Advances in SOVIET

MATHEMATICSVolume 2

A. A. Kirillov Editor

Topics in Representation

Theory

American Mathematical Society

Topics in Representation

Theory

Advances in SOVIET

MATHEMATICS

Editorial Committee

V. I. Arnold S. G. Gindikin V. P. M aslov

Advances inS ovietMathematicsVolume 2

Topics in Representation

Theory

A. A. Kirillov Editor

American M athem atical SocietyProvidence, Rhode Island

10.1090/advsov/002

Translated from the Russian by SERGE! GELFAND

Translation edited by A. B. SOSSINSKY

1980 Mathematics Subject Classification (1985 Revision). Primary 17B10, 17B65, 22E47, 22E65, 22E70; Secondary 17B65, 20C32, 20M30, 58A99.

Library of Congress Cataloging-in-Publication Data

Topics in representation theory/A. A. Kirillov, editorp. cm. - (Advances in Soviet mathematics, ISSN 1051-8037; v. 2)Includes bibliographical references.ISBN 0-8218-4101-71. Representations of groups. 2. Representations of algebras. I. Kirillov, A. A.

(Aleksandr Aleksandrovich), 1936- . II. Series.QA171.T664 1991 90-26017512'.2-dc20 CIP

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Contents

In Place of an Introduction vii

Representations of Infinite-Dimensional Classical Groups, Limits of Enveloping Algebras, and Yangians

G. I. OLSHANSKlI 1

On Semigroups Related to Infinite-Dimensional GroupsG. I. OLSHANSKlI 67

Infinite-Dimensional Groups, Their Mantles, Trains, and Representations

YU. A. NERETIN 103

On Finite-Dimensional Representations of the Lie Algebra L =sl2(R) • R

R. S. ISMAGILOV 173

Unitarizability of Some Enright-Varadaraj an u(p, #)-modulesA. I. MOLEV 199

Lie Derivatives of Order n on the Line. Tensor Meaning of the Gelfand-Dikii Bracket

O .D . OVSIENKO and V. YU. OVSIENKO 221

The Vocabulary of Geometry and Harmonic Analysis on the Infinite-Dimensional Manifold Diff+(S'1)/S '1

D. V. YUR'EV 233

V

In Place of an Introduction

The papers in this volume were written by members of the Seminar on Representation Theory which has been working at Moscow University since 1961. I think that the publication of this volume is an appropri­ate occasion to speak a little bit about the current state of representation theory.

Some 20 years ago in the book [1] I wrote that representation theory enters the fourth stage of its evolution, when the three main directions of its development will presumably be the following ones:

— representations of algebraic groups over local fields and adele rings;— representations of infinite-dimensional Lie groups and Lie algebras;— representations of Lie supergroups and Lie superalgebras.These directions, together with the application of the orbit method,

formed the main topic of discussion at our seminar.Of course, real life corrected this prediction. Two new areas of math­

ematical physics, namely the theory of completely integrable systems and string theory, have had a great influence on mathematics in general, and on representation theory in particular.

One must also mention here the growing influence of the idea of super- symmetry. Originally this idea appeared in theoretical physics as the means to “put on an equal footing” fermions and bosons, but later it became a general mathematical principle. According to this principle, each ordi­nary (“even”) object of a mathematical theory has an “odd counterpart”, which, being combined with the initial even prototype, forms a unified “superobject” whose properties resemble the properties of the prototype, but on a somewhat higher level. Important examples of this principle for representation theory are the following transitions:

— the passage from the algebra C°°(M) of smooth functions on a manifold M to the algebra A(M) of differential forms on M ;

— the passage from the Laplace and Klein-Gordon equations to the Cauchy-Riemann and Dirac equations;

VII

IN PLACE OF AN INTRODUCTIONviii

— the passage from the Virasoro algebra to the Neveu-Schwarz and Ramon superalgebras.

A characteristic property of these new areas is the previously unseen diversity of mathematical notions and methods involved. Practically each major mathematical theory, from 19th century classical analysis and geom­etry to the newest abstract constructions of category theory, have recently acquired a “physical flavor”, The basis of modem mathematical physics is mathematics as a unified science, not as the disjoint union of arithmetics, algebra, analysis, geometry, differential equations, etc., as often happened in the past. This fact drastically increases the requirements to mathemati­cians, forcing them to reconsider both mathematics and its role in human culture.

Exactly how has this new situation influenced representation theory? First of all, the three directions that we mentioned earlier turned out to have a much closer relationship than one might have assumed previously.

The “adelic” approach, borrowed from number theory, was tradition­ally considered to be more or less unrelated to practical applications. It appears, however, that this approach is an efficient tool for constructing and analyzing representations of infinite-dimensional Lie algebras of vec­tor fields on manifolds (see [2, 3]).

The characters of certain representations of Kac-Moody and Virasoro algebras display some “modular” properties (see [4, 5]). Flag manifolds, well-known in the representation theory of compact groups and in geom­etry, happen to have infinite-dimensional analogs that are related to inte­grable equations of KdV type [6] and to such classical objects of complex analysis as spaces of univalent functions (see [7, 8]).

The canonical symplectic structure on orbits of the coadjóint represen­tation allows one to consider these orbits as generalized phase spaces of classical physics, and the construction of the unitary representation from an orbit as a generalization of the quantization procedure. B. Kostant remarked that this procedure does not require homogeneity, and can be defined for an arbitrary symplectic manifold. This remark gave birth to a new field of research, called geometric quantization (see [9, 10, 11]). One must mention, however, that the main results obtained by this approach deal with the homogeneous situation (for example, Duflo’s construction in [12] which gives a unified construction of unitary irreducible representa­tions for an arbitrary Lie group).

Almost all the main infinite-dimensional Lie groups and Lie algebras are naturally embedded into the corresponding Lie supergroups and su­peralgebras. Superalgebras corresponding to the Virasoro and Kac-Moody

IN PLACE OF AN INTRODUCTION IX

algebras, have already established their positions in the mathematical and physical literature, and now an extensive search and study of their multi­dimensional analogs is under way (see, for example, the paper by O. D. and V. Yu. Ovsienko in this volume).

The unification of the ideas of symplectic geometry and of noncommu- tative geometry gave birth to a completely new mathematical object, the quantum group (see [13, 14, 15]). Due to the fortunate choice of their name and to their relationship with various areas of modem mathematics and mathematical physics, quantum groups are now very popular among both physicists and mathematicians, less than five years after their inven­tion.

A new approach to representation theory of “large” groups (A. M. Ver- shik’s terminology ) was successfully developed. These groups include, among others, groups of operators in infinite-dimensional spaces, groups of automorphisms of vector bundles, and the infinite-dimensional sym­metric group.

The main idea of this new approach is that each large group G can be embedded into a semigroup A and is related to some category K . Namely, G is the automorphism group of a certain object from the cat­egory K and the semigroup A can be identified with the semigroup of endomorphisms of this object. A fascinating experimental fact is that the most interesting representations of G can be extended to (in general, projective) representations of the category K , so that linear spaces are associated to objects and linear operators to morphisms.

Details can be found in the papers by Yu. A. Neretin and by G. I. Olshan- skii in this volume.

Together with this wave of new connections and new trends in represen­tation theory, a more traditional activity has also flourished. This activity mostly deals with the study of classical objects, such as representations of finite and compact groups, finite-dimensional representations of Lie groups, unitary representations of locally compact groups, and harmonic analysis on manifolds.

However, even in these “evergreen” areas the modem outlook has re­sulted in new approaches. A good example is the representation theory of the symmetric group S{k) which, in a sense, initiated representation theory more than a century ago.

First of all, this theory can now be viewed as a special case of the representation theory of the Hecke algebra H (n) . which is a “quantum deformation” or a “ ^-analog” of the group algebra C[S(«)] and turns into

X IN PLACE OF AN INTRODUCTION

this algebra when q = 1. This approach clarifies, in particular, the theory of Young bases and helps to answer some other classical questions [16].

Second, the family of groups S{n) and its inductive limit S(oo) is one of the examples of a “large” group which is embedded into the category of finite spaces and partial bijections (see the papers by Yu. A. Neretin and by G. I. Olshanskil in this volume, and [17]).

Next, polynomial covariants of the symmetric group happen to have nice superanalogs [18], which can be thought of as elements of a new (still nonexistent) (super-)theory of invariants.

Finally, projective representations of S{n) studied by Schur in the be­ginning of this century turn out to be related to a very interesting projective factor representation of the group S(oo) (see [19]).

In conclusion, a few words about the papers in this volume.R. S. Ismagilov describes indecomposable representations of the affine

unimodular group of the plane. This is a part of a very complicated, but still rather mysterious, area of representation theory which includes mod­ular representations of finite groups, finite-dimensional nonunitary repre­sentations of Lie groups and representations of infinite-dimensional Lie groups and Lie algebras with highest weight type conditions.

A. I. Molev has achieved significant progress in the description of dual objects for the real reductive Lie group U{p, q) .

The papers by Yu. A. Neretin and G. I. Olshanskil were mentioned earlier.

O. D. and V. Yu. Ovseenko suggest a geometrical interpretation of a very interesting infinite-dimensional Lie algebra introduced by I. M. Gelfand and L. A. Dikii. (Presumably, this Lie algebra cannot be realized as a Lie algebra of vector fields on a finite-dimensional manifold, and its geomet­rical interpretation requires new ideas.)

The original paper by D. V. Yurev will, I hope, be useful in attempts to set up bridges between physicists and mathematicians in the area of infinite-dimensional geometry.

R e f e r e n c e s

1. A. A. Kirillov, Elements o f the theory o f representations, “Nauka”, Moscow, 1972; English transl., Springer, Berlin and New York, 1976 .

2. B. L. Feigin and D. B. Fuks, Cohomology o f some nilpotent subalgebras o f the Virasoro and Kac-Moody Lie algebras, J. Geom. Phys. 5 (1988).

3. E. Witten, Free fermions on an algebraic curve, Proc. Sympos. Pure Math., 48, Amer­ican Mathematical Society, Providence, R.I., 1988.

4. V. G. Kac and M. Wakimoto, Modular and conformal invariance constraints in rep­resentation theory o f affine algebras, Adv. in Math. 70 (1988), no. 2, 156-236.

IN PLACE OF AN INTRODUCTION xi

5. V. G. Kac, R. V. Moody, and M. Wakimoto, On E l0 , Preprint, 1989.6. G. Segal and G. Wilson, Loop groups and equations o f KdV, Inst. Hautes Études Sci.

Publ. Math. 63 (1985), 1-64.7. A. A. Kirillov, Kàhler structure on K-orbits o f the group o f diffeomorphisms o f the

circle, Funktsional. Anal, i Prilozhen. 21 (1987), no. 2, 40-45; English transi, in Functional Anal. Appl. 21 (1988), no. 2.

8. A. A. Kirillov and D. V. Yur'ev, Kàhler geometry o f the space M = DifF+(5 1)/R ot(51) , Funktsional. Anal, i Prilozhen. 21 (1987), no. 4, 35-46; English transi, in Functional Anal. Appl. 21 (1988), no. 4.

9. B. Kostant, Quantization and unitary representations, Lecture Notes in Math. vol. 170, Springer, 1970, pp. 87-208; 1977, pp. 177-306.

10. J.-M. Souriau, Systèmes dynamiques, Dunod, Paris, 1970.11. A. A. Kirillov, Geometric quantization, Itogi Nauki i Tekhniki. Sovremennye Prob-

lemy Matematiki. Fundamental'nye Napravleniya, VINITI, Moscow, 1985, vol. 4; English transi., Encyclopaedia of Math. Sciences, vol. 4, Springer-Verlag, 1989.

12. M. Duflo, Construction des représentations unitaires des groupes de Lie, Cours d’été du C.I.S.M., Cortona, 1980.

13. V. G. Drinfeld, Quantum groups, Proc. Int. Congress Math., Berkeley, 1986, Amer­ican Mathematical Society, Providence, R.I., 1987, pp. 798-820.

14. L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 4, 178-206; English transi, in Leningrad Math. J. 1 (1990), no. 4.

15. M. Jimbo, A q-analog o f U( g l ( N+ 1 )), Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.

16. I. V. Cherednik, A new interpretation o f Gelfand-Tsetlin bases, Duke Math. J. 54 (1986), no. 2, 563-577.

17. A. M. Vershik and S. V. Kerov, Characters and factor representations o f the infinite symmetric group, Dokl. Akad. Nauk SSSR 257 (1981), no. 5, 1037-1040; English transi, in Soviet Math. Dokl. 25 (1981); see also Characters and realizations o f infinite-dimensional Hecke algebra and invariants o f knots, Dokl. Akad. Nauk SSSR 301 (1988), no. 4, 777-780; English transi, in Soviet Math. Dokl. 38 (1989), no. 1.

18. A. A. Kirillov and I. M. Pak, Covariants o f the symmetric groups and o f some o f its analogs in the Weyl algebra, Funktsional. Anal, i Prilozhen. 24 (1990), no. 3, 9-13; English transi, in Functional Anal. Appl. 24 (1990), no. 3.

19. M. L. Nazarov, Factor representations o f the infinite spin-symmetric group, Uspekhi Mat. Nauk 43 (1988), no. 4, 221-222. (Russian).

A. A. Kirillov

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