψ) : = γ(s, r ρ, ψ).

(12.14)

In the case where G = GLn and the map r : LGLn → GLn(C) is just theprojection to the neutral component r is omitted from notation, i.e.

L(s, ρ) := L(s, ρ, r).

12.3 The archimedean Langlands correspondence

For this section F is an archimedean local field. Over archimedean local fieldsthe Langlands correspondence between L-packets of representations of G(F )and L-parameters into LG was proven by Langlands [Lan89]. We should notethat there is a substantial literature refining his result in various importantways, but we will not describe it.

Instead we will focus on the G = GLn case and describe the Langlandscorrespondence explicitly. Let

K := Kn := g ∈ GLn(F ) : gg∗ = In

where g∗ = gt if F is real and g∗ = gt if F is complex. This is the standardmaximal compact subgroup of GLn(F ). Let

g := gn := ResF/Rgln.

We recall from §10.5 that any irreducible admissible (g,K)-module π canbe written as an isobaric sum

π ∼= ki=1πi

where the πi are irreducible and essentially square integrable. It turns outthat the possibilities for essentially square integrable π are very limited. IfG is a semisimple group over R we say that G is equal rank if there is amaximal torus T ≤ G that is anisotropic. In other words, T ≤ G is maximaland T (R) is compact.

Harish-Chandra’s fundamental theorem on the existence of discrete seriesrepresentations follows:

Theorem 12.3.1 (Harish-Chandra) Let G be a reductive group over R.There are essentially square-integrable irreducible representations of G(R) ifand only if Gder is equal rank. ut

We refer to [Wal88, §7.7.1] for the proof.


By the theorem, if F is complex then GLn(F ) admits irreducible essen-tially square-integrable representations if and only if n = 1 (in fact one-dimensional representations are always essentially square-integrable). If F isreal then GLn(F ) admits irreducible essentially square-integrable represen-tations if and only if 1 ≤ n ≤ 2 (see Exercise 12.11).

We now explicitly define the Langlands correspondence. The admissiblerepresentation corresponding to a 1 dimensional representation WF → C× isthe quasi-character defined by the composite

F×ArtF−→ W ab

F −→ C×.

If WF → GLn(C) is irreducible and n > 1 then n = 2 by Proposition 12.2.2.In this case the representation is of the form IndWR

WC(χs,m) by the same propo-

sition. The associated irreducible (g,K)-module, by definition, is | · |sπm,where πm is the discrete series representation (or limit of discrete series rep-resentation) of §4.7.

In general, if ρ : WF → GLn(C) is a representation, it is isomorphicto ⊕i=1ρi where the ρi are all irreducible of dimension 1 or 2, and we letthe corresponding irreducible (g,K)-module be π(ρ) := ki=1π(ρi), whereπ(ρi) is the irreducible admissible (g,K)-module corresponding to ρi. UsingProposition 12.2.2 and the discussion of isobaric representations given in §10.5we deduce the following very special case of the archimedean local Langlandscorrespondence:

Theorem 12.3.2 The map ρ 7→ π(ρ) described above is a bijection betweenisomorphism classes of irreducible admissible representations of GLn(F ) andequivalence classes of representations WF → GLn(C). ut

Now suppose m,n ∈ Z≥1 and we are given an irreducible admissible(gn,Kn)-module π and an irreducible admissible (gm,Km)-module π′ withassociated representations ρ(π) : WF → GLn(C) and ρ(π′) : WF → GLm(C).For any nontrivial character ψ : F → C× we let

L(s, π × π′) : = L(s, ρ(π)⊗ ρ(π′)),

ε(s, π × π′, ψ) : = ε(s, ρ(π)⊗ ρ(π′), ψ),

γ(s, π × π′, ψ) : = γ(s, ρ(π)⊗ ρ(π′), ψ).

This is the definition of the archimedean Rankin-Selberg L-factors. Giventhis indirect definition, it is remarkable that Jacquet, Piatetski-Shapiro andShalika were able to prove that it satisfies the same analytic properties (e.g.functional equation) that the nonarchimedean local L-function does (see The-orem 11.5.4). The fact that they were able to prove this indicates that thearchimedean Langlands correspondence given above has meaning. Withoutproving that some other property is preserved by the correspondence it israther ad-hoc. Another indication of the utility of the correspondence is thatit can be used to compute characters of representations. We refer to the

12.4 Local Langlands for GLn 239

foundational work of Shelstad and Langlands [Lan89, She82] for more onthis matter.

12.4 Local Langlands for GLn

Let F be a local field. Let Π(GLn) be the set of isomorphism classes ofirreducible admissible representations of GLn(F ) and let Φ(GLn) be the setof GLn(C)-conjugacy classes of L-parameters ρ : W ′F → LGLn.

By a local Langlands correspondence for GLn over F one means a collec-tion, for every n ≥ 1, of bijections

rec := recF : Π(GLn)−→Φ(GLn) (12.15)

such that

(a) If π ∈ Π(GL1) then rec(π) = π Art−1F .

(b) If π1 ∈ Π(GLn1) and π2 ∈ Π(GLn2) then

L(s, π1 × π2) = L(s, rec(π1)⊗ rec(π2))

andε(s, π1 × π2, ψ) = ε(s, rec(π1)⊗ rec(π2), ψ).

(c) If π ∈ Π(GLn(F )) and χ ∈ Π(GL1(F )) then

rec(π ⊗ (χ det)) = rec(π)⊗ rec(χ).

(d) If π ∈ Π(GLn) and π has central quasi-character χ then

det rec(π) = rec(χ).

(e) If π ∈ Π(GLn) then rec(π∨) = rec(π)∨ (where ∨ denotes the contragre-dient).

Theorem 12.4.1 (Local Langlands correspondence) There is a localLanglands correspondence for GLn for every local field F . ut

When n = 1 this is essentially local class field theory. When F has positivecharacteristic (and hence is the completion of a global function field) thetheorem is due to Laumon, Rapoport and Stuhler [LRS93]. In the mixedcharacteristic case the proved by Harris-Taylor first [HT01], then a simplifiedproof was found by Henniart [Hen00].

There are many additional desiderata that one might ask of the local Lang-lands correspondence. We mention a few. First, the correspondence sendsisobaric sums to direct sums. In other words,

rec(π1 π2) = rec(π1)⊕ rec(π2). (12.16)


Second, supercuspidal representations of GLn(F ) correspond bijectively toirreducible representations of W ′F that are trivial on the SL2(C)-factor. Fi-nally, a representation π is essentially square integrable if and only if rec(π)is irreducible. Indeed, in the archimedean case this is clear from our explicitdescription of the correspondence in §12.3. In the nonarchimedean case thetheorem of Bernstein, Theorem 8.4.3, asserts that a representation is essen-tially square integrable if and only if it is of the form J(σa, λa) with notationas in Theorem 8.4.3 (here σ is supercuspidal). Such representations corre-spond to representations of W ′F = WF × SL2(C) of the form rec(σ)⊗ Syma.All of these requirements are more or less built into the classification. In thenonarchimedean setting, the first paper to discuss how the whole correspon-dence could be reduced to the supercuspidal case was [Zel80].

12.5 The local Langlands conjecture

The (conjectural) Langlands correspondence for arbitrary reductive groupsG over local fields F is far more complicated. The general shape involves arelationship between

Π(G) :=

isomorphism classes of irreducibleadmissible representations of G(F )

(12.17)

and

Ψ(G) := G(C)-conjugacy classes of L-parameters into LG (12.18)

We will state the conjecture for quasi-split groups G in this section. Ourexposition follows that of [Kal16].

The conjectural correspondence is simplest to state in the tempered caseand we will restrict to this setting. We have already defined the notion of atempered representation in Definition 4.6. An L-parameter

ρ : W ′F −→ LG

is tempered if its image under the projection to G(C) is bounded. Note

that the projection LG→ G(C) is just a set-theoretic projection; it is only ahomomorphism if G is split. It is clear the notion of being tempered dependsonly on the equivalence class. We denote by Φ(G) the set of all equivalenceclasses of L-parameters and by

Φt(G) (12.19)

the set of equivalence classes of tempered L-parameters. We denote by

12.5 The local Langlands conjecture 241

Πt(G)

the set of isomorphism classes of tempered irreducible admissible representa-tions of G(F ). The basic form of the local Langlands conjecture for quasi-splitgroups is the following:

Conjecture 12.5.1 (Local Langlands correspondence) Assume G is quasi-split.

(a) There is a surjective map

LL : Πt(G) −→ Φt(G)

with finite fibers Π(ρ) := LL−1(ρ).(b) If one element of Π(ρ) is essentially square-integrable then they all are,

and this is the case if and only if the image of ρ is not contained in aproper parabolic subgroup of LG.

(c) If ρ ∈ Φt(G) is the image of ρM ∈ Φt(M) for a proper Levi subgroup M ≤G, then Π(ρ) consists of the irreducible constituents of the representationsthat are parabolically induced from elements of Π(ρM ).

For the notion of a parabolic subgroup of LG, we refer to §7.4. In the lastassertion we are implicitly using the fact that there is a natural map LM →LG for every Levi subgroup M ≤ G. We will discuss the compatibility ofthe formulation above with the known case of G = GLn at the end of thissection.

Let us describe the structure of an L-packet in an important special case.

Definition 12.4. An L-parameter is unramified if F is nonarchimedean, Gis unramified and the parameter is trivial on IF × SL2(C).

Let ρ : W ′F → LG be an unramified parameter. Given a hyperspecialsubgroup K ≤ G(F ), the coset

ρ(Fr) ∈ (Go Fr)Fr−ss(C)

gives rise to an unramified representation via the Satake correspondence asin Corollary 7.5.2. Specifically, the coset ρ(Fr) gives rise to λ ∈ X∗(T )F ⊗ Cvia Lemma 7.6.1. The corresponding K-spherical representation is the uniqueK-spherical subquotient J(λ). Now if G is semisimple then there is a uniqueGad(F )-conjugacy class of hyperspecial subgroups of G(F ) [Tit79, §2.5 and3.8]. However (even for semisimple G) there is in general more than oneG(F )-conjugacy class of hyperspecial subgroups of G(F ). Thus the definitionof J(λ) depends on the choice of K; to indicate this we temporarily write itas J(λ,K). By definition

Π(ρ) :=∐

J(λ,K)


where the disjoint union is over the G(F )-conjugacy classes of hyperspecialsubgroups of G(F ) with K a representative of the class. This is an unrami-fied L-packet.

Given a tempered L-parameter ρ, the isomorphism classes of representa-tions in Π(ρ) are by definition a local L-packet. The conjecture, as stated,leaves open the question of the structure of the packet Π(ρ). To describe alittle of what is expected, let

Sρ := CG(C)(ρ(W ′F )) (12.20)

be the centralizer of ρ(W ′F ) in G and let

Sρ := Sρ/ZGalFG(C)

. (12.21)

Here we are using the fact that G comes equipped with an action of GalF ,the same action used to define LG.

Lemma 12.5.2 For G = GLn the group Sρ is connected.

Proof. It suffices to show that Sρ is connected. For irreducible ρ Schur’slemma implies that Sρ = GmIn (the scalar matrices in GLn). In general, thefact that ρ is Frobenius semisimple implies that it is completely reducible(see Exercise 12.4) so Sρ is isomorphic to a product of Gm’s, one for eachirreducible subrepresentation of ρ. ut

Conjecture 12.5.3 For a tempered L-packet ρ there is an injection

ι : Π(ρ) −→ Irr(π0(Sρ))

that is bijective if F is nonarchimedean.

Here Irr(π0(Sρ)) is the set of isomorphism classes of irreducible representa-tions of the finite group π0(Sρ), the group of components of Sρ.

The theory of generic representations gives a means of isolating a basepointin Π(ρ). Consider the set of pairs (B,ψ) where B ≤ G is a Borel subgroupand ψ is a generic character of the F -points of its unipotent radical. There is anatural action of G(F ) on this set by conjugation, and a Whittaker datumis a G(F )-orbit. Let w be a Whittaker datum and say that an admissibleirreducible representation π is w-generic if it is ψ-generic for some (henceall) pairs (B,ψ) ∈ w.

The following is a version of a conjecture of Shahidi [Sha90, §9]:

Conjecture 12.5.4 (Shahidi) Every tempered L-packet Π(ρ) contains aunique w-generic member.

Assume for this paragraph that conjectures 12.5.1, 12.5.3 and 12.5.4 arevalid. Given a Whittaker datum w we assume that the injection ι : Π(ρ)→

12.5 The local Langlands conjecture 243

Irr(π0(Sρ)) can be normalized so that the generic element of the packet (whichis unique by Conjecture 12.5.4) is sent to the trivial representation. We callthis normalized injection ιw. Thus we have a pairing

〈 , 〉 : Π(ρ)× π0

(Sρ)−→ C

(π, s) 7−→ tr(ιw(π)(s)).

It will play a crucial role in Conjecture 12.6.3 below.Important progress towards the conjectures 12.5.1, 12.5.3, and 12.5.4 has

been made. We very briefly indicate some results and refer to [Kal16] for amore detailed discussion. The following is Arthur’s main theorem on the localLanglands correspondence [Art13, Theorem 1.5.1].

Theorem 12.5.5 Assume the stabilization of the twisted trace formula andthat F is of characteristic zero. Parts (a) and (b) of Conjecture 12.5.1 andConjecture 12.5.3 are valid for the split groups Sp2n and SO2n+1. A slightlyweaker version of these conjectures is valid for the quasi-split (or split) specialorthogonal groups of even-dimensional quadratic spaces. ut

Let G = SO∗2n, which we take to be either the split form of SO2n or aquasi-split inner form, and let O∗2n be the corresponding orthogonal group.In the theorem the slighly weaker version of the correspondence is the corre-spondence obtained by replacing equivalence classes in Φt(G) and Πt(G) by

equivalence classes up to conjugation by O∗2n(C) and O∗2n(F ), respectively.We point out that technically speaking we have only defined L-groups forreductive groups, not disconnected algebraic groups with reductive neutralcomponent like O∗2n. The definition must be extended to make sense of the

expression O∗2n(C).We will not say precisely what is meant by the stabilization of the trace

formula. Moeglin and Waldspurger [MW16a, MW16b] have proven this sta-bilization under the assumption of the twisted weighted fundamental lemma.There does not seem to be any theoretical obstacle to obtaining this lastresult, but it has not been completed. Thus the work of Arthur is still con-ditional at this point.

There is an analogous statement for quasi-split unitary groups over localfields [Mok15, Theorem 2.5.1]:

Theorem 12.5.6 Assume the stabilization of the twisted trace formula andthat F is of characteristic zero. Let E/F be a quadratic extension and let G bethe corresponding quasi-split unitary group. Parts (a) and (b) of Conjecture12.5.1 and Conjecture 12.5.3 are valid for G. ut

In this section we have limited our focus to the tempered case. In prac-tice one needs to refine this in two qualitatively different manners. First, oneoften has to enlarge the set of equivalence classes of tempered L-parametersto a set of almost tempered L-parameters Φat(G) and define a set of al-most tempered representations Πat(G) such that there is a local Langlands


correspondence relating the two sets. This enlargement is necessary becausewe do not know that the local components of automorphic representationsthat are expected to be tempered are in fact tempered (see §12.6 below forfurther discussion). For general representations stating a version of the localLanglands correspondence requires the introduction of Arthur’s A-packets(see [Art89, Art90]).

We close this section by discussing the compatibility of conjectures 12.5.1,12.5.3, and 12.5.4 and our discussion of the unramified correspondencewith the local Langlands correspondence of Theorem 12.4.1. For GLn orResE/FGLn where E/F is a field extension take Φ(G) to be the set of allequivalence classes of L-parameters. Then for each ρ we can let Π(ρ) bethe unique irreducible admissible representation of G(F ) corresponding toρ under the local Langlands correspondence. To check that under the localLanglands correspondence of Theorem 12.4.1 irreducible tempered represen-tations are sent to tempered parameters is Exercise 12.9. The fact that theLanglands correspondence in this case is compatible with the unramified cor-respondence discussed above is a consequence of the fact that all hyperspecialsubgroups of GLn(F ) are GLn(F )-conjugate to GLn(OF ) (see [Ser06, II.IVAppendix 1]). Condition (b) in Conjecture 12.5.1 is valid by the discussion atthe end of §12.4. To check consistency with condition (c) of Conjecture 12.5.1

let M be the Levi subgroup of GLn, so M ∼=∏ki=1 GLni is a product of gen-

eral linear groups. We note that if ρM ∈ Φt(M) =∏ki=1 Φt(GLni) then the

parabolic induction of the unique representation π =∏ki=1 πni ∈ Π(ρM ) to

GLn(F ) is irreducible by Exercise 12.10 and equal to ki=1πni . Given that thelocal Langlands correspondence sends isobaric sums to direct sums we deducecompatibility of the local Langlands correspondence with requirement (c). Tocheck compatibility with Conjecture 12.5.3 we use Lemma 12.5.2. Finally, tocheck compatibility with Conjecture 12.5.4 we recall that every irreducibletempered representation of GLn(F ) is generic [JS83, §1.2].

12.6 Global Langlands functoriality

Let F be a global field and let G be a connected reductive group over F . Asbefore, let GalF := Gal(F sep/F ) and for all places v of F let

GalFv := Gal(F sepv /Fv).

Then for each place v of F there is a GalF -conjugacy class of injective ho-momorphisms GalFv −→ GalF . Choosing such a homomorphism one obtainsan injection

LGFv := G(C) o GalFv −→ G(C) o GalF =: LG.

12.6 Global Langlands functoriality 245

Motivated by this define

Φ(GAF ) : =

(ρv) ∈

∏v

Φ(GFv ) : ρv is unramified for a.e. v

. (12.22)

Let Φt(GAF ) ⊂ Φ(GAF ) be the subset of ρ = (ρv) such that ρv ∈ Φt(GFv )for all v. We call Φ(GAF ) the set of adelic L-parameters into LG and callΦt(GAF ) the set of tempered adelic L-parameters into LG. We will oftendrop the adjective “adelic.”

Assume momentarily that G is quasi-split and that we know the localLanglands conjecture for GFv for all places v. Thus, for example, we couldtake G to be GLn. Then given a tempered L-parameter ρ ∈ Φt(GAF ) we canform the global L-packet

Π(ρ) := π = ⊗′vπv : πv ∈ Π(ρv) for all v . (12.23)

This is a set of admissible representations of G(AF ). The set of all L-packetsof G(AF ) up to equivalence is denoted Π(G). Here two L-packets Π(ρ) andΠ(ρ′) are said to be equivalent if there is a bijection

j : Π(ρ) −→ Π(ρ′)

such that j(π) is isomorphic to π as a representation of G(AF ). In generalL-packets are infinite. We say a global L-packet is automorphic if one of itselements is an automorphic representation. The set of automorphic L-packetsis denoted Πaut(G). There is a natural commutative diagram

Πaut(G) −−−−→ Πw,aut(G)y yΠ(G) −−−−→ Πw(G)

where the subscript w denotes the weak L-packets of (7.22). The verticalarrows above are injective, but the horizontal arrows are not in general.

We now return to the general situation; we do not assume anything aboutG other than it is reductive over a global field F . Let H be another reductivegroup over F and let

r : LH −→ LG

be an L-map. We then obtain a commutative diagram

LHFvr−−−−→ LGFvy y

LHr−−−−→ LG

.


Given any ρ ∈ Φ(HAF ) we can define

r ρ = (r ρv) ∈ Φ(GAF ). (12.24)

Thus given an L-map LH → LG we obtain maps

Φt(HAF ) −−−−→ Φt(GAF )y yΦ(HAF ) −−−−→ Φ(GAF ).

Assume for the remainder of this section that G and H are quasi-splitand that conjectures 12.5.1, 12.5.3 and 12.5.4 are known for GFv and HFv

for all places v. This is a reasonable working assumption, as the local Lang-lands correspondence is known in many cases. The remainder of this sectionis devoted to using the local Langlands correspondence to state the globalLanglands functoriality conjecture. We begin with the simplest case. Assumethat G = ResE/FGLn for some field extension E/F of finite degree. In thiscase local L-packets are singletons (see §12.5). Thus global L-packets aresingletons in this setting and the following conjecture is natural:

Conjecture 12.6.1 Suppose that G = ResE/FGLn. Let π be an everywheretempered cuspidal automorphic representation of H(AF ) with L-parameterρ ∈ Φt(HAF ). The unique admissible representation in Π(r ρ) is automor-phic.

We invite the reader to compare this with the weaker Conjecture 7.7.1. Atleast in the cases where the local Langlands conjecture is known for H Con-jecture 12.6.1 does not rely on a chain of conjectures for its statement.

For general (even non-quasi-split) H and G the statement of the weakerConjecture 7.7.1 doesn’t even require any unknown conjecture in its state-ment. However, Conjecture 7.7.1 leaves much to be desired. There could bemany L-packets in a weak L-packet, and it does not explain which elementsof an L-packet are automorphic.

Giving a more specific conjecture seems to require the introduction of theconjectural Langlands group LF . This first appeared in [Lan79a] and isalso discussed in [Art02], where a conjectural construction of the group isgiven. The theory of Tannakian categories also indicates that such a groupought to exist if we assume Langlands functoriality. For a discussion of thispoint of view and the obstacles one runs into if one pursues it we refer to[Ram94].

The Langlands group LF of a global field F is supposed to be an extensionof the Weil group WF by a locally compact group. For reductive F -groupsG one should be able to define global L-parameters LF → LG as in thelocal case; they ought to be continuous homomorphisms commuting withthe projection to GalF along the maps LF → WF → GalF . Two global L-parameters are equivalent if they are G(C)-conjugate. For every place v one

12.6 Global Langlands functoriality 247

should have homomorphisms

W ′Fv −→ LF ,

unique up to conjugacy. We let

Φt(G)

be the set of isomorphism classes of L-parameters LF → LG such that theimage of the parameter under the set-theoretic projection map LG → G(C)is bounded. Similarly we let Πt,aut(G) be the set of isomorphism classes ofautomorphic representations of G(AF ) that are tempered at every place.

One is supposed to have a global analogue of the local parametrization ofConjecture 12.5.1:

Conjecture 12.6.2 (Global Langlands reciprocity) There is a surjec-tive map

Πt,aut(G) −→ Φt(G) (12.25)

such that the diagram

Πt,aut(G) −−−−→ Φt(G)y y⊗vΠt(GFv ) −−−−→ Φt(GAF )

commutes.

In practice one hypothesizes the existence of LF , derives consequences from it,rephrases those consequences in terms that do not depend on the existence ofLF , and then proves those consequences. For example, given an L-parameterρ : LF → LG one should obtain an adelic L-parameter Π(ρ). As in the localsetting, one defines

Sρ := CG(ρ(LF ))

the centralizer of ρ(LF ) in G and sets

Sρ := Sρ/ZGalFG

.

Then global L-packets ought to be controlled by the component group of Sρ)as in the local setting. Remarkably, in [Art13] Arthur was able to use thisyoga to give an unconditional definition of a group π0(Sρ) which, if LF exists,ought to be the component group of Sρ. It then played a crucial role in thestatement of his results.

Assume that π is a given tempered automorphic representation of H(AF )(which is to say that πv is tempered for all v). Then there is a ρ ∈ Φt(HAF )such that π ∈ Π(ρ). It is natural to ask if there is an automorphic represen-


tation in Π(r ρ), where r ρ is defined as in (12.24). Langlands functorialityposes a conjectural answer to this question.

Let B ≤ G be a Borel subgroup with unipotent radical N and let ψ :N(AF ) → C× be a character trivial on N(F ) such that ψv is generic for allv. This provides us with a Whittaker datum wv for all v and hence pairings

〈 , 〉v : Π(r ρv)× π0(Sρv ) −→ C.

Let π′ be the unique element of Π(r ρ) such that π′v is generic for all v.Assume that π′ occurs in L2

disc([G]), the largest closed subspace on which therepresentation of G(AF ) decomposes discretely. We assume moreover that forevery v there is a homomorphism π0(Srρ) −→ π0(Srρv ). Finally a globalL-parameter

ρ : LF −→ LG

is said to be discrete generic if it is semisimple, its projection to G(C) isbounded and its image is not contained in a proper parabolic subgroup ofLG.

Conjecture 12.6.3 Under the assumptions above, the multiplicity of a π ∈Π(r ρ) in L2

disc([G]) is∑ρ

1

|π0(Srρ)|

∑s∈π0(Srρ)

∏v

〈πv, s〉,

where ρ runs over the equivalence classes of discrete generic global parameterssatisfying πv ∈ Π(r ρv) for all places v.

A few comments are in order. Theoretically, one should be able to re-duce whatever one wants to know about automorphic representations to thetempered case. The A-packets of Arthur and his conjectures regarding themwere introduced to make this precise [Art89, Art90] (see also [Clo07] and[Sha11] where the relationship between A-packets and Conjecture 10.6.4, theRamanujan Conjecture, is discussed). However, even when we expect rep-resentations to be tempered, in most cases we can only prove bounds thatestablish that they are close to being tempered. Moreover, the only meanscurrently known to prove that they are indeed tempered seems to be usingLanglands functoriality (compare [Lan70, Sar05]). To overcome this prob-lem, as in the local case, instead of just dealing with tempered L-parametersand representations one enlarges the set of parameters involved in the localLanglands correspondence to a set of almost tempered parameters and rep-resentations. This is supposed to be a set that is restrictive enough that thelocal Langlands correspondence is still correct, but general enough to includeall representations that can occur as local components of the most temperedpart of the discrete series of L2([G]). For examples of such an enlargementwe refer to [Art13].

12.7 Langlands L-functions 249

12.7 Langlands L-functions

Assume that we are given a quasi-split reductive G over a local field F suchthat the local Langlands correspondence is known for G(F ). Given a repre-sentation

r : LG −→ GL(V )

an irreducible tempered representation π of G(F ), and a nontrivial characterψ : F → C× one can then use (12.14) to define the local Langlands L-function, ε-factor, and γ-factor as follows:

L(s, π, r) : = L(s,LL(π), r)

ε(s, π, r, ψ) : = ε(s,LL(π), r, ψ)

γ(s, π, r, ψ) : = γ(s,LL(π), r, ψ).

(12.26)

By definition admissible representations in the same L-packet necessarilyhave the same L-functions, ε-factors, and γ-factors. This is the reason forthe terminology “L-packet.” Sometimes one says that elements in the sameL-packet are L-indistinguishable. When G = GLn ×GLm and

LG −→ GLnm(C)

is the tensor product Rankin-Selberg theory as discussed in Chapter 11 al-ready furnishes us with a definition of these factors. Part of the content ofthe local Langlands correspondence for GLn of Theorem 12.4.1 is that thetwo definitions coincide.

Now assume that F is a global field, ψ : F\AF → C× is a nontrivial char-acter, and G is a quasi-split reductive F -group such that the local Langlandscorrespondence for G(Fv) is known for every place v. Assume moreover thatwe are given a representation

r : LG −→ GL(V ).

The global Langlands L-function and global ε-factor are then

L(s, π, r) : =∏v

L(s, πv, r),

ε(s, π, r) : =∏v

ε(s, πv, r, ψv).

As the notation indicates, the global ε-factor is (conjecturally) independentof the choice of ψv. The basic conjectures regarding these L-functions are asfollows:

Conjecture 12.7.1 The L-function L(s, π, r) is meromorphic as a func-tion of s, is bounded in vertical strips, and satisfies the functional equationL(s, π, r) = ε(s, π, r)L(1− s, π∨, r∨).


This is in fact a consequence of Conjecture 12.6.1 and Theorem 11.7.1.In addition to the Rankin-Selberg theory discussed in the previous chapter,

many important cases of this conjecture have been proven using Langlands-Shahidi theory [Sha10, CKM04]. However in most cases the conjecture isopen. The most that is known in general is that L(s, π, r) converges for Re(s)sufficiently large (see [Lan71] and [Sha10, §2.5]).

12.8 Algebraic representations

Let F be a number field. The hypothetical Langlands group LF is expectedto admit a morphism

LF −→ GalF .

In particular, suitable morphisms

GalF −→ LG

commuting with the projection to GalF should give rise to L-packets of au-tomorphic representations of G(F ). It is natural to try and describe classesof L-packets that are of this form.

In [Clo90b] isolated a class of automorphic representations on G = GLnthat he called “algebraic” and then conjectured that they are associated toparticular types of Galois representations (not just representations of LF ).Buzzard and Gee [BG14] later extended Clozel’s conjectures to the case of ar-bitrary reductive G. We recall Buzzard and Gee’s conjectures in this section,following the exposition in [BG14] closely.

Until otherwise specified we work locally at an archimedean place v of Fwhich we omit from notation, writing F := Fv. Let T ≤ G be a maximaltorus, and let BC > TC be a Borel subgroup of GC. The local Langlandscorrespondence is known in the archimedean case [Lan89]. It provides uswith a set-theoretic map

LL : Π(G) −→ Φ(G)

π 7−→ LL(π)

with notation as in (12.17) and (12.18).

Fix a maximal torus T in G with an identification X∗(T ) = X∗(T ). We

can and do assume that LL(π)(C×) ⊆ T (C). Let

HomR(C,C) = σ, τ.

Then one has that ρπ(z) = σ(z)λστ(z)λτ for z ∈ F× and for λσ, λτ ∈ X∗(T )⊗C such that λσ − λτ ∈ X∗(T ). This follows from the fact that all quasi-characters of C× are of the form (12.8) (see Lemma 12.2.1).

12.8 Algebraic representations 251

The group

(X∗(T )⊗ C)HomR(C,C) (12.27)

has a diagonal action of the Weyl group W (G,T )(C). The action of GalF onX∗(T )⊗ C is W (G,T )(C)-semilinear, i.e. for

(ξ, w, x) ∈ GalF ×W (G,T )(C)×X∗(T )⊗ C

one has ξ(wx) = ξ(w)ξ(x). The group (12.27) also has an action of GalF de-fined using both the action of GalF on X∗(T ) and the action on HomR(F ,C).Explicitly, if we think of an element of (12.27) as a morphism

φ : HomR(C,C) −→ X∗(T )⊗ C

then ξ.φ := (t 7→ ξφ(tξ)). These considerations impliy that (λσ, λτ ) give us awell-defined element of(

(X∗(T )⊗ C)HomR(F,C)/W (G,T )(C))GalF

The W (G,T )(C)-orbit of (λσ, λτ ) in (X∗(T ) ⊗Z C)HomR(C,C) is an invariantattached to ρπ.

Definition 12.5. An L-parameter ρ : WF → LG is L-algebraic if λσ ∈X∗(T ). An irreducible admissible representation π of G(F ) is L-algebraic ifLL(π) is L-algebraic.

Recall that we have fixed T ≤ B ≤ GC and hence we have the notion ofthe positive roots. As usual, denote by ρB the half sum of positive roots.

Definition 12.6. An L-parameter ρ : WF → LG is C-algebraic if λσ−ρB ∈X∗(T ). An irreducible admissible representation π of G(F ) is C-algebraic ifLL(π) is C-algebraic.

The assertion λσ ∈ X∗(T ) is independent of the choice of B and of the iso-morphism F ∼= C. Likewise the assertion λσ−ρB ∈ X∗(T ) is also independentof such choices. If ρB ∈ X∗(T ), then two notions coincide.

The expectation (which will be made precise below) is that L-algebraic au-tomorphic representations should correspond to representations of GalF withimage in G(Qp) (what we mean by this last symbol will be explained below).By Corollary 15.5.2 cohomological automorphic representations in the senseof Definition 15.6 are C-algebraic (hence the terminology C-algebraic). Theknown techniques for proving that L-algebraic representations correspond torepresentations of GalF invariably involve cohomlogical representations, soL-algebraic and C-algebraic representations ought to be studied in tandem.The two definitions differ by half the sum of the positive roots. For G = GLn,the notion of C-algebraic coincides in the isobaric case with Clozel’s notationof algebraic used in [Clo90b, Clo16].


It is clear that the notion of L-algebraicity (resp C-algebraicity) for an

L-parameter ρ depends only on the restriction of ρ to F×

. Moreover thenotion of L-algebraicity (resp. C-algebraicity) of an irreducible admissiblerepresentation π of G(F ) depends only on the infinitesimal character of thisrepresentation. More precisely:

Lemma 12.8.1 An irreducible admissible representation π of G(R) is L-algebraic (resp. C-algebraic) if and only if the infinitisimal character χλ of πsatisfies λ ∈ X∗(T ) ⊂ (tC)∨ (resp. λ− δB ∈ X∗(T )).

Proof. In the notation of §4.6, the infinitisimal character of π is χλ withλ ∈ (tC)∨, uniquely defined up to the action of W . Now

(tC)∨ = X∗(T )C = X∗(T )C

where the identifications are W -equivariant, so we may regard λ as an elementof X∗(T ). The W -orbit of λ contains λσ. For a sketch of the proof of this lastassertion see [Vog93, Proposition 7.4]. ut

For the global definitions of algebraic representations, we let G be a con-nected reductive groups defined over a number field F . Fix an algebraic clo-sure F of F and form the L group LG = GoGalF . For each place of v of F ,fix an algebraic closure F v of Fv and an embedding F → F v. Let π = ⊗′vπvbe an automorphic representations of G(AF ).

Definition 12.7. We say that π is L-algebraic (resp. C-algebraic) if πv isL-algebraic (resp. C-algebraic) for all infinite places of F .

In the remainder of this section we state an important conjecture on ex-istence of Galois representations attached to L-algebraic automorphic repre-sentations and a related result (see Theorem 12.8.3 below).

To state the conjecture, we let p be a fixed prime and fix an injectionQ → C. We fix once and for all a choice of algebraic closure Qp of Qp and

isomorphism ι : C −→ Qp. Therefore the inclusion Q → C together with ι

induces an embedding Q −→ Qp.We note that in the construction of G and the action of GalF on it we

could replace the base field C by any algebraically closed field. Thus we couldconsider G as a group over Q = F , for example, and ι induces an isomorphism

ι : G(C) −→ G(Qp)

We can also form

G(Qp) o GalF (12.28)

and we have an isomorphism

ι : LG −→ G(Qp) o GalF .

12.8 Algebraic representations 253

This construction is used in the following conjecture due to Buzzard and Gee.It is a generalization of a conjecture of Clozel in the case G = GLn.

Conjecture 12.8.2 If π is L-algebraic, then there is a finite set S of placesof F containing all infinite places, all places dividing p, and all ramifiedplaces, and a continuous Galois representation

ρπ = ρπ,ι : GalF −→ G(Qp) o GalF

such that the composite of ρπ and the projection to GalF is the identity andif v 6∈ S then the Frobenius semisimplification of ρπ|WFv

is G(Qp)-conjugateto ι(LL(πv)).

There is additional important information about π at the infinite places andthe places dividing p predicted in the full version of Conjecture 12.8.2 givenin [BG14] but we omit a discussion.

Though in general Conjecture 12.8.2 is open there are important specialcases that are known.

Theorem 12.8.3 Suppose that F is CM or totally real and that π has thesame infinitisimal character as an irreducible representation of ResF/QGLnand in particular is L-algebraic. Then Conjecture 12.8.2 is true. ut

There are in fact two different proofs of Theorem 12.8.3 available, one due toHarris, Lan, Taylor and Thorn [HLTT16] and another due to Scholze [Sch15].

Exercises

12.1. With notation as in Example 12.2, prove that if p 6= ` then χ`(Frp) = p.

12.2. Given a map G → H of reductive groups with normal image, provethat there is an induced map H → G.

12.3. For reductive groups H and G, give an example of a morphism H → Gthat is not induced by a map H → G as in the previous problem.

12.4. Prove that a representation ρ : W ′F −→ GLn(C) (or more properly aFrobenius semisimple representation) is completely reducible.

12.5. Prove (12.6).

12.6. Prove (12.7).


12.8. Prove Proposition 12.2.2.


12.9. Prove that tempered representations are sent to tempered parametersunder the local Langlands correspondence for GLn.

12.10. For 1 ≤ i ≤ k let πi be an irreducible tempered representation ofGLni(F ). Let π = Ind(π1 ⊗ · · ·πk, 0). Prove that π is irreducible.

12.11. Let G be a semisimple group over C. Show that ResC/RG is equalrank if and only if G is trivial. Show that SLn (as a group over R) is equalrank if and only if n ≤ 2 (here we interpret SL1 as the trivial group.

12.12. Let G be a reductive group over C. State and prove an analogue ofLemma 12.8.1 for an irreducible admissible representation π of G(C).

Chapter 13

Known Cases of Global LanglandsFunctoriality

...I had not recognized in 1966,when I discovered after manymonths of unsuccessful search apromising definition ofautomorphic L-function, what afortunate, although, and thisneeds to be stressed, unforeseenby me, or for that matter anyoneelse, blessing it was that it lay inthe theory of Eisenstein series.

R. P. Langlands

Abstract In the number field case the global Langlands functoriality con-jecture is wide open. Despite this several important cases have established.We survey some of the hard-earned progress in this chapter.

13.1 Introduction

What is now known as the Langlands functoriality conjecture was posed ina seventeen page handwritten letter [Lan] that Langlands wrote in 1967 toAndre Weil (see [Lan70]). In Chapter 12 we recalled the conjecture with somedegree of precision. There has been decisive progress on the conjecture in thelocal setting in both equal and mixed characteristics (see §12.4 and §12.5)and in the global setting when the base field F is a function field (see §13.9).However for number fields the conjecture remains mostly open.

Despite this there has been hard-earned progress. We will survey some ofthis progress in the current chapter. Throughout F is a global field with ringof adeles AF . The symbol G will denote a reductive group over F .

255

256 13 Known Cases of Global Langlands Functoriality

13.2 Parabolic induction

Let P be a parabolic subgroup of G. A choice of Levi subgroup M ≤ P yieldsa Levi decomposition P = MN where N is the unipotent radical of P . LetT ≤MF sep be a maximal torus and let

(X∗(T ), X∗(T ), ∆,∆∨)

be a based root datum for G (see §7.3). It follows from the theory of parabolicsubgroups reviewed in §1.9 that there are subsets of ∆1 ⊆ ∆ and ∆∨1 ⊆ ∆∨

such that(X∗(T ), X∗(T ), ∆1, ∆

∨1 )

is a based root datum for M . Using this fact one can define an L-map

LM −→ LG. (13.1)

In the special case where G = GLn the conjugacy classes of Levi subgroupsof parabolics are indexed by tuples n1, . . . , nd ∈ Z>0 such that

∑di=1 ni = n

as explained in (1.10). The map (13.1) is given by the identity on the Galoisfactor and the natural block diagonal embedding

d∏i=1

GLni(C) −→ GLn(C)

(x1, . . . , xd) 7−→

x1

. . .

xd

on the neutral components.

Langlands functoriality for the L-map (13.1) was essentially fully estab-lished by Langlands himself before Langlands even posed his functorialityconjecture. It is merely one manner of interpreting the theory of Eisensteinseries and its relationship to the spectral decomposition of L2([G]) recalledin §10.4.

13.3 L-maps into general linear groups

Let G be a quasi-split reductive group over F and let E/F be a finite sepa-rable extension. Suppose we are given an L-map

r : LG −→ LResE/FGLn

13.4 The strong Artin conjecture 257

for some n. Assume moreover that we know the local Langlands correspon-dence for G. Then for every cuspidal automorphic representation π of G(AF )and every place v of F we can associate an L-parameter

ρ(πv) : W ′Fv −→LG.

We therefore obtain, for each v, an L-parameter

r ρ(πv) : W ′Fv −→LResE/FGLn.

By the local Langlands correspondence for ResE/FGLn(Fv) (which is known,see §12.3 and §12.4) associated to r ρ(πv) there is a unique irreducibleadmissible representation r(πv) with this L-parameter. We denote by

r(π) := ⊗vr(πv).

This is an irreducible admissible representation of GLn(AE). By the versionof Langlands functoriality explicated in Conjecture 12.6.1, at least if π istempered r(π) ought to be automorphic. In practice one does not want toassume that π is tempered, but merely that it is “almost tempered” in thesense that the matrix coefficients of πv are suitably close to being essentiallysquare integrable. We refer to [Art13] for an example of how to make thisprecise.

The following definition is extremely useful:

Definition 13.1. An automorphic representation π′ of ResE/FGLn(AF ) is aweak transfer of π with respect to r if

r(πv) ∼= π′v

for all v not in some finite set of places S of F .

Weak transfers are also often called weak lifts. If one wishes to be morespecific, one can say that the functorial transfer of π with respect to r iscompatible with the local Langlands correspondence outside of S. If a weaktransfer exists then it is unique by strong multiplicity one for GLn (Theo-rem 11.7.2). In practice one constructs weak transfers first and then checkscompatibility of the transfer with the local Langlands correspondence at thefinite set of places S.

13.4 The strong Artin conjecture

Let E/F be a (nontrivial) finite degree Galois extension of number fields.Any homomorphism

r : Gal(E/F ) −→ GLn(C)


defines a representation

r : LGL1 −→ GLn(C) (13.2)

by tensoring with the trivial representation of LGL1 = GL1(C). Temporar-ily denote by 1 the trivial representation of GL1(AF ), which is obviouslyautomorphic. The Langlands functoriality conjecture predicts that r(1) isautomorphic. One has

L(s, r(1)) = L(s,1, r) = L(s, r)

where the L-function at the right is the usual Artin L-function. Artin andBrauer proved that the L-function on the right admits a meromorphic contin-uation to the plane, satisfies a functional equation, and has a finite number ofpoles [MM97]. Artin conjectured the following refinement of this statement:

Conjecture 13.4.1 (Artin) If r is nontrivial and irreducible then L(s, r)is holomorphic.

The only irreducible representation that not covered by this conjecture is thetrivial representation. If r is the trivial representation then L(s, r) = ΛF (s),the completed Dedekind zeta function of F . It is certainly not holomorphic;in fact it has simple poles at s ∈ 0, 1. Artin’s conjecture implies that this isthe only irreducible Galois representation whose Artin L-function has a pole.

Thus Langlands functoriality implies the following conjecture, which isoften called the strong Artin conjecture:

Conjecture 13.4.2 (Strong Artin) If r is irreducible, then there exist acuspidal representation π of GLn(AF ) such that

πv = r(1)v

for all places v.

This is in fact a much deeper conjecture than the Artin conjecture, and onecan make precise the difference between Conjecture 13.4.1 and Conjecture13.4.2 using Theorem 11.9.1.

The strong Artin conjecture for nilpotent groups is known to be true[AC89, Theorem 7.1]:

Theorem 13.4.3 (Arthur-Clozel) Assume that E/F is a Galois exten-sion of number field with nilpotent Galois group Gal(E/F ). Then if r is anyirreducible complex representation of Gal(E/F ), the strong Artin conjectureis true for r. ut

Even when Gal(E/F ) is solvable the strong Artin conjecture is still not knownin general.

In a different direction, one could ask for results for representations r :Gal(E/F )→ GLn(C) for a fixed n. The case n = 1 is of course true by class

13.4 The strong Artin conjecture 259

field theory. Even the case n = 2 is not known in general. It is convenient torecall the following theorem of Klein:

Theorem 13.4.4 (Klein) A finite subgroup of PGL2(C) is isomorphic toone of the following:

a cyclic group,a dihedral group,the tetrahedral group A4,the octahedral group S4,the icosahedral group A5.

ut

A subgroup of GL2(C) mapping to a cyclic group in PGL2(C) acts re-ducibly on C2, so we can ignore these groups. We say a representation

r : Gal(E/F ) −→ GL2(C)

is dihedral if the image of the composite map

r : Gal(E/F ) −→ GL2(C) −→ PGL2(C)

is a dihedral group. We use the analogous convention when dihedral is re-placed by tetrahedral, octahedral or icosahedral. In all but the icosahe-dral case the strong Artin conjecture is known to be true:

Theorem 13.4.5 (Langlands-Tunnell) The strong Artin conjecture is truefor dihedral, tetrahedral, or octahedral representations r. ut

In [Lan80] Langlands derived the dihedral and tetrahedral cases from thetheory of base change which will be reviewed in the following section. Tunnellcompleted the octahedral case in [Tun81]. Interestingly this theorem was usedin Wiles’ proof of Fermat’s last theorem.

The key difference with the icosahedral case is that it is the only casewhere the Gal(E/F ) is nonsolvable. If one had the theory of base changefor nonsolvable extensions one could make decisive progress on this last case[Get12].

Though the theory of base change for nonsolvable extensions is unavailableat the time of this writing, one can still make progress under special assump-tions. Namely, assume that the base field F is totally real. For every infiniteplace v of F complex conjugation defines a conjugacy class cv of elementsσv ∈ Gal(F/F ). The representation r extends to Gal(F/F ), and we say it isodd if det(r(cv)) = −1 for all infinite places v.

Theorem 13.4.6 If F is a totally real field and r is odd then the strongArtin conjecture is true for r. ut

This theorem was proven by Pilloni and Stroh [PS16] after the case F = Qwas settled by Khare and Wintenberger [KW09a, KW09b] as a consequence of


their proof of Serre’s conjecture (Kisin made a decisive contribution [Kis09]).There was a body of work prior to this spearheaded by Taylor; see [PS16]for references. The oddness assumption allows one to apply the powerfulmachinery of Galois deformation theory that played so pivotal a role in Wiles’proof of Fermat’s last theorem.

13.5 Base change

Let E/F be a finite degree Galois extension of number fields. Then

LResE/FGLn := GL[E:F ]n (C) o GalF

where GalF acts via its quotient Gal(E/F ), which in turn acts by permutingthe factors. There is an L-map

rE/F : LGLn −→ LResE/FGLn (13.3)

defined by stipulating that it is the diagonal embedding on the neutral com-ponent and the identity on the Galois factor. Suppose that ρ ∈ Φ(GLnFv ) forsome place v of F (see §12.6 for the definition of Φ(GLnFv )). Then

L(s, rE/F ρ) =∏w|v

L(s, ρ|W ′Ew ) (13.4)

where the product is over all places w of E dividing v. Here W ′Ew is theWeil-Deligne group as (12.2). In other words, rE/F corresponds to restric-tion of Galois representations. The L-map rE/F and the functorial trans-fers it induces (conjectural or not) are known as base change. It con-jecturally allows us to relate automorphic representations of GLn(AF ) andResE/FGLn(AF ) := GLn(AE). It is customary to write

πE := rE/F (π).

Consider the representation

ρreg : GalF → Gal(E/F )→ GL[E:F ](C)

where the first map is the quotient and the second is the regular representa-tion. We define an L-map

AIE/F : LResE/FGLn −→ LGLn[E:F ] (13.5)

by stipulating that its composite with the quotient map LGLn[E:F ] −→GLn[E:F ](C) is the tensor product of the standard representation of GLn(C)with ρreg. This L-map and the funtorial transfers (conjectural or not) it in-

13.5 Base change 261

duces are called automorphic induction. It allows us to relate automorphicrepresentations of GLn(AE) = ResE/FGLn(AF ) and GLn[E:F ](AF ). We notethat

L(s, π,AIE/F ) = L(s, π). (13.6)

Using the fact that the regular representation of Gal(E/F ) decomposes as⊕ρ

ρ⊕d(ρ)

where the sum is over the isomorphism classes of irreducible representationsρ of Gal(E/F ) and d(ρ) is the dimension of the space of ρ, one has

L(s, π, rE/F ) = L(s, π,AIE/F rE/F ) =∏ρ

L(s, π, rst ⊗ ρ)m(ρ). (13.7)

Here rst is the standard representation rst : LGLn −→ GLn(C) and we asso-ciate to ρ the representation LGL1 −→ GLd(ρ)(C) as in (13.2).

If E/F is a prime degree cyclic extension, then base change and auto-morphic induction are known. The case n = 2 is due to Langlands [Lan80],following a crucial special case discovered by Saito [Sai75] and rephrasedrepresentation-theoretically by Shintani [Shi79]. Besides placing the theoryin the correct level of generality, Langlands was able to use it and conversetheory to establish the strong Artin conjecture for 2-dimensional Galois rep-resentations in many solvable cases (see Theorem 13.4.5).

For GLn the result was established by Arthur and Clozel [AC89]. To stateit, let E/F be a prime degree Galois extension. Let θ be a generator ofGal(E/F ). It acts on the set of automorphic representations π′ of GLn(AE)via

π′θ(g) := π′(θ(g)).

Finally, let NE/F : A×E → A×F be the norm map.

Theorem 13.5.1 (Base change) For every cuspidal automorphic repre-sentation π of GLn(AF ) the base change πE exists; it is an automorphicrepresentation πE of GLn(AE) that is cuspidal if and only if π 6∼= π ⊗ η forany Hecke character η ∈ (F×\A×F /NE/F (A×E))∧. Conversely, if a cuspidal

automorphic representation π′ of GLn(AE) satisfies π′ ∼= π′θ then π′ = πEfor some cuspidal automorphic representation π of GLn(AF ). ut

Theorem 13.5.2 (Automorphic induction) If π is a cuspidal automor-phic representation of GLn(AF ), then the automorphic induction AI(π) is anisobaric automorphic representation of GLn[E:F ](AF ). It is cuspidal if and

only if π 6∼= πθ. A cuspidal automorphic representation of GLn[E:F ](AF ) is theautomorphic induction of a cuspidal automorphic representation of GLn(AF )if and only if π ∼= π ⊗ η for some η ∈ (F×\A×F /NE/F (A×E))∧. ut


Implicitly in this theorem we are using the local Langlands correspondencefor the general linear group to define base change and automorphic induc-tion. This was not available when Arthur and Clozel proved their result.Arthur and Clozel instead used the representation-theoretic analogue definedby Shintani [AC89, Definition 6.1]. The compatibility of the two notions ischecked in [HT01, Lemma VII.2.6].

By breaking an arbitrary solvable extension E/F into a sequence of prime-degree cyclic extensions one deduces the existence of base changes of isobaricautomorphic representations of GLn(AF ) along E/F . Characterizing the im-age of the base change in this case is more subtle, however, see [LR98, Raj02].

The case of an arbitrary extension E/F is open. The first author hasdescribed a possible approach to this case in [Get12] and explained how itimplies the remaining open cases of the Artin conjecture for two-dimensionalcomplex representations of GalF , at least up to an abelian twist. The workin [Get12] can be seen as an refinement and explication of Langlands’ beyondendoscopy idea [Lan04] in a special case.

13.6 The Langlands-Shahidi method

As explained in §10.4, Langlands proved that L2([G]) could be completelydescribed in terms of cuspidal representations of Levi subgroups of G bymeans of Eisenstein series. Some time after completing this he realized thatthe constant terms of Eisenstein series (which naturally depend on certaincomplex parameters) admitted functional equations [Lan71]. This led him tothe definition of what are now called Langlands L-functions and to posing hisfunctoriality conjecture. Langlands only obtained coarse functional equationsfor these L-functions. His results were subsequently completed and refinedby Shahidi [Sha88]. This method of proving the analytic continuation of L-functions is known as the Langlands-Shahidi method. Combining theseresults with converse theory in the sense of Theorem 11.9.1 one can obtaincases of Langlands functoriality that, at present, can not be obtained byany other method. The book [Sha10] and the surveys [CKM04] are usefulreferences for this theory.

We record some important results obtained via the Langlands-Shahidimethod. We assume throughout this section that F is a number field. Thefirst case concerns the symmetric powers of cuspidal automorphic represen-tations of GL2. More precisely, let π = ⊗vπv be a cuspidal automorphicrepresentation of GL2(AF ) and let

Symk : LGL2 −→ LGLk+1 (13.8)

be the representation that is the identity on the Galois factor and the sym-metric kth power representation on the neutral components. By the local

13.6 The Langlands-Shahidi method 263

Langlands correspondence explained in §12.5, Symk(πv) is a well-defined ir-reducible admissible representation of GLk+1(Fv) for all places v of F . Hence

Symk(π) = ⊗vSymk(πv)

is an irreducible admissible representation of GLk+1(AF ).

Theorem 13.6.1 If π is a cuspidal automorphic representation of GL2(AF ),then the admissible representation Symk(π) of GLk+1(AF ) is automorphic fork ≤ 4. ut

The proof of Theorem 13.6.1 was given by Gelbart-Jacquet [GJ78] for k = 2,by Kim-Shahidi [KS02] for k = 3, and by Kim [Kim03] for k = 4. As of thiswriting the corresponding statement for k ≥ 5 remains unproven. Gelbartand Jacquet’s work occurred before the local Langlands correspondence wasproven. In [Kim03] Kim gives an alternate proof that includes compatibilitywith the local Langlands correspondence.

The second case we would like to mention here is the Langlands functori-ality for Rankin-Selberg products of cuspidal automorphic representations ofGLn. For any m,n ∈ Z≥0, let

r⊗ : L(GLm ×GLn) −→ LGLmn (13.9)

be the representation that is the tensor product on the neutral componentsand the identity map on the Galois factor. This family of L-maps is extremelyimportant; taking tensor products is arguably the most basic procedure forconstructing new representations out of old representations.

For every place v of F and every admissible representation π1v ⊗ π2v ofGLm(Fv)×GLn(Fv) we have

L(s, π1v ⊗ π2v, r⊗) = L(s, π1v × π2v)

where the L-function on the right is the Rankin-Selberg L-function appearingin §11.5. This identity is built into the local Langlands correspondence. It iscustomary to write

π1v π2v := r⊗(π1v ⊗ π2v).

Thus for automorphic representations π1, π2 of GLm(AF ) and GLn(AF ), re-spectively, we have an admissible representation

π1 π2 :=∏v

π1v π2v

of GLmn(AF ). Now we know that the standard L-function of this represen-tation behaves as predicted under the Langlands functoriality conjecture byRankin-Selberg theory (see Chapter 11). In general this is very far from theassertion that π1 π2 is automorphic. Converse theory in the sense of The-orem 11.9.1 makes the difference precise. For small m and n the difference is


not as significant and thus using converse theory and the Langlands-Shahidimethod one can prove the following theorem:

Theorem 13.6.2 For 1 ≤ k ≤ 3, let π1, π2 be cuspidal automorphic repre-sentations of GL2(AF ) and GLk(AF ), respectively. Then π1π2 is automor-phic. ut

For the GL2 × GL2 case this was proven in [Ram00a]. For the GL2 × GL3

case this is [KS02, Theorem 5.1]. The paper [Ram00a] was written before theproof of the local Langlands correspondence. Compatibility of the transferwith the local Langlands correpondence was checked in [Kim03].

13.7 Functoriality for the classical groups

We continue to assume that F is a number field. Let

Jn :=

1

. ..

1

∈ GLn(Z)

be the antidiagonal matrix and let

J ′2n : =

(Jn

−Jn

)

J ′2n+1 : =

Jn1

−Jn

.

We take our “standard” orthogonal and symplectic groups to be the groupsover F whose points in an F -algebra R are given by

On(R) : = g ∈ GLn(R) : gJngt = Jn

Sp2n(R) : = g ∈ GL2n(R) : gJ ′2ngt = J ′2n.

(13.10)

We also define some quasi-split groups as follows. Let E/F be a quadraticextension. We then define

Un(R) := g ∈ GLn(E ⊗F R) : gJ ′ngt = J ′n

where the bar denotes the action of the nontrivial element of Gal(E/F ).This is the quasi-split unitary group attached to the extension E/F . Theextension E/F also defines a quasi-split orthogonal group as follows: View Eas a vector space over F equipped with the quadratic form NE/F . Choosinga basis we obtain a symmetric matrix γE ∈ GL2(F ) corresponding to thequadratic form in the usual manner. Then

13.7 Functoriality for the classical groups 265

O∗2n(R) :=

g ∈ GLn(R) : g

Jn−1

γEJn−1

gt =

Jn−1

γEJn−1

.

We define SOn < On and SO∗n < O∗n to be the subgroups of determinant 1as usual. We refer to the groups above as the quasi-split classical groupsof rank n defined over R and denote by Gn.

Following [CPSS11] we have the following table of groups and embeddingsof L-groups:

Gn r : LGn → LHN HN

SO2n+1 Sp2n(C)×GalF → GL2n(C)×GalF GL2n

SO2n SO2n(C)×GalF → GL2n(C)×GalF GL2n

SO∗2n SO2n(C) o GalF → GL2n(C)×GalF GL2n

Sp2n SO2n+1(C)×GalF → GL2n+1(C)×GalF GL2n+1

Un GLn(C) o GalF → GLn(C)2 o GalF ResE/FGLn

We call the representation

r : LGn −→ LHN (13.11)

listed above the standard representation of LGn. The L-maps in the first,second, and fourth row are given by the natural inclusions. For the third, let

w :=

In−1

0 11 0

In−1

.

Then GalF acts on SO2n(C) via its quotient Gal(E/F ), with the nontrivialelement σ of Gal(E/F ) acting by conjugation by w. The map

SO2n(C) o GalF → GL2n(C)×GalF

is given by the natural embedding on the neutral component and sends 1 oσ to w o σ. Finally when Gn = Un and HN = ResE/FGLn the absoluteGalois group of F again acts on GLn(C) via its quotient Gal(E/F ), with thenontrivial element σ acting via


LUn = GLn(C) −→ GLn(C)

g 7−→ J ′ng−tJ ′−1

n

LHN = GLn(C)2 −→ GLn(C)2

(h1, h2) 7−→ (h2, h1)

(13.12)

and the embedding r is given by

r(g o σ) = (g, J ′ng−tJ ′−1

n ) o σ. (13.13)

There is a great deal known about Langlands functoriality for this collectionof L-maps. Definitive results were obtained for generic representations in[CKPSS04, CPSS11]. We focus on this case in the current section. LaterArthur [Art13] established the same result for arbitrary representations (inthe orthogonal and symplectic cases) under the assumption that the twistedweighted trace formula can be stabilized (see §13.8 for details). The unitarycase was completed by Mok [Mok15] under this same assumption.

The following theorem is [CKPSS04, Theorem 1.1]. It uses the notation ofthe table above. It represents the culmination of an impressive body of workby Cogdell, Kim, Piatetski-Shapiro and Shahidi.

Theorem 13.7.1 Let π be an irreducible globally generic cuspidal automor-phic representation of Gn(AF ). Then there exists a functorial transfer of πto HN (AF ) with respect to the L-map r. ut

The main tools used in the proof of this theorem are the Langlands-Shahidimethod and the converse theorem (Theorem 11.9.1). Let π be as in the state-ment of the theorem. For every integer m ≥ 1 and every cuspidal auto-morphic representation τ of GLm(AF ) Langlands-Shahidi theory provides anL-function L(s, π × τ) that admits a functional equation and meromorphiccontinuation to the plane. One constructs an admissible representation r(π)of HN (AF ) such that one has

L(s, πv × τv) = L(s, r(π)v × τv) (13.14)

for all archimedean places v and nonarchimedean places where π, τ and F areunramified. It is harder to obtain information at the ramified places. For thispurpose one uses stability of γ-factors, which says roughly that certain γfactors related to those defined in Theorem 11.5.4 depend only on the centralcharacter of πv after taking a suitably ramified by a character of F×v . Atthe end one then applies converse theorems analogous to Theorem 11.9.1 todeduce the automorphy of r(π).

Assuming Theorem 13.7.1, Ginzburg, Rallis and Soudry [GRS01] had pre-viously used their descent technique to characterize the image of the functo-rial transfer attached to r. The characterization is given in terms of poles ofLanglands L-functions. To state it, we define a family of representations r′

and characters χGn : [Gm]× → C×. The representations r′ listed below are

13.7 Functoriality for the classical groups 267

representations of HN , where HN is the group attached to Gn in the tableabove.

Gn r′ χGn

SO2n+1 ∧21

SO2n, n ≥ 2 Sym21

SO∗2n Sym2 ηE/F

Sp2n Sym21

U2n AsE/F ⊗ ηE/F 1

U2n+1 AsE/F 1

Here 1 denotes the trivial character, and ηE/F is the character attached toE/F by class field theory. The only representation that is not self-explanatoryis the Asai representation

AsE/F : LResE/FGLn = GL2n(C) o GalF −→ GLn2(C)

that is defined by stipulating that for ξ ∈ HomF (E,F ),

AsE/F (((gξ) o 1))(⊗ξvξ) = ⊗ξgξvξ and AsE/F ((1)ξ o σ)(⊗ξvξ) = ⊗ξvσξ.

This plays an important role in §14.5.Conjecture 12.7.1 on the analytic properties of Langlands L-functions is

more or less known in the cases above. In more detail, let ψ : F\AF → C× bea nontrivial character. Langlands-Shahidi theory produces L- and ε-factors

LLS(s, πv, r′)

εLS(s, πv, r′, ψv)

(13.15)

for every place v of F that are equal to the L- and ε-factors

L(s, πv, r′)

ε(s, πv, r′, ψv)

(13.16)

when v is archimedean or v lies outside of some finite set of nonarchimedeanplaces S. One sets

LLS(s, π, r′) :=∏v

LLS(s, πv, r′)

εLS(s, π, r′) :=∏v

εLS(s, πv, r′, ψv)

(13.17)


The global ε-factor is independent of ψ. These factors are originally definedfor Re(s) large. One has the following theorem, due to Shahidi following workof Langlands [Sha90, Theorem 7.7]:

Theorem 13.7.2 The L-function LLS(s, π, r′) admits a meromorphic con-tinuation to the plane to a function that is bounded in vertical strips. It sat-isfies the functional equation

LLS(s, π, r′) = ε′LS(s, π, r′)LLS(s, π∨, r′)

The boundedness of the L-function in vertical strips is in fact a joint theoremof Gelbart and Shahidi [GL06]. Shahidi’s book [Sha10] and Kim’s article in[CKM04] are also very useful references.

Generalizations of the Rankin-Selberg theory discussed in §11.7 provideyet another definition of the L-functions and ε-factors and one can prove atheorem analogous to Theorem 13.7.2 using these L-functions in most cases(see [Bum05] for a survey). The definition of the L-functions in terms ofRankin-Selberg integrals is important because in practice it provides infor-mation about the poles of L-functions that is complementary to that obtainedvia the Langlands-Shahidi method.

Of course, one wants to know that all of these L and ε factors agree. Thishas been proven in important cases. For the agreement of Langlands-Shahidiand Rankin-Selberg L-factors in many cases we refer to [Kap15]. We alsohave the following result [CST17]:

Theorem 13.7.3 When r′ is ∧2 or Sym2 then

LLS(s, πv, r′) = L(s, πv, r

′)

εLS(s, πv, r′) = ε(s, πv, r

′).

After this digression on L-functions we turn back to our characterization ofthe image of the functorial transfer. The group ResE/FGLn admits an outerautomorphism σ induced by the action of the nontrivial element of GalE/F .For irreducible admissible representations π′ of ResE/FGLn(AF ) let

π′σ(g) := π′(gσ)

and letπ′∗ := (π′σ)∨.

This is sometimes known as the conjugate dual. We say that π is conjugateself-dual if π ∼= π∗.

For the purposes of stating the following theorem it is convenient to letπ′∗ := π′∨ when HN = GLN and let π′∗ be defined as above otherwise. Letωπ′ be the central quasi-character of π′ (see [GRS01, Sou05] for more details):

Theorem 13.7.4 Let π be a globally generic cuspidal automorphic represen-tation of Gn(AF ). The functorial lift r(π) satisfies ωr(π)|A×F = χGn and is of

13.8 Endoscopic classification of representations 269

the formπ′ = Ind(π′1 ⊗ · · · ⊗ π′d) = π′1 · · · π′d,

where each πi is a unitary cuspidal automorphic representation of HNi(AF )satisfying

(a) π′∗i∼= π′i

(b) π′i∼= π′j if and only if i = j,

(c) LS(s, π′i, r′) has a pole at s = 1 for a sufficiently large finite set of places

S of F including all infinite places.

Moreover, any π′ as above satisfying ωπ′ |A×F = χGn and (a)-(c) is of the

form r(π) for some globally generic cuspidal automorphic representation π ofGn(AF ). ut

Analogues of 13.7.1 and 13.7.4 are known for spin similitude groups by workof Asgari and Shahidi [AS06, AS14].

In this section we have concentrated on generic representations. This is be-cause originally the converse theory approach explained above relied cruciallyon this assumption. Recently the genericity assumption has been removed inwork of Cai, Friedberg, Ginzburg, Kaplan. We refer to [CFK18] (which isbased on [CFGK17]) for a precise statement. This provides a new, uncondi-tional, and independent proof of a portion of the endoscopic classification ofrepresentations on classical groups that we discuss in the next section.

13.8 Endoscopic classification of representations

In this section we continue to use the notational conventions of the previoussection. In particular F is a number field. The great accomplishment of Arthurin [Art13] is to prove the existence of functorial transfer with respect to thethe L-maps r of the previous section (at least when Gn 6= Un). In particularthere is no genericity assumption in his work. Moreover, he gave a preciseenough description of the fibers of the functorial transfer that he could classifythe discrete spectrum of L2([Gn]) in terms of automorphic representations onHN . We explain these results in the current section. The main tool used toestablish Arthur’s result is the theory of twisted endoscopy (see §19.5), soone refers to Arthur’s work and subsequent refinements as the endoscopicclassification of representations.

In his book [Art13] Arthur gives a careful account of how to replace objectsattached to the conjectural global Langlands group by well-defined, uncon-jectural objects. For the sake of clarity and brevity we will not reproduce thisdiscussion. Instead, we will try to state Arthur’s main results as directly aspossible.

There are two points the reader should bear in mind. First, we have notstated all of Arthur results, and we have not included Mok’s results in the


Gn = Un case since this would require more notation. We refer to [Mok15].Second, all of the work in this section (and in Mok’s work) is conditional onthe stabilization of the twisted weighted trace formula. Moeglin and Wald-spurger have proven this under the assumption of the twisted weighted funda-mental lemma [MW16a, MW16b]. The original fundamental lemma of Shel-stad and Langlands was proved in the breakthrough work of Ngo in [Ngo10b]for which he received the Fields medal. Ngo’s work is based on the work ofmany over decades and his survey [Ngo10c] is a good place to obtain somehistorical perspective. Other versions of the fundemental lemma have beenproven by Laumon and Chaudouard using nontrivial generalizations of Ngo’stechniques [CL10, CL12]. In principle there should be no new obstacle inproving the twisted weighted fundamental but as of this writing it is notcomplete.

For the remainder of this section we assume Gn 6= Un. The first maintheorem is the following:

Theorem 13.8.1 Every irreducible subrepresentation of L2([Gn]) admits afunctorial transfer to HN with respect to r. ut

This result is implicit in the main theorems stated in [Art13]. Since irreduciblesubrepresentations of L2([Gn]) need not be tempered, to make precise whatone means by a functorial transfer one needs more than the theory outlinedin §12.5. We will not make this precise and instead refer the reader to [Art13].

There is a wrinkle which will continue to play a role below. When Gn isSO2n or SO∗2n let θ : Gn → Gn be an automorphism of order 2 preservingthe “standard splitting” of SO2n [Art13, p. 41]. It is unique if n > 3. Thenthe functorial transfer given in Theorem 13.8.1 is insensitive to replacing arepresentation π by πθ, where πθ(g) := π(θ(g)). With this in mind let

C∞c (Gn(AF )) (13.18)

be C∞c (Gn(AF )) except in the special case where Gn is SO2n or SO∗2n, inwhich case it is the subalgebra of C∞c (Gn(AF )) invariant under the outerautomorphism θ.

LetL2

disc([Gn]) < L2([Gn])

be the largest closed subspace that decomposes discretely under C∞c (Gn(AF ))(see §3.8 for details). Alternately L2

disc([Gn]) is the Hilbert space direct sumof all irreducible subrepresentations of L2([Gn]). In view of Theorem 13.8.1it is natural to partition L2

disc([Gn]) into packets consisting of fibers of thefunctorial transfer to HN (AF ) and then try to describe the fibers, preferablyin terms of local data. This is precisely what Arthur accomplishes.

We state the main classification result first. The remainder of the sectionis devoted to defining the notation that appears.

Theorem 13.8.2 (Arthur) There is an C∞c (Gn(AF ))-module isomorphism

13.8 Endoscopic classification of representations 271

L2disc([Gn]) ∼=

⊕ψ∈Ψ2(Gn)

⊕π∈Πψ(εψ)

π⊕mψ ,

where mψ = 1 or 2. ut

This is [Art13, Theorem 1.5.2]. For a cuspidal automorphic representationτ of HN (AF ) and m ∈ Z≥1 let (τ,m) be the Speh representation of §10.7.One says that τ is of orthogonal type if LS(s, τ, Sym2) has a pole at s = 1and of symplectic type if LS(s, τ,∧2) has a pole at s = 1, where S is a finiteset of places containing all infinite places and all places where τ is ramified.Since

LS(s, τ, Sym2)LS(s, τ,∧2) = LS(s, τ × τ)

it follows from Theorem 11.7.1 that τ cannot be both orthogonal and sym-plectic, and it is orthogonal or symplectic if and only if τ ∼= τ∨. We definethe type of the representation (τ,m) is determined as in the table below:

τ m (τ,m)

orthogonal even symplectic

symplectic even orthogonal

orthogonal odd orthogonal

symplectic odd symplectic

The set Ψ2(Gn) is the set of automorphic representations of HN (AF ) of theform

di=1(τi,mi)

where

(1) τi is a cuspidal automorphic representation of HNi(AF )

(2)∑di=1Nimi = N

(3) τ∨i∼= τi

(4) τi ∼= τj if and only if i = j.(5) If LGn is orthogonal (resp. symplectic) then (τi,mi) is orthogonal (resp. sym-

plectic).

We note that condition (5) actually implies condition (3). The set Ψ2(Gn) isknown as the set of discrete global A-packets of Gn (see [Art13, page33-34 in §1.4]). The multiplicity mψ in Theorem 13.8.2 is defined to be 1unless N is even, LGn = SON (C), and Nimi is even for all i, in which case itis 2. The discrete global A-packet is said to be generic if mi = 1 for all i. Inthis case we also refer to the packet as a discrete global generic L-packet.

For every ψ ∈ Ψ2(Gn) Arthur defines a finite 2-group Sψ and a character


εψ : Sψ −→ ±1.

We omit the definition. For every place v of F and every ψ ∈ Ψ2(Gn) wedefine a representation

ψv : W ′Fv × SL2(C) −→ LHN

byψv := ⊕di=1rec(τiv)⊗ Symm

where rec is the local Langlands reciprocity map of (12.15) in §12.4. We notethat ψv is not an L-parameter, but one can obtain an L-parameter from itas follows:

ρ(ψv)(g) := ψv

(g,(|g|1/2

|g|−1/2

)).

The extra SL2 factor occurring in ψv is colloquially known as the ArthurSL2 and plays a role similar to the representation of SL2 that appears inHodge theory. One proves that ψv factors through the L-map r and hencedefines a homomorphism

ψv : W ′Fv × SL2(C) −→ LGn.

These are examples of A-parameters, although we will not formally definethis concept. One shows in addition the existence of maps

Sψ −→ π0(Sψv )

whereSψv := CGn(C)(Im(ψv))/Z

GalFGn(C)

(the component group π0(Sψv ) is denoted by Sψv in [Art13]). Compare with(12.20) and (12.21) in §12.5.

Arthur then define sets Π(ψv) of admissible irreducible representations ofGn(Fv) attached to each such parameter satisfying certain desiderata. This isalready a substantial theorem as it amounts to a proof of the local Langlandsclassification for Gn. The sets Π(ψv) are the local A-packets attached to

ψv. The use of Π(ψv) instead of the notation Π(ρv) of §12.5 is intentional.First, the tilde is a reminder that one is only classifying representations up tothe action of the outer automorphism θ when Gn is SO2n or SO∗2n. Second,

the packets Π(ψv) are not L-packets, but an enlargement of them requireddue to the fact that the representations in question are often nontempered.In the special case where ψv is trivial on the Arthur SL2 the packet Π(ψv)is an (almost tempered) L-packet in the sense of §12.5. In particular ψv istrivial on the Arthur SL2 for all v for any discrete global generic L-packetΠ(ψ). As in §12.5, any πv ∈ Π(ψv) comes equipped with a character

〈·, πv〉 : π0(Sψv ) −→ C×.

13.9 The function field case 273

and thus a we obtain a character

〈·, π〉 :=∏v

〈·, πv〉.

This allows us to form the global adelic A-packet

Π(ψ) :=⊗vπv : πv ∈ Π(ψv) and 〈·, πv〉 = 1 for almost all v

. (13.19)

It consists of admissible representations of Gn(AF ). Not all of them are au-tomorphic, let alone occur in L2

disc([Gn]). The last piece of the classificationtheorem is a device for selecting which occur in L2([Gn]). This is providedby εψ. One defines

Πψ(εψ) := π ∈ Π(ψ) : 〈·, π〉 = εψ. (13.20)

This completes our discussion of the objects in Theorem 13.8.2.

13.9 The function field case

In the function field case much more is known. Let F be a function field ofcharacteristic p. For GL2, the Langlands correspondence for a function fieldwas proved by Drinfeld [Dd80, Dd87b, Dd87a, Dd88] and for higher rankGLn, this was proved by L. Lafforgue following Drinfelds’ argument.

In fact, L. Lafforgue, in Fields’ medal winning work [Laf02], proved theexistence of a correspondence between the set of equivalence classes of cus-pidal automorphic representations of GLn(AF ) and the isomorphism classesof continuous irreducible `-adic representations

GalF −→ GLn(Q`)

that are unramified almost everywhere. One can view this as establishingthe automorphic to Galois direction of the global Langlands correspondence.However, for GLn, this implies the both directions and hence establishes theglobal Langlands correspondence. More precisely, the other direction can bededuced from the converse theory of GLn of Weil [Wei67], Piatetski-Shapiroand Cogdell [CPS94, CPS99] as well as Grothendieck’s functional equationand the product formula for ε-factors of Laumon [Lau87].

More recently, V. Lafforgue (L. Lafforgue’s brother) gave a decompositionof the space of cusp forms on an arbitrary reductive group in terms of Lang-lands parameters that is compatible with the local Langlands correspondenceat almost all places [Laf18].

All of the work above involves an object known as a shtuka (Russian for“thing”). The very definition of a shtuka involves the scheme


Spec(OF ⊗Fp OF ). (13.21)

This sort of self product also comes up in Deligne’s proof of the Riemannhypothesis in the function field case. The analogue of this scheme does notexist in the number field case, since the various residue fields of the numberfield all have different characteristics.

There has been a great deal of thought about what can be used to replace(13.21) in the number field case. The idea is that instead of Fp one should usethe field with one element. This remains a very interesting but mostly specu-lative prospect. The motivation comes not only from the link with Langlandsfunctoriality but also from the link to the Riemann hypothesis.

Exercises

13.1. Construct the morphism (13.1).

13.2. Let r : LG → LResE/FGLn be an L-map and let π be a cuspidalautomorphic representation of G(AF ). Assume that a weak transfer π′ of πto ResE/FGLn(AF ) = GLn(AE) with respect to r exists and π′ is isobaric.Prove that π′ is the unique isobaric automorphic representation of GLn(AE)that is a weak transfer of π with respect to r.

13.3. Prove (13.4).

13.4. Prove (13.6).

13.5. Prove that a continuous homomorphism

GalF −→ GLn(C)

has finite image.

13.6. Let π be a cuspidal automorphic representation of AGL2\GL2(AF ). Let

rk := LGL2 → GL2(k+1)

be the representation given by Symk⊗(Symk)∨. Assume that L(s, π, rk) con-verges absolutely for Re(s) > 1. Prove the unramified Ramanujan conjecturefor π. In other words, for all places v of F such that πv is unramified, sayπv ∼= J(λ), the Langlands class (or Satake parameter) eλ ∈ GL2(C) haseigenvalues of complex norm equal to 1.

Chapter 14

Distinction and Period Integrals

Je serais reconnaissant a toutepersonne ayant compris cettedemonstration de me l’expliquer.

Pierre Deligne

Abstract In this chapter we define the notion of a distinguished represen-tation in local and global contexts. The global notion is defined in termsof period integrals. The relative trace formula was developed precisely tounderstand distinguished representations.

14.1 Introduction

The notion of distinction of automorphic representations with respect to sub-groups of the ambient group was introduced in a global setting by Harder,Langlands and Rapoport in their work on the Tate conjecture for Hilbertmodular surfaces [HLR86]. Jacquet later developed a tool for studying dis-tinction, namely the relative trace formula, which includes the usual traceformula as a special case. This will be treated in Chapter 18. In retrospect,the notion of distinction has been an integral part of representation theoryfor some time, as it often isolates models of representations of considerableinterest. For example, the whole theory of generic representations can bethought of as a special case of the study of distinguished representations.

In this chapter we discuss the notion of distinction, starting from thelocal setting and moving to the global. The special case of spherical sub-groups is discussed in §14.4 and important conjectures of Sakellaridis andVenkatesh are described in vague terms. We give a fairly thorough treatmentof symmetric subgroups of the general linear group in §14.5 and then dis-cuss the exciting prospect of generalizing this to classical groups in §14.6. Abrief introduction to the Gan-Gross-Prasad conjecture will follow after that.

275

276 14 Distinction and Period Integrals

This can be viewed as a precursor to the Sakellaridis-Venkatesh conjectures.Much is known but there are still important cases to be completed. Finallywe complete the chapter with some negative results in §14.8.

14.2 Distinction in the local setting

Let G be a reductive group over a local field F and let H ≤ G be a subgroup.Assume we are given a quasi-character χ : H(F )→ C×. Let Vχ be the spaceof χ (i.e. the one-dimensional complex vector space on which H(F ) acts viaχ).

Definition 14.1. An irreducible admissible representation (π, Vπ) of G(F )is (H,χ)-distinguished if

HomH(F )(Vπ, Vχ) 6= 0.

If χ is trivial, then we say that the representation is H-distinguished, orsimply distinguished if H is understood. An element of

HomH(F )(Vπ, Vχ)

is a relative character or an (H,χ)-character if we wish to be specific.

The definition above has the advantage that it is simple, but sometimes aslightly more sophisticated perspective is advantageous. Let

F(H(F )\G(F ), χ) : = f : G(F )→ C : f(hg) = χ(h)f(g) for h ∈ H(F ),C∞(H(F )\G(F ), χ) : = f ∈ C∞(G(F )) : f(hg) = χ(h)f(g) for h ∈ H(F ).

Thus the first space is just a space of functions, and the second is a space ofsmooth functions. The first space can be regarded as the space of sections onthe line bundle over H(F )\G(F ) defined by χ and the second can be regardedas the space of smooth sections.

Lemma 14.2.1 There is a C-linear isomorphism

HomH(F )(Vπ, Vχ)−→HomG(F )(Vπ,F(H(F )\G(F ), χ)). (14.1)

Proof. To a relative character λ and ϕ ∈ Vπ we associate the function

Iλ(ϕ)(g) := λ(π(g)ϕ).

To an intertwining map I : Vπ → F(H(F )\G(F ), χ) we associate the relativecharacter

λI(ϕ) := I(ϕ)(1)

where 1 is the identity of G(F ). ut

14.3 Global distinction and period integrals 277

Corollary 14.2.2 An irreducible admissible representation (π, Vπ) of G(F )is (H(F ), χ)-distinguished if and only if there is a nonzero intertwining map

Vπ −→ F(H(F )\G(F ), χ).

ut

The lemma allows us to describe natural subspaces of the space of relativecharacters HomH(F )(Vπ, Vχ). For example, let

HomH(F ),sm(Vπ, Vχ), (14.2)

the inverse image of C∞(H(F )\G(F ), χ) under the bijection of Lemma 14.2.1.We say that the elements of (14.2) are smooth relative characters. We saythat π is smoothly (H,χ)-distinguished if (14.2) is nonzero. More sophisti-cated notions of smoothness must sometimes be employed in the archimedeancase (see, for example, §11.3).

It is instructive to observe that one can view the theory of generic repre-sentations discussed in §11.3 as a special case of the more general conceptof a distinguished representation. Assume for simplicity that F is nonar-chimedean. If G is quasi-split, N ≤ G is the unipotent radical of a Borelsubgroup, and ψ : N(F ) → C× is a generic character then an irreducibleadmissible representation π of G(F ) is ψ-generic if and only if it is smoothly(N,ψ)-distinguished. The space HomN(F ),sm(Vπ, Vψ) is the space of Whit-taker functions, and HomG(F )(Vπ, C

∞(N(F )\G(F ), ψ)) is the space of Whit-taker models. Both are at most one-dimensional. For many interesting cases,the phenomenon that (14.2) is at most one dimensional (or at least finitedimensional) persists. We discuss this in more detail in §14.4 below.

14.3 Global distinction and period integrals

The global version of distinction involves period integrals of cusp forms. LetG be a reductive group over a global field F and let H be a subgroup of G.Let

χ : H(AF ) −→ C×

be a quasi-character trivial on (AG∩AH)H(F ). If ϕ : [G]→ C is a continuousfunction then we define the period integral

Pχ(ϕ) :=

∫(AG∩AH)H(F )\H(AF )

ϕ(h)χ(h)dh (14.3)

whenever this integral is absolutely convergent.


For a cuspidal automorphic representation π of AG\G(AF ) let L2cusp(π)

be the π-isotypic subspace of L2cusp([G]). We assume that Pχ(ϕ) is absolutely

convergent for all smooth ϕ ∈ L2cusp([G]).

Definition 14.2. A cuspidal automorphic representation π of AG\G(AF ) is(H,χ)-distinguished if Pχ(ϕ) is nonzero for some smooth ϕ in L2

cusp(π).

If H and/or χ are understood then we often speak of H-distinguished, χ-distinguished or simply distinguished representations. If χ is trivial, then weoften write P for Pχ.

Assume F is a number field. Let K∞ ≤ G(F∞) be a maximal compactsubgroup. It is often useful to work with K∞-finite functions in L2

cusp(π)instead of merely smooth ones. With this in mind we prove the followinglemma:

Lemma 14.3.1 Let π be a cuspidal automorphic representation of AG\G(AF )such that Pχ(ϕ) is absolutely convergent for all smooth ϕ ∈ L2

cusp(π). Therepresentation π is (H,χ)-distinguished if and only if there is a K∞-finitesmooth function ϕ ∈ L2

cusp(π) such that Pχ(ϕ) 6= 0.

Proof. The “if” direction is obvious. We prove the “only if.”Let U be a neighborhood of 1 in AG\G(F∞) and let ε > 0. By the

proof of Proposition 4.4.2 we can choose a nonnegative K∞-finite f∞ ∈C∞c (AG\G(F∞)) with support in K∞U such that∫

AG\G(F∞)

f(g)dg = 1 and

∫AG\G(F∞)−U

f(g)dg < ε.

Choose a smooth function ϕ in L2cusp(π) such that Pχ(ϕ) 6= 0. It is fixed by

some compact open subgroup K∞ ≤ G(A∞F ). Let

f := measdg∞(K∞)−1f∞ 1K∞ .

Then π(f)ϕ is K∞-finite, and

|Pχ(π(f)ϕ)− Pχ(ϕ)| =

∣∣∣∣∣∫AG\G(AF )

f(g) (Pχ(π(g)ϕ)− Pχ(ϕ)) dg

∣∣∣∣∣ .Here we have used the absolute convergence of the period integral to justifyswitching the order of integration. The above is bounded by∫

AG\G(AF )

f(g) |Pχ(π(g)ϕ)− Pχ(ϕ)| dg

which can be made as small as we want by adjusting U and ε. ut

To continue our example from the previous section, if G is quasi-split,H = N is the unipotent radical of a Borel subgroup, and ψ : N(AF ) → C×

14.3 Global distinction and period integrals 279

is a generic character then a cuspidal representation is ψ-generic if and onlyif it is (H,ψ)-distinguished.

We now develop criteria for Pχ(ϕ) to be convergent. We assume for theremainder of this section that

• The neutral component of H ≤ G is the direct product of a unipotent anda reductive group.

We begin with the following lemma:

Lemma 14.3.2 If H is reductive then the quotient (AG∩AH)H(F )\H(AF )is closed in [G].

Proof. By definition of the quotient topology, we are to show that

AG ∩AH\G(F )H(AF ) = AG\G(F )AGH(AF ) ⊂ AG\G(AF )

is closed. For this it suffices to show that AG\G(F )AGH(AF )/H(AF ) isclosed in AG\G(AF )/H(AF ). We in fact show that the image of G(F ) inthe quotient AG\G(AF )/H(AF ) is discrete in Lemma 17.6.4. ut

A version of the following proposition is proven in [AGR93]:

Proposition 14.3.3 If f ∈ C∞c (AG\G(AF )) and ϕ ∈ L2([G]) is smooththen the integral defining Pχ(ϕ) is absolutely convergent.

Proof. We first reduce the proposition to the case where H is connected. LetK∞ ≤ H(A∞F ) be a compact open subgroup such that ϕ and χ are left K∞-invariant. We recall that H(AF )/H(AF ) is compact by a theorem of Borel[Con12a, Proposition 3.2.1], and hence

H(F )H(AF )\H(AF )/K∞

is finite. For appropriate choices of Haar measures dh on H(AF ) and dh onH(AF ) one has∫

(AG∩AH)H(F )\H(AF )

ϕ(h)χ(h)dh

=

∫(AG∩AH)H(F )\H(AF )/K∞

ϕ(h)χ(h)dh

=∑

h∈H(F )H(AF )\H(AF )/K∞

∫(AG∩AH)H(F )\H(AF )

ϕ(hh)χ(hh)dh.

Thus we are reduced to the case where H is connected.We now assume H is connected. Write H = Hr×Hu where Hr is reductive

and Hu is unipotent. We then have

Pχ(ϕ) :=

∫(AG∩AHr )Hr(F )\Hr(AF )

∫[Hu]

ϕ((hr, hu))χ(hr, hu)dhrdhu


(see Exercise 2.16). The set [Hu] is compact by Theorem 2.6.2. Thus applyingthe Fubini-Tonelli theorem, we see that it suffices to establish the propositionin the special case where H is reductive.

Assume first that we are in the function field case. In this setting Theorem9.4.1 asserts that any smooth function in L2

cusp([G]) is compactly supported.Thus the proposition follows from Lemma 14.3.2.

Now assume that we are in the number field case. Let TH0 ≤ Hder andT0 ≤ Gder be maximal F -split tori in Hder and Gder, respectively. We canand do assume T0 ∩Hder = TH0. Let PH be a minimal parabolic subgroup ofH containing TH0. It admits a Levi decomposition MHNH = PH , where MH

is the centralizer of TH0 in H (it is a Levi subgroup of P [Bor91, Proposition20.4] or [BT65, Theorem 4.15(b)] ) and NH < PH is the unipotent radical ofPH . By Theorem 2.7.2 there is an 0 < r < 1 such that

H(AF ) = AHH(F )ATH0(r)ΩKH

where Ω is relatively compact subset of NH(AF )MH(AF )1, KH ≤ H(AF ) isa maximal compact subgroup, and ATH0

(r) is defined with respect to the setof simple roots ∆H of TH in H attached to the parabolic subgroup PH . Werefer to §2.7 for notation.

The Haar measure on H(AF ) decomposes as

d(anmk) = dadndmdk

for (a, n,m, k) ∈ AHATH0(r)×NH(AF )×MH(AF )1×KH by the Iwasawa de-

composition (Theorem 2.7.1) and Proposition 3.2.1. Thus the integral Pχ(ϕ)converges absolutely provided that∫

(AG∩AH)\AHATH0(r)×Ω×KH

|ϕ(anmk)|dadndmdk (14.4)

converges.Let ∆ be a set of simple roots for T0 in G. We can then form the corre-

sponding setAT0(r0)

for r0 ∈ R>0. There is a Weyl chamber C ⊂ Lie(AT0) such that the closureof the image of C under the exponential map is AT0(1). Weyl chambers inLie(AT0

) are permuted simply transitively by W (G,T )(F ) [Bor91, Theorems14.7, 21.2, and 21.6]. It follows that

AT0 =⋃

w∈W (G,T )(F )

wAT0(1)w−1

and the intersection of any two W (G,T )(F )-conjugates of AT0(1) is a set ofmeasure zero with respect to any Haar measure on AT0 . Hence (14.4) is equalto

14.4 Spherical subgroups 281∑w∈W (G,T )(F )

∫(AG∩AH)\AHATH0

(r)∩wAT0(1)w−1×Ω×KH

|ϕ(anmk)|dadndmdk.

Changing variables a 7→ w−1aw and using the left invariance of ϕ under wwe see that this is the sum over w ∈W (G,T )(F ) of∫

(AG∩w−1AHw)\w−1AHATH0(r)w∩AT0

(1)×Ω×KH|ϕ(awnmk)|dadndmdk.

The function ϕ is rapidly decreasing as a function on G(AF ) by Corollary9.7.2. It follows immediately from this that the integral above is finite. ut

One can also give a reasonable definition of distinction for representationsthat are not necessarily cuspidal, see [JLR99, LR03, Off06, Zyd19].

14.4 Spherical subgroups

We continue to let G be a reductive group over a global field or a localfield F and let H ≤ G be a subgroup. We assume for this section that thecharacteristic of F is zero because the majority of references restrict to thiscase.

For general H, one does not even have a conjectural answer to the ques-tion of which representations are H-distinguished. However, there is an im-portant class of subgroups for which one can conjecturally characterize H-distinguished representations:

Definition 14.3. A connected subgroup H ≤ G is said to be spherical ifHomH(V, Vtriv) is at most one dimensional for all irreducible representationsρ : G→ GL(V ), where Vtriv denotes the trivial representation.

Note that here ρ is a representation in the category of algebraic groups.Now assume that G and H are split reductive groups. The group H acts

on O[G], the ring of regular functions (see §17.1 for details on this), and onedefines

X := G/H := Spec(O[G]H). (14.5)

Gaitsgory and Nadler [GN10] defined a dual group GX equipped with anembedding

GX × SL2 −→ G (14.6)

of reductive algebraic groups over C. There are antecedents of this definitionin the special case where H is symmetric subgroup in the sense of §14.5 be-low, see [Ric82, JLR93]. These antecedents did not include the SL2 factor.


This factor is crucial for understanding nontempered distinguished represen-tations.

Assume that G and H are both split and that the local Langlands corre-spondence is known for tempered representations of G(F ). In this case thebasic local principle (due to Sakellaridis and Venkatesh) is the following:

Principle 14.4.1 A tempered L-parameter ρ : W ′F → LG factors throughthe natural map

GX(C)×GalF → LG

if and only if some π ∈ Π(ρ) is H-distinguished.

Here we have ignored the SL2 factor in (14.6). This is to avoid discussingArthur packets, which are necessary to understand the SL2(C) factor. Wehave stated this as a principle instead of a conjecture because it might befalse as stated. For example, it might be necessary to enlarge the L-packet to aso-called Vogan L-packet that includes representations of pure inner formsof G. There are other more serious problems that can occur, for examplein the case where H is an orthogonal group On and G is the general lineargroup GLn (see §14.5 below). However we do not wish to delve into the detailsfor which we refer to [SV12, §16]. There are antecedents of this conjecturein the special case of symmetric varieties in the work of Jacquet and hiscollaborators [JLR93, Jac05a].

One knows the most about Principle 14.4.1 in the case where the L-parameters is unramified. The most general statements currently known arecontained in work of Sakellaridis [Sak08, Sak13]. We point out that even inthe unramified case there is not a robust understanding of distinguished rep-resentations for groups that are quasi-split and not split. Developing such atheory is an important (and probably accessible) open problem.

Now assume that F is a global field and G is a reductive group over F .Let H ≤ G be a spherical subgroup. By Exercise 14.5 if π = ⊗′vπv is acuspidal automorphic representation of G(AF ) that is H-distinguished thenπv is H-distinguished for all places v. In general, the converse statement isfalse. There is usually a global condition that is also necessary that can of-ten be phrased in terms of L-functions. We will illustrate this with examplesnext few sections. Under suitable assumptions, in [SV12] Sakellaridis andVenkatesh have a conjecture characterizing cuspidal automorphic representa-tions π distinguished by spherical subgroups in these terms, that is, in termsof local distinction of the components of π together with a criterion in termsof L-functions. There is even a conjectural formula for the period integralin terms of L-functions that is known as the Ichino-Ikeda conjecture. Itwas first posed in a special case in [II10]. The power and beauty of the con-jectures of Sakellaridis and Venkatesh lies in the fact that they provide auniform perspective on many proved and unproved conjectures in local rep-resentation theory and period integrals and at the same time suggest manynew avenues of research. They are currently the most precise enunciation ofwhat has become known as the relative Langlands program.

14.5 Symmetric subgroups 283

Before moving to examples of spherical subgroups in the next few sectionswe mention that Knop and Schalke [KS17] have defined a dual group GXattached to any G-scheme X equipped with a morphism GX → G. It wouldbe interesting to investigate whether or not this generalized dual group canbe used to generalize Principle 14.4.1.

14.5 Symmetric subgroups

Assume for the moment that F is a characteristic zero field. Let σ : G → Gbe an automorphism over F of order 2. For F -algebra R, let

Gσ(R) := g ∈ G(R) : gσ = g (14.7)

be the subgroup fixed by σ and denote by H = (Gσ) its connected compo-nent.

In this case we refer to X := G/H as a symmetric variety and H as asymmetric subgroup. We have the following theorem of Vust [Vus74, §1.3]:

Theorem 14.5.1 A symmetric subgroup is a spherical subgroup. ut

To classify symmetric subgroups it suffices to classify involutions of G.This question has been studied by several authors; we mention in particularthe work of Helminck [Hel00]. The classification is somewhat complicated. Toget a feel for what is going on it is illuminating to consider the involutionsof the classical groups over an algebraically closed field. When the field is C,there is a nice discussion in [GW09, §11.3.4].

We now describe some results on distinction by symmetric subgroups ofGLn or ResE/FGLn when E/F is a quadratic extension of number fields. Theupshot of the discussion is that there is a great deal known in this setting.

One of the motivations for Jacquet’s development of the relative traceformula was to prove the following theorem [Jac10]:

Theorem 14.5.2 Let E/F be a quadratic extension, let G = ResE/FGLnand let H be a quasi-split unitary group attached to the extension E/F . Thena cuspidal automorphic representation π of G(AF ) is H-distinguished if andonly if π ∼= πσ, where 〈σ〉 = Gal(E/F ). ut

See also [FLO12] for more refined versions of this theorem. This is sometimesknown as the Jacquet-Ye case since its proof was more or less reduced to alocal statement known as the Jacquet-Ye fundamental lemma in [JY96] andlater proved in two different manners by Ngo [Ngo99b, Ngo99a] and Jacquet[Jac04, Jac05b] (see §19.6 for more details). The condition that π ∼= πσ isequivalent to π being a base change lift from GLn(AF ), by work of Arthurand Clozel [AC89] (see Theorem 13.5.1). Thus this theorem is consonant with


the expectation of the principles laid out in §14.4, although the dual groupof the relevant spherical variety is not defined here as G is not quasi-split.

Similarly, using a variant of the Rankin-Selberg theory we discussed in§11.7 Flicker and Zinoviev proved the following theorem [Fli88, FZ95]:

Theorem 14.5.3 For G as in Theorem 14.5.2 an automorphic respresenta-tion π of AG\G(AF ) is distinguished by GLn if and only if the Asai L-functionL(s, π,AsE/F ) has a pole at s = 1. ut

Here the Asai L-function is the Langlands L-function attached to the Asairepresentation. To define it, let E/F be an arbitrary (not just quadratic)extension with d := [E : F ]. Then the Asai representation

AsE/F : LResE/FGLn = GLdn(C) o GalF −→ GLnd(C)

is defined by stipulating that for τ ∈ HomF (E,F ),

AsE/F (((gτ ) o 1))(⊗τvτ ) = ⊗τgτvτ , AsE/F ((1)τ o σ)(⊗τvτ ) = ⊗τvστ .

A representationρ : GalE −→ GLn(C)

extends uniquely to a homomorphism

ρ : GalF −→ LResE/FGLn

commuting with the projections to GalF on the L-group side. Thus to eachsuch ρ we can associate the representation

AsE/F (ρ) := AsE/F ρ : GalF −→ GLnd(C).

Note that for all field extensions L ≥ E one has

AsE/F (ρ)∣∣GalL

∼= ⊗τ∈HomF (E,F )ρτ∣∣GalL

.

Thus the Asai representation AsE/F (ρ) is a canonical extension of

⊗τ∈HomF (E,F )ρτ

to GalF . Suppose now that E/F is quadratic and that τ is the gen-erator of Gal(E/F ). Let π be a cuspidal automorphic representation ofAResE/FGLn\ResE/FGLn(AF ). In this case the analytic properties of the AsaiL-function are understood by Theorem 13.7.2. The discussion above (withglobal Galois groups replaced by local Weil-Deligne groups) implies that

L(s, π × πτ ) = L(s, π,AsE/F )L(s, π ⊗AsE/F ⊗ ηE/F ) (14.8)

where ηE/F is the character attached to Gal(E/F ) by class field theory.In particular, by Theorem 11.7.1 a pole of L(s, π,AsE/F ⊗ ηiE/F ) for some

14.5 Symmetric subgroups 285

i ∈ 0, 1 implies that π ∼= π∨τ . Moreover, if L(s, π,AsE/F ⊗ηiE/F ) has a poleat s = 1 then π is a functorial transfer from a quasi-split unitary group withrespect to an L-map depending on the parity of n and i [Mok15, Theorem2.5.4(a), Remark 2.5.5]. This is again consonant with the expectations in§14.4.

If n = 2, F = Q and E is a real quadratic extension then Theorem 14.5.3was proven in [HLR86]. It was then used to prove many cases of the Tate con-jecture for Hilbert modular surfaces. As mentioned in the introduction, thispaper is significant because it was where the notion of a distinguished repre-sentation was first defined. Extensions of Harder, Langlands and Rapoports’results are contained in [Ram04, GH14].

We now move on to symmetric subgroups of GLn (not ResE/FGLn). Let

σm,n : GLm+n −→ GLm+n

be conjugation by(Im−In

). Then let

H := GLσm+n = GLm ×GLn.

The first result is the following [FJ93, Proposition 2.1]:

Proposition 14.5.4 If m > n then no cuspidal automorphic representationof GLm+n(AF ) is distinguished by (GLm, χ det) for any quasi-characterχ : F×\A×F → C×. ut

Here we view GLm as the subgroup GLm × 1 of GLm × GLn = H. In par-ticular, no cuspidal automorphic representation of GLm+n(AF ) is (H,χ)-distinguished for any χ.

Thus from the point of view of cuspidal representations the only interestingcase is when m = n. In this case consider the quasi-character

µs : H(AF ) −→ C×(ab

)7−→

∣∣∣∣det a

det b

∣∣∣∣s−1/2

χ

(det a

det b

)η(det b),

where χ and η are characters [Gm] (in particular they are trivial on AGm). Inthis setting Jacquet and Friedberg [FJ93, Theorem 4.1] prove the following:

Theorem 14.5.5 Let π be a cuspidal automorphic representation of GL2n(AF )with central character ωπ. Assume that ηnωπ = 1. The cuspidal representa-tion π on GL2n(AF ) is (H,µs0)-distinguished if and only if L(s, π,∧2 ⊗ η)has a pole at s = 1 and L(s0, π ⊗ χ) 6= 0. ut

In fact Friedberg and Jacquet prove that for ϕ in the space of π the pe-riod integral Pµs0 (ϕ) is a holomorphic multiple of L(s0, π ⊗ χ) (under the

assumption that L(s, π,∧2 ⊗ η) has a pole at s = 1). Ash and Ginzburg useTheorem 14.5.5 to construct p-adic L-functions under a technical hypothesis,see [AG94].


If η = 1 and π has trivial central character, the condition that L(s, π,∧2)has a pole at s = 1 is equivalent to the statement that π is a lift of anautomorphic representation of SO2n+1(AF ) by Theorem 13.7.4. We note that

SO2n+1 = Sp2n, and in this case the Lie algebra of GX is sp2n (see [KS17,Table 3]). This again agrees with the expectations of §14.4.

If we take G = GLn and take σ(g) = J−1g−tJ for a symmetric matrixJ ∈ GLn(F ) then Gσ is an orthogonal group On. In the case n = 2 one has acomplete characterization of representations of GL2 distinguished by O2 viawork of Waldspurger reinterpreted and reproved in [Jac86]. In general the rep-resentations of GLn(AF ) distinguished by On should be functorial lifts fromthe topological double cover of GLn(AF ), which is not the adelic points of areductive group. This expectation is stated as a precise conjecture in [Mao98].In the same paper it is more or less reduced to a local conjecture known asthe Jacquet-Mao fundamental lemma. The Jacquet-Mao fundamental lemmais now known [Do15] so it seems likely that the expected characterization ofdistinguished representations mentioned above will be proven in the near fu-ture. It would be interesting to see how the Sakellaridis-Venkatesh conjecturesmentioned §14.5 could be generalized to handle this setting.

If we take G = GL2n and take σ(g) := J−1g−tJ for a skew-symmetric ma-trix J ∈ GL2n(F ) then Gσ is a symplectic group Sp2n. It is not hard to checkthat all of these subgroups are conjugate under GLn(F ), so the choice of Jis irrelevant (see Exercise 14.1). It is known that there are no cuspidal au-tomorphic representations of GL2n that are distinguished by Sp2n by [JR92,Proposition 1]. However there are discrete automorphic representations ofGL2n that are distinguished by Sp2n; these are classified in [Off06].

We close by noting that in the discussion above we have treated everysymmetric subgroup H of GLn up to conjugation by GLnF , see [GW09,§11.3.4]. Of course, this is not the same as conjugation up to GLnF , but weomit a discussion for brevity.

14.6 Relationship with the endoscopic classification

We continue to let F be a number field and E/F be a quadratic extension inthis section. In the previous section we explained many results on distinctionof cuspidal automorphic representations of GLn(AF ) and ResE/FGLn(AF )with respect to symmetric subgroups. These results are very interesting, butthey all suffer a serious drawback. Since the locally symmetric space attachedto GLn(AF ) is never hermitian for n > 2, there is no obvious connection be-tween algebraic and arithmetic geometry and distinction implied by the re-sults of the previous section in higher rank. In order to obtain such results, thetheory of Shimura varieties exposed in §15.8 below indicates that we shouldreally obtain analogous results for automorphic representations of classical

14.6 Relationship with the endoscopic classification 287

groups (see also §15.6). Of coures this is also of interest independently ofapplications to algebraic or arithmetic geometry.

The first author and Wambach [GW14] provide one means of relating dis-tinction of automorphic representations of GLn to distinction of automorphicrepresentations on unitary groups (including those defining hermitian locallysymmetric spaces). It is suggestive of broader phenomena that merit study.

To state the result, let E/F be a quadratic extension of totally real numberfields, let 〈σ〉 = Gal(E/F ), and let M/F be a CM extension. Let H be aquasi-split unitary group in n variables attached to the extension M/F andlet G := ResE/FH. It is the restriction of scalars down to F of a quasi-splitunitary group attached to the extension ME/E. The following is the maintheorem of [GW14] under some simplifying assumptions:

Theorem 14.6.1 Let π be a cuspidal automorphic representation of G(AF ).Suppose that π satisfies the following assumptions:

(a) There is a finite-dimensional representation V of GF∞ such that π isV -cohomological.

(b) There are finite places v1 6= v2 of F totally split in ME/F such that πv1

and πv2 are supercuspidal.(d) For all places v of F such that ME/F is ramified one of the extensionsM/F or E/F is split.

The representation π admits a base change π′ to ResM/EG(AE) := GLn(AME)with respect to the L-map of §13.7:

r : LG −→ ResM/EGLn.

If the partial Asai L-function LS(s, π′,AsME/F ) has a pole at s = 1 thensome cuspidal automorphic representation π′′ of G(AF ) in the L-packet of π′

is H-distinguished. Moreover, we can take π′′ to be V -cohomological. ut

Here S is a sufficiently large finite set of places of M including all infiniteplaces. The notion of V -cohomological is explained in §15.5 below.

The heart of the proof is based on a particular case of a general principlewhich we now explain. Let G be a reductive group, let 〈τ〉 = Gal(M/F ), andlet

G := ResM/FG.

Let σ be an automorphism of G of order 2 and let H := Gσ ≤ G and Gσ ≤ Gbe the subgroups fixed by σ. Let θ = σ τ , viewed as an automorphism of G,and let Gθ ≤ G be the subgroup fixed by θ. Let π be a cuspidal automorphicrepresentation of G(AF ). Assume that π admits a weak transfer π′ to G(AF ).In favorable circumstances the machinery developed in [GW14] together withsuitable fundamental lemmas should imply the general principle:

The representation π′ is Gσ-distinguished and Gθ-distinguished if andonly if a cuspidal automorphic representation in the L-packet of π is H-distinguished.


We have not asserted this principle as a conjecture because it may be false asstated. In particular one might need to work with, e.g. (H,χ)-distinguishedrepresentations for characters χ or incorporate pure inner forms of the groupsinvolved (see §14.7 for the definition of pure inner forms). In Theorem 14.6.1

the group G is a unitary group and G is the restriction of scalars of a generallinear group. Even restricting G to this setting there are more interestingcases to consider. More generally, the hope is that generalizing the principleand explicating it will allow one to classify the distinguished representationsof classical groups in terms of their transfer to general linear groups much aswe have a classification of the whole discrete spectrum of classical groups interms of the general linear group due to the theory of twisted endoscopy (see§13.8 and [Art13, Mok15]). We remark that in many case one direction of theprinciple can be obtained as a consequence of known results on Langlandsfunctoriality and the so-called residue method. For a family of exampleswe refer to [PWZ19].

14.7 Gan-Gross-Prasad type period integrals

The Gan-Gross-Prasad conjecture is an characterizing representations of apair of groups distinguished by a subgroup embedded diagonally [GGP12a,GGP12b]. It can be viewed as a special case of the Sakellaridis-Venkateshconjectures mentioned in §14.5, though the original version of the conjecture,due to Gross and Prasad, is older [GP92].

For simplicity, we will only state the conjectures of Gan, Gross and Prasadin a restricted setting, referring to [GGP12a, GGP12b] for the more generalsetup. Let F be a local or global field and let E be an etale F -algebra ofdegree 1 or 2. In other words E = F , E = F ⊕F , or E/F is a quadratic fieldextension. We let

σ : E −→ E

be the generator of Gal(E/F ). Let V be a finite rank free E-module. Welet V0 be the fixed points of σ acting on V ; it is an F -vector space. AnE-Hermitian form 〈 , 〉 on V is a pairing

〈 , 〉 : V × V −→ E

that is E-linear in the first variable and satisfies

〈v, u〉 = σ(〈u, v〉). (14.9)

We assume that 〈 , 〉 is nondegenerate. We let GV be the group whose pointsin an F -algebra R are given by

GV (R) := g ∈ GLV0(E ⊗F R) : 〈gv, gw〉 = 〈v, w〉 for all v, w ∈ V ⊗F R.

14.7 Gan-Gross-Prasad type period integrals 289

Let GV be the neutral component of GV . Then GV is a classical group. Welist the possibilities in the following table:

E GV

E = F OV

E = F ⊕ F GLV0

E/F quadratic UV

Here OV and UV are the orthogonal and unitary group of V , respectively.In order to proceed it is useful to use the definition of a pure inner form

of an algebraic group G. We will give two equivalent definitions, one usingGalois cohomology, and an ad-hoc definition using the specific features ofour given situation. The reader unfamiliar with Galois cohomology shouldfeel free to skip to the ad-hoc definition below.

A canonical reference for Galois cohomology is [Ser02]. Two algebraicgroups G and G′ over F are said to be forms of eachother if GF sep ∼= G′F sep .For an algebraic group H over a field k we let

H1(k,H) := H1(Galk, H(ksep)) (14.10)

denote the usual Galois cohomology set. The forms of G are in bijection withH1(F,Aut(G)). The so called inner forms are those forms correspondingto the subgroup of inner automorphisms. This means that the correspondingcocycles are in the image of the map

H1(F,G/Gder) −→ H1(F,Aut(G)).

The pure inner forms are those corresponding to cocycles in the image ofthe map

H1(F,G) −→ H1(F,G/Gder) −→ H1(F,Aut(G)).

Two pure inner forms are said to be equivalent if the corresponding cocyclesin H1(F,G) are equivalent. It turns out that the pure inner forms of GV areall of the form GV ′ for free E-modules V ′ with E-rank equal to V . Morespecifically, if E and V are fixed the side conditions on V ′ that are equivalentto GV ′ being a pure inner form of GV are given in the following table (see[GGP12b, §2]):


E conditions

E = F rankEV′ = rankEV and disc(V ) = disc(V ′)

E = F ⊕ F rankEV′ = rankEV

E/F quadratic V ′ ∼= V as hermitian spaces

We recall the following:

Proposition 14.7.1 If G and G′ are reductive groups over a global field Fand are inner forms of eachother then there exists a finite set of places S ofF such that for v 6∈ S one has an isomorphism GFv→G′Fv .

Proof. There is a finite set of places S of F such that GFv and G′Fv arequasisplit for v 6∈ S [Spr79, Lemma 4.9]. On the other hand two inner formsof the same quasi-split reductive group are necessarily isomorphic [Mil17,Corollary 23.53]. ut

Thus for S as in the proposition we can unambiguously identify isomorphismclasses of representations of G(ASF ) and G′(ASF ). We say that automorphicrepresentations π of G(AF ) and π′ of G′(AF ) are nearly equivalent if πS ∼=π′S for any finite set of places S as above. By design, the notion of nearequivalence is unchanged if we enlarge S.

Let W < V be a subspace nondegenerate with respect to 〈 , 〉 such thatrankEV/W = 1. Then the restriction of 〈 , 〉 toW is naturally an E-Hermitianform and we can form the group GW as above. Moreover one has a naturalembedding ι : GW → GV and hence an embedding of GW into

G := GV ×GW

given on points in an F -algebra R by

GW (R) −→ GV (R)×GW (R)

g 7−→ (ι(g), g).(14.11)

We let H be the image of GW . If G′ is a pure inner form of G, so G′ =GV ′ ×GW ′ with V ′ and W ′ as in the table above we let H ′ be the image ofGW ′ under the analogous map GW ′ → GV ′ ×GW ′ .

Assume that GW and GV are quasi-split. We then have standard repre-sentations rV of LGV and rW of LGW given in the table (13.11) and we canform the tensor product representation rV ⊗ rW . In the special case whereGV and GW are general linear groups we take the standard representationto be the identity map.

Assume for the remainder of the section that F is a global field. Let πis a cuspidal automorphic representation of AG\G(AF ) that lies in a globaldiscrete generic L-packet in the sense of §13.8. In the special case where GV

14.7 Gan-Gross-Prasad type period integrals 291

and GW are general linear groups this simply means that π is a cuspidalautomorphic representation.

Conjecture 14.7.2 (Gan-Gross-Prasad) One has

L( 12 , π, rV ⊗ rW ) 6= 0

if and only if there is a pure inner form G′ of G and a cuspidal automorphicrepresentation π′ of G′(AF ) nearly equivalent to π that is H ′-distinguished.

There is a local version of the conjecture which we omit. The import ofthe local version for the global conjecture 14.7.2 is that it allows one todetect fairly explicitly which π′ in the near equivalence class of π is distin-guished. Stating this in a reasonable form seems to require one to collectrepresentations on the adelic points of all the pure inner forms of G into aso-called Vogan L-packet. We prefer to not make this precise and insteadrefer the reader to the lucid explanation in [GGP12b]. Assuming certain ex-pected properties of Vogan L-packets the local Gan-Gross-Prasad conjecturehas been proven (for tempered representations) in the orthogonal case byMœglin and Waldspurger [MW12] and by Beuzart-Plessis in the unitary case[BP16].

We state now what is known about the global conjecture. When E = F⊕Fthe groups GV and GW are general linear groups. In this case the conjectureis true (see Exercise 14.9). In fact the general linear group admits only oneisomorphism class of pure inner form. Thus in the statement of the conjecturewe may take G = G′ and H = H ′ in this case.

Under suitable assumptions on G (in particular that it is orthogonal)Ginzburg, Jiang and Rallis proved one implication, namely if some repre-sentation in the near equivalence class of π is distinguished then the centralL-value is nonzero [GJR05, Theorem A] (see also [GJR04]). The proof is ele-gant in its simplicity. It is a clever application of the theory of automorphicdescent which explicitly constructs the inverse of the functorial lift from clas-sical groups to the general linear group. It is this theory that underlies oneproof of Theorem 13.7.4.

Currently the most definitive result on the global Gan-Gross-Prasad con-jecture is in the unitary case:

Theorem 14.7.3 Assume that E/F is a quadratic extension of number fields(so that G is a product of unitary groups). Assume that there is a nonar-chimedean place v of F such that the transfer (rV ⊗rW )(πv) is supercuspidal.Then Conjecture 14.7.2 is true. ut

The method of proof is quite beautiful and was suggested by Jacquetand Rallis [JR11]. One reduces the unitary case of the Gan-Gross-Prasadconjecture to the general linear case, which, as mentioned above, is known,using a comparison of relative trace formulas that in turn relies on a localconjecture called the Jacquet-Rallis fundamental lemma that was proven by


Z. Yun [Yun11]. The approach is an antecedent of the principle enunciatedin §14.6. It is interesting to note that, unlike in the settings considered in§14.5 and in Theorem 14.6.1, allowing pure inner forms of G is necessary forthe theorem to be true. It is expected that including pure inner forms will benecessary in proving cases of the principle in §14.6.

Theorem 14.7.3 is the culmination of the work of quite a few mathemati-cians. After Z. Yun proved the Jacquet-Rallis fundamental lemma, W. Zhangestablished another important local result (the smooth transfer) in the nonar-chimedean case, and then proved a slightly weaker version of Theorem 14.7.3[Zha14]. The archimedean case was proven by H. Xue [Xue17]. Other nec-essary local results were completed by Bleuzart-Plessis [Beu16]. Finally, aswith most trace formula comparisons, to obtain a complete result requiresdelicate arguments deal with the fact that the geometric and spectral sidesof trace formulae are rarely convergent. In the current setting this work wasundertaken by Chaudouard and Zydor. The theorem in the form stated aboveis [CZ16, Theoreme 1.1.6.2].

The work above does not generalize in an obvious way to the orthogonalcase of Conjecture 14.7.2. However, an approach to this case has recentlybeen proposed and completed in a low rank setting [Kri19].

We close this section by mentioning an exciting analogue of Conjecture14.7.2 that (in certain cases) gives a formula for L′( 1

2 , π, rV ⊗ rW ) whereL( 1

2 , π, rV ⊗ rW ) = 0. The derivative can in some cases be related to heightsof algebraic cycles, and the relationship represents an important link be-tween representation theory and arithmetic geometry. A formula of this typewas conjectured in [GGP12b] and is known as the arithmetic Gan-Gross-Prasad conjecture. It generalizes the famous Gross-Zagier formula [GZ86],which is a key input into the best known results on the Birch and Swinnerton-Dyer conjecture in the number field case. In the unitary setting, the arithmeticGan-Gross-Prasad conjecture was essentially reduced to a local conjectureknown as the arithmetic fundamental lemma by W. Zhang [Zha12b]. Z. Yunand W. Zhang together used related ideas to prove the best known resultstowards the Birch and Swinnerton-Dyer conjecture in the function field case[YZ17, YZ19].

14.8 Necessary conditions for distinction

Assume for the moment that F is a global field. One should be aware that forgeneral reductive groups G and connected subgroups H ≤ G there do not ex-ist cuspidal automorphic representations of G(AF ) that are H-distinguished.A somewhat trivial example is given by taking H to be the unipotent radi-cal of a parabolic subgroup. In this case there are no cuspidal automorphicrepresentations of G(AF ) that are H-distinguished by the very definition ofa cuspidal representation.

14.8 Necessary conditions for distinction 293

A more substantial result was obtained by Ash, Ginzburg, and Rallis in[AGR93]:

Theorem 14.8.1 If (G,H) is one of the following pairs of reductive groupsover a number field F then there are no cuspidal automorphic representationsof G(AF ) that are H-distinguished:

(GLn+k,SLn × SLk) for n 6= k,(GL2n,Sp2n),(GL2n+1,Sp2n),(SO(n, n),SLn) for n odd,(Sp2(n+k),Sp2n×Sp2k),(Sp2n,Sp2(n−l)) with 4l < n.

ut

There is an additional case in [AGR93] which we omit since it involves cuspforms on (disconnected) orthogonal groups and we have not defined cuspforms on such groups. We also point out that the theorem is stated in thecase F = Q in [AGR93] but this is not used in the proofs. We refer to [AGR93]for the definition of the embeddings of H into G.

Let F be a nonarchimedean local field of characteristic not equal to 2 andlet G be a quasi-split reductive group over F equipped with an automorphism

σ : G −→ G

of order 2. LetH := ((Gσ))der.

In this setting Prasad [Pra19, Theorem 1] gave a beautifully simple criterionfor there to exist generic representations of G(F ) distinguished by H(F ):

Theorem 14.8.2 If there is an irreducible admissible generic representationof G(F ) distinguished by H(F ) then there is a Borel subgroup B < GF suchthat B ∩ σ(B) is a maximal torus of GF . ut

One also has the following partial converse [Pra19, Proposition 11]:

Proposition 14.8.3 If there is a Borel subgroup B < G such that B ∩σ(B)is a maximal torus of G then there exists an irreducible generic representationof G(F ) distinguished by Gσ(F ). ut

In this proposition we are assuming B is a subgroup of G, not GF .

Exercises

14.1. Let G be a reductive group over a local or global field F and letH,H ′ ≤ G be subgroups that are G(F )-conjugate. If F is local prove that an


irreducible admissible representation of G(F ) is H-distinguished if and onlyif it is H ′-distinguished. If H is the direct product of a reductive and a unipo-tent group prove that a cuspidal automorphic representation π of G(AF ) isH-distinguished if and only if it is H ′-distinguished.

14.2. Let H be a reductive group over the nonarchimedean local field F ,viewed as a subgroup of G := H ×H via the diagonal embedding. Show thatan irreducible admissible representation π1 × π2 of G(F ) is H-distinguishedif and only if π1

∼= π∨2 .

14.3. Assume that G is a reductive group over a global field F and thatH ≤ G is a subgroup. Prove that

AG ∩H(AF ) = AG ∩AH .

14.4. Let H be a reductive group over the global field F viewed as a subgroupof G := H × H via the diagonal embedding. Let π1 and π2 be cuspidalautomorphic representations of AH\H(AF ) and let π = π1 ⊗ π2 be theirexterior tensor product, viewed as a cuspidal automorphic representation ofAG\G(AF ). Show that π1 ⊗ π2 is H-distinguished if and only if π1

∼= π∨2 .

14.5. Let G be a reductive group over a global field F , let H ≤ G be asubgroup, and let χ : H(AF ) → C× be a quasi-character trivial on H(F ).Let π = ⊗′vπv be a cuspidal automorphic representation of G(AF ) that is(H,χ)-distinguished. Prove that for all nonarchimedean v the representationπv is smoothly (H,χv)-distinguished.

14.6. Give an example of a subgroup H ≤ GL2×GL2 (over a global field F )a quasi-character χ : H(AF ) → C×, and a smooth function ϕ in a cuspidalautomorphic representation of (AGL2\GL2(AF ))2 such that the integral∫

[H]

|ϕ(h)χ(h)|dh

diverges.

14.7. Assume the notation in the proof of Proposition 14.3.3. Give an ex-ample of a pair of reductive subgroups H ≤ G such that the closure ofAH − (AH ∩AT0(r0)) is noncompact for any choice of r0 > 0 and any choiceof simple roots ∆ of T0 in G.

14.8. Prove (14.8).

14.9. Prove the Gan-Gross-Prasad conjecture when E = F ⊕ F (and henceGV and GW are general linear groups) using Rankin-Selberg theory.

Chapter 15

The Cohomology of Locally SymmetricSpaces

We are in a forest whose treeswill not fall with a few timidhatchet blows. We have to takeup the double-bitted axe and thecross-cut saw, and hope that ourmuscles are equal to them.

R. P. Langlands

Abstract In this chapter we discuss how automorphic representation theorycontrols the cohomology of arithmetic locally symmetric spaces.

15.1 Introduction

A fruitful application of the theory of automorphic representations is to givethe decomposition of the cohomology of locally symmetric spaces as a mod-ule under certain algebras of correspondences. We survey this connection inthis chapter. The primary technical tool one uses to make the connection is(g,K∞)-cohomology. The definition is given in §15.4.

One case that is particularly important is when the locally symmetric spaceis the C-points of a quasi-projective scheme over a number field. Shimura wasthe first to study this case systematically, and in his honor such varieties arecalled Shimura varieties. We recall Deligne’s reformulation of the notion ofa Shimura variety at the end of this chapter. Part of the reason that this caseis so important is it is one of the few cases where there is a direct relationshipbetween automorphic representations and Galois representations. We pointthe reader to some of the vast body of work that has been done elucidatingthis relationship in §15.8.

Our real motivation for this chapter, however, is to explicitly discuss therelationship between distinguished representations and cycles on locally sym-

295

296 15 The Cohomology of Locally Symmetric Spaces

metric spaces. This was in fact the original motivation for Harder, Langlands,and Rapoport’s introduction of the notion of a distinguished automorphicrepresentation [HLR86]. In this paper they related period integrals of au-tomorphic representations to algebraic cycles on Hilbert modular varieties.Using this relation they proved cases of the Tate conjecture for these varieties.There has been work generalizing their results, but even very basic questionsremain. We point out one such question explicitly in §15.6. We hope that inthe near future the knowledge one now has about the etale cohomology ofShimura varieties can be combined with the relative trace formula to system-atically study the Tate and Beilinson-Bloch conjectures for these varieties.It is beyond the scope of this book to make this completely precise, but wemention some promising results that have been obtained in §15.8.

Understanding the material in this chapter requires some knowledge ofsheaves and cohomology. In §15.8 we even briefly discuss etale cohomology ofShimura varieties. The results will not be used in the remainder of the book,so the reader should feel free to skip this chapter on a first reading. However,we encourage the reader to at some point investigate the fascinating interplaybetween arithmetic geometry and automorphic forms that is the subject ofLanglands’ quote in the epigraph.

15.2 Locally symmetric spaces

Unless otherwise specified throughout this chapter G is a connected reductivegroup over Q. We let A := AQ be the adeles of Q, let K∞ ≤ G(R) be acompact subgroup containing the maximal connected compact subgroup (inthe real topology) and, as before, let

AG ≤ G(R)

be the identity component in the real topology of the maximal Q-split torus inthe center of G. To ease notation, we write K for K∞ ≤ G(A∞), a compact-open subgroup. Finally we set

X := AG\G(R)/K∞.

Any connected component of X is a symmetric space. One can essentiallytake this to be the definition of a symmetric space, although it is not themost natural definition. We let

ShK := Sh(G,X)K := G(Q)\X ×G(A∞)/K (15.1)

and refer to it as a Shimura manifold. As explained in (2.14) this is a finiteunion of locally symmetric spaces. Indeed, if we take a set of representatives(gi)i∈I for the finite set G(Q)\G(A∞)/K then

15.2 Locally symmetric spaces 297∐i∈I

Γi\X−→ShK (15.2)

Γix 7−→ G(Q)(x, gi)K

where Γi = giKg−1i ∩G(Q).

To relate this setting within the traditional framework of locally symmetricspaces it is useful to have in mind the notion of arithmetic and congruencesubgroups. These were already defined in §2.6, but we recall the definitionfor the convenience of the reader:

Definition 15.1. A subgroup Γ ≤ G(Q) is arithmetic if there is somefaithful representation ρ : G → GLn such that ρ(Γ ) is commensurable withρ(G(Q)) ∩GLn(Z). It is congruence if it is of the form G(Q) ∩K for somecompact open subgroup K ≤ G(A∞).

Congruence subgroups are arithmetic (see Exercise 15.3). The famous con-gruence subgroup problem asks whether every arithmetic subgroup is con-gruence. The answer is in general no, but the failure of arithmetic subgroupsto be congruence can be controlled via techniques that originated in the in-fluential paper [BMS67] (see also [Men65]). In particular, every arithmeticsubgroup of SLnQ, n ≥ 3, and Sp2nQ, n ≥ 2, is congruence (see loc. cit.). Thesurvey by Prasad and Rapinchuk in [Mil10] is a source for more up to dateinformation.

A useful assumption on a subgroup Γ ≤ G(Q) is that it is torsion-free.This implies, for example, that

Γ\X (15.3)

is a manifold, as opposed to an orbifold (see Exercise 15.2). Unfortunately, theproperty of being torsion free is not preserved under certain constructions,such as intersecting with a parabolic subgroup and mapping to the Leviquotient. A more robust condition is that of being neat:

Definition 15.2. An element g ∈ GLn(Q) is neat if the subgroup of Q×

generated by its eigenvalues is torsion-free. An arithmetic subgroup Γ ≤G(Q) is neat if given any (equivalently, one faithful) representation ρ : G→GLn, ρ(g) is neat for all g ∈ Γ .

Clearly, if Γ is neat, then all its subgroups and homomorphic images areneat.

Lemma 15.2.1 Any congruence subgroup Γ ≤ G(Q) contains a congruencesubgroup of finite index that is neat.

Proof. When G = GLn, we can take the neat subgroup to be of the form

g ∈ GLn(Z) : g ≡ I mod N

for a sufficiently divisible integer N . The general case follows from this. ut


Definition 15.3. A compact open subgroup K ≤ G(A∞) is neat if

G(Q) ∩ g−1Kg

is neat for all g ∈ G(A∞).

In fact, one only has to check this condition for all g ∈ G(Q)\G(A∞)/K,which is a finite set. An elaboration of the proof of Lemma 15.2.1 implies thefollowing (see Exercise 15.4):

Lemma 15.2.2 If K ≤ G(A∞), then K contains a neat subgroup of finiteindex. ut

Our motivation for introducing this notion is the following:

Lemma 15.2.3 If K is neat then ShK is a smooth manifold.

Proof. Since K is neat gKg−1 ∩ G(Q) is torsion-free for all g ∈ G(A∞). Itfollows that the Γi occurring in the decomposition (15.2) are all torsion free.As mentioned above, this implies that the Γi\X are all manifolds. ut

In §5.2 we discussed the Hecke algebra C∞c (G(A∞)). We recall that this isthe space of C-linear combinations of characteristic functions of double cosetsKgK for g ∈ G(A∞). We now explain how these operators can be realizedgeometrically as correspondences. Let K and K ′ be compact open subgroupsof G(A∞), and g ∈ G(A∞) be such that K ′ ⊂ gKg−1. Then we have a map

Tg : ShK′−→ ShK (15.4)

G(Q)(x, hK ′) 7−→ G(Q)(x, hgK).

These are finite etale maps if K and K ′ are neat.The geometric realization of the characteristic function of KgK is the

correspondence

ShK∩gKg−1

TIyy Tg %%ShK ShK .

(15.5)

It acts on functions via pullback T ∗I along TI followed by pushforward Tg∗along Tg. In other words if we set K ′ := K ∩ gKg−1 then for functions ϕ on

ShK and ϕ′ on ShK′

T ∗I (ϕ)(x, hK ′) : = ϕ(x, hK)

Tg∗(ϕ′)(x, hK) : =

∑k′∈K/gK′g−1

ϕ′(x, hk′gK ′).

15.3 Local systems 299

Set

T (g) := Tg∗ T ∗I (15.6)

Then for any function ϕ : ShK → C,

T (g)ϕ = R(1KgK)ϕ, (15.7)

where

R(1KgK)ϕ(x) :=

∫G(A∞)

1KgK(g)ϕ(xg)dg

with the Haar measure dg normalized so that dg(K) = 1.

15.3 Local systems

A well-known classical fact is that if Γ is a congruence subgroup of GL2Qcontained in the subgroup of matrices with positive determinant and H is theupper half-plane then the cohomology group

H1(Γ\H,C)

can be decomposed as a direct sum of three summands, one isomorphic to thespace of weight 2 cusp forms on Γ namely S2(Γ ) (see §6.6 for the definition),one isomorphic to the space

f(z) : f ∈ S2(Γ )

of antiholomorphic forms, and one isomorphic to a certain space of Eisensteinseries. In order to give a geometric interpretation of modular forms of weightbigger than 2, it is necessary to introduce local systems. We now recall theirconstruction in the case at hand. Our treatment is largely modeled on thatof the thesis of S. Morel [Mor08].

Until further notice Γ ≤ G(Q) is a neat arithmetic subgroup. Let V be arepresentation ofG equipped with the discrete topology and form the quotient

VΓ := Γ\(V ×X) (15.8)

by the diagonal action of Γ on the product. We say that the diagram givenby the natural projection

VΓ −→ Γ\X (15.9)

is a local system. For example, if V is a representation of Γ over C, thenthis is the total space of a locally constant sheaf of C-vector spaces. For each


open set U ⊆ Γ\X we let

VΓ |U := s : U −→ VΓ (15.10)

be the abelian group of sections of the map (15.9). Then the functor

U ⊆ Γ\X −→ Ab (15.11)

U 7−→ VΓ |U

from the category of open sets of Γ\X with morphisms given by inclusionsto the category of abelian groups Ab is a sheaf. It is called the sheaf ofsections of VΓ . Its sheaf cohomology is denoted

H•(Γ\X,VΓ ). (15.12)

We now turn to the adelic setting. As above, let V be a representation ofG. For K ≤ G(A∞) a compact open neat subgroup, define

VK := G(Q)\V ×X ×G(A∞)/K, (15.13)

where for γ ∈ G(Q),

γ(v, x, gK) = (γ.v, γ.x, γgK).

We thus have a natural map VK → ShK . This is again a local system, andwe define the associated sheaf of sections as above.

The relationship between the two constructions given above can be de-scribed as follows. Fix g ∈ G(A∞) and let Γ = gKg−1 ∩G(Q). We then havean embedding

ι : Γ\X −→ ShK (15.14)

Γx 7−→ G(Q)(x, g)K.

If V is a representation of G(Q) then ι−1VK ∼= VΓ .We now describe how to associate to each g ∈ G(A∞) a homomorphism

T (g) : H•(ShK ,VK) −→ H•(ShK ,VK). (15.15)

We first note that for any neat compact open subgroups K,K ′ ≤ G(A∞)with K ′ ≤ gKg−1 one has an isomorphism of sheaves

θ : VK′−→T−1

g VK = VK ×ShK ShK′

(15.16)

(v, (x, hK ′)) 7−→ ((v, (x, hgK)), (x, hK ′)).

Here T−1g VK is the inverse image sheaf and the object on the right is the fiber

product over the canonical projection VK → ShK and the morphism Tg :

15.3 Local systems 301

ShK′→ ShK . The isomorphism θ is called a lift of the correspondence.

This isomorphism canonically induces a morphism (15.15). Concretely theprocedure is to take a cocycle on ShK with values in VK , pull it back to

ShK′, use the isomorphism θ to produce a cocycle on ShK

′with values in

T−1g VK , and then push it down along Tg while summing over the fibers of

T−1g VK → VK to obtain a cocycle on ShK .

Alternately, one can work at the level of sheaves as follows. By generalitieson sheaf cohomology [Har11, (4.33)] one has a homomorphism

H•(ShK ,VK) −→ H•(ShK′, T−1I V

K) = H•(ShK′,VK

′). (15.17)

Here the equality results from the identification T−1I VK = VK′ . The isomor-

phism θ induces an isomorphism

θ : H•(ShK′,VK

′)−→H•(ShK

′, T−1g VK). (15.18)

To finish we construct a “trace” homomorphism

H•(ShK′, T−1g VK) −→ H•(ShK ,VK). (15.19)

Composing (15.17), (15.18), and (15.19) then gives

T (g) : H•(ShK ,VK) −→ H•(ShK ,VK). (15.20)

To construct (15.19) we follow [Ive86, §VII.4], to which we refer for moredetails. We note that the assertions below all heavily depend on the fact thatTg is a covering map. For any sheaf F on ShK and any y ∈ ShK one has

Tg∗T−1g Fy = ⊕x∈T−1

g (y)Fy.

One can check that there is a unique morphism of sheaves

tr : Tg∗T−1g F −→ F

given on the fiber over y by taking the sum. Now let

VK −→ I•

be an injective resolution of VK . We then obtain a morphism

tr : Tg∗T−1g I• −→ I•. (15.21)

We apply the “cohomology of global sections” functor to obtain a map

H•Γ (ShK , Tg∗T−1g I•) −→ H•(ShK ,VK).

But T−1g transforms injectives into injectives, and Tg∗ is exact, so


H•Γ (ShK , Tg∗T−1g I•) = H•(ShK

′, T−1g VK).

15.4 (g,K∞)-cohomology

As above let V be a finite-dimensional representation of G. A fundamentaltool for describing H•(ShK ,VK) as a Hecke module is (g,K∞)-cohomology.We explain its definition in this section. The basic reference is [BW00].

Let g ≥ k be the Lie algebras of GR and K∞ respectively. Let A be ag-module (not necessarily of finite dimension). For q ∈ Z≥0, let

Cq(g, k;A) = Homk(∧q(g/k),A). (15.22)

Here the action of k on ∧q(g/k) is the adjoint action. Explicitly Homk(∧q(g/k),A)is the subspace of HomC(∧q(g/k),A) consisting of functions f satisfying

q∑i=1

f(x1, . . . , [x, xi], . . . , xq) = x.f(x1, . . . , xq),

for x ∈ k, where [ , ] is the Lie bracket of g. We define a differential

d : Cq(g, k;A)→ Cq+1(g, k;A)

on the complex by

(df)(x0, . . . , xq) =

q∑i=1

(−1)ix · f(x0, . . . , xi, . . . , xq)

+∑

1≤i<j≤q

(−1)i+jf([xi, xj ], x0, . . . , xi, . . . , xj , . . . , xq).

Here a hat denotes an omitted argument. One checks that this is well-defined(i.e. independent of the choice of representative of the coset x+ k) and thatd d = 0. We then let H•(g, k;A) denote the cohomology of this complex:

Hq(g, k;A) := Hq(C•(g, k;A), d) :=ker d : Cq(g, k;A)→ Cq+1(g, k;A)

im d : Cq−1(g, k;A)→ Cq(g, k;A).

This is called the (g, k)-cohomology of A. This cohomology is insensitive tothe connected components of K∞. It is therefore desirable to refine it.

Assume now that A is a (g,K∞)-module. The group K∞ acts in a naturalmanner on C•(g, k;A) via the action of K∞ on A and the adjoint action ong. For this action define

C•(g,K∞;A) = C•(g, k;A)K∞ .

15.5 The cohomology of Shimura manifolds 303

Denote by H•(g,K∞;A) the cohomology of this complex.

Definition 15.4. The (g,K∞)-cohomology of the (g,K∞)-module A isH•(g,K∞;A).

15.5 The cohomology of Shimura manifolds

Let V be a finite dimensional representation of G. We now relate (g,K∞)-cohomology and the cohomology of ShK with coefficients in VK . We refer to[BW00, §VII.2] for proofs.

Let m := dimR(X) for X = AG\G(R)/K∞. Fix a basis ωi, 1 ≤ i ≤ m, ofleft invariant 1-forms on X. For

I = i1, . . . , iq ⊂ 1, . . . ,m

with ij < in for j < n set

ωI := ωi1 ∧ · · · ∧ ωiq .

LetA := C∞(G(Q)\G(A)/K∞).

Denote byAq(ShK ,VK)

the space of differential q-forms on ShK with coefficients in VK . Any elementof Aq(ShK ,V) can be written as

η =∑I

fIωI

with fI ∈ (A⊗ V )AG . Let

aG := Lie(AG). (15.23)

The discussion above yields an identification

A•(ShK ,VK) = C•(aG\g,K∞; (A⊗ V )AG) (15.24)

commuting with the differentials, which in turn yields an isomorphism

H•(ShK ,VK) ∼= H•(aG\g,K∞; (A⊗ V )AG). (15.25)

This gives an explicit link between cohomology and automorphic representa-tions. This isomorphism is Hecke equivariant in the sense that for g ∈ G(A∞)the action of the correspondence T (g) is intertwined with the action of 1KgK .


We leave the proof of this Hecke equivariance assertion to the reader (see Ex-ercise 15.7).

This whole construction motivates the following definition:

Definition 15.5. A vector ϕ ∈ A is cohomological if there exists a repre-sentation V of G, v ∈ V and ωI on G(R) such that

ϕωI ⊗ v ∈ C•(aG\g,K∞; (A⊗ V )AG)

defines a non-zero class in H•(aG\g,K∞; (A⊗ V )AG).

If we wish to specify the representation V we could speak of a V -cohomological vector instead.

Now A is naturally a (g,K∞)×G(A∞)-module. Let us decompose it underthe action of G(A∞). There is a quasi-character

ξ : AG → R>0

such that the restriction of V to AG is ξ−1. Let

A(ξ) := ϕ ∈ A : ϕ(ag) = ξ(a)ϕ(g).

Then(A⊗ VK)AG = A(ξ)⊗ VK .

As usual [G] := AGG(Q)\G(A). There is a G(A)1-intertwining map

A(ξ) −→ C∞([G])K

ϕ 7−→ (g 7→ ξ′(g)ϕ(g))(15.26)

for an appropriate quasi-character ξ′ : G(A)→ R>0. We let

A(ξ)cusp

be the inverse image of L2cusp([G])K ∩ C∞([G])K under the map (15.26).

By Corollary 9.1.2

A(ξ)cusp =⊕π

π⊕m(π)

where the sum is over isomorphism classes of cuspidal automorphic represen-tations π of G(A) and m(π) is the multiplicity of π in A(ξ)cusp. Hence

H•(ShK ,VK) ⊃ H•cusp(ShK ,VK) = H•(aG\g,K∞;A(ξ)cusp ⊗ V ) (15.27)

=⊕π

H•(aG\g,K∞; (π ⊗ V )K

)⊕m(π).

The group H•cusp(ShK ,VK) is known as the cuspidal cohomology. Its com-plement is described in terms of so called Eisenstein cohomology; for some

15.6 The relation to distinction 305

information about the decomposition of the whole of the cohomology see[BLS96]. It follows that, as a module under C∞c (G(A∞) //K), one has

H•cusp(ShK ,VK) =⊕π

(H•(aG\g,K∞; (π∞ ⊗ V )⊗ (π∞)K

)m(π). (15.28)

This decomposition is sometimes called Matsushima’s formula. For a cus-pidal automorphic representation π of AG\G(A) we refer to

H•cusp(ShK ,VK)(π∞) :=⊕

π′:π′∞∼=π∞

(H•(aG\g,K∞; (π′∞ ⊗ V )⊗ (π∞)K

)m(π′)

as the π∞-isotypic component of the cohomology.

Definition 15.6. A cuspidal automorphic representation π of AG\G(A) iscohomological if there is a representation V of G such that

H•(aG\g,K∞;π∞ ⊗ V ) 6= 0.

A vector ϕ in the space of π is cohomological if there exists an embeddingfrom the space of π into L2

cusp([G]) such that the image of ϕ is cohomological.

Thus if there is a vector in the space of π that is cohomological, then πitself is cohomological. If we wish to be more specific, we could speak ofV -cohomological representations or V -cohomological vectors.

The condition that π is cohomological is very restrictive. An easy way toquantify this is given by the following theorem [BW00, Corollary I.4.2]:

Theorem 15.5.1 If H•(aG\g,K∞;π∞⊗V ) 6= 0 then the infinitisimal char-acter of π∨∞ is equal to the infinitisimal character of V . ut

A more or less equivalent way of phrasing this assertion is the followingcorollary:

Corollary 15.5.2 A cohomological cuspidal representation π of G(A) is C-algebraic in the sense of Definition 12.6.

Proof. This follows from Theorem 15.5.1 and Lemma 12.8.1. ut

15.6 The relation to distinction

Usually in the literature one finds references to cohomological representationsbut no references to cohomological vectors. Despite this, the notion of acohomological vector is of great importance.

Suppose that H ≤ G are connected reductive Q-groups and that theircorresponding symmetric spaces are chosen so that XH → X. Then there isan embedding


Sh(H,XH)K∩H(A∞) −→ Sh(G,X)K =: ShK .

For the remainder of this section we assume for simplicity that ShK is com-pact , which is to say that Gder has no nontrivial Q-split subtorus by Theorem2.6.2. We also assume ShK is orientable.

Let V be a representation of G and let V ∨ be the dual representation.We denote by VH the local system on Sh(H,XH)K∩H(A∞) attached to V |H .Let n be the real dimension of ShK and let q be the real codimension ofSh(H,XH)K∩H(A∞) in ShK . There is a cycle class map

H0(Sh(H,XH)K∩H(A∞),VH) −→ Hq(ShK ,VK) (15.29)

defined as follows. For a class [Z] ∈ H0(Sh(H,XH)K∩H(A∞),VH) we have alinear functional

Hn−q(ShK , (VK)∨) −→ C

η 7−→∫Z

η.(15.30)

Poincare duality [Har11, Corollary 4.8.10] furnishes a perfect pairing

Hn−q(ShK , (VK)∨)×Hq(ShK ,VK) −→ C (15.31)

so [Z] defines an element of Hq(ShK ,VK), yielding the cycle class map(15.29).

Describing the C∞c (G(A∞) //K∞)-span of the image of (15.29) is an im-portant (and mostly open) problem. This is especially true in the situationwhere ShK and Sh(H,XH)K∩H(A∞) are Shimura varieties, not just manifolds,due to the connection with the Tate conjecture (see §15.8).

We now explain the precise connection between the image of (15.29) anddistinguished representations and point out an important technical problemthat deserves more study. By Matsushima’s formula (15.28) to describe theC∞c (G(A∞) //K∞)-span of the image of (15.29) it suffices to describe itsprojection to

H•(ShK ,VK)(π∞)

for all automorphic representations π of G(A). Here we have used that ShK iscompact, and hence every automorphic representation of G(AF ) is cuspidal,to simplify the notation: H• = H•cusp.

Unwinding the considerations of §15.5 and the definition of the cycle classmap we see that the projection of (15.29) to H•(ShK ,VK)(π∞) is nonzero ifand only if there is

• a cuspidal automorphic representation π′ of G(A) with π∞ ∼= π′∨,• (v∨, v) ∈ V ∨ ×H0(Sh(H,XH)K∩H(A∞),VH),• a V ∨-cohomological vector ϕ in the space of π′, and a ωI on G(R) such

that

15.7 More on (g,K∞)-cohomology 307∫[Sh(H,XH))K∩H(A∞)]

ϕωIv∨(v) 6= 0. (15.32)

This motivates the following question:

Question 15.6.1 If π is H-distinguished and V ∨-cohomological, then isthere

(v∨, v) ∈ V ∨ ×H0(Sh(H,XH)K∩H(A∞),VH),

an ωI on G(R), a V ∨-cohomological vector ϕ in the space of π such that(15.32) holds?

Recall that π is H-distinguished if and only if∫AG∩H(A)H(Q)\H(A)

ϕ(h)dh 6= 0

for some smooth ϕ ∈ L2(π), the π-isotypic subspace of L2([G]). The pointof the question is that, a priori, it could happen that no such ϕ satisfies theadditional stipulations of the question. For example, it could be that no suchϕ is V ∨-cohomological.

We do not know, even conjecturally, the answer to Question 15.6.1 in anydegree of generality. In special cases the question above can be answered inthe affirmative (see [KS15, Sun17] for example).

In this section we have assumed that ShK is compact. The considerationsabove are of course still of interest in the noncompact case, but even definingthe cycle class map (15.29) in any level of generality is complicated. Extendingthe cycle class map to the noncompact case is another open problem thathas not received the scrutiny it deserves. We refer to [GG12] for an examplewhere the difficulties caused by noncompactness of ShK are overcome in avery special case.

15.7 More on (g,K∞)-cohomology

We have seen that the question of whether or not a given automorphic rep-resentation contributes to the cohomology of a Shimura manifold with coeffi-cients in a local system is completely determined by the (g,K∞)-cohomologyof its factor at infinity. One thing that makes this so useful is the fact that(g,K∞)-cohomology is a very pleasant object with which to work. In thissection we list some properties of these groups; the canonical reference is[BW00].

Let G1, G2 be reductive groups over R and let Ki∞ ≤ Gi(R) be maximalcompact subgroups. Moreover let Ai be admissible (gi,Ki∞)-modules. Thenone has a Kunneth forumla:


Theorem 15.7.1 (Kunneth formula) For any n ∈ Z≥0 one has naturalisomorphisms

Hn(g1⊕g2,K1∞×K2∞;A1⊗A2) =⊕p+q=n

Hp(g1,K1∞;A1)⊗Hq(g2,K2∞;A2).

For the rest of this section let G be a reductive group over Q (we only usethe Q-structure to define AG). Let g be the Lie algebra of GR and let K∞ ≤G(R) be a maximal compact subgroup. Let A be an admissible (aG\g,K∞)-module.

Theorem 15.7.2 (Poincare duality) One has that

Hq(a\g,K∞;A) ∼= H(dimRX)−q(a\g,K∞;A∨)∨.

ut

Thus, in particular, to compute the (g,K∞)-cohomology of a (aG\g,K∞)-module it suffices to understand the case where g is simple over R and whenq ≤ dimRX/2.

We now consider the cohomology of admissible representations of the form

π∞ ⊗ V

where π∞ is an irreducible unitary (g,K∞)-module and V is an irreduciblerepresentation of G trivial on AG. Let

θ : g −→ g (15.33)

be the Cartan involution. Its fixed point subalgebra is k, the Lie algebra ofK. Let p be its −1 eigenspace.

When π∞ is trivial we have (see [BW00, Corollary II.3.2]) the followingproposition:

Proposition 15.7.3 One has that

H•(g,K∞;V ) =

0 if V is nontrivial,

(∧•p)K∞ if V is trivial.

ut

We can interpret this proposition as saying that much of the cohomol-ogy vanishes when V is nondegenerate. This phenomenon persists for otherπ∞. We now explain some vanishing theorems for cohomology in this con-text. Combined with the Poincare duality above these theorems indicate thatcohomology of ShK is concentrated around the degree equal to 1

2dimRShK .For each θ-stable parabolic subalgebra q ≤ g⊗C let n(q) be its nilradical.

Let

15.8 Shimura varieties 309

P(V ) := θ-stable parabolic subalgebras q : dimV n(q) = 1

and letc(V ) := mindim(n(q) ∩ p) : q ∈ P(V ).

Theorem 15.7.4 Assume that the kernel of π∞ is contained in k. Then

Hi(aG\g,K∞;π∞ ⊗ V )

vanishes for i < c(V ∨).

Proof. See [BW00, §II.10]. ut

We close this section by pointing out that Vogan and Zuckerman haveprovided a classification of all (g,K∞)-modules with nonzero cohomologyand computed the cohomology of these modules (see [VZ84]).

15.8 Shimura varieties

In this section we recall the notion of a Shimura variety. Although the founda-tions of Shimura varieties were established by Shimura, a simpler formulationof the theory was introduced by Deligne in [Del71]. Deligne’s formulation hasbeen almost universally adopted in the literature and we will use it here. Anintroduction to [Del71] is contained in [Mil05]. Proofs for the results we statebelow can be found in these references.

The Deligne torus is

S := ResC/R(Gm). (15.34)

Note that S(R) = C×. The following definition is due to Deligne.

Definition 15.7. Let G be a connected reductive group over Q and let Xbe a G(R)-conjugacy class of homomorphisms h : S → GR. The pair (G,X)is a Shimura datum if

(a) For h ∈ X, only the characters z/z, 1, and z/z occur in the representationof S on Lie(Gad)C defined by h. Here Gad := G/ZG is the adjoint group;

(b) θ = ad(h(√−1)) is a Cartan involution of Gad;

(c) Gad has no Q-factors on which the projection of h is trivial.

The condition (b) is equivalent to the statement that

Gad,(θ)(R) := g ∈ Gad(C) : g = θ(g)

is compact. Conditions (a) and (b) together imply that X is a Hermitiansymmetric space for G.


Since ShK is a complex manifold in this case, a natural question is whetherShK can be realized as the complex points of some variety. This is indeedthe case by a basic theorem of Baily and Borel which states that if K isneat then ShK can be given the structure of the complex points of smoothquasi-projective scheme over C in a canonical manner. One can say evenmore.

For each x ∈ X, we have a cocharacter

ux(z) := hxC(z, 1),

where xC denotes be base change of x to C. Its image in

G\X∗(G)(C)

is independent of the choice of x. Here G acts on X∗(G) via conjugation andG\X∗(G) is the quotient in the sense of geometric invariant theory (see §17.1for more information on this construction).

Let E := E(G,X) ⊂ C be the field of definition of ux. It is a number field,independent of the choice of x ∈ X. It is called the reflex field of (G,X).

Theorem 15.8.1 For each neat K ≤ G(A∞) there exists a smooth quasi-projective variety M(G,X)K defined over the reflex field E of (G,X) suchthat

ShK = M(G,X)K(C)

and all the correspondence T (g) are defined over E. Furthermore, there is acanonical such model, characterized by the Galois action on certain specialpoints. ut

Definition 15.8. The Shimura variety attached to (G,X) is the projectivelimit

M := M(G,X) := lim←−K

M(G,X)K

of the canonical models of ShK .

Definition 15.9. A morphism of Shimura data (G,X)→ (G′, X ′) is a mor-phism of algebraic groups G→ G′ such that the induced map G(R)→ G′(R)sends X to X ′.

Definition 15.10. A morphism of Shimura varietiesM(G,X)K →M(G′, X ′)K

is an inverse system of regular maps compatible with the action of Hecke cor-respondences.

Theorem 15.8.2 A morphism of Shimura data induces a morphism ofShimura varieties over the compositum E(G,X)E(G′, X ′). Moreover, it isa closed immersion if G→ G′ is injective. ut

Example 15.1. Let G = GL2,Q and set h to be h(a+ b√−1) =

(a b−b a

). In this

case each Sh(G,X)K is a finite union of modular curves (see the end of §2.6).


Example 15.2. Let G = ResF/QGL2 where F/Q is totally real. Take

h =∏σ

(aσ bσ

−bσ aσ)

for embedding σ : F −→ R. Then the associated Shimura varieties are knownas Hilbert modular varieties.

Example 15.3. Let G = GSp2n. For a Z-algebra R, we have that

GSp2n(R) = g ∈ GL2n(R) : gtJg = c(g)J for some c(g) ∈ R×

for J =(

1

1

). Take h to be h(a + b

√−1) =

(aI bJ−bJ aI

). The associated

Shimura varieties are known as Siegel modular varieties. This can begeneralized to totally real number fields as in the previous example, yieldingHilbert-Siegel modular varieties.

Example 15.4. Let F be a totally real number field and k/F a totally imagi-nary quadratic extension. Let D be a simple algebra with center k and assumeD admits an antiautomorphism ∗ that induces the nontrivial automorphismof k/F . For Q-algebras R set

G(R) := x ∈ D ⊗Q R : xx∗ ∈ R×.

Assume we are given a homomorphism

h0 : C −→ D ⊗Q R

such that h0(z)∗ = h0(z). The restriction of h0 to C× then defines

h : S −→ GR.

Let X be the G(R)-conjugacy class of h. Then (G,X) is a Shimura datum.The associated Shimura varieties are sometimes called Kottwitz varietiessince Kottwitz employed them to fantastic effect in the construction of Galoisrepresentations attached to automorphic representations [Kot92]. There aretwo key properties that make these varieties pleasant objects to consider.First, if D is chosen appropriately ShK is compact. Second, the group Gis a form of a unitary similitude group, and thus for essentially half theprimes p the group GQp is a product of general linear groups. This fact wasexploited in the original proofs of the local Langlands correspondence forGLn [HT01, Hen00].

Suppose that (G,X) is a Shimura datum with the reflex field E. Let K ≤G(A∞) be a neat compact open subgroup. For simplicity assume that ShK

is compact. To ease the notation, let

MK := M(G,X)K ,


where M(G,X)K is as in Theorem 15.8.1. We can then consider the etalecohomology groups

H•et(MKE,Q`) := H•et(M

K ×E E,Q`). (15.35)

We can also consider versions of these groups with coefficients in a localsystem, but let us take trivial coefficients for simplicity. We refer the readerto [Mil80] for proofs of the properties of etale cohomology we use below.

These groups are modules under the Hecke algebra C∞c (G(A∞) //K) andGalE . Upon choosing an isomorphism Q`−→C and an embedding E → Cone obtains a comparison isomorphism

H•et(MKE,Q`)−→H•(ShK ,C) (15.36)

that is C∞c (G(A∞) //K)-equivariant.Combining this with (15.28) we obtain a decomposition

Hiet(M

KE,Q`) =

⊕π∞

Hiet(π

∞)⊗ π∞K

as GalE×C∞c (G(A∞) //K)-modules. Here Hiet(π

∞) is a GalE representationon which C∞c (G(A∞) //K) acts trivially and GalE acts trivially on π∞K . Thesum is over all irreducible admissible G(A∞)-representations π∞ such thatthere is an irreducible admissible representation π∞ of AG\G(R) such thatπ∞⊗π∞ occurs in L2([G]). The GalE-representation Hi

et(π∞) is of dimension∑

π∞

m(π∞ ⊗ π∞) dimHi(aG\g,K∞;π∞)

where the sum is over irreducible admissible representations of AG\G(R) andm(π∞ ⊗ π∞) is the multiplicity of π∞ ⊗ π∞ in L2([G]).

One has a conjectural description of the GalE×C∞c (G(A∞) //K)-modulesHi

et(π∞) ⊗ π∞K in terms of the L-parameter of π. A nice survey is given

in [BR94b]. In many cases these conjectures have now been proven (see[Shi11, CHLN11] for instance). This whole picture has an analogue whenMK is not assumed to be compact. However the technical complications inthe noncompact setting are enormous and have only recently been overcomein key cases [Mor08, Mor10].

The emphasis so far has been on using these results to relate Galois rep-resentations to automorphic representations. As we mentioned in §12.5, oneexpects packets of tempered automorphic representations of G(A) to be pa-rameterized by L-parameters LF → LG. The work on Shimura varieties de-scribed above can sometimes allow one to establish this correspondence whenthe L-parameter is C-algebraic in the sense of §12.8.

Lest the reader be misled, the description of the etale cohomology groupH•et(M

KE,Q`) in terms of automorphic representations is useful for more than


proving cases of Langlands functorality. In fact, this description is only thebeginning of the story. These etale cohomology groups are intimately tied totwo of the deepest conjectures in algebraic and arithmetic geometry, namelythe Tate [Tat94] and Beilinson-Bloch conjectures [Sch88, Nek94].

Let CHp(X)Q` denote the Chow group of codimension p cycles on X (with

Q` coefficients). There is a cycle class map

cl : CHp(X)Q` −→ H2pet (XE ,Q`(p))

GalE . (15.37)

Here (p) is the Tate twist.

Conjecture 15.8.3 (Tate) The map (15.37) is surjective.

The Beilinson-Bloch conjecture gives a description of the kernel of cl. Wewill not make the conjecture precise, but note that it contains the Birch andSwinnerton-Dyer conjecture as a special case.

Now for general X it is difficult to compute H2pet (XE ,Q`(p))GalE , and

even more difficult to exhibit any elements in CHp(X)Q` . Shimura vari-eties provide an example where one can compute the Galois representationH2p

et (XE ,Q`(p)) and produce at least some elements of CHp(X)Q` as we nowexplain.

Assume that (H,XH) is a Shimura datum with reflex field EH :=E(H,XH) that is equipped with a morphism

(H,XH) −→ (G,X).

For simplicity assume that H is a subgroup of G and the morphism is theinclusion. Write n for the real codimension of Sh(H,XH)KH in ShK for K <G(A∞) (it is independent of the choice of K). In the etale setting we againhave a cycle class map

H0et(M(H,XH)

K∩H(A∞)

EH,Q`) −→ Hn

et(MKE,Q`(n/2))GalEEH . (15.38)

The image of this map is contained in cl(CHn/2(X)Q`). The map is com-patible with the cycle class map of (15.29) under the relevant comparisonisomorphisms [DMOS82, §I]. Motivated by the Tate conjecture one is lead tothe following question:

Question 15.8.4 What part of the cohomology of

Hnet(M

KE,Q`(n/2))GalEEH

is in the C∞c (G(A∞) //K)-span of the image of (15.38)?

As explained in §15.6 this question can be rephrased in terms of the exis-tence of distinguished representations satisfying certain local assumptions.This was, in fact, the original motivation for the definition of a distinguished


representation as mentioned in the beginning of this chapter, and the an-swer is still not well-understood. However, many interesting cases have beenstudied [GH14, GW14, Kud97, MR87, Ram04, Yun11, Zha14].

There is also a connection between distinguished representations and theBeilinson-Bloch conjecture mentioned above, though it is not as direct. Formany years S. Kudla and his collaborators have investigated this conjecturein the context of Shimura varieties using a version of the Weil representation[Kud04, KRY06]. Remarkably, W. Zhang discovered that the relative traceformula can also be brought to bear [Zha12a]. In the function field setting,analogues of the relative trace formula can even be used to give the strongestresults known towards the Birch and Swinnerton-Dyer conjecture [YZ17].

Exercises

15.1. Let Y be a Hausdorff topological space and let Γ be a discrete groupacting freely and properly discontinuously on it. Then there is a unique topol-ogy on the set theoretic quotient Γ\Y such that the quotient map Y → Γ\Yis a continuous local homeomorphism. In other words, for every y ∈ Y there isan open neighborhood of y mapping homeomorphically onto an open neigh-borhood of Γy.

15.2. Let Γ and X be as in (15.3). Deduce that there is a unique manifoldstructure on Γ\X such that the quotient map X → Γ\X is a local homeo-morphism.

15.3. Prove that a congruence subgroup is arithmetic.


15.5. Compute the (gl2,O2(R))-cohomology of all irreducible admissible(gl2,O2(R))-modules.

15.6. Prove the formula (15.7).

15.7. Prove that the isomorphism (15.25) is Hecke equivariant.

Chapter 16

Spectral Sides of Trace Formulae

A felicitous but unprovedconjecture may be of much moreconsequence for mathematicsthan the proof of many arespectable theorem.

A. Selberg

Abstract An important tool for the study of automorphic forms is theArthur-Selberg trace formula, or more generally, the relative trace formula.We consider the spectral side of these identities in this chapter.

16.1 The automorphic kernel function

The remaining chapters of this book are devoted to stating and proving simpleversions of trace formulae. These formulae all have a geometric side involvingintegrals of a test function along certain orbits, and a spectral side involvingperiod integrals of automorphic forms.

In any of these formulae the first step in this is to employ a fundamentalidea first applied to the study of automorphic forms by Selberg [Sel56] thatwe will now describe. Let G be a connected reductive group over a globalfield F and as in §2.6 let

[G] := G(F )AG\G(AF ). (16.1)

For x ∈ AG\G(AF ), one has the regular representation

R(x) : L2([G]) −→ L2([G]) (16.2)

ϕ 7→ (g 7→ ϕ(gx)) .

315

316 16 Spectral Sides of Trace Formulae

Forf ∈ C∞c (AG\G(AF )),

consider the integral operator, the smooth version of the regular representa-tion,

R(f) : L2([G]) −→ L2([G]) (16.3)

ϕ 7→

(g 7→

∫AG\G(AF )

f(x)ϕ(gx)dx

).

One has

R(f)ϕ(x) =

∫AG\G(AF )

f(y)R(y)ϕ(x)dy

=

∫AG\G(AF )

f(x)ϕ(xy)dy

=

∫AG\G(AF )

f(x−1y)ϕ(y)dy

=

∫[G]

∑γ∈G(F )

f(x−1γy)ϕ(y)dy.

Here to justify the last step one uses the unfolding lemma 9.2.4. In otherwords, R(f) is an integral operator with kernel

Kf (x, y) :=∑

γ∈G(F )

f(x−1γy). (16.4)

This is the automorpic kernel function attached to f . It is a smoothfunction of x and y. In fact, for (x, y) in a fixed compact set, the sum is finiteas we now check: Let

Ω1 ×Ω2 ⊂ AG\G(AF )×AG\G(AF )

be compact subsets. Let(x, y) ∈ Ω1 ×Ω2

then the only nonzero summands in Kf (x, y) correspond to γ satisfying γ ∈Ω1Supp(f)Ω2.

The kernel Kf (x, y) can also be expanded spectrally. The easiest contri-bution to describe is that given by the cuspidal spectrum as we now explain.By Theorem 9.1.1 the restriction Rcusp(f) of R(f) to L2

cusp([G]) has kernel

Kf,cusp(x, y) :=∑π

Kπ(f)(x, y), (16.5)

16.1 The automorphic kernel function 317

where the sum is over isomorphism classes of cuspidal automorphic represen-tations π of AG\G(AF ) and

Kπ(f)(x, y) =∑ϕ∈Bπ

π(f)ϕ(x)ϕ(y) (16.6)

for Bπ an orthonormal basis of L2cusp(π), the π-isotypical subspace of L2

cusp([G]).Here π(f) is just the restriction of R(f) to the space of π. We note that π(f)need not be finite rank in general, but it is if f is K∞-finite by admissibilityof π (see Exercise 16.1).

We thus arrive at the identity that underlies all trace formulae:∑γ∈G(F )

f(x−1γy) =∑π

Kπ(f)(x, y) + ∗ (16.7)

where the sum is over isomorphism classes of cuspidal automorphic represen-tations of AG\G(AF ), and the ∗ denotes the contribution of the orthogonalcomplement of L2

cusp([G]) in L2([G]). This contribution will be made precisein §16.3. The left hand side is the geometric expansion of the kernel. Theright hand side is the spectral expansion of the kernel.

The key point here is that the right hand side manifestly contains allautomorphic representations of AG\G(AF ) whereas the left hand side, atleast a priori, does not involve automorphic representations at all. It packagesautomorphic information in an entirely different manner. The goal is to playthese two different manners of encoding automorphic representations off ofeachother.

An evident method of extracting information from (16.7) is taking tracesas we now explain. As defined, Kπ(f)(x, y) is only a function in the L2 sense;a priori it is meaningless to evaluate it at a point. But

(ϕ1, ϕ2) −→∫

[G]

ϕ1(g)ϕ2(g)dg

is the pairing defining the metric on L2([G]). It follows that integratingKπ(f)(x, y) along [G] embedded diagonally in [G]× [G] is always well-defined,and ∑

π

∫[G]

Kπ(f)(x, x)dx =∑π

m(π)trπ(f), (16.8)

where m(π) is the multiplicity of π in L2cusp([G]). Thus (16.8), in principle, is

equal to the integral of the left hand side of (16.7) over the diagonal (minusthe contribution of ∗). We say in principle because this integral is rarelyconvergent. In any case, at the expense of a massive amount of work one canmake sense of the integral over the diagonal of the left hand side of (16.7)and one obtains the Arthur-Selberg trace formula.


For the moment we leave the geometric expansion of the kernel and focuson the spectral side. We return to the geometric side in §18.2 when we discusstrace formulae in simple settings.

16.2 Relative traces

Let π be a cuspidal automorphic representation of AG\G(AF ). As just men-tioned, the integral of Kπ(f)(x, y) over the diagonal copy of [G] gives the traceof the operator π(f).

There is no need to just integrate over the diagonal copy of [G] however.Jacquet was the first to systematically study the integrals of Kπ(f)(x, y) oversubgroups other than the diagonal copy of [G] (apart from twisted versionsof the diagonal embedding that appears in the twisted trace formula) [Jac97,Jac05a]. These new integrals are called relative traces. They are introducedin this section.

We start by clarifying the status of the function Kπ(f)(x, y). As it stands,it is only defined up to a measurable set. It is easy to remedy this. Forany closed subspace V ≤ L2

cusp([G]) let BV be an orthonormal basis of thissubspace. For f ∈ C∞c (AG\G(AF )), R(f)|V is of trace class by Theorem 9.1.1and hence its kernel admits an L2 expansion

Kf,V (x, y) =∑ϕ∈BV

R(f)ϕ(x)ϕ(y) (16.9)

(see (9.12)).

Lemma 16.2.1 There is a unique function that is smooth as a function of(x, y) ∈ (AG\G(AF ))2 that is equal to the kernel Kf,V (x, y) almost every-where.

Given the lemma, we can and do identify the kernel of R(f) restricted to aclosed subspace of L2

cusp([G]) with a smooth function.

Proof. By the Dixmier-Malliavin lemma, Theorem 4.2.5, we can write f as afinite sum of functions of the form

f1 ∗ f2 ∗ f3

for f1, f2, f3 ∈ C∞c (AG\G(AF )). It clearly suffices to prove the lemma for f ofthis special form, so we assume that f = f1 ∗f2 ∗f3. For f ∈ C∞c (AG\G(AF ))let

f∨(g) := f(g−1) and f∗(g) := f(g−1), (16.10)

where the bar denotes the complex conjugate. If ϕ1, ϕ2 ∈ BV then

16.2 Relative traces 319∫[G]×[G]

∑ϕ∈BV

R(f)ϕ(x)ϕ(y)ϕ2(x)ϕ1(y)dxdy

= (R(f)ϕ1, ϕ2)L2([G])

= (ϕ1, R(f∗)ϕ2)L2([G])

=

∫[G]×[G]

∑ϕ∈BV

ϕ(x)ϕ(y)R(f∨)ϕ2(x)ϕ1(y)dxdy.

We deduce that ∑ϕ∈BV

R(f)ϕ(x)ϕ(y) =∑ϕ∈BV

ϕ(x)R(f∨)ϕ(y). (16.11)

Thus

Kf,V (x, y) =∑ϕ∈BV

R(f1 ∗ f2 ∗ f3)ϕ(x)ϕ(y)

=∑ϕ∈BV

R(f2 ∗ f3)ϕ(x)R(f∨1 )ϕ(y)

= (R(f2)×R(f∨1 ))∑ϕ∈BV

R(f3)ϕ(x)ϕ(y). (16.12)

The function (16.12) is smooth as a function of (x, y) by Proposition 4.2.3.This is the unique smooth function representing the kernel in the lemma. ut

We now prepare to define relative traces. Let

H ≤ G×G

be a subgroup. We will assume that H is connected and that it is the directproduct (not semidirect product) of a unipotent and a reductive group. Weset

AG,H := AH ∩ (AG ×AG). (16.13)

Note that the containment AG,H ≤ AH is often proper. For example, it isproper when G = GL2 and H is the maximal torus of diagonal matrices.Implicit in the definition of R(f) is a choice of measure on AG\G(AF ). Todefine relative traces we additionally choose a Haar measure d(h`, hr) onAG,H\H(AF ). It induces a measure on AG,HH(F )\H(AF ).

Letχ : H(AF ) −→ C×

be a quasi-character trivial on AG,HH(F ). We define the relative trace ofπ(f) with respect to H and χ to be


rtrH,χ π(f) =

∫AG,HH(F )\H(AF )

Kπ(f)(h`, hr)χ(h`, hr)d(h`, hr). (16.14)

This is well-defined by the following lemma:

Lemma 16.2.2 For any closed subspace V ≤ L2cusp([G]) one has∫

AG,HH(F )\H(AF )

|Kf,V (h`, hr)χ(h`, hr)|d(h`, hr) <∞.

Proof. As in the proof of Lemma 16.2.1 we may assume f = f1 ∗f2 ∗f3. Thenby (16.12)

Kf,V (x, y) = (R(f2)×R(f∨1 ))∑ϕ∈BV

R(f3)ϕ(x)ϕ(y). (16.15)

The right hand side is rapidly decreasing by Theorem 9.7.1 and hence theintegral in the lemma converges by Proposition 14.3.3. ut

The key property of relative traces is contained in the following proposi-tion:

Proposition 16.2.3 Let π be a cuspidal automorphic representation of G(AF )that is trivial on AG. The representation π ⊗ π∨ of (AG\G(AF ))2 is (H,χ)-distinguished if and only if rtrH,χ π(f) 6= 0 for some f ∈ C∞c (AG\G(AF )).

In the proof of Proposition 16.2.3 we are about to give we will apply Lemma16.2.2 in the special case where V is a cuspidal automorphic representationof AG\G(AF ). In this case we have two natural notions of smoothness of anelement of ϕ ∈ L2

cusp([G]). The first is that ϕ is a smooth vector, that is, itis fixed by a compact open subgroup K∞ ≤ G(A∞F ) and it is smooth underthe action of G(F∞) in the sense of §4.2. The second is that ϕ is a smoothfunction, that is, it is (equal a.e., and hence can be identified with) an elementof C∞(G(AF )) invariant on the left by AGG(F ). It is obvious that if ϕ is asmooth function then it is a smooth vector. At least when ϕ is K∞-finite wehave a converse:

Lemma 16.2.4 An element ϕ ∈ L2cusp(π) that is K∞-finite is equal to a

smooth function almost everywhere.

Proof. Since ϕ is K∞-finite it is equal almost everywhere to an automorphicform by Theorem 6.5.2. ut

Proof of Proposition 16.2.3: Recall the definition of the period integral Pχfrom (14.3). One has

Pχ(Kπ(f)

)= rtrH,χ π(f),

16.2 Relative traces 321

where the convergence is justified by Lemma 16.2.2. Certainly Kπ(f) lies inthe π ⊗ π∨ isotypic subspace of L2([G]2) so we deduce the “if” direction.

Now assume that π ⊗ π∨ is (H,χ)-distinguished. By Lemma 14.3.1 andExercise 16.3 there are K∞-finite smooth functions ϕ,ϕ′ ∈ L2

cusp(π) such thatPχ(ϕ′ ⊗ ϕ) 6= 0. Upon renormalizing we can and do assume that ϕ has L2-norm 1. Using Proposition 4.5.5 and Exercise 5.6 choose f ∈ C∞c (AG\G(AF ))such that π(f) maps ϕ to ϕ′ and sends any vector in L2(π) orthogonal to ϕto zero. Then

rtrH,χ π(f) = Pχ(ϕ′ ⊗ ϕ).

2

Thus investigating which representations of (AG\G(AF ))2 are (H,χ)-distinguished is equivalent to studying the linear forms

C∞c (AG\G(AF )) −→ Cf 7−→ rtrH,χ π(f).

The main result of this chapter relates the kernel function Kf,cusp(x, y) tothese distributions. It amounts to the computation of the cuspidal contribu-tion to the spectral side of the relative trace formula:

Theorem 16.2.5 Let f ∈ C∞c (AG\G(AF )). One has∫AG,HH(F )\H(AF )

Kf,cusp(h`, hr)χ(h`, hr)d(h`, hr) =∑π

rtrH,χ π(f).

Moreover, the integral on the left and the sum on the right are absolutelyconvergent. In particular, if R(f) has cuspidal image then∫

AG,HH(F )\H(AF )

Kf (h`, hr)χ(h`, hr)d(h`, hr) =∑π

rtrH,χ π(f).

After the proof of Theorem 16.2.5 we describe the general spectral expansionof Kf (x, y) and then, in §16.4, describe how to construct function f such thatR(f) has cuspidal image.

Proof. By the Dixmier-Malliavin lemma, Theorem 4.2.5, we can and do as-sume f = f1 ∗ f2 ∗ f3. Using (16.11) we have∑

π

Kπ(f)(x, y) =∑π

π(f2)× π(f∨1 )Kπ(f3)(x, y)

≤∑π

|π(f2)× π(f∨1 )Kπ(f3)(x, y)|.

By Proposition 9.6.1 and with the notation of that proposition this is boundedby a constant depending on B1, B2 ∈ R>0 and f1, f2 times

322 16 Spectral Sides of Trace Formulae∑π

maxα1,α2∈∆

(α(sx))−B1(α(sy))−B2‖Kπ(f3)(x, y)‖2

=∑π

maxα1,α2∈∆

(α(sx))−B1(α(sy))−B2trπ(f∗3 ∗ f3).(16.16)

Since the operator Rcusp(f∗3 ∗f3) is trace class by Theorem 9.7.1 this convergesabsolutely. We conclude that the sum∑

π

Kπ(x, y) (16.17)

converges absolutely and uniformly on compact subsets of [G]×[G] and henceis a continuous function. On the other hand the operator Rcusp(f) is traceclass by Theorem 9.1.1 therefore Kf,cusp(x, y) is equal to (16.17) in the L2

sense. Since Kf,cusp(x, y) is smooth by Lemma 16.2.1 we conclude that it isequal to (16.17) pointwise. We can now use the estimate (16.16) togetherwith Proposition 14.3.3 to deduce the theorem. ut

16.3 The full expansion of the automorphic kernel

Using Langlands’ decomposition of the entire spectrum L2([G]) we now givethe full spectral expansion of Kf (x, y), not just its restriction to the cuspidalsubspace. Assume for this section that F is a number field.

We will use the notation and terminology developed in Chapter 10. Let f ∈C∞c (AG\G(AF )). The automorphic kernel functionKf (x, y) has an expansion

Kf (x, y) =∑P

n−1P

∫ia∗P

∑σ

∑ϕ∈Bσ

E(x, I(σ, λ)(f)ϕ, λ)E(y, ϕ, λ)dλ (16.18)

where the sum on P is over all standard parabolic subgroups of G, the sumon σ is over all irreducible representations of G(AF ) occurring in the discretespectrum of HP , and Bσ is over an orthonormal basis of the σ-isotypic sub-space of σ in HP . This expression, a priori, only converges in L2. However,one can make sense of it pointwise by an analogue of the argument provingTheorem 16.2.5 (see [Art78, §4]).

16.4 Functions with cuspidal image

The general spectral expansion of Kf (x, y) is formidable, and it becomesmore serious when one tries to integrate the kernel. In particular it is notintegrable over the diagonal copy of [G]. Arthur has spent much of his careerinvestigating truncated versions of the kernel. In particular these truncated

16.4 Functions with cuspidal image 323

versions are integrable over the diagonal and can be given spectral interpre-tations ([Art05] is a nice introduction).

If we place assumptions on the test function f then the expansion can bemade much simpler. One such simplifying assumption is that the operatorR(f) has cuspidal image. In this case Kf,cusp(x, y) = Kf (x, y). Lindenstraussand Venkatesh [LV07] have defined a large class of functions with purelycuspidal image that essentially have no kernel when restricted to the cuspidalspectrum, provided one is only interested in functions spherical at infinity. Itwould be interesting to remove this assumption.

Before their work, there was a standard example of such functions thatcan be used to good effect in studying local factors of automorphic represen-tations. We recall it now.

Definition 16.1. Let v be a nonarchimedian place of a global field F . A func-tion fv ∈ C∞c (G(Fv)) is said to be F -supercuspidal or simply supercuspidalif ∫

N(Fv)

fv(gnh)dn = 0

for all proper parabolic subgroups P < G (defined over F ) with unipotentradical N and all g, h ∈ G(Fv).

Lemma 16.4.1 If f ∈ C∞c (AG\G(AF )) is of the form f = fvfv for some

finite place v and fv is supercuspidal then R(f) has cuspidal image.

Proof. Let P ≤ G be a proper F -rational parabolic subgroup with unipotentradical N . As usual let [N ] := N(F )\N(AF ). For ϕ ∈ L2([G]), R(f)ϕ issmooth and hence can be integrated over any compact subset. For all x ∈AG\G(AF ) we have∫

[N ]

R(f)ϕ(nx)dn =

∫[N ]

∫AG\G(AF )

f(g)ϕ(nxg)dgdn

=

∫[N ]

∫AG\G(AF )

f(x−1n−1g)ϕ(g)dgdn

=

∫[N ]

∫AGN(F )\G(AF )

∑δ∈N(F )

f(x−1n−1δg)ϕ(g)dgdn.

Taking a change of variables g 7→ δ−1g this becomes∫AGN(F )\G(AF )

∫[N ]

∑y∈N(F )

f(x−1n−1g)ϕ(g)dndg

=

∫AGN(F )\G(AF )

∫N(AF )

f(x−1n−1g)ϕ(g)dndg

= 0,

where the last equality follows from the fact that the inner integral

324 16 Spectral Sides of Trace Formulae∫N(AF )

f(x−1n−1g)ϕ(g)dn

vanishes since fv is supercuspidal. ut

Essentially all examples of supercuspidal functions are obtained using thefollowing lemma (see Exercise 16.4):

Lemma 16.4.2 Assume that ZG(Fv) is compact for some v and that (πv, V )is an irreducible supercuspidal representation of G(Fv). If fv is a matrixcoefficient of πv then fv is F -supercuspidal.

The assumption that ZG(Fv) is compact is not essential; see the discussionof truncated matrix coefficients in [HL04]. This lemma indicates the intrinsiclimitation of supercuspidal functions: supercuspidal functions can only beused to study representations that are supercuspidal at some place. For aprecise statement see Exercise 16.5.

Proof. Let P be a proper parabolic subgroup of G with unipotent radical N .Let V/V (N) be the space of coinvariants defined as (8.11) in §8.3. If∫

N(Fv)

fv(gnh)dn 6= 0

for some g, h ∈ G(Fv) then upon realizing V as a subspace of C∞c (G(Fv)) asin the proof of Proposition 8.5.3 we obtain a nonzero map

V −→ V/V (N)

fv 7−→

(g 7→

∫N(Fv)

fv(gnh)dn

)

contradicting the supercuspidality of πv. ut

Exercises

16.1. Let K∞ be a maximal compact subgroup of G(F∞). If f is K∞-finite,then prove that π(f) has finite image.

16.2. Prove that Kf,V (x, y) is independent of the choice of orthonormal basis.

16.3. For 1 ≤ i ≤ 2 let Gi be a reductive group over a number field F , letKi∞ ≤ AGi\Gi(F∞) be maximal compact subgroup, and let πi be a cuspidalautomorphic representation of AGi\Gi(AF ). Then a smooth function in

L2cusp([G1 ×G2])(π1 ⊗ π2) (16.19)

16.4 Functions with cuspidal image 325

need not be in the algebraic tensor product

L2cusp([G1])(π1)⊗ L2

cusp([G2])(π2).

Despite this show that a K1∞ ×K2∞-finite smooth function in (16.19) is afinite sum of functions of the form ϕ1 ⊗ ϕ2 where ϕi is a Ki∞-finite smoothfunction in L2

cusp([Gi])(πi) for 1 ≤ i ≤ 2.

16.4. Let v be a finite place of the number field F and let G be a reductivegroup over F . Assume that ZG(Fv) is compact. Prove that any supercuspidalfunction is a finite sum of matrix coefficients of supercuspidal representationsof G(Fv).

16.5. Under the assumptions of Exercise 16.4 let f = fvfv ∈ C∞c (G(AF ))

where fv is a matrix coefficient of an irreducible supercuspidal representationπ′v. Prove that if π is a cuspidal automorphic representation of G(AF ) suchthat Kπ(f)(x, y) 6= 0 then πv ∼= π′v.

16.6. Let K∞ be a maximal compact subgroup of G(F∞). If f is K∞-finiteprove that

rtrH,χ π(f) =∑ϕ∈Bπ

Pχ(π(f)ϕ× ϕ)

where we take the basis Bπ to consist of K∞-finite forms (and hence smoothby Proposition 4.4.2).

Chapter 17

Orbital Integrals

When I suggested the termendoscopy to Shelstad I didn’tknow it had a medical meaning.

A. Ash

Abstract In this chapter we define and study orbital integrals. We then usethem to describe geometric sides of trace formulae.

17.1 Group actions, orbits and stabilizers

This chapter requires more serious algebraic geometry and Galois cohomologythan other chapters in this book. Moving forward, the main concepts thatwe will require are various constructions and definitions related to relativeclasses contained in the current section and the definition of relative orbitalintegrals, given in §17.4 in the local setting and §17.7 in the adelic setting. Ifdesired, the other sections of this chapter can be omitted, although they willbe used in the proofs of results in §17.4 and §17.7.

We recall the notion of an algebraic group action and some construc-tions involving it. These show up constantly in the study of trace formu-lae, implicitly or explicitly, so we have summarized some useful results.Our basic references for the geometric constructions in this chapter are[MFK94, Mil17, Poo17].

Let k be a Noetherian ring, let H be a smooth (affine) group scheme overk, and let X be an affine scheme of finite type over k. A morphism

a : X ×H −→ X (17.1)

is a (right) action of H on X if the following diagram

327

328 17 Orbital Integrals

X ×H ×H a×1H−−−−→ X ×H

a×my a

yX ×H a−−−−→ X

commutes and the composite

X1X×e−−−−→ X ×H a−−−−→ X

is the identity. Here m is the multiplication map and e : Spec(k) → H isthe identity section. These assumptions imply, in particular, that for everyk-algebra R

a : X(R)×H(R) −→ X(R)

is a right action. One can formulate the notion of a left action in the analogousmanner. Let a : X ×H → X and a′ : X ′ ×H → X ′ be right actions of H onaffine schemes X and X ′ of finite type over k, respectively. A morphism ofright actions is a morphism of schemes b : X → X ′ such that

X ×H a−−−−→ X

b×1H

y ybX ′ ×H −−−−→

a′X ′

commutes. It is an isomorphism if b is an isomorphism.For γ ∈ X(k) we have a morphism

a(γ, ·) : H −→ X.

We denote by Hγ the stabilizer of γ. Thus Hγ is a subgroup scheme that canbe constructed as a fiber product

Hγ = Spec(k)×X H

where the map H → X is a(γ, ·) and the map from Spec(k) to X is given byγ. In terms of points, for k-algebras R one has

Hγ(R) = h ∈ H(R) : a(γ, h) = γ. (17.2)

When k is a field the map Spec(k)→ X defined by γ is a closed immersion,and hence Hγ is a closed subgroup scheme of H.

Assume for the remainder of this section that k is a field. If I ≤ His an algebraic subgroup, then one can always define a quotient scheme I\H.It is a separated scheme of finite type over k [Mil17, Theorem 7.18]. In factit is quasi-projective in the sense that it admits an immersion into projectivespace (this is proven in the proof of [Mil17, Theorem 7.18]). It admits a

17.1 Group actions, orbits and stabilizers 329

basepoint b ∈ I\H(k) and is equipped with a morphism given on points in ak-algebra R by

H(R) −→ I\H(R)

h 7−→ bh.(17.3)

The fibers of this morphism are the right cosets of I(R) inH(R) [Mil17, §5(c)].It is important to point out that the map (17.3) need not be surjective for agiven R. For more information we refer to Proposition 17.1.7. Of course wecan also form a right quotient H/I in the analogous manner.

Example 17.1. If I is a parabolic subgroup of a reductive group H then I\His projective. When H = GL2 and I = B is a Borel subgroup

B\GL2∼= P1.

The following is [Mil17, Proposition 7.17]:

Proposition 17.1.1 The natural map

Hγ\H −→ X

is an immersion. ut

For affine schemes Y = Spec(A) for a ring A, denote by |Y | the set ofprime ideals of A equipped with the Zariski topology. In other words |Y |is the underlying topological space of Y . The morphism a(γ, ·) induces acontinuous map of topological spaces |H| → |X|. The image

a(γ, |H|) ⊆ |X|

is open in its closure [Mil17, Proof of Proposition 1.65]. We define

O(γ) ⊂ X (17.4)

to be the subscheme with this image as its underlying set, given the reducedinduced scheme structure.

Proposition 17.1.2 The map a(γ, ·) induces an isomorphism

Hγ\H −→ O(γ).

The scheme O(γ) is smooth. Moreover, the map

H −→ O(γ)

is smooth if Hγ is smooth.


Proof. See [Mil17, Corollary 7.13 and Proposition 7.17] for the first assertionand see [Mil17, Proposition 7.4 and Proposition 7.15] for the last two asser-tions. ut

For technical reasons later we will require the following complementaryresult:

Lemma 17.1.3 If I ≤ H is a closed subgroup scheme then the morphism

I\H −→ I\H

is finite and flat (and in particular proper).

Proof. The quotient I\I is finite etale by [Mil17, Proposition 5.58] (Milnetakes etale to mean finite etale [Mil17, §A(i)]). Thus the result follows from[Mil17, Proposition 7.15]. ut

Definition 17.1. IfO(γ) ⊂ X is closed we say that γ is relatively semisim-ple. We say that γ is relatively regular if the dimension of O(γ) is maximalamong all γ ∈ X(k), where k is an algebraic closure of k.

Here the adjective relatively is short for relative to the action of H.Relative semisimplicity has important consequences for Hγ :

Theorem 17.1.4 Assume that H is reductive. The orbit O(γ) is affine ifand only if (Hγ) is reductive. In particular, if O(γ) is closed then (Hγ) isreductive. ut

This was proven by Matsushima in characteristic zero [Mat60] and Haboush[Hab78] and Richardson [Ric77] in arbitrary characteristic. We point out thatin general if the neutral component of Hγ is reductive it is not the case thatO(γ) is closed (see Exercise 17.2).

Example 17.2. Consider the situation when X = H and H acts via conjuga-tion. Thus for k-algebras R the action is given by

X(R)×H(R) −→ X(R)

(γ, h) 7−→ h−1γh.

We refer to this as the group case. In this setting O(γ) is the conjugacyclass of γ and Hγ is the centralizer of γ. Assume that k is perfect. Then anelement γ ∈ X(k) is relatively semisimple if and only if it is semisimple inthe usual sense of Definition 1.12 (see [Ste65, Corollary 6.13]). Moreover γ isrelatively regular if and only if γ is regular in the sense of [Ste65]. This is thereason for the terminology relatively semisimple and relatively regular.

Example 17.3. Let X = h, where h := LieH, and let H act on X via theadjoint action. Even if one is primarily interested in the group case, it has


proven crucial to use this infinitisimal model of conjugation. For example,the proof of the fundamental lemma was reduced to a Lie algebra versionby Waldspurger [Wal97, Wal06, Wal08, Wal09a, Wal09b]. It was this versionthat was proved by Ngo [Ngo10b].

Example 17.4. Let G be a reductive group, let X = G, and let H ≤ G × Gbe a subgroup. Then we have a natural action

G(R)×H(R) −→ G(R)

(g, (h`, hr)) 7−→ h−1` ghr. (17.5)

We can recover (17.2) by taking H to be G embedded diagonally into G×G.

Definition 17.2. Let R be an k-algebra. A class is an element of

Γ (R) := X(R)/H(R).

A relatively semisimple class is an element of

Γss(k) := γ ∈ X(k) : γ is relatively semisimple/H(k). (17.6)

We use the notation [γ] for the class of γ.

From the point of view of algebraic geometry, the notion of a class istoo refined. In fact it turns out to be too refined even for some purposes inautomorphic representation theory. One needs to consider the coarser notionof a geometric class. This entails some understanding of quotients of affineschemes by reductive group actions, so we digress to discuss this topic.

The action of H on X corresponds to a morphism of k-algebras

a : O(X) −→ O(X)⊗k O(H), (17.7)

where O(Y ) is the coordinate ring of the affine scheme Y (see §1.2). In otherwords, a defined in (17.1) is just the map of affine schemes induced by themap a of k-algebras. Call an element r ∈ O(X) invariant if a(r) = r ⊗ 1.The set of invariant elements is a k-subalgebra, and by abuse of notation welet

O(X)H ≤ O(X) (17.8)

be this subalgebra. If H is reductive we set

X/H := Spec(O(X)H). (17.9)

This is the GIT quotient, Geometric Invariant Theory quotient, of Xby H. The inclusion (17.8) induces a morphism

p : X −→ X/H.


Often in the literature the GIT quotient is denoted by X //H and the stacktheoretic quotient is denoted by X/H or [X/H]. Since we only use GITquotients in this work and the doubleslash could be confused with notationwe use for double quotients (see (5.5) for example) we adopt the notationabove.

Let use explain why X/H deserves to be regarded as a quotient. Let p1 :X×H → X denote the projection to the first factor. A categorical quotientof X by H is a k-scheme Y and a morphism p : X → Y such that

X ×H a−−−−→ X

p1

y ypX

p−−−−→ Y

(17.10)

commutes and for any morphism q : X → Z from X to another k-scheme Zsuch that

X ×H a−−−−→ X

p1

y yqX

q−−−−→ Z

(17.11)

commutes there is a unique morphism χ : Y → Z such that q = χ p.Concretely, the assertion that (17.11) commutes is a way to make precise theassertion that q is constant on H-orbits. The universal property states thatany morphism constant on H-orbits factors through the categorical quotient.As usual with universal properties, the definition immediately implies thatthe categorical quotient is unique up to isomorphism if it exists.

As the reader probably has guessed the GIT quotient is a categorical quo-tient [MFK94, Theorem 1.1], and this explains why it is reasonable to call ita quotient. In fact, X/H is a uniform categorical quotient, , which im-plies in particular that for all separable field extensions E/k the base change(X/H)E is a categorical quotient of XE by HE .

For our purposes we will need to understand the points of X/H. This issomewhat complicated. Let k ≤ ksep ≤ k be a separable closure and algebraicclosure of k, respectively. We recall that the closed points of the topologicalspace underlying a scheme Y of finite type over k can be identified withAutk(k)-orbits of elements of Y (k) [GW10, Proposition 5.4]. Using this stan-dard fact we can give a fairly concrete description of the points of X/H overk:

Proposition 17.1.5 Assume that H is reductive. There is a bijection

X/H(k)−→O(γ) :

γ ∈ X(ksep) is relatively semisimple andO(γ)(ksep) is Galk-invariant


given by sending x ∈ X/H(k) to the closed H-orbit O in X such that Oksep

is the unique closed Hksep-orbit in the fiber of p : Xksep → (X/H)sep over x.Under this identification the map X(k) → X/H(k) sends an H(k)-orbit

γH(k) to the unique closed orbit in the Zariski closure of γH(ksep).

We will use the following standard fact in the proof of this proposition andrepeatedly below [Sta16, Lemma 32.25.6]:

Lemma 17.1.6 If Y is a smooth scheme over a field k then Y (ksep) is densein the underlying topological space of Y . ut

Proof of Proposition 17.1.5: Using the action of Galk (see [GW10, §5.2] forexample) one easily reduces the proposition to the case where k = ksep.

We henceforth assume that k = ksep. The morphism X → X/H is surjec-tive (even universally submersive) [MFK94, Chapter 1.2, Theorem 1.1]. Weclaim that the induced map

p : X(ksep) −→ X/H(ksep) (17.12)

is also surjective. An element of X/H(ksep) defines a closed point x of(X/H)ksep . Consider the fiber F of the morphism Xksep → (X/H)ksep overx. One has that F(k) is nonempty. Thus Fk is smooth [Mil17, Proposition7.4(b)]. This implies that F(ksep) is nonempty by Lemma 17.1.6. Thus (17.12)is surjective.

Now let x ∈ (X/H)(ksep) and let F denote the fiber of p over x. It isa closed subscheme of X and is a union of H-orbits. Let O be the orbit ofsmallest dimension. We claim that is closed. Indeed, let O be the closure of O.Then O−O, if nonempty, is a union of H-orbits of dimension strictly smallerthan the dimension of O. Thus O is closed. Any two closed H-orbits in Xcan be separated by an element of O(X)H (see Chapter 1.2, Corollary 1.2and Appendix C, Corollary A.1.3 of [MFK94]). Thus O is the unique closedH-orbit in F . This implies the first claim of the proposition. The secondfollows from the first. 2

We now drop our running assumption that H has reductive neutral com-ponent. The following is a useful analogue of Proposition 17.1.5:

Proposition 17.1.7 For any smooth affine algebraic subgroup I ≤ H,

I\H(k) = I(ksep)h ∈ I(ksep)\H(ksep) : hξ(h−1) ∈ I(ksep) for all ξ ∈ Galk.

Proof. Using the action of Gal(ksep/k) (see [GW10, §5.2] for example) oneeasily reduces the proposition to the case where k = ksep.

We henceforth assume that k = ksep. The morphism H → I\H is faithfullyflat and hence surjective [Mil17, Proposition 7.4(b)]. Arguing as in the proofof Proposition 17.1.5 we deduce that

p : H(ksep) −→ I\H(ksep) (17.13)


is also surjective. Since the fibers are the cosets of I(ksep) in H(ksep) (see thediscussion around (17.3)) we deduce the proposition. ut

We make the following definition:

Definition 17.3. Two relatively semisimple elements γ1, γ2 ∈ X(k) arein the same geometric relative class if O(γ1) = O(γ2). We denote byΓgeo,ss(k) the set of relatively semisimple geometric classes.

By Proposition 17.1.5, when H has reductive neutral component one hasa bijection

Γgeo,ss(k)−→Im(X(k)→ X/H(k)

). (17.14)

Still assuming H has reductive neutral component one has a natural sequenceof maps

Γss(k) −→ X(k)/H(k) −→ Γgeo,ss(k). (17.15)

If k = ksep, then the composite is an isomorphism by Proposition 17.1.5.However, in general the first map is not surjective and the second map is notinjective (even if k = ksep) (see Exercise 17.4). If k 6= ksep then the secondmap is not in general surjective.

For use in §17.7 we record two lemmas.

Lemma 17.1.8 If γ ∈ X(k) is relatively semisimple with respect to H thenit is relatively semisimple with respect to H.

Proof. We can and do assume k = k. Let O(γ) be the H-orbit of γ in Xand let O(γ)′ be the H-orbit of γ. Then using bars to denote underlyingtopological spaces,

|O(γ)| =⋃

h∈H(k)\H(k)

|O(γ)′h|.

Therefore |O(γ)| is a finite union of closed subspaces of |X| and hence isclosed. ut

Assume that

H := Hr ×Hu (17.16)

with Hr reductive and Hu unipotent.

Lemma 17.1.9 An element γ ∈ X(k) is relatively semisimple with respectto the action of H if it is relatively semisimple with respect to the action ofHr.

Proof. We can and do assume k = k. By Lemma 17.1.8 it suffices to treatthe case H = H. Throughout this proof, γ is relatively semisimple. We first


deal with the case where γ is relatively regular with respect to the action ofHr.

The subset Z ⊆ X(k) of points that are relatively regular with respect tothe action of Hr can be identified with the set of closed points in an opensubset of the underlying topological space of X. This is a consequence of theupper semicontinuity of the dimension of stabilizers (see the discussion above[MFK94, Definition 0.9], but be aware that the notion of regular in loc. cit. isnot the same as ours). Let

Xreg ⊂ X

be the open subscheme of X with underlying topological space Z given thereduced induced subscheme structure. Thus Xreg(k) = Z. It is easy to seethat Xreg is an H-invariant subscheme.

Assume for the moment that γ ∈ Xreg(k). Now consider the GIT quotient

p : X −→ X/Hr

Using the universal property of categorical quotients and the fact that Hu

and Hr commute we see that the action of Hu on X induces an action of Hu

on X/Hr.Let O(p(γ)) be the Hu orbit of p(γ). By [Mil17, Theorem 17.64] the orbit

O(p(γ)) is closed. Thus p−1(O(p(γ))) is closed. But using Proposition 17.1.5and our assumption that γ is regular one sees that p−1(O(p(γ))) is the orbitof γ under H.

Thus we have proven the proposition for γ ∈ Xreg(k), in other words,when γ is relatively regular with respect to Hr. We now reduce the generalcase to this one. Let X0 = X, X1 = X − Xreg, and for i > 1 let Xi =Xi−1 − (Xi−1)reg. These subschemes are all preserved by H. Thus we havea sequence of closed subschemes

X = X0 ⊇ X1 ⊇ · · · .

Since X is Noetherian this sequence stabilizes. It follows that there exists ann such that (Xn)reg = Xn. Thus our sequence is

X = X0 ) X1 ) · · · ⊃ Xn ) ∅

where for each i one has Xi−1 − Xi = X(i−1)reg. Now let γ ∈ X(k) berelatively semisimple with respect to Hr. Let i be the largest index such thatγ ∈ Xi(k). Then γ is relatively regular in Xi(k) with respect to the action ofHr. Let O(γ) be the Hr-orbit of γ in X. Its intersection with Xi is O(γ) itself,and it is closed (being the intersection of two closed subschemes). Moreoverthe H-orbit of γ in X is contained in Xi. Thus we have reduced to the casewhere γ is relatively regular with respect to Hr, as desired. ut


17.2 Luna’s slice theorem

We assume until §17.4 below that H is a smooth algebraic group over a fieldk with reductive neutral component that acts (on the right) on an affine k-scheme X. The structure of X/H around sufficiently nice points γ ∈ X(k) hasa simple structure. Luna’s slice theorem, which we will state in this section,makes this precise. Luna’s slice theorem will only be used in passing below,so the reader should feel free to omit this section. However, it is useful fordeveloping an understanding of the quotient X/H.

If H acts on an affine scheme Y of finite type over k we let

X ×H Y := X × Y/H. (17.17)

Here the action of H on X × Y is the diagonal action. Let γ ∈ X(k) be rela-tively semisimple and let S ⊂ X be a Hγ-stable affine subscheme containingγ. We then have a right action

S ×H ×Hγ −→ S ×H

given on points by (s, h)·h0 = (sh0, h−10 h). The natural action map S×H −→

X is constant on Hγ-orbits and hence induces a map

ϕ : S ×Hγ H −→ X.

Moreover, we have a natural isomorphism

(S ×Hγ H)/H−→S/Hγ . (17.18)

In the setting above we therefore have a diagram

S ×Hγ Hϕ−−−−→ Xy y

(S ×Hγ H)/H ∼= S/Hγϕ/H−−−−→ X/H

We recall that a morphism X → Y of affine k-schemes of finite type over afield k is etale if and only if it is smooth of relative dimension 0 (which is tosay that the dimension of X and Y are equal). One should think of an etalemap as an algebro-geometric analogue of a covering map.

Definition 17.4. We say that S as above is an etale slice of the action ofH on X at γ if

(a) The morphism ϕ is etale with open image U .(b) The morphism ϕ/H is etale and the induced morphism

S ×Hγ H −→ S/Hγ ×U/H U/H

17.2 Luna’s slice theorem 337

is an isomorphism.

Luna’s slice theorem asserts the existence of etale slices under mild hy-potheses (see [MFK94, Appendix D]):

Theorem 17.2.1 (Luna’s slice theorem) Assume that k is a character-istic zero field. Etale slices S exist for any γ ∈ X(k) with closed orbit alongwhich X is normal. Moreover if X is smooth at γ then S can be taken tobe smooth with image a neighborhood of 0 in the normal vector space to theorbit O(γ) at γ. ut

One can use this theorem to give a nice description of the quotient X/Hitself. In more detail, for k-algebras R and subgroups H ′ ≤ H let

XH′(R) := x ∈ X(R) : xh = x for h ∈ H ′(R′) and R-algebras R′.(17.19)

This is a closed subscheme of X [Mil17, Theorem 7.1].We have the following theorem, which is an amalgam of results of Luna

and Richardson (see [MFK94, Appendix D]):

Theorem 17.2.2 Assume that X is normal and that k has characteristiczero. Let H ′ ≤ H be a subgroup with reductive neutral component. Then forγ ∈ XH′(k) the orbit of γ under H is closed if and only if the orbit of γunder the normalizer NH(H ′) is closed. The map

XH′/NH(H ′) −→ X/H (17.20)

is finite. If γ is a representative for a generic closed orbit in X, H ′ = Hγ

and XH′ is irreducible then the map (17.20) an isomorphism. utThere is a slightly more precise description of what is meant by a generic

closed orbit in [LR79], but yet more precision is needed in practice. This canbe achieved in a nice family of examples. We elaborate in the remainder ofthe section.

When we consider the action of H on itself (X = H) via conjugationthe Chevalley restriction theorem provides a definitive description of thequotient X/H [Ste65, §6]:

Theorem 17.2.3 Assume H is reductive and let T ≤ H be a maximal torus.Then the inclusion T → H induces an isomorphism

T/W (H,T )−→H/H,

where the action of H on H is via conjugation. ut

In this case the stabilizer Hγ of γ ∈ H(k) is called the centralizer of γ andis denoted Cγ,H . We refer to §1.7 for the definition of a maximal torus andthe corresponding Weyl group W (H,T ) of T in H.

The following is [Ste65, 2.11]:


Theorem 17.2.4 Assume H is reductive and that γ ∈ H(k) is semisimple.Then γ is regular if and only if (Cγ,H) is a maximal torus in H. ut

In practice it is useful to know that one can place assumptions on H sothat Cγ,H itself is connected. To make this precise we recall some foundationalwork of Steinberg. Let

σ : H −→ H

be an automorphism.

Definition 17.5. The automorphism σ is semisimple if there is a smoothalgebraic group G containing Hk and a semisimple γ ∈ NG(H)(k) such thatσ is equal to the restriction of conjugation by γ to Hk.

This is an intuitive definition, but the following more technical definition isalso quite useful in practice

Definition 17.6. The automorphism σ is quasi-semisimple or quass if σfixes a Borel subgroup B of Hk and a maximal torus of Hk contained in B.

In both of these definitions the conditions on σ are phrased in terms of theautomorphism it induces on Hk. A semisimple automorphism is quass by[Ste68, Theorem 7.5]. Consider

Hσ(R) := h ∈ H(R) : σ(h) = h. (17.21)

This is an algebraic subgroup of H. We have the following basic result ofSteinberg [Ste68, Theorem 9.1]:

Theorem 17.2.5 If H is a semisimple simply connected group and σ is aquass automorphism of H then Hσ is reductive (so it is connected). ut

Corollary 17.2.6 If H is a reductive group with simply connected derivedgroup and γ ∈ H(k) then Cγ,H is connected.

The Jordan decomposition is badly behaved over imperfect fields. There-fore we will make the following convention for the remainder of this section:we say that an element γ ∈ H(k) is semisimple if its image under thecanonical map H(k)→ H(k) is semisimple.

Proof. This is clear if H is semisimple. In the general case let Hder ≤ H bethe derived group of H, let ZH ≤ H the center of H, and let ZH,t be themaximal torus contained in ZH . We then have ZH,tH

der = H [Mil17, Proofof Proposition 21.60(c)]. Assume that γ ∈ Hder(k). Then one checks that

Cγ,H = ZH,tCγ,Hder

and the lemma follows in this case. In general we have γ = zγ′ where (z, γ′) ∈ZH,t(k)×Hder(k) and one checks that

17.2 Luna’s slice theorem 339

(Cγ,H)k = Cγ′,Hk ,

so we deduce the corollary in this case as well. ut

Without any additional assumptions on σ we can at least deduce the fol-lowing result:

Theorem 17.2.7 If σ is semisimple, then (Hσ) is reductive.

Proof. We may assume that k = k. Choose G and γ as in the definitionof a semisimple automorphism as in Definition 17.5. The orbit of γ underconjugatin by H is closed in G by [BT65, Theoreme 10.2(a)]. The stabilizerof γ is Hσ, so we conclude by Theorem 17.1.4. ut

Corollary 17.2.8 Assume that σ has finite order and that either the char-acteristic of k is zero or the order of σ is coprime to the characteristic of k.Then (Hσ) is reductive.

Proof. The automorphism σ is semisimple (see Exercise 17.6). ut

Assume for the remainder of this section that the characteristic of k is not2. Let G be a reductive group over k and let

σ : G −→ G

be an automorphism of order 2. There is an action of H := Gσ × Gσ onX = G given on points in a k-algebra R by

X(R)×H(R) −→ X(R)

(g, (g`, gr)) 7−→ g−1` ggr.

(17.22)

In this case the set Γ (R) is often known as the set of relative conjugacyclasses (see Example 17.4). They are a generalization of the notion of con-jugacy classes in a strict sense (see Exercise 17.9).

There is a great deal of helpful geometry available in this special case thataids in the study of the set of classes Γ (R) of Definition 17.2. It is useful tointroduce the map

Bσ : G −→ G

given on points by g 7→ g−σg. We denote by Q the scheme-theoretic image ofBσ. Then Q is a closed affine k-subscheme of G.

We have an isomorphism [Ric82, Lemma 2.4] of k-schemes

Bσ : Gσ\G −→ Q (17.23)

which intertwines the right action of G with the following action of G on Q:

Q(R)×G(R) −→ Q(R)

(g, q) 7−→ g−1qgσ.(17.24)


In particular one has an isomorphism of affine k-schemes

X/H = G/H = Gσ\G/Gσ−→Q/Gσ,

where the action on the right is given by (17.24).For γ ∈ G(R) let

Cγ,Gσ := Cγ,G ∩Gσ.

We leave the following lemma as an exercise (see Exercise 17.11).

Lemma 17.2.9 There is an isomorphism

Hγ(R) −→ CBσ(γ),Gσ (R)

(g1, g2) 7−→ g2.

utWe then have the following proposition:

Proposition 17.2.10 An element γ ∈ X(k) = G(k) is relatively semisimplewith respect to the action of H if and only if Bσ(γ) is semisimple in the usualsense.

Proof. See [Ric82, §7]. utWe have seen that the notion of a maximal torus in a reductive group is ab-

solutely crucial. In the case of symmetric subgroups, the following definitionis a substitute:

Definition 17.7. A torus T ⊆ G is said to be σ-split if for all k-algebras Rand all t ∈ T (R) one has t−1 = tσ.

Beware that in [Ric82] a σ-split torus is called a “σ-anisotropic torus.”It is not hard to see that if T is any σ-split torus then T ≤ Q. Indeed,

every element of T (k) is in the image of the isogeny t 7→ t2 = t−σt. Moreover,σ-split tori exist (see [Ric82, (2.6)] and [Vus74, §1]).

Let Tσ ≤ Q be a maximal σ-split torus. We have a corresponding Weylgroup

W (G,Tσ) = NG(Tσ)/ZG(Tσ), (17.25)

sometimes called the little Weyl group (see §1.7). HereNG(Tσ) (resp. ZG(Tσ))is the normalizer (resp. centralizer) of Tσ in G. This is a finite etale groupscheme over k. We have the following generalization of the Chevalley restric-tion theorem [Ric82, Corollary 11.5]:

Theorem 17.2.11 (Richardson) Let Tσ ⊆ G be a maximal σ-split torus.The inclusion Tσ → Q induces an isomorphism

Tσ/W (G,Tσ)∼−→ Q/Gσ.

ut

17.3 Local geometric classes 341

17.3 Local geometric classes

Let k be a field. Suppose that γ ∈ X(k) has closed orbit under H and thatHγ is smooth (we do not assume that H has reductive neutral component).We then have a sequence

1 −→ Hγ(k) −→ H(k) −→ Hγ\H(k)

and we can identify the last object with O(γ)(k) by means of Proposition17.1.2. We now briefly explain how to compute the image using a little Galoiscohomology. The reader should feel free to omit this discussion and refer backto it later if necessary. Our primary reference is [Ser02, §I.5].

Recall that for a smooth affine algebraic group H over a field k one definesZ1(k,H), the set of 1-cocycles of Galk in H(ksep) to be the set of settheoretic maps

c : Galk −→ H(ksep)

σ 7−→ c(σ)

that satisfy the cocycle condition

c(σ1σ2) = c(σ1)σ1(c(σ2)) (17.26)

for all σ1, σ2 ∈ Galk. Two cocycles c and c′ are called cohomologous ifthere is a h ∈ H(ksep) such that c′(σ) = h−1c(σ)σ(h). This is an equivalencerelation, and the set of equivalence classes is denoted by

H1(k,H). (17.27)

It is the first Galois cohomology set of Galk in H (or more preciselyin H(ksep)). This is not a group unless H is commutative, but it does comewith a distinguished element, called the neutral element. It is the classof the cocycle sending every element of Galk to 1. The cohomology sets arefunctorial in H in the obvious sense. We also mention, though this is not atall obvious from the definition, that the sets H1(k,H) can be computed inmany interesting cases. We refer to [Ser02] for more details.

When H is commutative the set H1(k,H) has a group structure, and usingthis fact Borovoi was able to give a group structure on H1(k,H) all reductiveH (at least if k is of characteristic zero) [Bor98]. A nice reference is [Lab99].We will not need this refinement, but it is extremely useful for the stabi-lization of the trace formula [Lab99, KS99]. In fact, Borovoi’s constructionsborrow heavily from Kottwitz’s work on the stabilization of the trace formula.

If I → H is a morphism of smooth group schemes over k let

D(k, I,H) := ker(H1(k, I)→ H1(k,H)). (17.28)


Here the kernel simply means all classes that are sent to the neutral elementof H1(k,H).

Now suppose that I ≤ H is a smooth subgroup scheme. We leave thefollowing lemma as Exercise 17.12:

Lemma 17.3.1 The map

cl : I\H(k) −→ D(k, I,H)

I(ksep)h 7−→ (σ 7→ hσ(h−1))

defines a bijection between the H(k)-orbits in I\H(k) and D(k, I,H). ut

We refer to the map cl in the lemma as the class map. The lemma impliesthat there is an exact sequence of pointed sets

1 −→ I(k)\H(k) −→ I\H(k) −→ D(k, I,H) −→ 1. (17.29)

In other words the set of classes in the geometric class of γ is in bijectionwith D(k, I,H). This observation is the beginning of the theory of endoscopy,first introduced by Langlands in [Lan83]. The groundwork of the theory waslaid by Kottwitz in [Kot84, Kot86b] and more recent useful references include[KS99, Lab99].

To proceed further we require a description of the other H(k)-orbits inI\H(k). By Proposition 17.1.7 any element of I\H(k) is a coset I(ksep)hfor some h ∈ H(ksep) satisfying hσ(h−1) ∈ I(ksep) for all σ. Let Ih be thestabilizer of I(ksep)h under the action of H. Then tautologically we have

I\H(k) =∐h

Im(Ih(k)\H(k)→ I\H(k)), (17.30)

where the disjoint union is over the set of H(k)-orbits in I\H(k) and theimplicit maps are given by

Ih(k)\H(k) −→ I\H(k)

x 7−→ I(ksep)hx.

We note thatcl(Im(Ih(k)\H(k)→ I\H(k)))

is simply the class of the cocycle hσ(h−1).The groups Ih are not isomorphic as h varies, but they are related in an

important manner. Two smooth algebraic groups Q and Q are said to beforms of each other if there is an isomorphism

ϕ : Qksep−→Qksep .

The concept of a form arises naturally in the classification of algebraic groups;one can proceed in many circumstances by classifying groups of a certain type

17.4 Local relative orbital integrals 343

over ksep and then classifying all of the groups over k that base change to agiven group over ksep. A form is called an inner form if for all σ ∈ Galk theautomorphism ϕ−1 σ ϕ of Qksep is an inner automorphism. It is easy tocheck that the Ih are all inner forms of I (see Exercise 17.13).

Now suppose that k is a local field. In keeping with our usual notation forlocal and global fields we let k = F . We record the following useful result(see [Poo17, Proposition 3.5.73]):

Proposition 17.3.2 If X → Y is a smooth map of separated schemes offinite type over F then X(F )→ Y (F ) is open. ut

Using this proposition we prove the following lemma:

Lemma 17.3.3 Let I ≤ H be a smooth algebraic subgroup. The map

cl : I\H(F ) −→ D(F, I,H)

is continuous if we give D(F, I,H) the discrete topology. In particular theimage of I(F )\H(F ) in I\H(F ) is closed.

Proof. The last assertion follows from the first and Lemma 17.3.1. To provethe first assertion it suffices to show that the preimage under cl of any pointin D(F, I,H) is open. This is equivalent to the assertion that the maps

Ih(F )\H(F ) −→ I\H(F )

of (17.30) have open image. By definition of the quotient topology onIh(F )\H(F ) this will follow if we show that the map H(F ) −→ I\H(F )given by the action of H(F ) on the orbit of I(ksep)h is open. This map isinduced by a morphism of schemes φ : H → I\H. By Proposition 17.3.2 itsuffices to check that φ is smooth. The fibers of φ are isomorphic (as schemes)to Ih, which is smooth since it is an inner form of the smooth group I. ut

The following theorem gives conditions under which D(F, I,H) is finite:

Theorem 17.3.4 Assume I is a smooth algebraic group over F . If I is re-ductive or F has characteristic zero then H1(F, I) is finite.

Proof. If F has characteristic zero this is contained in [Ser02, §III.4.2 and4.3]. In the reductive case see [Ser02, §III.4.3, Remarks]. ut

17.4 Local relative orbital integrals

Let F be a local field and let H be a smooth algebraic group defined over Facting on a smooth affine scheme X. We define local relative orbital integralsin this section.

Assume we are given a quasi-character χ : H(F ) −→ C×.


Definition 17.8. A relatively semisimple element γ ∈ X(F ) is χ-relevantif χ is trivial on Hγ(F ).

If χ is understood we often omit it from notation and speak simply of relevantelements. We require the notion of relevance to define local orbital integrals,and only relevant γ will contribute to the global trace formula (see Theo-rem 17.7.2 below). If χ is trivial then all relatively semisimple elements arerelevant.

Since we have assumed X is affine and smooth over F , if F is archimedeanthen X(F ) is a smooth manifold, so C∞c (X(F )) is defined. If F is nonar-chimedean we let C∞c (X(F )) be the space of compactly supported locallyconstant functions on X(F ).

We require one more assumption on γ ∈ X(F ) before defining local relativeorbital integrals.

Definition 17.9. We say γ ∈ X(F ) is relatively unimodular if δH |Hγ =δHγ .

See §3.2 for definition of the modular character δH . In particular if H isreductive then relatively semisimple γ are unimodular by Theorem 17.1.4.

Now suppose we are given f ∈ C∞c (X(F )) and a relevant relative semisim-ple and relatively unimodular γ ∈ X(F ). We fix right Haar measures drh onH(F ) and drhγ on Hγ(F ). Because γ is relatively unimodular one can thenform the quotient measure drh

drhγ(see Theorem 3.2.2). We can then form the

local relative orbital integral

ROγ(f) := ROχγ (f) =

∫Hγ(F )\H(F )

f(γh)χ(h)drh

drhγ. (17.31)

Note that we have used the fact that γ is χ-relevant to define this integral.This integral depends on the choice of measures, but traditionally they arenot encoded in the notation. In some applications it may be more natural toreplace Hγ by some subgroup of Hγ containing (Hγ).

Theorem 17.4.1 If γ ∈ X(F ) is relevant relatively unimodular and rela-tively semisimple then the integral defining ROγ(f) is absolutely convergent.

Proof. Since the measure drhdrhγ

is a Radon measure on Hγ(F )\H(F ), to show

the integral is well-defined and absolutely convergent it is enough to constructa pull-back map

C∞c (X(F )) −→ C∞c (Hγ(F )\H(F ))

attached to the map

Hγ(F )\H(F ) −→ X(F )

Hγ(F )h 7−→ γh.(17.32)

17.5 Torsors 345

Thus it suffices to show that this map is proper. The orbit O(γ) of γ is closedin X. Hence O(γ)(F ) is closed in X(F ) by Theorem 2.2.1.

It therefore suffices to treat the case where X = O(γ) = Hγ\H. In this casewe invoke Lemma 17.3.3 to see that the image of Hγ(F )\H(F ) in Hγ\H(F )is closed. ut

17.5 Torsors

In this section we discuss torsors, which play a key role in §17.6. It is con-venient to return to the more general setting of §17.1. Thus H is a smoothgroup scheme over a Noetherian ring k acting (on the right) on an affinescheme X of finite type over k.

For example one could take X = H with the natural right H action. This isknown as the trivial H-torsor. This example can be profitably generalized:

Definition 17.10. Assume X is faithfully flat over a Noetherian ring k. Itis an H-torsor if the map

1X × a : X ×H −→ X ×X

is an isomorphism.

An equivalent (and more intuitive) way to define a H-torsor is to say thatit is a scheme flat and of finite type over k equipped with an action of Hthat becomes isomorphic to the trivial H- torsor after a faithfully flat finitetype base extension. For more details we refer the reader to [Poo17, §6.5.1].Our definition is the specialization of the definition in [Poo17, §6.5.1] to thecase of affine Noetherian base schemes. The set of all isomorphism classes ofH-torsors is denoted by

qH1fppf(k,H) = qH1

fppf(Spec(k), H). (17.33)

It is a pointed set with the class of the trivial torsor as neutral element. Thefppf stands for fidelment plat de presentation finie, since torsors are requiredto be faithfully flat and locally of finite presentation over k. Since we haveassumed H is smooth, H-torsors will in fact be smooth over Spec(k) [Poo17,Remark 6.5.2]. Assume that k′ a faithfully flat k-algebra of finite type. Thenone has a natural map

qH1fppf(k,H) −→ qH1

fppf(k′, H) (17.34)

given by sending an H-torsor Y to its base extension Yk′ .Suppose that I ≤ H is a smooth closed subgroup. Let Y be an I-torsor.

One has an action of I ×H on Y ×H given on points in a k-algebra R by


Y (R)×H(R)× I(R)×H(R) −→ Y (R)×H(R)

(y, h, i, h′) 7−→ (yi, i−1hh′).(17.35)

The actions of I and H commute.

Definition 17.11. The contracted product of Y and H

Y ∧I H := Y ×H/I

is the quotient of Y ×H by the action of I.

The contracted product Y ∧I H is an H-torsor, where the action is inheritedfrom the action of H in (17.35). We will not enter into a detailed discussion ofwhat is meant by a quotient in this context. Briefly, the action of I on Y ×Hdefines a quotient which is a presheaf of sets for the fppf site on Spec(k)([Poo17, §6] is a useful reference for this concept). The corresponding sheafis representable [Poo17, Theorem 6.5.10]. The contracted product providesus with a morphism of pointed sets

qH1fppf(k, I) −→ qH1

fppf(k,H) (17.36)

sending the isomorphism class of an I-torsor Y to the isomorphism class ofY ∧I H.

Assume momentarily that k is a field. Then there is a canonical bijection

qH1fppf(Spec(k), H)−→H1(k,H). (17.37)

It is constructed as follows. Suppose we are given an H-torsor X. Then, sinceX is smooth we can choose x ∈ X(ksep) by Lemma 17.1.6. Since the map

1X × a : X ×H −→ X ×X (17.38)

is an isomorphism we see that the map a(x, ·) : H(ksep) → X(ksep) is a bi-jection. In particular for each σ ∈ Galk there is c(σ) ∈ H(ksep) such thatσ(x) = x.c(σ). One checks that c(σ) is a 1 cocycle, and this is the defini-tion of the map (17.37). We omit the construction of the inverse map (see[Poo17, §5.12.4]). Using (17.37) we occasionally allow ourselves to identifyqH1

fppf(Spec(k), H) and H1(k,H) when k is a field.

Lemma 17.5.1 Assume k is a field and that I is a smooth closed subgroupof H. One has a commutative diagram

qH1fppf(k, I) −−−−→ qH1

fppf(k,H)

∼y y∼

H1(k, I) −−−−→ H1(k,H),

17.5 Torsors 347

where the vertical arrows are given by the bijection (17.37), the top arrow isgiven by (17.36) and the bottom arrow is given by observing that a cocyclewith coefficients in I(ksep) has coefficients in H(ksep).

Proof. We claim that there is a Galk-equivariant identificatiom

Y ∧I H(ksep) = Y (ksep)×H(ksep)/I(ksep). (17.39)

To see this choose y ∈ Y (k′) for some finite separable extension k′/k usingLemma 17.1.6. There is then an isomorphism (I ×H)k′→(Y ×H)k′ given onpoints in a k′-algebra R by

I(R)×H(R)−→Y (R)×H(R)

(i, h) 7−→ (yi−1, ih).

This intertwines the right action of I on I × H induced by its right actionon I with the left action of I on Y × H. Now I\I × H is, by construction,a quotient in the category of fppf sheaves. Using the definition of a quotientin the category of fppf sheaves given in [DG74, Expose IV, Definition 3.1.2]one checks that I\I ×H and Y ∧I H are isomorphic in the category of fppfsheaves over Spec(k′). This uses the well-known fact that schemes define fppfsheaves [Poo17, Proposition 6.3.19]. Thus for all separable extensions L/k′ offinite degree we obtain a commutative diagram

I ×H(L)∼−−−−→ Y ×H(L)y y

I\I ×H(L)∼−−−−→ Y ∧I H(L),

where the vertical arrows are induced by the quotient maps in the categoryof fppf sheaves over Spec(k′) and the horizontal arrows are bijections. Takingthe direct limit over all separable finite extensions ksep ≥ L ≥ k we deduce thesame diagram for L = ksep. In this case the horizontal arrows are not GalL =Galk equivariant. Despite this, one still deduces the claim from Proposition17.1.7.

Let y ∈ Y (ksep) and for each σ ∈ Galk let c(σ) ∈ I(ksep) be the elementssuch that σ(y) = yc(σ). Thus c is the cocycle attached to Y by (17.37). Wenow pick (y × 1)I(ksep) to be our basepoint in Y ∧I H(ksep). Then

σ(y × 1)I(ksep) = (yc(σ)× 1)I(ksep) = (y × c(σ))I(ksep).

ut


17.6 Adelic classes

Let F be a global field, let H be a smooth algebraic group over F acting on anaffine scheme X of finite type over F on the right. Thus we are in the settingof §17.1 in the special case where F = k. We now develop topological resultson H(AF ) classes in X(AF ) that are necessary to study global relative orbitalintegrals (see Proposition 17.7.1). Our exposition is based on the treatmentin [GH15], and generalizes some of the results of loc. cit.

Theorem 17.6.1 Suppose that γ is relatively semisimple. Then the map

Hγ(AF )\H(AF ) −→ X(AF )

is proper.

For γ as in the statement of the theorem one has that O(γ)(AF ) is closedin X(AF ) by Theorem 2.2.1. It is therefore no loss of generality to assumethat X = O(γ), which is to say that there is a smooth subgroup I of H suchthat X := I\H. For any finite set of places S of F including the infiniteplaces we set

OSF =∏v 6∈S

OFv and OSF := F ∩ OSF .

Thus OSF < F is the set of elements of F that are integral outside of thefinite places in S. Recall the notion of a model of an affine scheme reviewedin §2.4. For a sufficiently large set of places S of F we can choose models Hand X of H and X = I\H over OSF , respectively, such that there is a map

H → X

whose generic fiber is the canonical map H → X [Poo17, Theorem 3.2.1]. Welet I be the schematic closure of I in H.

The map OSF → ASF is an open topological embedding so

X (OSF ) −→ X (ASF ) = X(ASF ) (17.40)

is an open topological embedding by Exercise 2.2. Similarly we obtain, forevery finite place v, open topological embeddings

X (OFv ) −→ X (Fv) = X(Fv). (17.41)

We then have a homeomorphism of topological spaces

X(AF ) ∼=′∏v

X(Fv), (17.42)

17.6 Adelic classes 349

where the restricted direct product is taken with respect to the open setsX (OFv ). The isomorphism depends on the choice of model of X. This discus-sion is also valid with X and X replaced by H (resp. I) and H (resp. I). Infact we discussed the algebraic group case in §2.3. Moreover, there are againisomorphisms of topological groups

H(AF ) ∼=′∏v

H(Fv) and I(AF ) ∼=′∏v

I(Fv), (17.43)

where the restricted direct product is taken with respect to the subgroupsH(OFv ) (resp. I(OFv )) for all finite places v. These isomorphisms are inducedby the choice of the model of H and are compatible with the group actionsX(Fv)×H(Fv)→ X(Fv) andX(AF )×H(AF )→ X(AF ) because we assumedthat there is a morphismH → X whose generic fiber is H → X. It is temptingto try to define the quotient I\H and relate it to X , but we do not wish todiscuss representability of quotients over Dedekind rings. Fortunately comingto grips with this issue is unnecessary for our purposes.

Lemma 17.6.2 Assume that I is connected. For a large enough finite setof places S of F including the infinite places the inverse image of X (OSF ) in

H(ASF ) is equal to I(ASF )H(OSF ). The morphism H → X induces a bijection

I(OSF )\H(OSF )−→X (OSF ).

Proof. The mapH(AF )→ X(AF )

is open by [Con12b, Theorem 4.5]. For any finite set S of places of F contain-

ing the infinite places the subset H(OSF ) ⊂ H(ASF ) is open and so its image

is open in X(ASF ). Upon enlarging S we see that this image must be X (OSF )by definition of the restricted direct topology on X(AF ) (see (17.43)).

Let v 6∈ S be a place of F . If h ∈ H(Fv) is in the inverse image of X (OFv )then there is some h′ ∈ H(OFv ) and an x ∈ I(F sep

v ) such that xh′ = h byProposition 17.1.7 which implies that x = hh′−1 ∈ I(Fv). Thus we deducethe first assertion of the lemma.

Again applying Proposition 17.1.7, if h, h′ ∈ H(OFv ) are mapped to thesame point of X (OFv ) then there is some x ∈ I(F sep

v ) such that xh = h′. Itfollows that x = h′h−1 ∈ I(OFv ), and we deduce the second assertion of thelemma. ut

Proof of Theorem 17.6.1: As mentioned below the statement of Theorem17.6.1, we can and do assume that X = O(γ) = Hγ\H. Since the map

Hγ\H −→ Hγ\H

is proper by Lemma 17.1.3 the map


Hγ\H(AF ) −→ Hγ\H(AF )

is proper by [Con12b, Proposition 4.4]. Here we have also used the fact,mentioned above, that for any smooth algebraic subgroup I ≤ H the schemeI\H is separated and of finite type over F [Mil17, Theorem 7.18].

Thus we are reduced to showing that for any connected smooth subgroupI ≤ H the map

I(AF )\H(AF )→ I\H(AF )

is proper. To keep notation consistent with Lemma 17.6.2 we write X = I\Hand let X and H be defined as above that lemma.

Let Ω ⊂ X(AF ) be a compact set. We must show its inverse image inI(AF )\H(AF ) is compact. Let S be a finite set of places of F including theinfinite places. It suffices to treat the case Ω = ΩS ×ΩS where ΩS ⊂ X(FS)

and ΩS = X (OSF ) are compact since any compact subset admits a finite coverby open sets whose closure is of this form.

Taking S sufficiently large and invoking Lemma 17.6.2, we see that theinverse image of ΩS in H(ASF ) is I(ASF )H(OSF ). We conclude that the inverse

image of ΩS in I(ASF )\H(ASF ) is I(ASF )H(OSF ), hence compact. On the otherhand the inverse image of ΩS in I(FS)\H(FS) is compact by Lemma 17.3.3.2

We observe the following:

Lemma 17.6.3 If Y is an affine scheme of finite type over F then Y (F ) isdiscrete and closed in Y (AF ).

Proof. The set F < AF is discrete and closed by Lemma 2.1.3 and hence weconclude by Exercise 2.2. ut

Using the techniques above we can prove a result which was used earlierin Lemma 14.3.2:

Lemma 17.6.4 Let G be a reductive group over F and let H ≤ G be areductive subgroup. The image of G(F ) in AG\G(AF )/H(AF ) is discrete andclosed.

Here, of course, we have given AG\G(AF )/H(AF ) the quotient topology.

Proof. We begin by showing that the image of G(F ) in G(AF )/H(AF ) isclosed. Let

p : G(AF )/H(AF ) −→ G/H(AF )

be the natural map. The set G/H(F ) is discrete and closed in G/H(AF ) byLemma 17.6.3, so it suffices to show that for any x ∈ G/H(F ) the F -points ofthe fiber p−1(x) are discrete and closed in G(AF )/H(AF ). Under the naturalright action of H on G the stabilizer of any point is trivial, and every elementof G(AF ) is relatively semisimple. Applying propositions 17.1.5 and 17.1.7


we deduce that p−1(x)(AF ) = p−1(x)(F ) is a single point in G(AF )/H(AF )hence discrete and closed. Thus the image ofG(F ) inG(AF )/H(AF ) is closed.

Define G(AF )1 as in (2.10). The subset

G(AF )1/(H(AF ) ∩G(AF )1) ⊆ G(AF )/H(AF )

is a closed set containing G(F )(H(AF )∩G(AF )1), and hence G(F ) is discreteand closed in G(AF )1/(H(AF )∩G(AF )1). On the other hand the natural map

G(AF )1/(H(AF ) ∩G(AF )1) −→ AG\G(AF )/H(AF )

is a homeomorphism sending G(F )(H(AF ) ∩G(AF )1

)to AGG(F )H(AF ) so

we deduce the lemma. ut

Theorem 17.6.1 gives us useful topological information about a single rel-ative class. To proceed further we must analyze all of the H(AF )-orbits ofrelatively semisimple elements of X(F ) in X(AF ). In the function field casethere are serious difficulties having to do with the fact that H1(Fv, I) can beinfinite for non-reductive I. This motivates the following definition:

Definition 17.12. An element γ ∈ X(F ) is gcf (Galois cohomologicallyfinite) if

|H1(Fv, Hγ/(Hγ))| <∞

for all places v of F .

We emphasize that in the number field case all γ are gcf by Theorem 17.3.4.Clearly γ is gcf if and only if every element of its class is gcf so it makes senseto speak of a class being gcf.

In the local setting we analyzed classes in a geometric class using the classmap of Lemma 17.3.1. We now provide the adelic analogue.

Let I ≤ H be a smooth algebraic subgroup of the smooth algebraic groupH over F . We do not assume that H is reductive. Let X = I\H and let I,H and X be smooth models of I, H and X over OSF for a sufficiently largefinite set of places S of F including the infinite places as in the beginning ofthis section.

Recall the cohomology sets of (17.33). We let

Dfppf(OFv , I,H) := ker( qH1fppf(OFv , I) −→ qH1

fppf(OFv ,H)), (17.44)

where the map is given as in (17.34). There is a commutative diagram

qH1fppf(OFv , I) −−−−→ qH1

fppf(OFv ,H)y yH1(Fv, I) −−−−→ H1(Fv, H)

(17.45)


where the vertical arrows are given by (17.34), the top arrow is given by(17.36), and the map on the bottom row corresponds to the obvious morphismgiven by observing that a cocycle with coefficients in I(F sep

v ) tautologicallyhas coefficients in H(F sep

v ) (see Lemma 17.5.1). We let

D(OFv , I,H) := Im(Dfppf(OFv , I,H)→ D(Fv, I,H)). (17.46)

We then set

D(AF , I,H) :=

′∏v

D(Fv, I,H)

where the prime indicates that we take the restricted direct product withrespect to D(OFv , I,H) ⊆ D(Fv, I,H). We note that if S′ ⊇ S then thenatural map

D(AF , I,H) −→ D(AF , IOS′F ,HOS′F )

is an isomorphism.Assume now that D(Fv, I,H) is finite for all v. In this case we give

H1(AF ,H) a topology by stipulating that the open sets are all subsets thatare equal to

ΩS′ ×∏v′ 6∈S′

D(OFv , I,H) (17.47)

for some finite set of places S′ of F containing S, where ΩS′ is any subsetof∏v∈S′ D(Fv, I,H). With respect to this topology every set of the form

(17.47) is a compact open set.The most important and nontrivial properties of D(AF , I,H) are con-

tained in the following theorem:

Theorem 17.6.5 The image of the diagonal map D(F, I,H)→∏v D(Fv, I,H)

is contained in D(AF , I,H). If H1(Fv, I/I) is finite for all v then

D(F, I,H) −→ D(AF , I,H)

is proper if we give D(F, I,H) the discrete topology.

Proof. Using (17.37) we may regard an element of D(F, I,H) as the isomor-phism class of an I-torsor Y such that Y ∧I H is isomorphic to the trivialH-torsor. By spreading out [Poo17, Theorem 3.2.1], for a sufficiently largeset of places S′ ⊇ S the I-torsor Y is isomorphic as an I-torsor to the generic

fiber of a IOS′F -torsor Y such that Y ∧IOS′F HOS′F is isomorphic to a trivial

HOS′F -torsor. This implies the first assertion of the theorem.

We claim that the inverse image of∏v∈S′D(Fv, I,H)×

∏v 6∈S′D(OFv , I,H) (17.48)


in D(F, I,H) is finite for all S′ ⊇ S. This is enough to deduce the secondassertion of the theorem. It suffices to show that the inverse image of∏

v∈S′H1(Fv, I)×

∏v 6∈S′

Im( qH1fppf(Spec(OFv ), I)→ H1(Fv, I))

in H1(F, I) is finite. This set is denoted by H1S′(F, I) in [Poo17, Theorem

6.5.13], and we conclude by part (a) of that theorem. ut

For every nonarchimedean place v of F let OnrFv

be the ring of integers ofthe maximal unramified extension F nr

v of Fv in some algebraic closure. Wewill use the following lemma several times below:

Lemma 17.6.6 Let v be a nonarchimedean place of F . If Y is a smoothscheme over Spec(OFv ) then

Y (OnrFv ) 6= ∅.

Proof. Let F be the residue field of OFv and let F be its algebraic closure. Itis the residue field of Onr

Fv. The surjection Onr

Fv→ F induces a surjection

Y (OnrFv )→ Y (F)

by Hensel’s lemma [BLR90, §2.3, Proposition 5]. On the other hand Y (F) isnonempty since Y is smooth by Lemma 17.1.6. ut

Lemma 17.6.7 A class in H1(Fv, H) is in the image of the map fromH1(OFv ,H) if and only if it is cohomologous to a cocycle with values inH(OE) for some finite unramified Galois extension E/Fv.

Proof. Let H be an HOFv -torsor. It is smooth over OFv , and hence there isa finite unramified extension E/Fv such that Y(OE) is nonempty by Lemma17.6.6. The cocycle defined using this point y as in the definition of (17.37)will take values in H(OE). This implies the “only if” implication.

Conversely, let c : GalFv → H(OE) be a cocycle, where E/Fv is an unram-ified extension. This defines a descent datum on HOE and hence a H-torsorY (see [Poo17, Remark 4.4.7]). The cocycle attached to the generic fiber ofthis torsor is c, so we deduce the converse assertion. ut

For each place v we have a class map cl : I\H(Fv) → D(Fv, I,H) as inthe proof of Lemma 17.3.1.

Lemma 17.6.8 The local class maps induce a map

cl : I\H(AF ) −→ D(AF , I,H).

If H1(Fv, I/I) is finite for all v it sends compact sets to compact sets.

We do not know if cl is continuous.


Proof. Let OnrFv

be the ring of integers of the maximal unramified extensionF nrv in F sep

v . By spreading out, for S′ large enough the map HOS′F → XOS′Fis smooth [Poo17, Theorem 3.2.1] and hence the map H(Onr

Fv) → X (Onr

Fv) is

surjective for v 6∈ S′ by Lemma 17.6.6 applied to the fibers of the morphismHOFv → XOFv for v 6∈ S′. Thus for v 6∈ S′ given x ∈ X (OFv ) we can chooseh ∈ H(Onr

Fv) such that I(F sep)h = x. Thus cl(x) is contained in D(OFv , I,H)

by Lemma 17.6.7. This implies the first assertion of the lemma.For the second assertion in view of Lemma 17.3.1 it suffices to observe

that by the argument above cl(x) ∈ Dfppf(OFv , I,H) for v 6∈ S′. ut

Proposition 17.6.9 Let A′ ≤ AH be a subgroup. One has a commutativediagram

1 −−−−→ I(F )\H(F ) −−−−→ I\H(F )cl−−−−→ D(F, I,H)y y y a

y1 −−−−→ I(AF )\H(AF )/A′ −−−−→ I\H(AF )/A′

cl−−−−→ D(AF , I,H).

The top row is an exact sequences of pointed sets, and the map a is proper ifwe give D(F, I,H) the discrete topology.

We have incorporated the group A′ for use in the proof of Theorem 18.2.2below. One checks that cl(x) = cl(xa) for all x ∈ I\H(AF ) and a ∈ AH sothe map cl : I\H(AF )/A′ → D(AF , I,H) is well-defined.

Proof. The commutativity of the diagram is clear, the top line is exact since(17.29) is exact, and a is proper by Theorem 17.6.5. ut

We now prove that the set of relative classes inside a geometric class isfinite under suitable assumptions:

Theorem 17.6.10 Let A′ ≤ AH be a subgroup. Assume that H1(Fv, I/I) is

finite for all v. Let Ω ⊆ I\H(AF )/A′ be a compact set. There are only finitelymany H(F )-orbits xH(F ) ⊆ I\H(F ) such that xH(AF )/A′ intersects Ω.

Proof. We use the diagram in Proposition 17.6.9. Let x ∈ I\H(F ). If

xH(AF )/A′ ∩Ω 6= ∅

then cl(xH(AF )/A′)∩ cl(Ω) 6= ∅ and hence by commutativity of the diagramcl(xH(F )) ∩ a−1(cl(Ω)) 6= ∅. Using Lemma 17.3.1, we deduce that the set ofH(F )-orbits xH(F ) in I\H(F ) such that xH(AF )/A′ intersects Ω is mappedinjectively via cl into

cl(I\H(F )) ∩ a−1(cl(Ω)). (17.49)

The set cl(Ω) is compact by Lemma 17.6.8. Since a is proper, the set in(17.49) is finite. ut

17.7 Global relative orbital integrals 355

In §17.7 we will use Theorem 17.6.10 and the following finiteness result forgeometric classes:

Lemma 17.6.11 Assume H is reductive. Let Ω ⊂ X(AF ) be a compact set.There are only finitely many relatively semisimple geometric classes in X(F )whose H(AF )-orbit intersects Ω.

Proof. Letp : X(AF ) −→ X/H(AF )

be the canonical map. By Proposition 17.1.5 an element of X/H(F ) maybe regarded as a relatively semisimple class. The set of relatively semisimplegeometric classes in X(F ) whose H(AF ) orbit intersects Ω injects into

X/H(F ) ∩ p(Ω). (17.50)

But p(Ω) is compact and X/H(F ) is discrete and closed in X/H(AF ) byLemma 17.6.3, so we deduce that (17.50) is finite. ut

17.7 Global relative orbital integrals

For use in the relative trace formula we need to have global analogues of thelocal orbital integrals (17.31). Before this we must discuss what we mean bya smooth function in this setting.

For this section we continue to assume that F is a global field. Fix amodel of X over OSF in the sense of §2.4 and denote it again by X by abuseof notation. Here S is a sufficiently large set of places of F including allinfinite places. Then for all v 6∈ S of F the characteristic function 1X(OFv ) isan element of C∞c (X(Fv)). We define

C∞c (X(A∞F )) = ⊗′v-∞C∞c (X(Fv))

where the restricted direct product is taken with respect to the functions1X(OFv ). It turns out that C∞c (X(A∞F )), defined in this manner, is equal tothe C-vector space of smooth compactly supported functions on X(A∞F ) (seeExercise (17.14)).

If F is a global function field we let C∞c (X(F∞)) be the space of locallyconstant compactly supported functions on X(F∞), and if F is a numberfield we let C∞c (X(F∞)) be the usual space of compactly supported smoothfunctions on X(F∞), viewed as a real manifold. We then set

C∞c (X(AF )) := C∞c (X(F∞))⊗ C∞c (X(A∞F )).

Assume thatχ : H(AF ) −→ C×


is a quasi-character trivial on H(F ). For f ∈ C∞c (X(A∞F )), we then considerthe global relative orbital integral

ROγ(f) := ROχγ (f) =

∫Hγ(AF )\H(AF )

f(γh)χ(h)drh

drhγ. (17.51)

Definition 17.13. A relatively semisimple element γ ∈ X(F ) is χ-relevantif χ is trivial on Hγ(AF ).

Proposition 17.7.1 If γ ∈ X(AF ) is relevant relatively semisimple and rel-atively unimodular then the integral defining ROγ(f) is absolutely convergent.

Proof. As in the proof of Theorem 17.4.1 it suffices to show that the map

Hγ(AF )\H(AF ) −→ X(AF )

is proper. This is the content of Theorem 17.6.1. ut

In practice one needs more than this result. More precisely, one needsconditions under which the integral∫

H(F )\H(AF )

∑γ

f(γh)χ(h)drh (17.52)

converges. This is a very delicate question in general and we will only addressit in special cases. We first make the assumption that

H := Hu ×Hr (17.53)

whereHu is unipotent andHr has reductive neutral component. This assump-tion is necessary in order to obtain a convergent formula (Exercise 17.16).We also restrict to relatively unimodular elements so that our orbital inte-grals are well-defined, and to elements that whose Hr-orbit is closed, becauseotherwise even the contribution of a single orbital integral can diverge (see[Art05, p. 21]).

Let CX be the stabilizer of X under Hr, and let

A := ACX (17.54)

where ACX is defined as in (2.11). Clearly if A is nontrivial then (17.52) isnot absolutely convergent. Therefore we consider instead∫

AH(F )\H(AF )

∑γ

f(γh)χ(h)drh (17.55)

where f ∈ C∞c (X(AF )). There is an obvious obstruction for the contributionof a given γ to converge. Letting Hγ denote the stabilizer of γ in H, it ispossible that AHγ is not contained in A, which implies that the contribution


of the class of (17.55) is not absolutely convergent by the unfolding argumentin the proof of Theorem 17.7.2 below. We exclude this possibility using thenotion of an elliptic element.

Definition 17.14. A γ ∈ X(F ) is relatively elliptic if AHγ ≤ A.

If γ ∈ X(F ) is relatively elliptic the measure

τ(Hγ) := measdrhγ (Hγ(F )A\Hγ(AF )) (17.56)

is finite by Theorem 2.6.2. Here drhγ is a right Haar measure onHγ(F )A\Hγ(AF )(which, traditionally, is not encoded into the notation).

Finally, in the function field case, we need to assume that our classes aregcf in the sense of Definition 17.12 above. Upon imposing these restrictions,the corresponding piece of (17.55) does indeed converge absolutely:

Theorem 17.7.2 Assume (17.53) and that χ is trivial on A. The expression∫AH(F )\H(AF )

∑|f(γh)χ(h)|dh (17.57)

is convergent. Here the sum is over γ ∈ X(F ) that are gcf, relatively uni-modular and relatively elliptic, and such that the Hr-orbit of γ is closed.Moreover∫

AH(F )\H(AF )

∑f(γh)χ(h)dh =

′∑[γ]∈Γ (F )

τ(Hγ) ROγ(f) (17.58)

where the sum over γ on the left is as before and the prime indicates the sumon the right is over all classes that are relevant, gcf, relatively unimodularand elliptic and such that the Hr-orbit of γ is closed.

Several remarks are in order. First, the assumption that the Hr-orbit ofγ is closed implies that γ is relatively semisimple with respect to H (seeLemma 17.1.9). We will use this fact without further mention below. Theassumption that γ is relatively unimodular is equivalent to the assertion thatHγ is unimodular, which is implied by our assumption that the Hr-orbit ofγ is closed if H = Hr by Theorem 17.1.4. In the definition of ROγ(f) weuse the measure dh

dhγon

Hγ(AF )\H(AF ) = (A\Hγ(AF ))\(A\H(AF ))

where dhγ is a Haar measure on A\Hγ(AF ). We use the same measure dhγ inthe definition of τ(Hγ). Finally, the assumption that γ is gcf is automaticallysatisfied if Hγ is connected or F is a number field (Theorem 17.3.4).

We now handle the convergence claim in Theorem 17.7.2:

Proposition 17.7.3 The expression (17.57) is convergent.


Proof. We first reduce to the case H = H. One has∫AH(F )\H(AF )

∑|f(γh)χ(h)|dh

=

∫H(F )H(AF )\H(AF )

(∫AH(F )\H(AF )

∑|f(γhh)χ(hh)|dh

)dh

dh

where dh and dh are appropriately normalized Haar measures on H(AF )and H(AF ). The set H(AF )\H(AF ) is compact [Con12a, Proposition 3.2.1],so to prove that (17.57) is convergent we can and do assume H = H.

A relatively semisimple γ ∈ X(F ) has closed Hr-orbit if and only if anyelement of its geometric class has closed Hr-orbit. Moreover, by Exercise17.17, a relatively semisimple γ ∈ X(F ) is gcf (resp. relatively unimodular,relatively elliptic) if and only if any element of its geometric class is gcf(resp. relatively unimodular, relatively elliptic). Thus we can talk about ageometric class having closed Hr-orbit, etc. With this in mind one has∫

AH(F )\H(AF )

∑|f(γh)χ(h)|dh

=∑[γ0]

∫AH(F )\H(AF )

∑γ∈[γ0]

|f(γh)χ(h)|dh.(17.59)

Here the outer sum is over elements of Γgeo,ss(F ) with closed Hr-orbit that aregcf, relatively unimodular and relatively elliptic. Unfolding the inner integralfor a fixed geometric class [γ0] we obtain∫

AH(F )\H(AF )

∑γ∈[γ0]

|f(γh)χ(h)|dh

=∑[γ]

∫AHγ(F )\H(AF )

|f(γh)χ(h)|dh

where the second sum is over the relative classes [γ] in the geometric class[γ0]. This is equal to∑

[γ]

measdhγ (AHγ(F )\Hγ(AF ))

∫Hγ(AF )\H(AF )

|f(γh)χ(h)| dhdhγ

(17.60)

where for each relatively unimodular γ the measure dhγ is a Haar measureon A\Hγ(AF ). Here we are using the fact that γ is relatively unimodular toconclude that dhγ is a Haar measure, not merely a right Haar measure. ByTheorem 2.6.2 the measure

τ(Hγ) = measdhγ (AHγ(F )\Hγ(AF ))


is finite for all γ in (17.60) since all such γ are relatively elliptic. The sumover [γ] in (17.60) is finite by Theorem 17.6.10. Thus by Proposition 17.7.1we deduce that (17.60) is finite.

It therefore suffices to show that there are only finitely many relativelysemisimple geometric classes [γ0] that can contribute a nonzero summand to(17.59). Let us return to (17.59). It is equal to∑

[γ0]

∫AHr(F )\Hr(AF )×[Hu]

∑γ∈[γ0]

|f(γhrhu)χ(hrhu)|dhrdhu

=∑[γ0]

∫AHr(F )\Hr(AF )

∑γ∈[γ0]

|f(γhr)χ(hr)|dhr

where f(x) :=∫F |f(xhu)χ(hu)|dhu ∈ C∞c (X(AF )) with F is a compact

fundamental domain for Hu(F ) acting on Hu(AF ) (see Theorem 2.6.2). Herethe sums over [γ0] and γ are as in (17.59). We write this as∑

[γ0]

∑[γ′]

∫AHrγ′ (F )\Hr(AF )

|f(γ′hr)χ(hr)|dhr (17.61)

where the inner sum is over Hr(F )-orbits (not H(F )-orbits as above) in[γ0] consisting of elements that are relatively unimodular, elliptic and gcf.It follows from Lemma 17.6.11 that there are only finitely many closed Hr-orbits in X(F ) that can contribute a nonzero summand to (17.61). Thisimplies that there are in fact only finitely many [γ0] that can contribute anonzero summand to (17.61), and hence to (17.59). ut

Proof of Theorem 17.7.2: The convergence assertion of the theorem is thecontent of Proposition 17.7.3. We now prove the identity. All of the manip-ulations with integrals occurring in this proof are justified by Proposition17.7.3.

Consider ∫AH(F )\H(AF )

∑f(γh)χ(h)dh (17.62)

where the sum is over γ with closed Hr-orbit that are relatively semisimple,elliptic, unimodular and gcf. We unfold the integral to obtain∑

[γ]

∫AHγ(F )\H(AF )

f(γh)χ(h)dh

=∑[γ]

∫AHγ(F )\Hγ(AF )

χ(h′)dγh′ROγ(f)


=

′∑[γ]

τ(Hγ) ROγ(f)

where the first sum over γ is over H(F )-orbits in X(F ) with closed Hr-orbit that are gcf and relatively elliptic and unimodular. The quotientAHγ(F )\Hγ(AF ) is of finite volume by Theorem 2.6.2, and the integral overit in the second expression above is nonzero if and only if γ is relevant. Asin the statement of the theorem, the prime on the sum in third expressionindicates that the sum is over relevant relatively elliptic unimodular classesin X(F ) with closed Hr-orbit. 2

Exercises

17.1. Let H ×X → X be an action of an affine algebraic group on an affinescheme X over a field k. Prove that Hγ ≤ G is a closed subgroup scheme anddeduce that it is again an affine algebraic group.

17.2. Give an example of a reductive group H acting on an affine schemeX over a field k and a γ ∈ X(k) such that O(γ) is not closed but Hγ isreductive.

17.3. Let σ : H → H be an automorphism of the smooth algebraic groupH over a field k. Prove that Hσ, defined as in (17.21), is a closed algebraicsubgroup of H.

17.4. Give examples to show that the map Γss(k) −→ H(k)\X(k) need notbe surjective and that the map

H(k)\X(k) −→ Γgeo,ss(k)

need not be injective.

17.5. Give an example to show that if H is a reductive group with simplyconnected derived group and σ : H → H is a semisimple automorphism thenHσ need not be connected.

17.6. Let k be an algebraically closed field. Assume that γ ∈ GLn(k) hasfinite order and that either k is of characteristic zero or that the order ofγ is coprime to the characteristic of k. Prove that γ is semisimple. Give acounterexample to show that if the characteristic of k is positive and dividesthe order of γ ∈ GLn(k) then γ need not be semisimple.

17.7. Let k be a field with algebraic closure k. Prove that two elements ofGLn(k) are GLn(k)-conjugate if and only if they are GLn(k)-conjugate.


17.8. Let k be a field with separable closure ksep. Prove that a conjugacyclass in GLn(ksep) that is fixed under Gal(ksep/k) contains an element inGLn(k).

17.9. Let G be a reductive group over a field k and let H ≤ G × G be thediagonal copy of G. Construct a bijection between G(k)-conjugacy classes inG(k) and

H(k)\G(k)/H(k).

17.10. Prove that (17.18) is an isomorphism.

17.11. Prove Lemma 17.2.9.

17.12. Using Proposition 17.1.7 prove Lemma 17.3.1.

17.13. Prove that the groups Ih appearing in (17.30) are all inner forms ofI.

17.14. Prove that the two definitions of C∞c (X(A∞F )) given in §17.7 producethe same space of functions.

17.15. Prove that if G is a commutative smooth algebraic group over a fieldk and gσ, g

′σ are 1-cocycles, then

σ 7→ gσg′σ

is a 1-cocycle. Deduce that this gives a law of composition on H1(k,G) thatendows it with the structure of an abelian group with the neutral element asthe identity element.

17.16. Give an example to show that (17.55) is not absolutely convergent ifH does not satisfy the assumption of (17.53).

17.17. Let F be a global field and let H be a smooth algebraic group overF acting in an affine scheme X of finite type over F on the right. Prove thatγ ∈ X(F ) is relatively semisimple if and only if any element in O(γ)∩H(F )is relatively semisimple. Assuming γ ∈ X(F ) is relatively semisimple, provemoreover that γ satisfies the following conditions if and only if any elementof the geometric class of γ satisfies the following conditions:

(a) γ is relatively unimodular.(b) γ is relatively elliptic.(c) γ is gcf.

Chapter 18

Simple Trace Formulae

It is rare that you can take aproof in the geometric settingand convert it to the genuinenumber theoretic setting. That iswhat has transpired throughNgo’s achievement on thefundamental lemma, which hasprovided a bridge and noweverybody is using this bridge.

P. Sarnak

Abstract In this chapter we prove a simple relative trace formula andoverview its various specializations. This recovers (simple versions of) es-sentially all trace formulae that have appeared in the literature. Due to itscontinuing importance in analytic number theory we work out the case of thePetersson-Bruggeman-Kuznetsov trace formula is some detail.

18.1 A brief history of trace formulae

The trace formula is a powerful tool for understanding automorphic repre-sentations on reductive groups. It is an expression for the integral operatorR(f) introduced in §16.1 on the space of cusp forms in L2([G]) in terms oforbital integrals. Relative trace formulae, which include the trace formula asa special case, are formulae that describe period integrals of automorphicforms in terms of the relative orbital integrals of Chapter 17.

The original trace formula was first introduced by Selberg in his seminal1956 paper [Sel56]. He worked with an algebraic group G over the real num-bers and discrete subgroup Γ ≤ G(R). Selberg gave a general formula in thespecial case where Γ\G(R) is compact. Secondly he treated in detail certainspecial noncompact quotient cases such as SL2(Z)\SL2(R) using the theory

363

364 18 Simple Trace Formulae

of Eisenstein series. His greatest achievement was proving the existence ofcusp forms on SL2(R) invariant under SO2(R) on the right. In fact he provedan asymptotic for the number of such forms as a function of their Laplacianeigenvalue. Results of this type are known as Weyl laws since the correspond-ing asymptotics were proven by Weyl for general compact manifolds [Wey12].This is discussed in §19.2 in next chapter.

The primary difference between the compact and noncompact case is thenecessity of understanding the analytic continuation of Eisenstein series andtheir contribution to the trace formula. As explained in §10.4, Eisensteinseries allow one to describe the entire spectrum of L2([G]) in terms of cuspidalrepresentations on Levi subgroups of G. It is a bit of an understatement tocall this a generalization of Selberg’s work, because the low rank case treatedby Selberg offered very little insight into the general picture. The generaltheory was established by Langlands in the 1960’s in groundbreaking work.Incidentally, this work on Eisenstein series led Langlands to the formulationof his functoriality conjecture. All of this earned him an Abel prize in 2018.

Giving a decomposition of L2([G]) in terms of Eisenstein series is a nec-essary first step to generalizing Selberg’s trace formula, but it is not enoughby itself. Over a lifetime of work Arthur used the theory of Eisenstein seriesas the basis for a completely general trace formula for an arbitrary reductivegroup. The technical difficulties he overcame were so immense that the traceformula is often called the Arthur-Selberg trace formula in honor of hiscontributions.

Let f ∈ C∞c (AG\G(AF )). The basic idea behind the proof of the traceformula is to integrate the automorphic kernel function Kf (g, h) of (16.4)along the diagonal copy of [G] and evaluate the expression in two differentmanners. Later, in [Jac86] (see also [Jac05a]), Jacquet realized that the traceformula could be generalized in an extremely useful manner. The idea behindJacquet’s relative trace formula is to replace the diagonal integration inthe usual trace formula by integration along a pair of subgroups H1 ×H2 ≤G×G.

Our goal in this chapter is to give a simple version of the relative traceformula. This is a matter of collecting our preparatory work in Chapter 16and Chapter 17. In the proof we impose assumptions on the test functionf to ensure that various analytic difficulties disappear. This is the reasonwe call the trace formula simple. The disadvantage is that in most casesthese assumptions restrict the class of automorphic representations one cantreat. In the context of the usual trace formula, Arthur’s work mentionedabove removes these assumptions. In the relative setting this is the subjectof ongoing work. See [Lap06, Zyd15, Zyd19].

18.2 A general simple relative trace formula 365

18.2 A general simple relative trace formula

Let F be a global field, let G be a reductive group over F , and let

H ≤ G×G

be a subgroup. We always assume that H is the direct product of a reductiveand a unipotent group. This implies in particular that H(AF ) is unimodular.In the setting of §17.1 we let X = G equipped with the right action of Hgiven on points in an F -algebra R by

G(R)×H(R) −→ G(R)

(g, (h`, hr)) −→ h−1` ghr.

(18.1)

Let

AG,H : = AH ∩ (AG ×AG)

A : = AH ∩∆(AG)(18.2)

where∆ : AG −→ AG ×AG

is the diagonal embedding. Thus A ≤ AG,H . Let CG ≤ H be the stabilizer(in H) of G. It is clear that A ≤ ACG . We assume throughout this sectionthat in fact A = ACG .

Letχ : AG,H\H(AF ) −→ C×

be a quasi-character trivial on H(F ) and let f ∈ C∞c (AG\G(AF )). As in(16.14) we define the relative trace of π(f) with respect to H and χ to be

rtrπ(f) := rtrH,χ π(f) =

∫AG,HH(F )\H(AF )

Kπ(f)(h`, hr)χ(h`, hr)d(h`, hr).

(18.3)

This depends on a choice of Haar measure d(h`, hr) on AG,H\H(AF ).We now define relative orbital integrals in this context. For every γ ∈ G(F )

that is relevant (see Definition 17.13) relatively semisimple (that is, has closedH-orbit) and relatively unimodular (so Hγ(AF ) is unimodular) we can definean orbital integral

ROγ(f) := ROχγ (f) :=

∫AG,H\H(AF )

A\Hγ (AF )

f(h−1r γh`)χ(h)

d(h`, hr)

dhγ. (18.4)

Note that this is slightly different from the definition in (17.51). The root ofthis is that it is more convenient for the spectral side of the trace formula towork with functions on AG\G(AF ) instead of G(AF ). In order for Theorem


18.2.2 below to be valid it is important that we use the same Haar measured(h`, hr) on AG,H\H(AF ) used in the definition of the relative trace, and dhγis a Haar measure on AG,H\Hγ(AF ). Thus d(h`,hr)

dhγis a left AG,H\H(AF )-

invariant Radon measure on

(A\Hγ(AF ))\(AG,H\H(AF )).

It depends on the choice of Haar measure dhγ on A\Hγ(AF ). We now checkthat this is well-defined by slightly modifying the proof of Proposition 17.7.1:

Proposition 18.2.1 If γ is relatively unimodular and semisimple then∫AG,H\H(AF )

A\Hγ (AF )

∣∣f(h−1r γh`)χ(h)

∣∣ dhdhγ

<∞.

Proof. Let

H(AF )′ = H(AF ) ∩G(AF )1 ×G(AF )1

Hγ(AF )′ = Hγ(AF ) ∩G(AF )1 ×G(AF )1.

We have a commutative diagram

Hγ(AF )\H(AF ) ←−−−− Hγ(AF )′\H(AF )′∼−−−−→ (A\Hγ(AF ))\(AG,H\H(AF ))

a(γ,·)y a(γ,·)

y ya(γ,·)

G(AF ) ←−−−− G(AF )1 ∼−−−−→ AG\G(AF )

wherea(γ, (h`, hr)) = h−1

` γhr

is the orbit map. The horizontal arrows from left to right are homeomor-phisms, and the horizontal arrows from right to left are injections onto closedsubsets. The leftmost vertical arrow is proper by Theorem 17.6.1, and it fol-lows that the middle, and hence right, vertical arrow is proper. As in theproof of Theorem 17.4.1 and Proposition 17.7.1 this suffices to establish theproposition. ut

Since we have assumed A = ACG the measure

τ(Hγ) := measdhγ (Hγ(F )A\Hγ(AF )) (18.5)

of (17.56) is finite if γ is relatively elliptic in the sense of Definition 17.14. Hereit is important that we take dhγ to be the same Haar measure on A\Hγ(AF )that we used in the definition of the relative orbital integral (18.4) in orderfor Theorem 18.2.2 below to be valid.

The following is the simple relative trace formula:


Theorem 18.2.2 Let f ∈ C∞c (AG\G(AF )) be a function such that R(f) hascuspidal image and such that if the H(AF )-orbit of γ intersects the support off then the γ is gcf, relatively elliptic, relatively unimodular, and the Hr-orbitof γ is closed. Then

∑π

rtrπ(f) =

′∑[γ]∈Γ (F )

τ(Hγ)ROγ(f)

where the sum on the left is over (H,χ)-distinguished cuspidal automorphicrepresentations π and the sum on the right is over relevant relative classes[γ] that are gcf, relatively unimodular, relatively elliptic, and such that theHr-orbit of γ is closed. Both sums are absolutely convergent.

We recall that if the Hr-orbit of γ is closed, then γ is relatively semisimpleby Lemma 17.1.9. Following Langlands, the left hand side of the trace for-mula is called the spectral side and the right hand side of the trace formulais called the geometric side. We call the trace formula of Theorem 18.2.2general because its geometric set-up includes all trace formulae studied inthe literature as special cases. Indeed the formula does not hold for a generalconnected algebraic subgroup H ≤ G × G without serious modification, soin some sense it is as general as possible. It is simple because we have im-posed assumptions on f to eliminate analytic difficulties on the spectral andgeometric sides.

In the proof of Theorem 18.2.2 we require the following analogue of Lemma17.6.11:

Lemma 18.2.3 Assume that H is reductive and let Ω ⊂ AG\G(AF ) be acompact set. The set of relatively semisimple geometric classes in G(F ) whoseH(AF )-orbit intersects Ω is finite.

Proof. As usual denote by X∗(G) := Hom(G,Gm) the (algebraic) charactergroup of G and similarly for H. For each χ ∈ X∗(G) we obtain a character

χ a(1, ·) : H −→ Gm

given on points by (h`, hr) 7→ χ(h−1` hr). We thus have a map

a(1, ·)∗ : X∗(G) −→ X∗(H)

χ 7−→ χ a(1, ·)

Let Y ≤ X∗(G) be the kernel of a(1, ·)∗. Let

G(AF )′ :=⋂χ∈Y

ker(χ | · | : G(AF )→ R>0)

Then G(F ) is contained in G(AF )′ and the action (18.1) of H(AF ) on G(AF )preserves G(AF )′. Moreover the inclusion G(AF )′ → G(AF ) induces a home-


omorphism

G(AF )′/H(AF )−→AG\G(AF )/H(AF ). (18.6)

(see Exercise 18.1). Thus it suffices to show that given a compact subset Ω ⊂G(AF )′/H(AF ) there are only finitely many relatively semisimple geometricclasses in G(F ) whose image in G(AF )′/H(AF ) intersects Ω. Let

p : G(AF )′/H(AF )→ G/H(AF )

denote the restriction of the natural map G(AF )/H(AF )→ G/H(AF ) to theGIT quotient. In view of Proposition 17.1.5 it suffices to observe that p(Ω)is compact and G/H(F ) is discrete and closed by Lemma 17.6.3, hence

p(Ω) ∩G/H(F )

is finite. ut

Proof of Theorem 18.2.2: Let

Kf (g, h) =∑

γ∈G(F )

f(g−1γh).

Since we have assumed that R(f) has cuspidal image one has

Kf (g, h) = Kf,cusp(g, h)

in the notation of (16.5). By Theorem 16.2.5 we have∫AG,HH(F )\H(AF )

Kf (h`, hr)χ(h`, hr)d(h`, hr) =∑π

rtrπ(f).

At this point if AG,H = A (which, by assumption, is ACG) then we can invokeTheorem 17.7.2 to finish the proof. In general we can still follow the argumentof the proof of Theorem 17.7.2. We now give the details. Unfolding the sumabove we have that it is equal to∑

[γ]

∫AHγ(F )\Hγ(AF )

χ((h`, hr))dhγ

∫AG,H\H(AF )

A\Hγ (AF )

f(h`, hr)χ((h`, hr))d(h`, hr)

dhγ

where the sum on [γ] is over classes in Γ (F ) that are gcf, relatively uni-modular, relatively elliptic, and such that the Hr-orbit of γ is closed. Theintegral ∫

AHγ(F )\Hγ(AF )

χ((h`, hr))dhγ


is absolutely convergent for γ in the sum because γ is relatively elliptic, andit is zero unless γ is relevant. Thus the sum above is equal to

′∑[γ]∈Γ (F )

τ(Hγ)

∫AG,H\H(AF )

A\Hγ (AF )

f(h`, hr)χ(h`, hr)d(h`, hr)

dhγ(18.7)

=

′∑[γ]∈Γ (F )

τ(Hγ) ROγ(f)

where the sum is over classes [γ] as in the statement of the theorem.To justify the manipulations above we must check that∑

[γ]

τ(Hγ)

∫AG,H\H(AF )

A\Hγ (AF )

|f(h`, hr)χ(h`, hr)|d(h`, hr)

dhγ(18.8)

is finite, where the sum is over classes [γ] that are gcf, relatively unimod-ular, relatively elliptic, and such that the Hr-orbit of γ is closed. The in-tegral in (18.8) converges absolutely by Proposition 18.2.1. It therefore suf-fices to prove that only finitely many classes [γ] can contribute a nonzerosummand to (18.8). Consider the contribution of a particular gcf, rela-tively semisimple geometric class [γ0]. By Theorem 17.6.10 for any com-pact set Ω ⊂ Hγ0

\H(AF )/AG,H there are only finitely many H(F )-orbitsxH(F ) ⊂ I\H(F ) such that xH(AF ) intersects Hγ\H(AF )/AG,H . This im-plies that there are only finitely many relative classes in the geometric class[γ0] that can contribute to the sum.

Lastly we show that there are only finitely many geometric classes that cancontribute to the sum. By the same argument used in the proof of Theorem17.7.2 we are reduced to showing that for any compact set Ω ⊂ AG\G(AF )there are only finitely many geometric classes in G with closed Hr-orbit whoseHr(AF )-orbit intersects Ω. This is the content of Lemma 18.2.3. 2

We now make some comments on the assumptions on f in Theorem 18.2.2.For more information on the assumption on f involving the geometric sideof the trace formula we refer to exercises 18.2, 18.3 and 18.4.

In practice in order to ensure that f has cuspidal image one assumesf = fvf

v with fv ∈ C∞c (G(Fv)) and fv ∈ C∞c (AG\G(AvF )) for some finiteplace v where fv is supercuspidal. This suffices to ensure R(f) has cuspidalimage by Lemma 16.4.1. Unfortunately, assuming that fv is supercuspidallimits the set of π that can be studied using the simple relative trace formula:

Lemma 18.2.4 Let v be a finite place of F . Assume that f ∈ C∞c (G(Fv))is supercuspidal (not just F -supercuspidal). If πv is an admissible irreduciblerepresentation of G(Fv) and πv(f) 6= 0 then πv is supercuspidal.

Proof. To ease notation we drop the subscript v, writing F := Fv, etc. Thiswas proven under the assumption that ZG(F ) is compact in Exercise 16.4. We


reduce to this case. This has the advantage of illustrating some techniquesuseful for considering restrictions of representations to derived subgroups.The restriction of π to ZG(F )Gder(F ) decomposes as a finite direct sum ofirreducible representations and hence the same is true of the restriction toGder(F ) since ZG(F ) acts via scalars on π:

π|Gder(F ) = ⊕ni=1π⊕mi

Here πi ∼= πj if and only if i = j and, as written, the multiplicity of eachπi is independent of i. The group G(F ) acts via conjugation on the set ofisomorphism classes of the πi, and if we let G(F )π1 ≤ G(F ) be the subgroupfixing the isomorphism class of π1 the group G(F )/G(F )π1 permutes the iso-morphism classes of the πi’s transitively. Finally, the πi are admissible. Forall of this see [GK82, §2]. Since supercuspidal representations are representa-tions whose matrix coefficients are square-integrable modulo the center π issupercuspidal if and only if πi is supercuspidal for all i, which is equivalentto πi being supercuspidal for a single i since G(F )/G(F )π1 acts transitivelyon the set of representations.

Since π(f) is nonzero there is a ϕ in the space of π such that∫G(F )

f(g)π(g)ϕ 6= 0

which implies that there exists an a ∈ G(F )/Gder(F ) such that∫Gder(F )

f(ga)π(ga)ϕ 6= 0.

Let fa(g) := f(ga). For some i the operator πi(fa) is nonzero. Since f issupercuspidal so is fa (as a function on Gder(F )) so we deduce that πi issupercuspidal by Exercise 16.4. As already mentioned, this implies that π issupercuspidal. ut

Thus if we assume that fv is supercuspidal we can only study cuspidalautomorphic representations π such that πv is supercuspidal. When F = Qand G is split and adjoint Lindenstrass and Venkatesh [LV07] found anothermethod to construct functions f ∈ C∞c (AG\G(AF )) that have cuspidal im-age. This method is interesting for at least two reasons. First it requires oneto work with test functions that are not factorable. Second, provided that thecuspidal automorphic representation π is spherical at infinity i.e. π∞ admitsa vector fixed under a maximal compact subgroup K∞ ≤ G(R) and is notnearly equivalent to a globally induced representation one can find many testfunctions f with cuspidal image such that π(f) 6= 0. The restriction to repre-sentations that are spherical at infinity (and the restriction to split adjoint Gover Q) does not seem to be essential, but significant work would be required

18.3 Products of subgroups 371

to remove it. This idea was applied again in [GH15], and it is likely that itcould be useful in much greater generality.

18.3 Products of subgroups

Assume that H = H1×H2 ≤ G×G, where both H1 and H2 are subgroups ofG, and for each i the neutral component Hi is either unipotent or reductive.Assume moreover that χ = χ1 ⊗ χ2, where χi is a character of Hi(AF ). Inthis case the formula in Theorem 18.2.2 simplifies somewhat. Let K∞ be amaximal compact subgroup of G(F∞).

Corollary 18.3.1 Let f ∈ C∞c (AG\G(AF )) be a K∞-finite function. As-sume that R(f) has cuspidal image and that if the H(AF )-orbit of γ inter-sects the support of f then γ is gcf, relatively elliptic, and the Hr-orbit of γis closed. Then

∑π

∑ϕ∈B(π)

Pχ1(π(f)ϕ)Pχ2

(ϕ) =

′∑[γ]∈H2(F )\G(F )/H1(F )

τ(Hγ)ROγ(f)

where B(π) an orthonomal basis of L2cusp(π) consisting of K∞-finite (and

hence smooth) vectors. Here the sum on the right is over relevant relativeclasses [γ] that are gcf, relatively elliptic, and such that the Hr-orbit of γ isclosed.

Here we are using the notation (14.3) for the period integrals occurringin the corollary. It is important to point out that the only π that contributea nonzero summand to the identity in Corollary 18.3.1 are π that are both(H1, χ1) and (H2, χ2)-distinguished. Moreover the assumption on relativeunimodularity in Theorem 18.2.2 can be (and is) dropped in the case athand. This sort of relative trace formula first appeared in work of Jacquet[Jac86], and has been applied many times in the intervening years.

Proof. Let π be a cuspidal automorphic representation of AG\G(AF ). If f isK∞-finite, then π(f) has finite image (see Exercise 16.1), so in the expansion

Kπ(f)(x, y) =∑

ϕ∈B(π)

π(f)ϕ(x)ϕ(y) (18.9)

of (16.9) only finitely many ϕ can contribute a nonzero summand. Thus

rtrπ(f) =∑

ϕ∈B(π)

Pχ1(π(f)ϕ)Pχ2

(ϕ).

To complete the proof we will show that a relatively semisimple element ofG(F ) is automatically relatively unimodular. If H1 and H2 are both reductive


then a relatively semisimple element has stabilizer with reductive neutralcomponent by Theorem 17.1.4. By Corollary 3.5.2 such a γ is unimodular.

Now assume that H1 is unipotent. For F -algebras R one has

Hγ(R) = (h, γ−1hγ) ∈ H1(F )×H2(F ) : h ∈ γH2γ−1(R)

The neutral component of this group is isomorphic to a subgroup of H1 .Hence it is unipotent, which implies that Hγ(F ) is unimodular (see Exercise(18.6)). ut

18.4 The simple trace formula

The famous simple trace formula of Deligne and Kazhdan [DKV84] (see also[Rog83, BDKV84]) is a special case of the simple relative trace formula ofTheorem 18.2.2. We explain this in the current section.

For γ ∈ G(F ) let

Cγ(R) := g ∈ G(R) : gγg−1 = γ (18.10)

be the centralizer of γ and let

Oγ(f) =

∫Cγ(AF )\G(AF )

f(g−1γg)dg

dgγ(18.11)

whenever this integral is well-defined. This is the usual orbital integral off along the conjugacy class of γ. In this context the assertion that γ is gcf isequivalent to the assertion that H1(Fv, Cγ/(Cγ)) is finite for all places v ofF . As usual, an element γ ∈ G(F ) is elliptic if AG = ACγ . This is equivalentto the assertion that Cγ/Z

G is anisotropic.

The following simple trace formula is a corollary of Theorem 18.2.2:

Corollary 18.4.1 Let f ∈ C∞c (AG\G(AF )). Assume that R(f) has cuspidalimage and that any element of G(F ) that is G(AF )-conjugate to the supportof f is gcf, elliptic and semisimple. Then∑

π

m(π)trπ(f) =∑γ

τ(Cγ)Oγ(f). (18.12)

where the sum on the left is over isomorphism classes of cuspidal automorphicrepresentations of AG\G(AF ), m(π) is the multiplicity of π in the cuspidalspectrum of L2([G]) and the sum on the right is over gcf, elliptic, semisimpleconjugacy classes in G(F ).

As in the statement of Theorem 18.2.2, some care must be taken withmeasures in order for the identity above to be correct. We choose a Haarmeasure dg′ on AG\G(AF ) and for each semisimple γ a Haar measure dg′γ

18.5 The simple twisted trace formula 373

on AG\Cγ(AF ). Then we use dg′ to define the operator π(f) (and hence itstrace) and use dg′γ to define τ(Cγ). Finally we normalize the Radon measuredgdgγ

on Cγ(AF )/G(AF ) used to define Oγ(f) so that it corresponds to dg′

dg′γ

under the natural isomorphism

Cγ(AF )\G(AF )−→(AG\Cγ(AF ))\(AG\G(AF )). (18.13)

Proof. In Theorem 18.2.2 we take χ to be trivial and take H to be thediagonal copy of G inside G×G. In this case A = AG,H = AG and Γ (F ) isjust the set of conjugacy classes in G(F ). Thus ROγ(f) = Oγ(f). All classesare trivially relevant, and all semisimple classes are relatively unimodular byTheorem 17.1.4 and Lemma 3.5.4. Moreover

rtrπ(f) :=

∫[G]

Kπ(f)(x, x)dx = m(π)trπ(f)

(see (16.8)). ut

18.5 The simple twisted trace formula

Let θ be an automorphism of a connected reductive group G over numberfield F . One then has a right action of G on itself by θ-conjugation, givenon points in an F -algebra R by

G(R)×G(R) −→ G(R)

(x, g) 7−→ g−1xθ(g).(18.14)

When θ is the identity this is simply conjugation. Following the lead of §18.4we let

Cθγ(R) := g ∈ G(R) : g−θγg = γ. (18.15)

be the θ-centralizer of γ. The intersection of AG and Cθγ(AF ) is AG∩AθG. Wedefine the twisted orbital integral

TOγ(f) :=

∫AG∩AτGCθγ(AF )\AG\G(AF )

f(g−1γθ(g))dg

dgγ(18.16)

. We say that γ is θ-gcf if H1(Fv, Cθγ/C

θγ)) is finite for all places v of F .

Similarly we say γ is θ-semisimple if Cθγ has reductive neutral component

and θ-elliptic if Cθγ is elliptic, which is to say that ACθγ ≤ AG.We now define the spectral counterpart of a twisted orbital integral. Let


Iθ : L2([G]) −→ L2([G])

ϕ 7−→ (g 7→ ϕ(θ−1(g)).(18.17)

Let π be a cuspidal automorphic representation of AG\G(AF ). If π is isomor-phic to πθ (where πθ(g) := π(θ(g))) then the π-isotypic subspace L2

cusp(π) ofL2

cusp([G]) is stable under Iθ, and thus we can consider the trace

tr(π(f) Iθ) := tr(π(f) Iθ : L2

cusp(π)→ L2cusp(π)

). (18.18)

This is the twisted trace of π(f) (with respect to θ). We observe that π(f)Iθ is acting on the whose π-isotypic space, so if π occurs with multiplicitygreater than 1 this is incorporated into the trace. In contrast, we separatedthe trace and the multiplicity in the statement of Corollary 18.4.1.

The following is a simple version of the twisted trace formula:

Corollary 18.5.1 Let f ∈ C∞c (AG\G(AF )). Assume that R(f) has cuspidalimage and that any element of G(F ) that is G(AF )-conjugate to the supportof f is θ-gcf, θ-elliptic and θ-semisimple. Then

∑π

tr(π(f) Iθ) =

′∑[γ]

τ(Cθγ)TOγ(f). (18.19)

where the sum over π is over isomorphism classes of cuspidal automorphicrepresentations of AG\G(AF ) such that πθ ∼= π and the sum on the right isover θ-conjugacy classes in G(F ) that are θ-gcf, θ-elliptic and θ-semisimple.

Proof. In Theorem 18.2.2 we take χ to be trivial and take H to be the twisteddiagonal copy of G in G times G; that is, for F -algebras R one has

H(R) = (g, θ(g)) : g ∈ G(R)

In this case AG = AG,H , A = AG∩AθG and Γ (F ) is just the set of τ -conjugacyclasses in G(F ). Moreover ROγ(f) = TOγ(f). All classes are trivially rel-evant, and all τ -semisimple classes are relatively unimodular by Theorem17.1.4. Moreover

rtrπ(f) :=

∫[G]

Kπ(f)(x, xθ)dx

This vanishes if π 6∼= πθ (see Exercise 14.2), and if π ∼= πθ it is equal totr(π(f) Iθ) (see the discussion around (16.8)). ut

Historically the twisted trace formula was first introduced by Saito [Sai75]and Shintani [Shi79]. For a completely general twisted trace formula we referto [LW13]. A relative version of the simple twisted trace formula (which isanother special case of Theorem 18.2.2) was introduced in [Hah09]. We discussthe relationship between the trace formula and the twisted trace formula inChapter 19.

18.6 A variant 375

18.6 A variant

It has proven to be beneficial to work with a variant of the simple relativetrace formula of Theorem 18.2.2. In this section we discuss the variant in im-pressionistic terms to aid the reader when it is encountered in the literature.

Lt X be a smooth affine scheme over the global field F equipped with anaction of the reductive F -group G. Since X is smooth the Schwartz spaceS(X(AF )) is defined (see §17.7).

Let H ≤ G be a subgroup such that H is the direct product of a reductivegroup and a unipotent group and let χ : H(AF )→ C× be a character trivialon (AG ∩ AH)H(F ). For each x ∈ X(F ) with closed orbit under G thestabilizer Gx has reductive neutral component by Theorem 17.1.4 and wecan consider the subgroup

Hx := Gx ×H ≤ G×G (18.20)

The character χ extends to a character

Gx(AF )×H(AF ) −→ C×

(g, h) 7−→ χ(h)

which we will continue to denote by χ. We are now in the setting of §18.2,and we can form the relative trace

rtrHx,χπ(f).

for any f ∈ C∞c (AG\G(AF )).Now let Φ ∈ C∞c (AG\X(AF )). We assume that∫

AG∩AHH(F )\H(AF )

∑x∈X(F )

|Φ(xh)χ|(h)dh

converges. For conditions ensuring convergence see Theorem 17.7.2. Thus theintegral ∫

AG∩AHH(F )\H(AF )

∑x∈X(F )

Φ(xh)χ(h)dh (18.21)

is well-defined.Assume that for each x ∈ X(F ) we can choose fx ∈ C∞c (AG\G(AF )) such

that

Φ(xy) =

∫Gx(AF )

fx(g−1y)dg (18.22)

for y ∈ G(AF ). This is not always possible, see Exercise 18.7. Then

376 18 Simple Trace Formulae∫AG∩AHH(F )\H(AF )

∑x∈X(F )

Φ(xh)χ(h)dh

=∑

x∈X(F )/G(F )

∫AHx,χHx(F )\Hx(AF )

∑γ∈G(F )

fx(h−11 γh2)χ(h2)dh1dh2.

(18.23)

Thus (18.21) is a sum over x ∈ X(F )/G(F ) of geometric sides (or spectralsides) of relative trace formula for the various Hx. Studying (18.21) amountsto studying distinction of automorphic representations by (Hx, χ) as x varies.In particular, if X is homogeneous, the Hx are all inner forms of eachother.This method of packaging distinction problems together has proven crucial,especially in work on the Gan-Gross-Prasad conjecture (see §14.7). We pointout that even when it is not possible to choose fx as in (18.22) it is stillpossible to expand the function∑

x∈X(F )

Φ(xg) ∈ C∞([G])

spectrally. More or less it will give something like the right hand side of(18.23), but the fx involved will not necessarily be compactly supported.

In some sense, this is the key point. Replacing G by X has the effectof changing the function space S(X(AF )). As an example, let E/F be aquadratic extension and let G = ResE/FGLn and let X be the space ofHermitian matrices with respect to E/F . Thus X(F ) is the space of n × nmatrices x with coefficients in E such that θ(x) = xt, where θ is a generatorof Gal(E/F ). Then there is a natural right action

X(R)×G(R) −→ X(R)

(x, g) 7−→ gtxθ(g)

The stabilizer of any invertible element of X(F ) is a unitary group. Jacquettook H to be the unipotent radical of a Borel subgroup of G and χ to be ageneric character and then studied (18.21) in order to prove Theorem 14.5.2[Jac05b, Jac10] (see also [Ngo99b, Ngo99a]). The fact that X is a linear space,and hence admits a Fourier transform, turns out to be crucial.

One can also ask what happens when X is no longer smooth. This isthe subject of important ongoing research. We refer to [BK00, BK02, GL19,Get14, Ngo14, Sak12].

18.7 The Petersson-Bruggeman-Kuznetsov formula

Given two nonzero integers m and n and a positive integer c, the classicalKloosterman sum is defined as

18.7 The Petersson-Bruggeman-Kuznetsov formula 377

K(m,n; c) :=∑

1≤a≤c(a,c)=1

e2πi(am+an)/c,

where aa ≡ 1 (mod c). Petersson [Pet32] expressed a certain sum of Fouriercoefficients of holomorphic weight k cusp forms in terms of the Bessel functionJk−1 and Kloosterman sums. Historically this was the first relative trace for-mula, although cosmetically Petersson’s proof appears different. It involvedconstructing a Poincare series and computing its coefficients in two differentmanners rather than integrating an automorphic kernel function. Brugge-man [Bru78] and Kuznetsov [Kuz80] later showed how to incorporate Maassforms into the formula. Traditionally, the formula is called the Kuznetsovformula.

Using the formula Kuznetsov was able to prove estimates for the sum

∞∑c=1

K(m,n; c)

cφ

(2π√mn

c

)for suitable φ ∈ C∞(R>0). We refer to [IK04, Chapter 16] for more details.There are many other analytic number theory papers that use the formula,for example [Iwa02, GS83]. It has also played a role in work on Langlands’beyond endoscopy proposal [Her11, Her12, Her16, Sar, Ven04].

Following Jacquet, we will derive the Kuznetsov formula as a relative traceformula (see [KL08, KL13] for a nice exposition for GL2, and [Ye00] for aslightly different take on the higher rank case). The beautiful fact about thistrace formula is that the general simple relative trace formula of Theorem18.2.2 (together with a small additional argument for the spectral side) canbe used to derive it in complete generality, with no additional assumptions onthe test functions involved. We state this precisely in Theorem 18.9.2 below.In this sense it is analytically the simplest trace formula. The Petersson-Kuznetsov formula remains an important tool in analytic number theory andhas not been fully exploited in higher rank (see, e.g. [BB15] for recent resultsfor GL3). We have therefore structured our treatment of the formula withapplications to analytic number theory in mind.

We now set notation. Let N := Nr ≤ GLr be the unipotent radical ofthe Borel B := Br of upper triangular matrices and let T := Tr ≤ B be themaximal torus of diagonal matrices. For this trace formula we take

H := N ×N ≤ GLr ×GLr.

Thus we have a right action

GLr ×N ×N −→ GLr

given on points by (n1, n2) · γ = n−11 γn2. For a global or local field F we

denote the stabilizer of a γ ∈ GLr(F ) under this action by


Nγ := Hγ ≤ N ×N.

Thus for F -algebras R one has

Nγ(R) :=

(n, γ−1nγ) : n ∈ N(R) ∩ γN(R)γ−1. (18.24)

Lemma 18.7.1 Every element of GLr(F ) is relatively semisimple, relativelyunimodular, and relatively elliptic.

Proof. For every γ ∈ GLr(F ) the stabilizer Nγ is unipotent, hence unimodu-lar by Exercise 18.6. Moreover the fact that Nγ is unipotent implies that ANγis trivial and we deduce that γ is relatively elliptic. Finally we conclude from[Mil17, Theorem 17.64] that every element of GLr(F ) is relatively semisim-ple. ut

The partial determinants

deti : GLr −→ Ga

given on points bydeti ( ∗ ∗Ai ∗ ) = det(Ai)

are invariant under the action of N × N for 1 ≤ i ≤ r. Here Ai is an i × imatrix.

For every representative w ∈ GLr(F ) of the Weyl group we define a mor-phism (of schemes, not group schemes)

Υw : Grm −→ GLr (18.25)

by

Υw(c1, . . . , cr) =

cr/cr−1

cr−1/cr−2

. . .

c2/c1c1

w. (18.26)

By the Bruhat decomposition [Mil17, Example 21.46] one has

GLr(F ) =∐

w∈W (GLr,T )(F )

N(F )T (F )wN(F )

and thus every element of N(F )\GLr(F )/N(F ) has a representative of theform Υw(c) for some c ∈ (F×)r. The orbit of a given w ∈ W (GLr, T )(F )under N ×N is called a Bruhat cell. Let

18.8 Kloosterman integrals 379

w0 =

1...

1

be (the standard representative of) the long Weyl element. The orbit of w0

is the unique open Bruhat cell [Mil17, Theorem 21.73]. Since it is possible tochoose test functions supported in this cell at a given place, elements of theopen Bruhat cell play a key role.

The map Υw0gives a useful description of the open Bruhat cell:

Lemma 18.7.2 The Υw0 induces a bijection

Υw0 : (F×)r −→ N(F )\B(F )w0B(F )/N(F ).

Moreover, the stabilizer Nγ of any element of B(F )w0B(F ) is trivial.

This lemma implies in particular that all elements of the open Bruhat cellare relevant.

Proof. It is clear that any element of N(F )\B(F )w0B(F )/N(F ) is in theimage of the map Υw0

. On the other hand the partial determinants deti areall N ×N -invariant and

deti(Υw0(c)) = ±ci,

where ci is the ith entry of c and the sign ± depends only on i and r. ThusΥw0

is injective. Thus Υw0induces a bijection as claimed.

For the claim on triviality of stabilizers we can assume γ = tw0 for t ∈T (F ) by what we have already proven. Then if (n1, n2) ∈ Nγ(F ) one has

w−10 t−1n1tw0 = n2. (18.27)

The left hand side is in the unipotent radical of the Borel opposite to B, soboth sides must equal to I. ut

18.8 Kloosterman integrals

The geometric side of the Kuznetsov formula is a sum of Kloosterman inte-grals. We define Kloosterman integrals in this section and discuss their basicproperties. The reader unfamiliar with integrals over totally disconnectedgroups may want to consult Appendix B and complete the exercises of thatappendix before reading this section.

We define a morphism

ν : Gr−1m −→ T (18.28)


via

ν(m) :=

∏r−1i=1 mi

. . .

mr−2mr−1

mr−1

1

. (18.29)

We note that

ν(m)−1

1 x1 ∗. . .

. . .

1 xr−1

1

ν(m) =

1 m−1

1 x1 ∗. . .

. . .

1 m−1r−1xr−1

1

. (18.30)

Let F be a global field and fix a nontrivial character

ψ : F\AF → C×.

It factors as ψ :=∏v ψv, where ψv : Fv → C× is a nontrivial character. For

m ∈ F r−1 we set

ψm

1 x1 ∗. . .

. . .

1 xr−1

1

:= ψ

(r−1∑i=1

mixi

). (18.31)

We also use the obvious local analogue of this notation.We now discuss which elements of N(F )\GLr(F )/N(F ) are relevant. One

hasSr−→W (GLr, T )(F ).

The map sends a permutation σ ∈ Sr to the class of the permutation matrixwσ such that wσ has nonzero entries in the (i, σ(i)) positions.

The following lemma was attributed to Piatetski-Shapiro in [Fri87, p. 175].

Lemma 18.8.1 Assume m1,m2 ∈ (F×)r−1. A class in N(F )\GLr(F )/N(F )is ψm1

ψm2-relevant only if it is a subset of N(F )T (F )wσN(F ) where σ−1(i) <

σ−1(i + 1) implies σ−1(i + 1) − σ−1(i) = 1 for 1 ≤ i < r. In other words aclass contained in N(F )T (F )wσN(F ) is relevant only if

wσ =

Ir1

Ir2

. ..

Irk


where∑ki=1 ri = r.

Proof. Consider u := I +∑r−1i=1 xiEi,i+1 where xi ∈ AF and Ei,j is the

elementary matrix with a 1 in the (i, j) position and zeros elsewhere. Onehas

wσuw−1σ = I +

r−1∑i=1

xiEσ−1(i),σ−1(i+1).

Suppose that σ−1(i) < σ−1(i+ 1). Then(I + xiEσ−1(i),σ−1(i+1), I + xiEi,i+1) : xi ∈ AF

≤ Nwσ (AF ).

In order for ψm1ψm2

to be trivial on this subgroup of Nwσ (AF ) it is necessary

that σ−1(i + 1) − σ−1(i) = 1. The proves that N(F )wσN(F ) is ψm1ψm2-

relevant only if σ−1(i) < σ−1(i + 1) implies σ−1(i + 1) − σ−1(i) = 1 for1 ≤ i < r. On the other hand for c ∈ (F×)r−1 the element ν(c)wσ is ψm1

ψm2-

relevant only if wσ is ψm1cψm2-relevant by a change of variables. Here

m1c := (m11c1, . . . ,m1(r−1)cr−1),

where m1 = (m11, . . . ,m1(r−1)) and c = (c1, . . . , cr−1). Thus we deduce thata general element of N(F )T (F )wσN(F ) is ψm1 ψm2 -relevant only if σ−1(i) <σ−1(i+ 1) implies σ−1(i+ 1)− σ−1(i) = 1 for 1 ≤ i < r. ut

For f ∈ C∞c (AGLr\GLr(AF )) and γ ∈ GLr(F ), we define the orbitalintegral

ROγ(f,m1,m2) :=

∫Nγ(AF )\N(AF )2

f(n−11 γn2)ψm1

(n1)ψm2(n2)dn1dn2.

(18.32)

It is sometimes called a Kloosterman integral. We use the analogous no-tation in the local setting. By Proposition 18.2.1 and Lemma 18.7.1 it isabsolutely convergent.

Proposition 18.8.2 The sum

′∑γ∈N(F )\GLr(F )/N(F )

τ(Nγ) |ROγ(f,m1,m2) |

is convergent and∫[N ]2

Kf (n1, n2)ψm1(n1)ψm2

(n2)dn1dn2

=

′∑γ∈N(F )\GLr(F )/N(F )

τ(Nγ)ROγ(f,m1,m2)


where the sum is over relevant classes. Here, as usual, [N ] := N(F )\N(AF ).

By Lemma 18.7.1 every γ is relatively unimodular, relatively elliptic, andrelatively semisimple. Thus in the number field case we could invoke Theorem18.2.2 to prove this proposition. In the function field case we do not knowif every γ ∈ GLr(F ) is gcf, and this is the obstruction to simply invokingTheorem 18.2.2 in the function field case. Fortunately the gcf assumptionwas just used to establish convergence in the proof of Theorem 17.7.2 and inthe current setting we can just proceed directly:

Proof. This follows from the same manipulations used in the proof of The-orem 18.2.2. The only point one must check is that these manipulations arejustified. This follows from the observation that∫

[N ]2

∑γ∈G(F )

|f(n−11 γn2)|dn1dn2

is absolutely convergent because [N ] is compact. ut

In the remainder of this section we assume that F is nonarchimedeanwith ring of integers OF . For γ in the open Bruhat cell we will relateROγ(1GLr(OF ),m1,m2) to classical Kloosterman sums, justifying callingthese orbital integrals Kloosterman integrals.

We start with some easy observations (compare [Ste87, Fri87]):

Lemma 18.8.3 The integral ROγ(1GLr(OF ),m1,m2) is only nonzero if det γ ∈O×F and deti(γ) ∈ OF for 1 ≤ i ≤ r.

Proof. This follows from the fact that the partial determinants deti are in-variant under the action of N ×N on GLr. ut

Let f∨(g) := f(g−1).

Lemma 18.8.4 One has that

ROΥw(c)(f,m1,m2) = ROΥw(c)−1(f∨,−m2,−m1).

Proof. We compute that∫(NΥw(c)(F )\N(F ))2

f(n−11 Υw(c)n2)ψm1(n1)ψm2

(n2)dn1dn2

=

∫(NΥw(c)(F )\N(F ))2

f∨((n−11 Υw(c)n2)−1)ψm1

(n1)ψm2(n2)dn1dn2

=

∫(NΥw(c)−1 (F )\N(F ))2

f∨(n−12 Υw(c)−1n1)ψm1

(n1)ψm2(n2)dn1dn2,

where the last equality is, by definition, ROΥw(c)−1(f∨,−m2,−m1). ut


It turns out that for suitable c the integral ROΥw(c)(f,m1,m2) defined in(18.32) can be expressed as the product of Kloosterman sums, see Propo-sition 18.8.6 below. To state this, we first prove a recursive relation forROΥw(c)(f,m1,m2). To ease on notation, for m1,m2 ∈ (F×)r−1, we let

m1 := (m11,m12, . . . ,m1(r−1)) and m2 := (m21,m22, . . . ,m2(r−1)).

Then under suitable assumptions the local orbital integral of (18.32) on GLrcan be written as that of GLr−1 in the following way:

Lemma 18.8.5 Assume that c = (c1, c2, . . . , cr) ∈ (OF ∩ F×)r and thatc1, cr ∈ O×F . Then one has that

ROΥw0(c)(1GLr(OF ),m1,m2)

= 1OF (m1(r−1))1OF (m21)ROΥw0(c)(1GLr−1(OF ), m1, m2),

where w0 is the long Weyl element for GLr−1,

c = (c2/c1, c3/c1, . . . , cr/c1) ∈ (OF ∩ F×)r−1,

and m1, m2 ∈ (F×)r−2 are obtained by removing the last entry m1(r−1) fromm1 and by removing the first entry m21 from m2, respectively.

Proof. To ease notation, let C be the (r − 1)× (r − 1) matrix given by

C :=

cr/cr−1

...

c2/c1

.

For n1, n2 ∈ Nr−1(F ), α1, α2 ∈ F r−1 (the latter viewed as column vectors)we compute(

n1 α1

1

)(C

c1

)(1 αt2n2

)=

(c1α1 n1Cc1

)(1 αt2n2

)=

(c1α1 Cc1 c1α

t2

).

HereC = c1α2α

t1 + n1Cn2

is an (r−1)×(r−1) matrix, c1α1 is a column vector, and c1αt2 is a row vector.

Since we assumed that c1 ∈ O×F and cr ∈ O×F , this matrix is in GLr(OF ) ifand only if α1, α2 ∈ Or−1

F and n1Cn2 ∈ GLr−1(OF ). The lemma follows. ut

Proposition 18.8.6 Assume that c = (c1, . . . , cr) ∈ OrF ∩ (F×)r where cr ∈O×F , cj 6∈ O×F , and ci ∈ O×F for all i 6= j. Then

ROΥw0(c)(1GLr(OF ),m1,m2) = 0

unless m1,m2 ∈ Or−1F , in which case it is equal to

384 18 Simple Trace Formulae∑x∈(OF /cj)×

ψ

(−m2jx+m1(r−j−1)cj+1cj−1x

cj

). (18.33)

Here we set cj−1 = 1 if j = 1.

Proof. By Lemma 18.8.5 iterated j−1 times we have that ROΥw0 (c)(1GLr(OF ),m1,m2)vanishes unless m1(r−1), . . . ,m1(r−j),m21, . . . ,m2(j−1) ∈ OF . Here m1i andm2i are the ith entry of m1 and m2 respectively. Assuming this is the case,the same lemma implies that

ROΥw0(c)(1GLr(OF ),m1,m2) = ROΥw0

(c)(1GLr−j(OF ), m1, m2).

Here w0 is the long Weyl element of GLr−j , and

c := (cj/cj−1, cj+1/cj−1, . . . , cr/cj−1),

m1 := (m11, . . . ,m1(r−j−1)),

m2 := (m2j , . . . ,m2(r−1)).

We now apply Lemma 18.8.4 to write this as

ROΥw0(c)−1(1GLr−j(OF ),−m2,−m1)

= ROΥw0(cj−1/cr,...,cj−1/cj)(1GLr−j(OF ),−m2,−m1).

Applying Lemma 18.8.5 this vanishes unless

m11, . . . ,m1(r−j−2),m2(j+1), . . . ,m2(r−1) ∈ OF

in which case it is

ROΥw0(cj/cj−1,cj+1/cj)(1GL2(OF ),−m2j ,−m1(r−j−1)).

We compute this directly. It is equal to∫1GL2(OF )

((1 −x

1

)(cj+1/cj

cj/cj−1

)(1 y

1

))ψ(m2jx)ψ(m1(r−j−1)y)dxdy

=

∫1GL2(OF )

(−cjx/cj−1 −cjxy/cj−1 + cj+1/cjcj/cj−1 cjy/cj−1

)ψ(m2jx)ψ(m1(r−j−1)y)dxdy

=1

|cj |2

∫1GL2(OF )

(−x/cj−1

−xy/cj−1+cj+1

cj

cj/cj−1 y/cj−1

)ψ

(−m2jx+m1(r−j−1)y

cj

)dxdy.

This vanishes unless m1(r−j),m2j ∈ OF , in which case it is equal to

∑x∈(OF /cj)×

ψ

(−m2jx+m1(r−j−1)cj+1cj−1x

cj

),

18.9 Sums of Whittaker coefficients 385

which completes the proof. ut

When F = Qp one can choose the character ψ so that (18.33) is a classicalKloosterman sums.

18.9 Sums of Whittaker coefficients

The spectral side of the Kuznetsov formula is a sum of Whittaker coefficients.We make this precise in the current section and then prove the Kuznetsovformula in Theorem 18.9.2 below.

Let f ∈ C∞c (AGLr\GLr(AF )). If π is a cuspidal automorphic representa-tion of AGLr\GLr(AF ) then we can form the relative trace

rtr(π(f),m1,m2) :=

∫[N ]2

Kπ(f)(n1, n2)ψm1(n1)ψm2

(n2)dn1dn2. (18.34)

Since [N ] is compact this is absolutely convergent. We then have the followingtheorem:

Theorem 18.9.1 Assume that R(f) has cuspidal image. Then one has anequality of absolutely convergent sums

∑π

rtr(π(f),m1,m2) =

′∑γ∈N(F )\GLr(F )/N(F )

τ(Nγ)ROγ(f,m1,m2) (18.35)

where the sum on the left is over isomorphism classes of cuspidal automorphicrepresentations of AGLr\GLr(AF ) and the sum on the right is over relevantclasses γ.

Proof. This is immediate from Theorem 16.2.5 and Proposition 18.8.2. ut

For the remainder of the section we assume that F is a number field. Wewill use Langlands’ spectral expansion of the automorphic kernel functionKf (x, y) from §16.3 to give a general version of Theorem 18.9.1 that requiresno assumption on f .

Let B ≤ GLr be the Borel subgroup of upper triangular matrices (in eachfactor) and call a parabolic subgroup P ≤ GLr standard if it contains B. Welet MP ≤ P be the unique Levi subgroup containing the maximal torus ofdiagonal matrices. We let AP ≤ ResF/QMP (R) be the connected componentof the identity (in the real topology) and let

aP := LieAP .

Then Langlands’ spectral decomposition of L2([GLr]) implies that


Kf (x, y) =∑P

n−1P

∑σ

∫iaP

KI(σ,λ)(f)(x, y)dλ (18.36)

where the sum on P is over standard parabolic subgroups of GLr, nP is theorder of the association class of P and the sum on σ is over isomorphismclasses of discrete automorphic representations of HP . Each σ occurs withmultiplicity 1 by the main theorem of [MgW89]. In §16.3 the sum over σ ison the inside of the integral over iaP , but they can be moved outside by theabsolute convergence statements proven in [Art78, §4]. Here

.KI(σ,λ)(f)(x, y) :=∑ϕ∈Bπ

E(x, I(σ, λ)(f)ϕ, λ)E(y, ϕ, λ) (18.37)

where Bπ is an orthonormal basis of the σ-isotypic subspace of HP consist-ing of vectors that are finite under a maximal compact subgroup K∞ ≤GLr(F∞). This expression is only an L2-expansion of the kernel. If f is finiteunder a maximal compact subgroup of GLr(F∞) then the sum can be takento be finite and hence the identity (18.37) is valid pointwise. Even if this isnot the case the sum is still represented by a smooth function of x and y. Werefer to [Art78, §4] for proofs of these assertions.

We set

rtr(I(σ, λ)(f),m1,m2) :=

∫[N ]2

KI(σ,λ)(f)(n1, n2)ψm1(n1)ψm2(n2)dn1dn2.

Since [N ] is compact this is absolutely convergent.The following is the full version of the Bruggemann-Petersson-Kuznetsov

formula:

Theorem 18.9.2 One has∑P

n−1P

∑σ

∫iaP

|rtr(I(σ, λ)(f),m1,m2)|dλ <∞.

Moreover ∑P

n−1P

∑σ

∫iaP

rtr(I(σ, λ)(f),m1,m2)dλ

=

′∑γ∈N(F )\GLr(F )/N(F )

τ(Nγ)ROγ(f,m1,m2),

where the prime indicates that the sum on the right is over relevant γ.

Proof. The absolute convergence statement follows from the fact that [N ] iscompact together with the estimates on the terms in the spectral expansion(18.36) contained in [Art78, §4]. Thus by (18.36) we conclude that

18.9 Sums of Whittaker coefficients 387∫[N ]2

Kf (n1, n2)ψm1(n1)ψm2(n2)dn1dn2

=∑P

n−1P

∑σ

∫iaP

rtr(I(σ, λ)(f),m1,m2)dλ.

The theorem now follows from Proposition 18.8.2. ut

It is important to observe that only generic representations can contributeto the spectral side of Theorem 18.9.2. In particular, for P = G only cuspidalrepresentations contribute by the following theorem:

Theorem 18.9.3 If π is a generic discrete automorphic representation ofAGLr\GLr(AF ) then π is cuspidal.

Proof. This is a combination of the main theorem of [MgW89] and [JL13,Theorem 5.5]. ut

In analytic number theory it is useful to understand the dependence ofthe quantities rtr(I(σ, λ),m1,m2) on m1 and m2. We now make this explicit.Assume I(σ, λ) is unramified outside a finite set S of places of F includingthe infinite places. Assume moreover that I(σ, λ) is generic.

Let ψ := ψ1, where ψm is defined in (18.31). Using the notation (11.3) wehave the following lemma:

Lemma 18.9.4 If I(σ, λ) is generic then there is a unique vector

WS ∈ W(I(σ, λ)S , ψS)GLr(OSF ) :=∏v 6∈S

W(I(σ, λ)v, ψv)GLr(OFv )

such that WS(I) = 1.

Proof. We first prove that I(σ, λ) is irreducible. This is equivalent to theassertion that I(σ, λ)v is irreducible for all v. Since I(σ, λ)v is generic it isirreducible for v|∞ (see below [Jac09, Lemma 2.5]). For nonarchimedean v therepresentation I(σ, λ)v is known to be irreducible even if we do not assumeit is generic [Ber84].

To complete the proof we recall that by Theorem 11.3.1 Whittaker modelsare unique. ut

For f ∈ C∞c (A\GLr(AF )) and m1S ,m2S ∈ (F×S )r−1 let

fm1S ,m2S(g) := f(ν(m1S)gν(m2)−1) ∈ C∞c (AGLr\GLr(AF )) (18.38)

were ν is defined as in (18.28).

Lemma 18.9.5 Assume that f and I(σ, λ) are unramified outside S. Onehas that

rtr(I(σ, λ)(fm1,m2),m1,m2) = WS(ν(m1))WS(ν(m2))rtr(πλ(f), 1, 1)


where WS is as in Lemma 18.9.4.

Here the 1 refers to the vector in (F×)r−1 that is identically 1 in all entries.The quantity WS(ν(m1)) can be computed in terms of the Satake parametersof I(σ, λ)S using Corollary 11.4.2.

Proof. To ease notation in the proof write

f := fm1S ,m2S.

We change variables

(n1, n2) 7→ (ν(m1)−1n1ν(m1), ν(m2)−1n2ν(m2))

and use (18.30) to see that

rtr(π(f),m1,m2)

=

∫[N ]2

KI(σ,λ)(f)(ν(m1)−1n1ν(m1), ν(m2)−1n2ν(m2))ψ1(n1)ψ1(n2)dn1dn2

=

∫[N ]2

KI(σ,λ)(f)(n1ν(m1), n2ν(m2))ψ(n1)ψ(n2)dn1dn2.

Here we have used the fact that KI(σ,λ)(f)(g1, g2) is left invariant under

G(F )×G(F ). This in turn is equal to∫[N ]2

KI(σ,λ)(f)(n1ν(mS1 ), n2ν(mS

2 ))ψ(n1)ψ(n2)dn1dn2. (18.39)

The function

(g1, g2) 7−→∫

[N ]2KI(σ,λ)(f)(n1g1, n2g2)ψ(n1)ψ(n2)dn1dn2

is a global Whittaker function for I(σ, λ)⊗I(σ, λ)∨ for the character (n1, n2) 7→ψ(n1)ψ(n2) that is unramified outside S. Thus the lemma follows fromuniqueness of Whittaker models in Theorem 11.3.1. ut

Exercises

18.1. Prove that (18.6) is a homeomorphism.

For the next three problems we assume that we are in the setting of §18.2.Let v be a finite place of the global field F and let

fv ∈ C∞c (AG\G(AvF )), fv ∈ C∞c (G(Fv)), f := fvfv.

18.9 Sums of Whittaker coefficients 389

Let γ ∈ G(F ) and let Ω ⊂ AG\G(AF ) be the H(AF )-orbit of γ.

18.2. Prove that if fv is supported in the set of elements of G(Fv) with closedHr-orbit and f(x) 6= 0 for some x ∈ Ω then γ is relatively semisimple.

18.3. Prove that if fv is supported in

γ′ ∈ G(Fv) : Hγ′ = Hγ′

and f(x) 6= 0 for some x ∈ Ω then γ is gcf.

18.4. Prove that if fv is supported in the set of elements of

γ′ ∈ G(Fv) : (ZH ∩∆(ZG))\Hγ′ is anisotropic

and f(x) 6= 0 for some x ∈ Ω then γ is relatively elliptic. Here ∆ : G→ G×Gis the diagonal embedding.

18.5. For f ∈ C∞c (AGLr\GLr(F∞)) and m1,m2 ∈ (F×∞)r−1 let

fm1∞,m2∞(g) := f(ν(m1)gν(m2)−1).

For c := (c1, . . . , cr) ∈ (F×∞)r, prove that

ROΥw(c)(fm1∞,m2∞ ,m1,m2) 6= 0

only if|cr det(ν(m1m

−12 ))|v f Y

for some Y ∈ R>0 independent of v|∞ and c satisfies

|deti(ν(m1)Υw(c)ν(m2)−1)|v f |cr det(ν(m1m−12 ))|i/rv

for all v|∞ and 1 ≤ i ≤ r. In particular, if w = w0, then

ROΥw0(c)(fm1∞,m2∞ ,m1,m2) = 0

unless

|ci|v f |cr det(ν(m1m−12 ))|i/rv |deti(ν(m−1

1 )w0ν(m2))|v

for all v|∞ and 1 ≤ i ≤ r.

18.6. Let H be a unipotent algebraic group over a local field F . Prove thatH(F ) is unimodular.

18.7. Let X be the space of n×n matrices, equipped with the natural actionof GLn on the right. Let Φ ∈ C∞c (X(AF )/AGLn). Prove that the restrictionof Φ to GLn(AF )/AGLn is smooth but not compactly supported.

Chapter 19

Applications of Trace Formulae

What we know is not much.What we do not know is immense

P.-S. de Laplace

Abstract In this chapter, we discuss applications of relative trace formulae.Our goal is to provide an introduction to the key motifs of existence andcomparison underlying these applications.

19.1 Existence and comparison

Selberg’s motivation for proving the trace formula for SL2 over Q was toprove the existence of cusp form that are fixed by SO2(R). In the process heactually estimated the number of such cusp forms in a suitable family andshowed that they exist in abundance. This was the first application of thetrace formula, and it can be viewed as a (quantifiable) existence result. Wediscuss this result in §19.2.

Motivated by the Langlands functoriality conjecture, Jacquet and Lang-lands [JL70] gave a very different type of application of the trace formula.They compared two different trace formulae on different groups in order to de-duce a relation between automorphic representations on the two groups. Thisidea has been extended to what is known as the theory of twisted endoscopyand continues to play a key role in automorphic representation theory. Wediscuss this in the second part of the chapter. All sections of the chapter aremore or less brief surveys. Our main aim is merely to whet the appetite ofthe reader and give them some guidance on further reading.

391

392 19 Applications of Trace Formulae

19.2 The Weyl law

Let M be a smooth compact Riemannian manifold with smooth bound-ary ∂M . Associated to this datum is a Laplace-Beltrami operator ∆. Calla function ϕ ∈ C∞(M) ∩ L2(M) a Laplacian eigenform with eigenvalue λ ifϕ|∂M = 0 and

∆ϕ = λϕ.

It is known that each eigenvalue λ is a positive real number, the set of eigen-values is discrete and that L2(M) is the Hilbert space direct sum of thecorresponding eigenspaces L2(M)(λ). Moreover dimL2(M)(λ) <∞ for eachλ. For X > 0, consider the counting function

N(X) := dimL2(M)(λ) :√λ ≤ X.

The following is the Weyl law (see [Wey12] and [MP49]):

Theorem 19.2.1 (The Weyl law) One has that

N(X) =vol(M)

(4π)dimM/2Γ (dimM2 + 1)

XdimM + o(XdimM ) (19.1)

as X →∞. ut

Consider the Shimura manifolds Sh(G,X)K of §15.2. These are finite unionsof Riemannian manifolds. When they are compact the Weyl law implies theexistence of an abundance of automorphic forms on them. However they areoften noncompact (see Theorem 2.6.2 for a precise statement). Even if onecompactifies Sh(G,X)K , it is still unclear in the noncompact case whetherthe eigenforms produced by the Weyl law correspond to the continuous orcuspidal spectrum. Selberg developed the trace formula for SL2 (over Q) toaddress precisely this question [Sel56], and he proved an analogue of (19.2.1)in this setting.

Muller [M07] established the analogue of the Weyl law for G = SL(n) overQ. His proof makes use of the full noninvariant trace formula of Arthur, andcan treat cuspforms that are not necessarily fixed by On(R). About the sametime Lindenstrauss and Venkatesh [LV07] proved an analogous result for allsplit semisimple adjoint groups over Q. Notably, their method is soft in thatit does not require the full trace formula, only a simple analogue. It is basedon a novel construction of test functions with purely cuspidal image. Theirmethod currently only treats cuspforms that are fixed by a maximal compactsubgroup at infinity, but it is likely that this can be removed.

Using the same technique Getz and Hahn [GH15] established a relativeanalogue of the Weyl law under some assumptions via the general simplerelative trace formula of Theorem 18.2.2. This relative Weyl law gives anasymptotic formula for the number of Laplacian eigenfunctions on Sh(G,X)K

19.3 The comparison strategy 393

as in (15.1) weighted by the L2-restriction norm over a locally symmetricsubspace associated to a subgroup H ≤ G.

The approach to the Weyl law via the full invariant Arthur-Selberg traceformula is more complicated, but it still has its merits. The results mentionedabove give no control on the error term in the Weyl law. Using the traceformula, a Weyl law for SLn over Q with remainder was established by Lapidand Muller in [LM09] using the approach of [DKV79b, DKV79a] combinedwith the Arthur-Selberg trace formula.

19.3 The comparison strategy

The Langlands’ functoriality conjecture predicts that for two reductive groupsG1 and G1 defined over a global field F and a L-map

r : LG1 −→ LG2,

there should be a transfer from L-packets of automorphic representations ofG1(AF ) to L-packets of automorphic representations ofG2(AF ). The questionbecomes how one should prove such a transfer. One option which has provento be effective is to study the geometric sides of suitable trace formulae forG1 and G2.

In more detail, suppose that we are given subgroups Hi ≤ Gi × Gi for1 ≤ i ≤ 2 and characters χi : AGi,Hi\Hi(AF ) −→ C×. For F -algebras R, thegroups Hi act on Gi via

Gi(R)×Hi(R) −→ Gi(R)

(g, (h`, hr)) 7−→ h−1` ghr

as in (18.1). We assume that

Hi = Hiu ×Hir

where Hiu is unipotent and Hir is reductive. Let CGi be the centralizer ofGi in Hi, and assume that ACi ≤ Ai, where Ai is the intersection of AGi,Hiand the diagonal copy of AGi . See §18.2 for this notation. Suppose that wecan construct a correspondence

χ2-relevant δ ∈ G2(F )/H2(F )! χ1-relevant γ ∈ G1(F )/H1(F ).

For example, one might have a bijection, but often the situation is morecomplicated. We refer to this as a geometric correspondence betweenrelevant relative classes. We say that the class of δ matches the class of γif the two classes correspond. Suppose that for matching δ and γ one canrelate the orbital integrals τ(G2δ)ROδ(f) and τ(G1γ)ROγ(fG1) for suitable


test functions f ∈ C∞c (AG2\G2(AF )) and fG1 ∈ C∞c (AG1

\G1(AF )). Thenone can hope to prove an identity of the form∑

δ∈G2(F )/H2(F )

τ(G2δ)ROδ(f) =∑

γ∈G1(F )/H1(F )

τ(G1γ)ROγ(fG1). (19.2)

Here the sums over γ and δ are over χ1 and χ2-relevant elements, respec-tively, that are additionally relatively regular, relatively elliptic, relativelyunimodular and such that the H1r-orbit of γ (resp. the H2r-orbit of δ) isclosed. We are assuming these things so that sums are convergent by The-orem 18.2.2. Assuming that an identity like (19.2) can be established for asufficiently rich set of test functions one can obtain a relationship between(H1, χ1)-distinguished representations of G1(AF ) × G1(AF ) and (H2, χ2)-distinguished representations of G2(AF ) × G2(AF ) and thereby deduce aninstance of Langlands functoriality (or its relative analogue due to Jacquet,Sakellaridis, and Venkatesh). We will explain this in more detail momentar-ily. Let us point out that in many cases one has to incorporate families of(Gi, Hi, χi) on both sides of the formula. We ignore this complication becausewe are just speaking in rough terms at the moment.

Let us elaborate what we mean by a relation between τ(G2δ)ROδ(f) andτ(G1γ)ROγ(fG1). We assume for simplicity that the geometric correspon-dence is a bijection, though this is rarely the case. One needs to be ableto relate the volumes τ(G1γ) and τ(G2δ) for matching γ and δ, but this isusually not the key issue.

For almost all finite places v there exists a hyperspecial subgroup Kiv ≤Gi(Fv) by Proposition 2.4.3. The map

r : LG1 −→ LG2

induces an algebra homomorphism

r : C∞c (G2(Fv) //K2v) −→ C∞c (G1(Fv) //K1v) (19.3)

via Theorem 7.5.1 (see also Exercise 19.2).One wants to prove that if γ and δ match then for almost all finite places

v one has the equality

ROδv (f) = ROγv (r(f)) (19.4)

for all f ∈ C∞c (G2(Fv) //K2v). This is often too naıve. The equality willonly hold up to an explicit function of δv and γv called a transfer factor,but we suppress this difficulty. Formulae of the form (19.4) are known asthe fundamental lemma for the Hecke algebra. The identity (19.4)for f = 1K2v is known as the fundamental lemma, or the fundamentallemma for the unit element of the Hecke algebra. In practice onededuces the fundamental lemma for the Hecke algebra from the fundamental

19.3 The comparison strategy 395

lemma. Finally one needs to know that for all places v there sufficiently largesupply of functions f ∈ C∞c (G2(Fv)) and fG1 ∈ C∞c (G1(Fv)) that match,meaning that when δ and γ match one has

ROδv (f) = ROγv (fG1). (19.5)

Statements of this type are known as transfer. Of course for an arbitraryf ∈ C∞c (AG2

\G2(AF )) we will have f ∈ C∞c (G(F2v) //K2v) with K2v hy-perspecial for almost all v, and in this case one can take fG1 = r(f). As inthe statement of the fundamental lemma, hoping for an equality as simple as(19.5) is often too naıve; it will only hold up to an explicit transfer factor.

Given the fundamental lemma for the Hecke algebra and transfer, theidentity (16.7) implies an identity∑

π′

rtrH2,χ2π′(f) =

∑π

rtrH1,χ1π(fG1), (19.6)

at least for f and fG1 satisfying the assumptions of Theorem 18.2.2. Herethe sum on the left is over isomorphism classes of cuspidal automorphic rep-resentations π′ of AG2\G2(AF ) such that π′ ⊗ π′∨ is (H2, χ2)-distinguishedand the sum on the right is over isomorphism classes of cuspidal automorphicrepresentations of AG1

\G1(AF ) such that π ⊗ π∨ is (H1, χ1)-distinguished.Let S be a finite set of places of F including all the infinite places. If f(resp. fG1) is fixed on the left and right under KS

1 (resp. KS2 ) where Kiv is

hyperspecial for v 6∈ S then this identity implies∑π′

trπ′S(fS) rtrH2,χ2π′(fS 1KS

1) =

∑π

trπS(fG1S) rtrG1,χ1π(fG1

S 1KS2

)

(see Exercise 19.3). Though the set of representations on each side of thisequation is in general infinite, in practice one can use a version of linearindependence of characters to refine it to an identity∑

π′:π′S∼=r(πS0 )

trπ′S(fS)rtrH2,χ2(π′(fS 1KS1

))

=∑

π:πS∼=πS0

trπS(fG1S)rtrH1,χ1(π(fG1

S 1KS2

))(19.7)

for each cuspidal automorphic representation π0 of AG1\G1(AF ). Of course,both sums can be zero, but if one sum is nonzero the other is also nonzero.If one can establish (19.7), then one can use it to establish a transfer from(weak) L-packets of automorphic representations of AG2

\G2(AF ) to (weak)L-packets of automorphic representations of AG1

\G1(AF ) and characterizethe image of the transfer. Often one can refine this comparison to the levelof L-packets (or even use it to define L-packets). We observe that in certain


circumstances it is better to start with a map of relative L-groups as explainedby Sakellaridis and Venkatesh [SV12].

To recap, we started with a geometric assumption of matching. Assumingthis we further assumed the fundamental lemma and transfer. Given thispile of assumptions we asserted that one could prove instances of Langlandsfunctoriality. Amazingly, it is possible to make this all rigorous and provefantastic results using it.

19.4 Jacquet-Langlands transfer and base change

In this section we describe some settings where the general strategy in §19.3has been successfully applied. They are related to the trace formula and thetwisted trace formula of §18.4 and §18.5. We assume for the remainder ofthe chapter that F is a number field because most of the results we will citebelow are proven in this setting (though it seems likely that the proofs wouldgeneralize).

We start by taking G1 to be an inner form of the quasi-split reductivegroup G2 over F . We assume that Gder

2 is simply connected so that we canapply Theorem 19.4.1 below. Then LG1 = LG2 and we take

r : LG1 −→ LG2 (19.8)

to be the identity map.We take Hi ≤ Gi × Gi to be the diagonal copy of Hi and we take χ to

be trivial. Then in view of Proposition 17.1.5 Gi/Hi(F ) is just the space ofsemisimple conjugacy classes in Gi(F ) that are Gal(F/F )-invariant. Usingthis and the definition of an inner form one can construct an isomorphism ofaffine F -schemes

G1/H1−→G2/H2. (19.9)

This is almost the first step of the procedure explained in the previous section.There is a slight wrinkle. Consider the map

semisimple γ ∈ Gi(F ) −→ Gi/Hi(F ) (19.10)

sending γ to its geometric class. Because G1 need not be quasi-split, for i = 1the map (19.10) need not be surjective. However, for i = 2 the group G2 isquasi-split and hence the map (19.10) is surjective by the following theorem:[Kot82]:

Theorem 19.4.1 (Kottwitz and Steinberg) Let G be a quasi-split groupover a characteristic zero field k with simply connected derived group. Asemisimple G(k)-conjugacy class fixed under Galk contains an element ofG(k). ut

19.4 Jacquet-Langlands transfer and base change 397

This turns out to be enough. In the current setting, the fundamental lemmafor the Hecke algebra is trivial because G1Fv and G2Fv are isomorphic foralmost every place v (see Exercise 19.5). Thus proving transfer of automorphicrepresentations is reduced to understanding transfer of functions. Jacquet andLanglands [JL70] were the first to introduce an argument of this type andthey used it to give a complete treatment of the special case where G1 is theunit group of a quaternion algebra and G2 = GL2. Because of this transfersattached to the maps (19.8) are known as Jacquet-Langlands transfers.Though the full theory of Jacquet-Langlands transfers is still not known,many results exist in the literature. In particular the theory for GLn is fairlycomplete. Let D be a simple algebra of dimension d2 over F . For F -algebrasR let

GLnD(R) := (D ⊗F Mn(R))×.

Let S be the (finite) set of places of F where Dv 6∼= Md(Fv). Here Mn(Fv) isthe space of n× n matrices with entries in Fv.

For every place v we have a diagram

semisimple δ ∈ GLnD(Fv) semisimple γ ∈ GLnd(Fv)

pD

y p

yGLnD/ ∼ (F )

t−−−−→ GLnd/ ∼ (F )

(19.11)

where the quotients on the bottom are the GIT quotients of GLnD and GLndby the conjugation action of GLnD and GLnd, respectively, and the verti-cal maps are the projections. The map p is surjective (Exercise 19.6), butpD is not. We say that δ and γ match if t(pD(δ)) = p(γ). An irreducibleadmissible representation π′ of GLn(Fv) is Dv-compatible if the characterΘπv (γ) is nonzero for some regular semisimple γ ∈ GLnd(Fv) that matchesa δ ∈ GLnD(Fv). See Theorem 8.5.1 for the definition of Θπ in the nonar-chimedean case, the archimedean case is similar [Kna86, Theorem 10.25]. Anautomorphic representation π′ of AGLn\GLn(AF ) is D-compatible if πv isDv-compatible for all v ∈ S.

The following theorem is due to Badulescu and Grbac [Bad08] followingseveral previous works [Rog83, BDKV84], going back to [JL70], where then = 2 case was treated.

Theorem 19.4.2 (Jacquet-Langlands correspondence) There exists aunique injection r from the set of isomorphism classes of discrete automorphicrepresentations of AGLnD\GLnD(AF ) to the set of isomorphism classes ofdiscrete automorphic representations of AGLnd\GLnd(AF ) such that r(π)v ∼=πv for v 6∈ S. The image consists of discrete automorphic representations π′

that are D-compatible. ut

Here we are using the isomorphism GLnD(Fv) ∼= GLnd(Fv) for v 6∈ S to makesense of the assertion that r(π)v ∼= π′v.


In the setting above the groups G1 and G2 were not that different fromeachother; indeed, they become isomorphic over the algebraic closure. Saitoand Shintani [Sai75, Shi79] discovered the first geometric correspondence in-volving groups that are not isomorphic over the algebraic closure. It was fullydeveloped for GL2 by Langlands in [Lan80]. Take G1 = G to be reductivewith simply connected derived group, let E/F be a cyclic field extension, andlet θ be a generator of Gal(E/F ). We then take G2 = ResE/FG, and let

r : LG −→ LResE/FG

be the map given by the diagonal embedding on the neutral component andthe identity on the Galois factor. To set up our geometric correspondence welet the χi be trivial, let H1 be the diagonal copy of G in G×G, and let H2

be the θ-twisted diagonal defined by

H2(R) := (g, θ(g)) : g ∈ ResE/FG(R) (19.12)

for F -algebras R. Thus G(F )/H1(F ) is the space of conjugacy classes andResE/FG(F )/H2(F ) is the space of θ-conjugacy classes in the sense of §18.5.

The geometric correspondence is defined as follows. A reference is [Kot82].Let δ ∈ ResE/FG(F ) and let

N(δ) := θ[E:F ]−1(δ) · θ[E:F ]−2(δ) · · · δ.

View N(δ) as an element of G(E). Then

θ(N(δ)) = δN(δ)δ−1

which implies that the G(E)-conjugacy class of N(δ) is fixed under Gal(E/F ).Thus there is an element of G(F ) in the conjugacy class by Theorem 19.4.1.We call such an element a norm of δ. This gives us a map

θ-semisimple θ-conjugacy classes in ResE/FG(F )ysemisimple conjugacy classes in G(F ).

(19.13)

We say that a θ-semisimple δ and a semisimple γ match if γ is a a norm ofδ (see §18.5 for the notion of a θ-semisimple class).

This geometric correspondence forms the basis for a comparison of thetrace formula of §18.4 and the twisted trace formula of §18.5. We point outthat one might anticipate such a comparison by consideration of the spectralsides of the two trace formulae. Indeed, the image of the functorial transferinduced by r is contained in the set of representations π′ that satisfy π′ ∼= π′θ,and the spectral side of the twisted trace formula is given in terms of suchrepresentations.

19.5 Twisted endoscopy 399

Unlike in the Jacquet-Langlands case, the fundamental lemma is nontrivialin this setting. However Kottwitz found a beautiful, direct, and simple argu-ment establishing it [Kot86a]. The fundamental lemma for the whole Heckealgebra was then deduced by Clozel [Clo90a] and Labesse [Lab90]. This wasthe basis for the work of Arthur and Clozel on cyclic base change for GLnstated in §13.5.

19.5 Twisted endoscopy

The twisted trace formula was formulated for an arbitrary automorphism θof an arbitrary reductive group G. There is no need to assume that Gder issimply connected and there is no need to assume that G is an inner formof GLn. Dealing with Gder that are not simply connected is a nontrivialproblem, but there are techniques available. The discussion of z-extensionsin [Kot82] gives the geometric comparison necessary to begin this extension.Dealing with the case G 6= GLn is more serious. The key point is that foressentially every reductive group other than GLn two semisimple elementsof G(F ) that are G(F )-conjugate need not be G(F ) conjugate. In particularthe cohomology sets D(F, I,G) of (17.28) need not be singletons.

In the settings described in §19.4 we did not really relate elements ofG1(F )/H1(F ) and G2(F )/H2(F ). We really related elements of G1/H1(F )and G2/H2(F ). For general Gi and Hi these really are different, and thedifference can be measured in terms of the sets D(F, I,Gi) of (17.28). Becauseof this subtlety, to understand the quotient G2/H2(F ) it is appropriate toplace G1 into a whole family of so-called twisted endoscopic groups. Thissubject is known as the theory of twisted endoscopy. It would take at leastanother book to discuss this fully, and fortunately such books are alreadyavailable. The reader might start with [Kot84, Kot86b] which is based on[Lan83, Lan79b]. More recent references include [KS99, Lab99, CHLN11].Perhaps the definitive treatment is [MW16a, MW16b].

For arbitrary θ, the geometric comparison was made precise fairly earlyin the references mentioned above, though it is far more complicated thanthe geometric comparisons mentioned in §19.4 for the general linear group.Originally it was expected that the fundamental lemma in this context wouldbe a routine combinatorial exercise as it was in the case of GL2, and this iswhy it was described it in unassuming terms as a lemma. In fact provingthe fundamental lemma, transfer, and then applying them became a struggleof Homeric proportions involving at least two generations of mathematiciansover a span of almost 30 years.

While the fundamental lemma was isolated and made precise by Shelstadand Langlands [LS87] Arthur began developing the trace formula in a formthat was designed to apply the lemma. It would take too much space to recordhis work and accomplishments here; fortunately there is a freely available


archive of his work online. Some of the fruits of his work are described in§13.5 and §13.8. We will also not give an overview of the “long march” (inthe words of Ngo, [Ngo10a]) to the fundamental lemma, preferring to leavethis to the survey [Ngo10a] written by the person who completed the proofin his Fields medal-winning work.

We would be remiss to point out that despite the amazing success oftwisted endoscopy, it has obvious drawbacks. As currently understood it canonly hope to establish cases of functoriality where the image of the functoriallift consists of representations that are isomorphic to their conjugates under asingle automorphism. A generalization to two automorphisms is proposed in[Get12, Get14]. These papers are based on interconnected ideas of Braverman,Kahzdan, L. Lafforgue Langlands, Ngo, and Sakellaridis [BK00, Laf14, Ngo14,Sak12].

19.6 The interplay of distinction and twisted endoscopy

The theory of twisted endoscopy discussed in the previous section has reacheda degree of maturity. One understands how to compare the twisted traceformulae and the trace formula. Since Jacquet has given us the new tool ofthe relative trace formula, we now have a much broader arena to apply theexperience gained in the theory of twisted endoscopy and prove new results.We mention a few geometric correspondences here and connect them to theresults on distinction reviewed in Chapter 14.

We start by explaining the comparison underlying Theorem 14.5.2 whichcharacterizes those representations of ResE/FGLn that are distinguished bya quasi-split unitary group. Here E/F is a quadratic extension. In this caseour G1 = GLn and G2 = ResE/FGLn and

r : LGLn −→ LResE/FGLn

is the base change map, given by the diagonal embedding on the neutralcomponent and the identity on the Galois factor.

In this case our basic outline from §19.3 must be modified along the linesof §18.6. Thus for F -algebras R we let

X(R) := x ∈Mn(E ⊗F R) : θ(x) = xt

where θ is the generator of Gal(E/F ). This admits an action of ResE/FGLn:

X(R)× ResE/FGLn(R) −→ X(R)

(x, g) 7−→ θ(g)txg.

19.6 The interplay of distinction and twisted endoscopy 401

The stabilizer of any x ∈ X(F ) ∩ ResE/FGLn(F ) is a unitary group. Thefunction ∑

x∈X(F )

f(xg)

is right G(F ) invariant. As in §18.6 it can be expanded spectrally in termsof automorphic representations of ResE/FGLn(AF ) that are distinguished bysome unitary group.

We take H2 = ResE/FNn where Nn is the unipotent radical of the Borelsubgroup of upper triangular matrices in GLn. It acts on X via restriction ofthe action of ResE/FGLn. We then let

χ2 : ResE/FNn(AF )→ C×

u 7−→ ψ(uΘ(u)),

where ψ : Nn(AF ) → C× is a generic character (trivial on Nn(F )). We donot need to modify the geometric setting for G1. We can simply take

H1(R) := (u−t1 , u2) : u1, u2 ∈ Nn(R)

for F -algebras R and let χ1(u1, u2) = ψ(u−11 u2). Note that this is precisely

the geometric setting of the Kuznetsov formula of §18.8, apart from slightlymodifying the action of the unipotent groups involved. Let T ≤ GLn be themaximal torus of diagonal matrices. We have a diagram

T (F )

p2

ww

p1

&&X(F )/ResE/FNn(F ) G1(F )/N2

n(F )

where the two arrows send an element of T (F ) to its class. We say thata χ2-relevant δ and and χ1-relevant γ match if there is a t ∈ T (F ) suchthat δ ∈ p2(t) and γ ∈ p1(t). In this setting the fundamental lemma wasconjectured by Jacquet and Ye [JY96] and proven in two different mannersby Ngo [Ngo99b, Ngo99a] and Jacquet [Jac04, Jac05b].

Jacquet and Rallis [JR11] later proposed a related comparison that even-tually led to the proof of the results on the Gan-Gross-Prasad conjecturediscussed in §14.7. A useful reference is [Cha19]. Let E/F be a quadraticextension of number fields let Un be the quasi-split unitary group of §13.7.We take

G1 = Un+1 ×Un and G2 = ResE/FGLn+1 × ResE/FGLn

and let

r : L(Un+1 ×Un) −→ L(ResE/FGLn+1 × ResE/FGLn)


be induced by the standard representations LUn+1 → LResE/FGLn+1 andLUn → ResE/FGLn (see §13.7). We take Hi ≤ Gi ×Gi to be the subgroupswith points in an F -algebra R given by

H2(R) : = (( g 1 ) , g) : g ∈ ResE/FGLn ×GLn+1 ×GLn(R)

H1(R) : = ι(Un)× ι(Un),

where ι : Un → Un+1 × Un is an appropriate “diagonal” embedding. ThusH2 is isomorphic to ResE/FGLn × GLn+1 × GLn and H1 is isomorphic toUn ×Un. Letting

η : F×\A×F /NE/F (A×E)→ ±1

be the character attached to E/F by class field theory we set

χ2 ((( g 1 ) , g) , (g1, g2)) = η(det g1)n+1η(det g2)n.

We take χ1 to be the trivial character. In this setting one has an isomorphismof GIT quotients

G2/H2∼= G1/H1

and we say that relatively regular relatively semisimple δ ∈ G2(F ) and γ ∈G1(F ) match if their geometric classes correspond under this isomorphism.It turns out that regular semisimple elements have trivial stabilizers so thequestion of relevance is irrelevant here.

In this setting the fundamental lemma was proven by Yun [Yun11].W. Zhang [Zha14] proved transfer and used it to deduce the global Gan-Gross-Prasad conjecture for unitary groups under certain local restrictions.For more details see §14.7.

Both of the comparisons mentioned above are in some sense much simplerthan the comparisons encountered in the theory of twisted endoscopy. Theorigin of this is easy to pinpoint. It is that stabilizers of relatively regularrelatively semisimple elements are trivial. As the reader can see from Chapter17, the Galois cohomology of centralizers creates a great deal of complicationin understanding orbital integrals. There is no reason to expect that one willalways be able to avoid nontrivial centralizers, and it is important to developrelative analogues of the theory of twisted endoscopy discussed in §19.5.

As an example where it does not seem possible to avoid centralizers, wediscuss the comparison underlying Theorem 14.6.1. As discussed in §14.6 themany variants of this setup are fertile ground for future research.

For a quadratic extension E/F we let UnE/F ≤ ResE/FGLn be the quasi-split unitary group defined in §13.7. We give ourselves a biquadratic extension


EM

E

σ

L

θ

M

τ

F

Thus σ is a generator for Gal(E/F ), etc.Let G1 = ResE/FUnM/F and G2 = ResEM/FGLn. The map

r : LResE/F (UnM/F )E −→ LResEM/FGLn

under consideration is induced by the standard representation LUnM/F →LResM/FGLn.

The automorphism σ induces an automorphism of ResE/FUn with Un

as its fixed point group, and hence an automorphism (still denoted σ) ofResEM/FGLn with ResM/FGLn as its fixed point group. Likewise the auto-morphism τ induces an automorphism of ResEM/FGLn with ResE/FUn asits fixed point group. We set θ = στ . Its fixed point group is ResL/FUnEM/L.For F -algebras R we take

H2(R) = ResL/FUnEM/L(R)× ResE/FGLn(R)

H1(R) = UnM/F (R)×UnM/F (R).

To relate these two quotients we use the constructions related to symmetricsubgroups recalled in §17.2. We have morphisms (of affine F -schemes, notgroups)

Bθ : ResEM/FGLn −→ ResEM/F

Bσ : ResE/F (UnM/F )E −→ ResE/F (UnM/F )E

defined as in (17.23). Let S be the scheme-theoretic image of Bθ and Q thescheme-theoretic image of Bσ. We then have isomorphisms

G2/H2 −→ S/ResE/FGLn (19.14)

G1/H1 −→ Q/UnM/F (19.15)

where ResE/FGLn acts via τ -conjugation on S and Un acts via conjugationon Q. This is formally similar to the geometric setting underlying base changediscussed in §19.4. One has a set of τ -conjugacy classes and a set of conjugacyclasses that one wishes to relate. For a relatively regular relatively semisimpleδ ∈ ResEM/FGLn(F ) we say that γ is a norm of δ if there is an h ∈ H2(F )such that

hγγ−σh−1 = δδ−θ(δδ−θ).


One can show that every relatively regular relatively semisimple δ admits anorm, and we say that

δ ∈ G2(F ) = ResEM/FGLn(F ) = GLn(EM)

andγ ∈ G1(F ) = ResE/F (UnM/F )E(F ) = UnEM/E(AE)

match if γ is a norm of δ. In this context the fundamental lemma can beobtained from the fundamental lemma for base change, and transfer for asufficiently rich supply of test functions was proven in [GW14]. This providesenough information to establish Theorem 14.6.1. However there is a relativeanalogue of twisted endoscopy lurking here that has yet to be developed.This theory of twisted relative endoscopy will probably be necessary to ob-tain a version of Theorem 14.6.1 valid for arbitrary cuspidal automorphicrepresentations of UnEM/E(AE).

Exercises

19.1. Relate the Laplace-Beltrami operator on a Shimura manifold Sh(G,X)K

and the Casimir operator of Exercise 4.6.

19.2. For i = 1, 2 let Gi be a reductive group over the nonarchimedean fieldF admitting a hyperspecial subgroup Ki ≤ Gi(Fv). If

r : LG1 −→ LG2

is an L-map prove that there is a unique algebra homomorphism

r : C∞c (G2(F ) //K2) −→ C∞c (G1(F ) //K1) (19.16)

such that if π is an irreducible admissible unramified representation of G1(F )with transfer r(π) to G2(F ) then

tr r(π)(f) = trπ(r(f))

for all f ∈ C∞c (G2(F ) //K2). Here we normalize the Haar measures implicit inthe traces so that the Ki are given measure 1. Observe that r(1K2v

) = 1K1v.

19.3. Let G be a reductive group over the global field F and let H ≤ G×Gand χ : AG,H\H(AF )→ C× be as in §18.2. Let S be a finite set of places ofF including the infinite places and let KS < G(ASF ) be a maximal compactopen subgroup such that Kv is hyperspecial for v 6∈ S. Let

f = fSfS ∈ C∞c (AG\G(AF ) //KS).


Prove thatrtr(π(f)) = trπ(fS)rtr(π(fS 1KS )).

19.4. Prove that (19.9) is an isomorphism of F -schemes.

19.5. Let G1 and G2 be reductive groups over a global field F that are innerforms of each other. Prove that G1Fv

∼= G2Fv for all but finitely many placesv of F .

19.6. Let k be a field. Prove that every semisimple conjugacy class inGLn(ksep) that is fixed by Galk has an element in GLn(k). Prove more-over that two elements γ1, γ2 ∈ GLn(k) that are GLn(ksep)-conjugate areGLn(k)-conjugate.

19.7. Let T ≤ GLn be a maximal torus over a field k. Prove thatD(k, T,GLn),defined as in (17.28), has only one element.

Appendix A

The Iwasawa Decomposition

Abstract The Iwasawa decomposition plays an important role in the theoryof automorphic representations. We prove the existence of the decomposi-tion and discuss its compatibility with the Levi decomposition of parabolicsubgroups.

A.1 Introduction

Let F be a local or global field and let P be a parabolic subgroup of areductive F group G. Fix a Levi subgroup M ≤ P and let N ≤ P be theunipotent radical. Thus P = MN . In the local case we say that a maximalcompact subgroup K ≤ G(F ) is in good position with respect to (P,M)if the Iwasawa decomposition

G(F ) = P (F )K (A.1)

holds and

P (F ) ∩K = (M(F ) ∩K)(N(F ) ∩K). (A.2)

Similarly, if F is a global field we say that a maximal compact subgroupK ≤ G(AF ) is in good position with respect to (P,M) if it is goodposition with respect to (PFv ,MFv ) for all places v of F .

Our goal in this appendix is to prove the following theorem:

Theorem A.1.1 Let G be a reductive group over a global or local field F andlet P ≤ G be a parabolic subgroup with Levi subgroup M . There exists a max-imal compact subgroup K of G(AF ) or G(F ), respectively, in good positionwith respect to (P,M). We can even assume that M(AF )∩K and M(F )∩Kare maximal compact subgroups of M(AF ) and M(F ), respectively.

407

408 A The Iwasawa Decomposition

The proof of this theorem, in general, invokes the formidable work ofBruhat-Tits. However, at least outside of a finite set S of places of F in-cluding all infinite places that in practice one can control, one can obtainthe decomposition using much less. Our aim in this appendix is to explainthis to readers who have some background in algebraic geometry. The proofwill also explain precisely how one chooses K, at least outside of finite set S.The reason for including this material is twofold. First, in most applicationsone needs at least a basic understanding of the relationship between the Kand P (AF ) occurring in the Iwasawa decomposition. Second, the proof givesus the opportunity to introduce concepts and techniques that are useful in abroad context.

A.2 The smooth case

In order to explain how one can choose K given P it is useful to discussnotions of reductive and parabolic subgroups over the base ring OF . Ourprimary reference for this is [Con14]. Before beginning we recall some ter-minology that is ubiquitous in algebraic geometry (specialized to the case ofschemes over an affine base).

Let k be a commutative Noetherian ring with identity. A geometric pointof an affine scheme Spec(k) is just a morphism

s := Spec(L)→ Spec(k)

where L is the algebraic closure of a field L. Recall from §1.2 that to givesuch a morphism is equivalent to giving a morphism of k-algebras k → L.A geometric fiber of a k-scheme X is the base change Xk where s is ageometric point of Spec(k)

Definition A.1. A reductive group scheme G over k is an affine smoothgroup scheme such that its geometric fibers are reductive groups.

This (trivially) recovers our original definition of reductive group when kis a field. Now suppose G is a reductive group scheme over k.

Definition A.2. A parabolic subgroup scheme P of G is a smooth affinek-group scheme equipped with a closed immersion P → G of group schemessuch that all geometric fibers of Ps are parabolic subgroups of Gs.

Assume we are given a reductive group scheme G and a parabolic subgroupscheme P ≤ G. Consider the functor which assigns to any k-algebra R the setof closed subgroup schemes of G that are etale locally conjugate to P. Thenthis functor is representable by a scheme called G/P over k. This scheme is notaffine, but it is smooth and proper and equipped with a canonical k-ample linebundle (and hence admits a canonical embedding into a projective space over

A.2 The smooth case 409

k) [Con14, Corollary 5.2.8]. We note that there is a natural transformationof functors G → G/P given on points in a k-algebra R by

G(R) −→ (G/P)(R) (A.3)

g 7−→ gP(R)g−1.

For fields this map (A.3) is surjective:

Lemma A.2.1 If k is a field then the map

G(k) −→ (G/P)(k)

is surjective and induces a bijection G(k)/P(k)→(G/P)(k).

Proof. Let k be an algebraic closure of k. By the generalities on parabolicsubgroups over fields recalled after Theorem 1.9.1 every parabolic subgroupof G(k) that is G(k)-conjugate to P is in fact G(k)-conjugate to P. ut

The following lemma is extremely useful:

Lemma A.2.2 Let G be a smooth affine group scheme of finite type over adiscrete valuation ring O with finite residue field κ and let X be a G-torsor.If the special fiber Gκ is connected then

X (O) 6= ∅.

Proof. Since G is smooth so is X [BLR90, §6.4]. Therefore, by Hensel’s lemma[BLR90, §2.3, Proposition 5] the map X (O)→ X (κ) is surjective. It thereforesuffices to show that X (κ) 6= ∅. But Xκ is a torsor under the connected affinealgebraic group Gκ, so X (κ) 6= ∅ by a theorem of Lang [Lan56]. ut

We can now prove a version of the Iwasawa decomposition if we assumethe existence of the parabolic subgroup scheme P:

Theorem A.2.3 Suppose that S is a finite set of places of the global fieldF containing all infinite places, that G is a reductive group scheme over OSFand P ≤ G is a parabolic subgroup scheme over OSF . Then

G(ASF ) = P(ASF )G(OSF ).

Proof. Suppose that g ∈ G(ASF ). Then g ∈ G(OFv ) for all but finitely manyplaces v, so it suffices to show that for every v 6∈ S one has

G(Fv) = P(Fv)G(OFv ).

Fix v 6∈ S and let $ ∈ OFv be a uniformizer.One has a commutative diagram


G(OFv ) −−−−→ G(Fv)

a

y y(G/P)(OFv )

b−−−−→ (G/P)(Fv).

The scheme G/P is projective over OFv , hence proper, and thus the arrowb is a bijection by the the valuative criterion of properness. Thus it sufficesto show that that the arrow a is surjective. The fiber of a over any point ofG/P(O) is a P-torsor, and hence has a point by Lemma A.2.2. ut

A.3 The dynamic method

Now in the statement of Theorem A.1.1 there is no mention of the groupscheme G or P; one has only the reductive group G and its parabolic subgroupP . To complete the proof of Theorem A.1.1 one must construct G and P givenG and P . One starts by choosing a faithful representation G → GLn. For asufficiently large finite set S of places of OF containing the infinite ones theschematic closure G of G in GLnOSF is then reductive by Proposition 2.4.3.Thus we can construct G. We now construct P.

Proposition A.3.1 Let S be a finite set of places of the global field F con-taining all infinite places and let G be a reductive group scheme over OSF . LetP be a parabolic subgroup of GF with Levi decomposition P = MN . Afterpossibly adding a finite set of places to S, the schematic closure P of P in Gis a parabolic subgroup scheme of G admitting a decomposition

P =MnN (A.4)

whereM is a reductive group scheme, N is a connected smooth group schemewith unipotent fibers, and MF = M , NF = N .

In the setting of the lemma we have P(OFv ) = M(OFv ) n N (OFv ) forv 6∈ S. This implies that

P (Fv) ∩ G(OFv ) = (M(Fv) ∩ G(OFv ))(N(Fv) ∩ G(OFv )),

the condition required in (A.2). We also note that M(OFv ) ≤ M(Fv) is ahyperspecial subgroup (and in particular is maximal, see Theorem 2.4.1).

To explain how to do this we will recall the so-called dynamic descriptionof parabolic subgroups. Many more details are contained in [Con14]. Supposewe are given an affine group scheme G of finite type over a ring k and acocharacter, i.e. a morphism of group schemes

λ : Gm −→ G.

A.3 The dynamic method 411

This defines a left action of Gm on G

λ : Gm × G −→ G

given on points byλ(t, g) := λ(t)gλ(t)−1.

We say that limt→0 λ(t, g) exists if the morphism λ extends to a morphism

λ : Ga × G −→ G.

Example A.1. Let λ : Gm −→ GL3 be the cocharacter given as

λ(t) =

t t−1

t−2

.

Then

λ(t)

a b cd e fh g i

λ(t)−1 =

a t2b t3ct−2d e tft−3g t−1h i

.

Thus limt→0 λ(t, g) exists if and only if g is a point of the Borel subgroup ofupper triangular matrices. Moreover limt→0 λ(t, g) = 1, the identity in GL3,if and only if g lies in the group of strictly upper triangular matrices, and gcommutes with the action of λ if and only if g is a diagonal matrix.

Let P (λ) be the group functor on k-algebras given on a k-algebra R by

P (λ)(R) := g ∈ G(R) : limt→0

λ(t, g) exists.

We also define

N(λ)(R) = g ∈ G(R) : limt→0

λ(t, g) = 1

and let Z(λ)(R) ≤ G(R) be the subgroup that commute with the action λ.We then have

P (λ)(R) = Z(λ)(R) nN(λ)(R). (A.5)

In the example above P (λ) turned out to be a parabolic subgroup of theambient group scheme. Moreover, the geometric fibers of N(λ) (resp. Z(λ))turned out to be the unipotent radical (resp. Levi subgroup) of the corre-sponding fiber of P (λ). This is a general phenomenon, and is the basis of theso-called dynamic description of parabolic subgroups, which is summarizedin Theorem A.3.2 and Theorem A.3.3 below. For the proof of Theorem A.3.2see [Con14, Theorem 4.1.7, Example 5.2.2] or [CGP10, §2.1]:


Theorem A.3.2 Assume that G is smooth. The functor P (λ) is representedby a closed smooth subgroup of G. If G has connected fibers then so does P (λ).If G is connected and reductive, T ≤ G is a maximal torus, and λ has imagein T then P (λ) is a parabolic subgroup scheme of G. ut

Here T ≤ G is a maximal torus if it is a flat, closed subgroup scheme of Gsuch that for all geometric points s of Spec(k) the fiber Ts is a maximal torusof Gs.

Theorem A.3.3 Assume that k is a field and that G is a reductive groupover k. Let P ≤ G be a parabolic subgroup with Levi decomposition P = MNand let T ≤M be a maximal torus of G. There is a cocharacter λ : Gm → Tsuch that P = P (λ), M = Z(λ), and N = N(λ).

Proof. The fact that every P is a P (λ) is [Mil17, Theorem 25.1] and N =N(λ) for any cocharacter λ such that P = P (λ). Since all Levi subgroupsof P are conjugate under P (k) we deduce that upon conjugating λ by anelement of P (k) we may assume that M = M(λ). ut

Using these theorems we can now prove Proposition A.3.1:

Proof of Proposition A.3.1: Let P be a parabolic subgroup of G containing amaximal torus T of G. Then there is a character λ : T → Gm such that P =P (λ) by Theorem A.3.3. Choose a faithful representation G → GLn. Uponchoosing as sufficiently large finite set of places S of F we can assume that theschematic closure G of G in GLnOSF is reductive (Proposition 2.4.3), that the

schematic closure T of T in G is a maximal torus of G [Con14, Corollary 3.1.8],and that the cocharacter λ : Gm → T extends to a cocharacter Λ : Ga −→ T .Let

P := P (Λ).

Then by Theorem A.3.2, P is a parabolic subgroup of G and we deduce theproposition. 2

A.4 Proof of Theorem A.1.1

We now complete the proof of Theorem A.1.1. In view of Theorem A.2.3 andProposition A.3.1 it suffices to prove Theorem A.1.1 in the local case. Thuswe assume for the remainder of the section that F is a local field.

We will actually prove something more precise than what is required byTheorem A.1.1. Fix a minimal parabolic subgroup P0 ≤ G and a Levi sub-group M0 ≤ P0. We write N0 for the unipotent radical of P0. We call apair (P,M) consisting of a parabolic subgroup P ≤ G and a Levi subgroupM ≤ P a standard parabolic pair if P0 ≤ P and M0 ≤M .

A.4 Proof of Theorem A.1.1 413

Theorem A.4.1 Assume F is archimedean. There is a maximal compactsubgroup K ≤ G(F ) such that for any standard parabolic pair (P,M) one hasP (F )K = G(F ), M(F ) ∩ K is a maximal compact subgroup of M(F ), andP (F ) ∩K = M(F ) ∩K.

Proof. By [Kna02, Proposition 7.31] the Iwasawa decomposition G(F ) =P (F )K holds for some maximal compact subgroup K ≤ G(F ). Since allmaximal compact subgroups are conjugate in the archimedean setting [Hel01,Chapter VI.2], it follows that G(F ) = P (F )K holds for all maximal compactsubgroups K.

LetG(F )+ ≤ G(F ) be the identity component in the natural topology. Theassertion that K is a maximal compact subgroup of G(F ) is equivalent to thestatement that K∩G(F )+ is maximal in G(F )+ and K meets all componentsof G(F ). In view of this, the assertion that M(F )∩K is a maximal compactsubgroup of M(F ) is part of [HC75, Lemma 8].

Consider the image of K in P (F )/N(F ) = M(F ). It is a compact subgroupcontaining M(F ) ∩K, hence equal to M(F ) ∩K. If mn ∈ K with (m,n) ∈M(F )×N(F ), we deduce that m ∈M(F )∩K, and hence n ∈ N(F )∩K = I.This completes the proof of the last assertion. ut

Theorem A.4.2 Assume F is nonarchimedean. There is a maximal compactsubgroup K ≤ G(F ) such that for any standard parabolic pair (P,M) one hasP (F )K = G(F ), M(F ) ∩ K is a maximal compact subgroup of M(F ), andP (F ) ∩K = (M(F ) ∩K)(N(F ) ∩K).

We do not know of a proof of this result that does not use Bruhat-Titstheory. We will have to assume some of this in the proof. A survey is givenin [Tit79] and the proofs are contained in [BT72, BT84]. The paper [Lan00]is also useful; there one can find the definition of the extended Bruhat-Titsbuilding. Let Be(G) denote the extended Bruhat-Tits building of G (overF ). It is a topological space equipped with an action of G(F ) and is a unionof subsets called (extended) apartments Ae(G,S) indexed by the maximallysplit tori S of G:

Be(G) :=⋃gAe(G,S). (A.6)

Usually Be(G) is denoted by Be(G,F ), etc, but since G is an F -group we haveomitted this from notation. The extended apartment Ae(G,S) is a X∗(S)R-torsor (affine X∗(S)R-space) equipped with a homomorphism

NG(S)(F ) −→ Aff(Ae(G,S)) (A.7)

where Aff(Ae(G,S)) is the group of affine transformations of Ae(G,S).

Proof. Assume that S ≤ M0 is a maximal split torus. Let x ∈ Ae(G,S) besuch a special point [Lan00, §2.2] and let K be the stabilizer of x. ThenK is a maximal compact subgroup of G(F ) and one has P0(F )K = G(F )


[BT72, 4.4.6(ii)]. It follows immediately that P (F )K = G(F ) for all standardparabolic subgroups P of G.

We now claim that we can choose x so that for each standard parabolicpair M(F )∩K is a maximal compact subgroup of M(F ). Assuming this therest of the theorem follows as in the proof of Theorem A.4.1. The inclusionM ≤ G induces an inclusion

Be(M) −→ Be(G)

that is equivariant with respect to the action of M(F ). Moreover, the inclu-sion sends Ae(M,S) bijectively to Ae(G,S). Let x ∈ Ae(G,S) be a specialpoint. Then one checks directly from the definition of a special point that theinverse image of x in Ae(M,S) is again a special point. The claim follows. ut

A.5 An addendum in the hyperspecial case

Suppose that G is a reductive group over a local field F that is unramified(quasi-split and split over an unramified extension). Thus there is a smoothmodel G of G over OF with reductive fibers by Theorem 2.4.1. The groupG(OF ) is a hyperspecial subgroup of G(F ).

Lemma A.5.1 There is a parabolic subgroup scheme B ≤ G and a maximaltorus T ≤ B such that BF is a Borel subgroup of G and BOF /$ is a Borelsubgroup of GOF /$.

Proof. We first check that a Borel subgroup exists. The functor assigningto an OF scheme S the set of Borel subgroups of GS is representable by asmooth proper OF -scheme, say X [DG74, XXII, Corollaire 5.8.3(i)]. We haveX (F ) = X (OF ) by the valuative criterion of properness, so there is a Borelsubgroup B ≤ G. The existence of T is [DG74, XXII, Corollaire 5.9.7] ut

Lemma A.5.2 For B and T as in Lemma A.5.1 the subgroup T (OF ) =T (F )∩G(OF ) is a maximal compact subgroup of T (F ) and G(OF ) is in goodposition with respect to (BF , TF ).

Proof. The assertion that T (OF ) ≤ T (F ) is a maximal compact subgroup is[Mac17, Proposition 4.6]. One has a decomposition

B = T oN

where N is a connected smooth group scheme with unipotent fibers [DG74,XXVI, Proposition 1.6]. In particular NF is the unipotent radical of BF . Thisimplies that G(OF ) is in good position with respect to (BF , TF ). ut

Appendix B

Poisson Summation

Abstract In this appendix we define the adelic version of the Fourier trans-form on a vector space and state the corresponding version of Poisson sum-mation

B.1 The standard additive characters

In this appendix we define the adelic version of the Fourier transform on avector space and state the corresponding version of Poisson summation. Allof the material below is either implicit or explicit in Tate’s thesis [Tat67]. Aleisurely account is given in [RV99, Chapter 7]

The goal of this section is to fix a standard additive character on everylocal field and an additive character of F\AF for every global field.

We first consider the case of completions of Q and Fp(t) for primes p. Welet

ψR : R −→ C×

a 7−→ e−2πia

and

ψQp : Qp −→ C×

a 7−→ e2πipr(a).

Here the principal part pr(a) ∈ Z[p−1] is any element such that

a ≡ pr(a) (mod Zp).

If F is a characteristic zero local field then it is an extension of a completionF0 of Q, and we let

415

416 B Poisson Summation

ψF := ψ trF/F0: F −→ C×.

We now treat the characteristic p case. We define the additive character

ψFp((t)) : Fp((t)) −→ C×r∑

n=−∞ant

n 7−→ e−2πia1/p.

Here on the right the ai are representatives in Z for ai ∈ Fp. If $ is anirreducible polynomial in Fp[t] and Fp(t)$ is the completion of Fp(t) at thecorresponding place and k is its residue field then we define

ψFp(t)$ : Fp(t)$ −→ C×∞∑n=r

an$n −→ e2πitrk/Fp (a−1)/p.

Here the ai are in k (or rather a fixed set of representatives for k = Fp[t]/$ inFp[t]). On the right we are implicitly identifying trk/Fp(a−1) with an elementof Z mapping to it under the quotient map Z → Fp. If F is a characteristicp local field then it is an extension of a completion F0 of Fp(t), and we let

ψF := ψ trF/F0: F −→ C×.

One checks that all of the ψF we have defined are indeed nontrivial characters.It is not hard to check the following lemma (see Exercise B.2):

Lemma B.1.1 If F is a local field and ψ : F → C× is any nontrivial char-acter then there is a bijection

F −→ F

α 7−→ (x 7→ ψ(αx)) .

ut

Here the right hand side is the unitary dual of F , which is simply the set ofcharacters of F , i.e. the set of continuous homomorphisms F → C×.

If F is a global field we let

ψF :=∏v

ψFv . (B.1)

This defines a nontrivial character of AF that is trivial on F . The globalanalogue of Lemma B.1.1 is the following lemma:

Lemma B.1.2 If F is a global field and ψ : F\AF → C× is any nontrivialcharacter then there is a bijection

B.3 Global Schwartz spaces and Poisson summation 417

F\AF −→ F\AFα 7−→ (x 7→ ψ(αx)) .

ut

Again the right hand side is the unitary dual of F\AF , which is simply theset of characters of F\AF , i.e. continuous homorphisms F\AF → C×.

B.2 Local Schwartz spaces and Fourier transforms

Let F be a local field. If F is nonarchimedean we define S(F ) = C∞c (F ), andif F is archimedean we let S(F ) denote the usual Schwartz space. We referto S(F ) as the Schwartz space of F .

If W is a finite-dimensional F -vector space we define S(W ) analogously.Let

〈 , 〉 : W ×W −→ F (B.2)

be a perfect pairing, let ψ : F → C× be a character, and let dy be a Haarmeasure on W . In the case where W = F the pairing is usually taken to be〈x, y〉 := xy. For f ∈ S(W ) one has a Fourier transform

f(x) :=

∫W

f(y)ψ (〈x, y〉) dy

which is also an element of S(W ). In the archimedean case this is standard,and in the nonarchimedean case it is Exercise B.5.

The Fourier transform depends on ψ, the pairing 〈 , 〉 and the Haar measuredy. Given ψ and 〈 , 〉 there is a unique way to normalize dy so that the Fourierinversion formula holds:

f(x) = (f)∧(−x). (B.3)

This Haar measure dy is known as the self-dual Haar measure (withrespect to the pairing and additive character).

B.3 Global Schwartz spaces and Poisson summation

Let F be a global field and let W be a finite dimensional F -vector space. Foran F -algebra R let W (R) := W ⊗F R. We define

S(W (AF )) := S(W (F∞))⊗C C∞c (W (A∞F )).

418 B Poisson Summation

When F is a number field we can view W (F∞) as an R-vector space to makesense of S(W (F∞)).

Let〈 , 〉 : W ×W −→ F

be a perfect pairing, let ψ : F\AF → C× be a nontrivial character, and let dybe a Haar measure on W (AF ). For f ∈ S(W (AF )) we then have an adelicFourier transform

f(x) :=

∫W (AF )

f(y)ψ (〈x, y〉) dy. (B.4)

Lemma B.3.1 One has f ∈ S(W (AF )).

Proof. It suffices to verify the lemma in the special case where W = F r and〈 , 〉 is the standard product

〈(x1, . . . , xr), (y1, . . . , yr)〉 7−→r∑i=1

xiyi.

Let S be a finite set of places of F including all the infinite places and allplaces where Fv is ramified. If S is large enough then

f = fS 1(OSF )r .

It also follows from (B.1.2) that ψ = ψSψS where ψS =

∏v 6∈S ψFv ; that

is, outside of S the character ψS is the standard character. Finally we canassume that dy = dySdy

S where dyS =∏v 6∈S dyv and dyv(OrFv ) = 1 for

v 6∈ S. Thusf = fS1(OSF )r .

We already know that fS ∈ S(W (F∞))⊗ S(W (F∞S )), so it suffices to check

that 1(OSF )r = 1OSF. This is the content of Exercise B.7. ut

Given 〈 , 〉, ψ and dy there is a unique choice of Haar measure dy on W (AF )such that the Fourier inversion formula holds:

f(x) = (f)∧(−x). (B.5)

This is the self-dual Haar measure (with respect to 〈 , 〉 and ψ).We can now state an adelic version of Poisson summation:

Theorem B.3.2 (Poisson summation) For f ∈ S(W (AF )) one has∑γ∈W (F )

f(γ) =∑

γ∈W (F )

f(γ).

ut

B.3 Global Schwartz spaces and Poisson summation 419

Exercises

B.1. Let F be a local field. Prove that the maps ψF defined in §B.1 arecharacters.

B.2. Prove Lemma B.1.1.

B.3. Let F be a global field. Prove that ψF is a character of AF trivial on F .

B.4. Prove Lemma B.1.2.

B.5. Let F be a nonarchimedean local field and let f ∈ C∞c (F ). Prove that

f ∈ C∞c (F ). Here f can be defined with respect to any Haar measure on Fand nontrivial character ψ : F → C×.

B.6. Prove that the Fourier inversion formula (B.3) holds when F is a nonar-chimedean local field.

B.7. Let F be a nonarchimedean local field that is unramified over its primefield. Show that

1OF = 1OF

where the Fourier transform is defined with respect to the standard additivecharacter ψF and the Haar measure on F such that dy(OF ) = 1. Concludethat this Haar measure is self-dual with respect to ψF and the pairing (x, y) 7→xy.

Hints to selected exercises

Chapter 1

1.12 Use the fact that parabolic subgroups would have unipotent elements.This would imply that B has zero divisors, contradicting the assumption thatit is a division algebra.

Chapter 2

2.10 Note that GLn(FS) is an open subset of gln(FS) for any finite set ofplaces S and use weak approximation for AF (see Theorem 2.1.4).

2.6 One approach uses the alternate definition of group schemes describedin Exercise 1.7.

2.14 In the archimedian case use the Gram-Schmidt process. In the nonar-chimedian case one can proceed in an elementary manner using induction onn. See [Bum97, Proposition 4.5.2].

2.15 Using the description of GL2(Q)\GL2(AQ)/K0(1) given at the end of§2.6 the exercise can be deduced from the following well-known fact: Everyelement of the complex upper half plane is SL2(Z)-equivalent (under Mobiustransformations) to an element z satifying −1

2 ≤ Re(z) ≤ 12 and |z| > 1.

Chapter 3

3.1 See [Fol95, Proposition 2.6].

3.2 Average a left Haar measure.

421

422 Hints to selected exercises

3.3 See [Fol95, §2.4], bearing in mind that δG(g) = ∆(g)−1 in the notationof loc. cit.

3.13 Prove first that V admits a G-invariant inner product.

Chapter 4

4.2 See [Bum97, Theorem 2.4.1].

4.9 The proof is an extension of the well-known construction of all thefinite-dimensional irreducible representations of the Lie algebra sl2.

Chapter 8

5.5 Show that V has a countable basis and invoke Lemma 4.1.

5.10 See [Car79].

5.11 To show that Vsm ≤ V is dense mimic the proof of Proposition 4.2.3.For the other claims one uses the following fact: if X,Y are topological vectorspaces, Y is an F -space, M ⊆ X is a dense subspace, and λ : M → Y is acontinuous map, then Λ admits a (unique) continuous extension to X → Y(see [Rud91, Chapter 1, Exercise 19]).

5.15 Use the fact that all hyperspecial subgroups are conjugate underG/ZG(F ) (see §2.4).

5.16 Use the fact that all hyperspecial subgroups are conjugate underG/ZG(F ) (see §2.4).

Chapter 6

6.7 Much of this amounts to computations and unwinding the definitions.The only points that are not formal are relating the notion of cuspidalityon both sides of the isomorphism and the related issue of proving that theautomorphic forms associated to classical cuspidal automorphic forms are ofmoderate growth. These issues are treated in detail in [Bum97, §3.2].

Hints to selected exercises 423

Chapter 8

8.5 Right exactness is the only point that is not obvious. Suppose thatone has a surjection A : V → W of smooth representations of M(F ). Letϕ : G→W be an element of IndGP (W ), and choose a compact open subgroupK ≤ G(F ) fixing ϕ. By Lemma 8.2.1 for all g ∈ G(F ) one has

ϕ(g) ∈ VM(F )∩gKg−1

.

Since VM(F )∩gKg−1 →WM(F )∩gKg−1

is surjective there is a vg ∈ VM(F )∩gKg−1

such that a(vg) = ϕ(g). Fix a minimal set of representativesX for P (F )\G(F )/K.We then define

ϕ : G −→ V

nmxk 7−→ m.vx

for (n,m, x, k) ∈ N(F )×M(F )×X×K. The function ϕ is fixed by K and istherefore smooth, and by construction A(ϕ) = ϕ. Thus the map IndGP (V )→IndGP (W ) is surjective.

8.6 It is clear that ResGP is additive. Assume that

0 −→ V1 −→ V2 −→ V3 −→ 0

is exact. It is easy to verify that V1N → V2N → V3N → 0 is exact. On theother hand

V1(N) = V1 ∩ V2(N),

and it follows that V1N → V2N is injective.

8.7 If ϕ ∈ V and n ∈ N(F ) then there is a compact open subgroup KN ≤N(F ) such that n′ ∈ KN . Choosing KN in this manner we see that∫

KN

π(n)(π(n′)ϕ− ϕ)dn = 0.

Conversely, suppose that ϕ ∈ V is such that∫KN

π(n)ϕdn = 0

for a compact open subgroup KN ≤ N(F ). Choose a compact open subgroupK ′N ≤ KN such that ϕ ∈ V K′N . Then

0 =

∫KN

π(n)ϕdn =1

[KN : K ′N ]

∑n∈KN/K′N

π(n)ϕ

424 Hints to selected exercises

which implies

ϕ =1

[KN : K ′N ]

∑n∈KN/K′N

(π(n)ϕ− ϕ).

8.9 For each i let mi ≤ A be the annihilator of Vi. It is a maximal ideal, andmi = mj if and only if i = j. Choose a1, . . . , an ∈ A such that ai ∈ mi and∑ni=1 ai = 1A, where 1A is the identity in A. Then 1 − ai is the identity on

Vi and zero on Vj for j 6= i.

8.15 Show that if J(σa, λa) and J(σa′, λa′) are linked then one of their

central characters is not unitary.

Chapter 9

9.4 See [Zhu93, Theorem 3.1.5].

Chapter 10

10.1 See [JZ87, §1].

10.2 See [JZ87, §1].

10.3 See [JZ87, §1].

Chapter 11

11.2 A Whittaker functional λ : Vsm → C× and a nonzero function ϕ ∈ Vsm

gives rise to a Whittaker function via

g 7−→ λ(π(g)ϕ).

Conversely, given a Whittaker model Vsm → W (ψ), we obtain a Whittakerfunctional by

ϕ 7−→Wϕ(I).

11.4 Consider the action of an unramified Hecke operator on J(λ). For µdominant relate the value of the Whittaker function at µ($) to the actionof the Hecke operator 1GL2(OF )µ($)GL2(OF ). Then use that the eigenvalues ofsuch an operator can be computed in terms of λ by Proposition 7.6.5.

11.5 See [JS81b, §2.5].

Hints to selected exercises 425

Chapter 12

12.10 Assume first that the πi are all square-integrable. Then since all the πiare tempered none of them are linked in the sense of §8.4. Hence we concludeby Theorem 8.4.4. The general case can be reduced to this case by Theorem8.4.5 and the transitivity of induction.

Chapter 15

15.4 LetK(N) := g ∈ GLn(Z) : g ≡ In (mod NZ).

Then GLn(Q) ∩K(N) is the group in the proof of Lemma 15.2.1 and henceis neat. Now observe that the set of eigenvalues of matrices in GLn(Q) ∩g−1K(N)g is independent of g ∈ GLn(A∞Q ).

Chapter 17

17.2 Consider the natural action of Gm on X = Ga.

Chapter 18

18.1 Since G(AF ) = AGG(AF )1 by §2.6 and G(AF )1 ≤ G(AF )′ the map

G(AF )′/H(AF ) −→ AG\G(AF )/H(AF )

is surjective. To check injectivity assume that ah`g1h−1r = g2 for some a ∈

AG, (h`, hr) ∈ H(AF ) and g1, g2 ∈ G(AF )′. For χ ∈ Y one has

1 = χ(ah`g1h−1r )χ(g2)−1 = χ(a)χ(h`h

−1r )χ(g1)χ(g2)−1 = χ(a)

By the duality between tori and their character groups (Theorem 1.7.1) thisimplies that a ∈ a(1, AH), which is to say that a = h′−1

` h′r for some (h′`, h′r) ∈

AH .

Index

(H,χ)-character, 276(H,χ)-distinguished

global, 278local, 276

(g,K)-module, 81admissible, 81

(g,K∞)-cohomology, 3031-cocycle, 341A-packet, 248C∗-algebra, 66Dv-compatible, 397H-distinguished


H-torsor, 345trivial, 345

K-finiteadelic, 117

K-finite vector, 80K-type σ, 80L-group, 133

parabolic subgroup, 134Levi subgroup, 135relevant, 135relevant Levi, 136standard, 135

L-indistinguishable, 249L-map, 133L-parameter, 233

almost tempered, 243equivalent, 234tempered, 240unramified, 241

R-valued points, 3V -cohomological vector, 304Z(g)-finite, 116w-generic representation, 242

σ-anisotropic torus, 340σ-isotypic subspace, 80σ-split torus, 340v-adic metric, 30etale slice, 336

Sakellaridis-Venkatesh conjecture, 282,286

A-parameters, 272absolute product

affine scheme, 5absolute value, 30

archimedean, 30nonarchimedean , 30trivial, 30

absolute valuesnormalized, 30

additive L-function, 235additive group, 6adeles ring, 32adelic L-parameter, 245

tempered, 245adelic automorphic form

over a function field, 119over a number field, 118

adelic Poisson summation, 418adelic quotient, 46adjoint group, 43adjoint representation, 13admissible Hilbert representation, 102admissible module, 102admissible representation, 80, 102affine group scheme, 6affine scheme, 3algebraic group, 9algebraic representation

427

428 Index

C-algebraic, 251L-algebraic, 251

almost simple group, 44anisotropic group, 23arithmetic subgroup, 47, 297Arthur SL2, 272Artin conjecture, 258Asai representation, 267, 284associated parabolic subgroup, 197automorphic form, 116

cuspidal, 119, 191automorphic induction, 261automorphic kernel, 322automorphic representation, 63, 114

cuspidal, 120cuspidal in the L2- sense, 120in the L2 sense, 114over a function field, 119over a number field, 119

automorphy factor, 122automorpic kernel function, 316

base change, 8, 260Bernstein-Zelevinsky classification, 157Borel subgroup, 22

Casimir element, 95Casselman and Shalika formula, 218categorical quotient, 332

uniform, 332central isogeney, 43central quasi-character of π, 94character, 15

of π, 91, 159Chevalley restriction theorem, 337class, 331class map, 342closed immersion, 4closed subscheme, 5co-character, 15cocycle condition, 341coefficient of π, 161cohomological representation, 305cohomological vector, 304, 305cohomologous cocycles, 341compact operator, 179complex dual, 21conductor, 220congruence subgroup, 47, 297conjugate dual, 268conjugate self-dual representation, 268connected

affine group scheme, 9affine scheme, 9

constant term of f along B, 138constant term of f along P , 163continuous spectrum, 68contracted product of Y and H, 346contragredient representation, 103Converse theory, 225coordinate ring, 3cuspidal cohomology, 304cuspidal function, 174cuspidal subspace, 120

decompositioncontinuous, 65discrete, 65

decomposition of representation, 65Deligne torus, 8, 309derived subgroup, 10dihedral representation, 259Dirac sequence, 179direct integral decomposition, 64discrete global A-packets of Gn, 271discrete series, 90, 91discrete spectrum, 68distinction, 275distribution, 159Dixmier-Malliavin lemma, 77dynamic, 410, 411

Eisenstein cohomology, 304Eisenstein series, 199equivalent inner forms, 289essentially square integrable, 91essentially tempered, 91

factorizable module, 108faithful representation

affine group scheme, 7Fell topology, 65, 66fiber product, 5finite type

affine scheme of, 3finiteness of class numbers, 45first Galois cohomology, 341Flath theorem, 108form of algebraic group, 342Fourier transform

adelic, 418local, 417

fppf, 345function field, 29fundamental lemma, 394

Garding subspace, 77Gan-Gross-Prasad conjecture, 288

Index 429

generic character, 212global, 213local, 213

generic fiber, 39generic representation, 213


geometric correspondence, 393geometric fiber, 408geometric point, 408geometric relative class, 334GIT quotient, 331global ψ-Whittaker function, 215global L-packet, 245

automorphic, 245global algebraic representation, 252C-algebraic, 252L-algebraic, 252

global field, 29global Langlands reciprocity, 247global Rankin-Selberg integral, 224global relative orbital integral, 356good position, 137, 163, 407

Harish-Chandra homorphism, 88Hecke algebra, 99

unramified, 128Hecke character, 128Hermitian contragredient, 104Hilbert modular variety, 311Hilbert representation, 102Hilbert-Schmidt, 174Hilbert-Schmidt norm, 174hull-kernel topology, 66hyperspecial subgroup, 40

Ichino-Ikeda conjecture, 282icosahedral representation, 259induced representation, 93, 149, 195inertia group, 233infinitesimal character, 89infinitesimally equivalent, 84inner form, 343intertwining operator, 197irreducible

affine scheme, 4irreducible principal series, 90isobaric representation, 145, 206isotropic group, 23Iwasawa decomposition, 47, 92, 138

Jacobi identity, 12Jacobson topology, 66Jacquet module, 154

Jacquet-Langlands transfer, 397Jacquet-Mao fundamental lemma, 286Jacquet-Ye fundamental lemma, 283Jordan decomposition Theorem, 10

Kunneth formula, 307kernel function, 176Kottwitz variety, 311Kuznetsov formula, 376

Langlands L-function, 249Langlands class, 137, 145Langlands classification, 94Langlands correspondence, 229

archimedean, 237local, 239

Langlands data, 93Langlands decomposition, 92Langlands dual group, 133Langlands functoriality conjecture

weak form, 145Langlands group, 246Langlands quotient, 93, 157

linked, 158Langlands reciprocity law, 229Langlands-Shahidi method, 262left Haar measure, 56Levi decomposition, 12Levi subgroup, 12Lie algebra, 12Lie bracket, 12lift of the correspondence, 301limit of discrete series, 90linear

affine group scheme, 7local A-packets, 272local L-packet, 242local field, 30local Langlands conjecture, 241local relative orbital integral, 344local system, 299locally symmetric space, 296Luna’s slice theorem, 337

matching, 393, 395, 397, 398matrix coefficient, 78, 150minimal parabolic, 24model, 39moderate growth function, 115

adelic, 117modular character, 140modular form, 122modular quasicharacter, 56, 149monomorphism

430 Index

affine algebraic group, 7morphism

affine group scheme of, 6affine scheme of, 3Lie algebra, 12

multiplicative group, 6multiplicity one, 216

natural action, 55neat arithmetic subgroup, 297neat compact open subgroup, 298neat element, 297neutral component, 10neutral element, 341nilpotent element, 10nondegenerate module, 101norm, 398number field, 29

octahedral representation, 259odd representation, 259orbital integral, 166

parabolic decent, 163parabolic induction, 150, 151parabolic subgroup, 22

minimal, 23opposite, 25proper, 22standard, 24, 157

parabolic subgroup scheme, 408period integral, 277Peter-Weyl Theorem, 79pinning, 132place, 30

archimedean, 30finite, 30infinite, 30nonarchimedean, 30

Plancherel measure, 92Poincare series, 177Poincare duality, 308positive roots, 26primitive ideal, 66principal series, 140

unramified, 142product formula, 32

quasi trivialgroup, 43torus, 43

quasi-semisimple automorphism, 338quasi-split classical groups, 265quasi-split group, 23

quasicuspidal representation, 150quotient map

affine algebraic group, 7

radical subgroup, 11Ramanujan conjecture, 205Ramanujan’s ∆-function, 124rank

algebraic group of, 16torus of, 14

Rankin-Selberg L-function, 211global, 222local, 218unramified, 221

rapidly decreasing function, 191reduced

affine scheme, 4root datum, 21

reduction theory, 48reductive group, 11reductive group scheme, 408reflex field, 310regular action, 55regular integral representation, 316regular representation, 173, 315relative character, 276relative conjugacy class, 339relative Langlands program, 282relative to the action of H, 330relative trace, 319relative Weyl law, 392relatively elliptic element, 357relatively regular element, 330relatively semisimple class, 331relatively semisimple element, 330relatively unimodular element, 344relevant element, 344, 356representable functor, 3representation, 54

affine group scheme, 7pre-unitarizable, 54pre-unitary, 54quotient, 54regular, 56unitarizable, 54unitary, 54

restricted direct product, 38restricted product topology, 32restricted tensor product, 107right Haar measure, 56ring of integers, 31root datum, 20

base, 131root group, 131

Index 431

root spaces, 19root system, 19

base, 19rank, 19reduced, 19

roots, 19

Sakellaridis-Venkatesh conjecture, 288Satake Isomorphsim, 129Satake parameter, 137, 145segment, 158

linked, 158self-dual Haar measure

adelic, 418local, 417

semisimple automorphism, 338semisimple element, 10, 338semisimple group, 10, 11sheaf of section, 300Shimura datum, 309Shimura manifold, 296Shimura variety, 295, 310Shintani formula, 218Siegel modular variety, 311Siegel set, 48simple general relative trace formula,

366, 371simple relative trace formula, 364simple trace formula, 372simple twisted trace formula, 373simply connected group, 43slowly increasing function, 115

adelic, 117smooth

affine scheme, 4algebraic group, 9

smooth O-scheme, 41smooth dual, 103smooth endomorphisms, 161smooth function, 98smooth relative characters, 277smooth representation, 100smooth vector, 75smoothly (H,χ)-distinguished, 277solvable group, 10special fiber, 39Speh representation, 207spherical Hecke algebra, 105spherical representation, 106spherical subgroup, 281split group, 17square integrable, 91stability of γ-factors, 266standard Borel space, 64

standard maximal compact, 237strong approximation, 34, 42strong Artin conjecture, 258strong multiplicity one, 223structure morphism, 6subgroup scheme, 8subquotient theorem, 93subrepresentation, 54

subquotient, 54supercuspidal representation, 150symmetric subgroup, 283symmetric variety, 283

Tamagawa measure, 62Tawagawa number, 62td-type group, 97tempered representation, 91tetrahedral representation, 259torus, 14

anisotropic, 16maximal, 16split, 14, 16

totally disconnected group, 97trace, 174trace class, 174trace norm, 174trace of π(f), 159transfer, 395twisted endoscopic groups, 399twisted endoscopy, 399type I group, 65, 67

unfolding, 177uniform moderate growth, 115

adelic, 117uniformizer, 31unimodular group, 56unipotent element, 10unipotent group, 10unipotent radical subgroup, 10unitarily equivalent, 54unitary dual, 65unramified L-packet, 242unramified group, 40, 105, 128unramified outside of S, 144unramified quasi-character, 140unramified representation, 106, 128

valuation, 29Vogan L-packet, 282, 291

weak approximation, 42weak global L-packet, 144weak lift, 257

432 Index

weak transfer, 257weakly globally L-indistinguishable, 143weight spaces, 18weights, 18Weil group, 230Weil restriction of scalars, 8Weil-Deligne group, 232Weyl chamber, 26

positive, 26Weyl group, 16

little, 340

root system of, 20Weyl law, 392Whittaker datum, 242Whittaker expansion, 215Whittaker functional

archimedean, 214nonarchimedean, 213

Whittaker model, 214

Zariski topology, 5

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