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ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS OFp-ADIC REDUCTIVE GROUPS I. DEFINITION AND FIRST

PROPERTIES

MATTHEW EMERTON

Contents

1. Introduction 12. Representations of p-adic analytic groups 23. Ordinary parts 164. Adjunction formulas 24References 33

1. Introduction

This paper is the first of two in which we define and study the functors of or-dinary parts on categories of admissible smooth representations of p-adic reductivegroups over fields of characteristic p, as well as their derived functors. These func-tors of ordinary parts, which are characterized as being right adjoint to parabolicinduction, play an important role in the study of smooth representation theory incharacteristic p. They also have important global applications: when applied inthe context of the p-adically completed cohomology spaces introduced in [5], theyprovide a representation-theoretic approach to Hida’s theory of (nearly) ordinaryparts of cohomology [9, 10, 11]. The immediate applications that we have in mindare for the group GL2: we will apply the results of our two papers both to theconstruction of a mod p local Langlands correspondence for the group GL2(Qp),and to the investigation of local-global compatibility for p-adic Langlands over thegroup GL2 over Q.

To describe our results more specifically, let G be (the Qp-valued points of) aconnected reductive p-adic group, P a parabolic subgroup of G, P an oppositeparabolic to P , and M = P

⋂P the corresponding Levi factor of P and P . If k

is a finite field of characteristic p, we let ModadmG (k) (resp. Modadm

M (k)) denote thecategory of admissible smooth G-representations (resp. M -representations) overk. Parabolic induction yields a functor IndG

P: Modadm

M (k) → ModadmG (k). The

functor of ordinary parts associated to P is then a functor OrdP : ModadmG (k) →

ModadmM (k), which is right adjoint to IndG

P.

In this paper, we define OrdP and study its basic properties. In the sequel [8] weinvestigate the derived functors of OrdP , and present the applications to the mod pLanglands correspondence for GL2(Qp). The applications to the study of local-global compatibility will form part of the arguments of [7]. We hope to discuss theapplications in the context of p-adically completed cohomology in a future paper.

The author was supported in part by NSF grants DMS-0401545 and DMS-0701315.

1

2 MATTHEW EMERTON

Let us only mention here that Theorem 3.4.4 below is an abstract formulation ofHida’s general principal that the ordinary part of cohomology should be finite overweight space.

1.1. Arrangement of the paper. In order to allow for maximum flexibility inapplications, we actually work throughout the paper with representations definednot just over the field k, but over general Artinian local rings, or even completelocal rings, having residue field k. This necessitates a development of the founda-tions of the theory of admissible representations over such coefficient rings. Sucha development is the subject of Section 2. We also take the opportunity in thatsection to present some related representation theoretic notions that do not seemto be in the literature, and which will be useful in this paper, its sequel, and futureapplications. The functors OrdP are defined, and some of their basic propertiesestablished, in Section 3. Their characterization as adjoint functors is proved inSection 4.

1.2. Notation and terminology. Throughout the paper, we fix a prime p, aswell as a finite extension E of Qp, with ring of integers O. We let F denote theresidue field of O, and $ a choice of uniformizer of O. Let Comp(O) denote thecategory of complete Noetherian local O-algebras having finite residue fields, andlet Art(O) denote the full subcategory of Comp(O) consisting of those objects thatare Artinian (or equivalently, of finite length as O-modules).

If A is an object of Art(O), and V is an A-module, then we write V ∗ :=HomO(V,E/O) to denote the Pontrjagin dual of V , equipped with its naturalprofinite topology. If V is a G-representation over A, for some group G, thenthe contragredient action makes V ∗ a G-representation over A (with each elementof G acting via a continuous automorphism).

If C is any O-linear category, then for any integer i ≥ 0 (resp. i = ∞), we letC[$i] denote the full subcategory of C consisting of objects that are annihilatedby $i (resp. by some power of $). We let Cfl denote the full subcategory of Cconsisting of objects that are flat over O, and let CE denote the E-linearizationof C.

2. Representations of p-adic analytic groups

Let G be a p-adic analytic group. Throughout this section, we will let A de-note an object of Comp(O), with maximal ideal m. We denote by ModG(A) theabelian category of representations of G over A (with morphisms being A-linear G-equivariant maps); equivalently, ModG(A) is the abelian category of A[G]-modules,where A[G] denotes the group ring of G over A. In this section we introduce, andstudy some basic properties of, various categories of G-representations, as well asof certain related categories of what we will call augmented G-representations.

2.1. Augmented G-representations. If H is a compact open subgroup of G,then as usual we let A[[H]] denote the completed group ring of H over A, i.e.

(1) A[[H]] := lim←−H′A[H/H ′],

where H ′ runs over all open subgroups of H. We equip A[[H]] with the projectivelimit topology obtained by endowing of the rings A[H/H ′] appearing on the righthand side of (1) with its m-adic topology. The rings A[H/H ′] are then profinite

ORDINARY PARTS OF ADMISSIBLE REPRESENTATIONS 3

(since each H/H ′ is a finite group), and hence this makes A[[H]] a profinite, andso in particular compact, topological ring.

2.1.1. Theorem. The completed group ring A[[H]] is Noetherian.

Proof. In the case when A = O, this is proved by Lazard [12]: after replacing H byan open subgroup, if necessary, Lazard constructs an exhaustive decreasing filtra-tion F • on O[[H]], which induces the $-adic filtration on O, such that associatedgraded ring is isomorphic to (Gr•$O) ⊗Fp

U(g), where Gr•$O denotes the gradedring associated to the $-adic filtration on O (which is a commutative F-algebraisomorphic to F[t]), and U(g) denotes the universal enveloping algebra of the Liealgebra g of H over Fp. Since this associated graded ring is Noetherian, so is O[[H]].

For general A, we may write A[[H]] := A⊗OO[[H]], and so equip A[[H]] withan exhaustive decreasing filtration F • obtained as the tensor product of Lazard’sfiltration on O[[H]] and the m-adic filtration on A. The associated graded ring isisomorphic to (Gr•mA) ⊗Fp

U(g), where Gr•mA denotes the graded ring associatedto the m-adic filtration on A. Since Gr•mA is Noetherian, the same is true of thistensor product, and thus A[[H]] is Noetherian, as claimed.

2.1.2. Proposition. (1) Any finitely generated A[[H]]-module admits a uniqueprofinite topology with respect to which the A[[H]]-action on it becomes con-tinuous.

(2) Any A[[H]]-linear morphism of finitely generated A[[H]]-modules is contin-uous with respect to the profinite topologies on its source and target givenby part (1).

Proof. Since A[[H]] is Noetherian, we may find a presentation

A[[H]]s → A[[H]]r →M → 0

of M , for some r, s ≥ 0. Since A[[H]] is profinite, it follows that the first arrowhas closed image, and hence that M is the quotient of the profinite module A[[H]]r

by a closed A[[H]]-submodule. If we equip M with the induced quotient topology,then it becomes a profinite module. The resulting profinite topology on M isclearly independent of the chosen presentation, and satisfies the requirements ofthe proposition.

2.1.3. Definition. If M is a finitely generated A[[H]]-module, we refer to thetopology given by the preceding lemma as the canonical topology on M .

2.1.4. Definition. By an augmented representation of G over A we mean an A[G]-module M equipped with an A[[H]]-module structure for some (equivalently, any)compact open subgroup H of G, such that the two induced A[H]-actions (the firstinduced by the inclusion A[H] ⊂ A[[H]] and the second by the inclusion A[H] ⊂A[G]) coincide.

2.1.5. Definition. By a profinite augmented G-representation over A we mean anaugmented G-representation M over A that is also equipped with a profinite topol-ogy, so that the A[[H]] action on M is jointly continuous 1 for some (equivalentlyany) compact open subgroup H over A.

1It is equivalent to ask that the profinite topology on M admits a neighbourhood basis at theorigin consisting of A[[H]]-submodules.

4 MATTHEW EMERTON

The equivalence of the conditions “some” and “any” in the preceding definitionsfollows from the fact that if H1 and H2 are two compact open subgroups of G, thenH := H1

⋂H2 has finite index in each of H1 and H2.

We denote by ModaugG (A) the abelian category of augmented G-representations

over A, with morphisms being maps that are simultaneously G-equivariant andA[[H]]-linear for some (equivalently, any) compact open subgroup H of G, and byModpro aug

G (A) the abelian category of profinite augmented G-representations, withmorphisms being continuous A-linear G-equivariant maps (note that since A[H] isdense in A[[H]], any such map is automatically A[[H]]-linear for any compact opensubgroup H of G). Forgetting the topology induces a forgetful functor

Modpro augG (A) −→ Modaug

G (A).

We let Modfg augG (A) denote the full subcategory of Modaug

G (A) consisting of aug-mented G-modules that are finitely generated over A[[H]] for some (equivalentlyany) compact open subgroup H of G. Proposition 2.1.2 shows that by equippingeach object of Modfg aug

G (A) with its canonical topology, we may lift the inclusionModfg aug

G (A)→ ModaugG (A) to an inclusion Modfg aug

G (A)→ Modpro augG (A).

2.1.6. Proposition. The category Modfg augG (A) forms a Serre subcategory of the

abelian category ModaugG (A). In particular, it itself is an abelian category.

Proof. Closure under the formation of quotients and extensions is evident, andclosure under the formation of subobjects follows from Theorem 2.1.1.

Write AE := E ⊗O A, and A[[H]]E := E ⊗O A[[H]].

2.1.7. Theorem. The ring A[[H]]E is Noetherian.

Proof. This follows directly from Theorem 2.1.1.

We introduce the following analogue of Definition 2.1.4, in the context of G-representations over AE .

2.1.8. Definition. An augmented representation ofG over AE consists of an AE [G]-module M equipped with an A[[H]]E-module structure for some (equivalently, any)compact open subgroup H of G, such that the two induced AE [H]-actions (thefirst induced by the inclusion AE [H] ⊂ A[[H]]E and the second by the inclusionAE [H] ⊂ AE [G]) coincide.

We let Modfg augG (AE) denote the category whose objects are augmented G-

modules over AE that are finitely generated over A[[H]]E for some (equivalentlyany) compact open subgroup H of G, and whose morphisms are maps that aresimultaneously (AE)[G] and A[[H]]E-linear.

2.1.9. Lemma. The functor M 7→ E ⊗O M induces an equivalence of categoriesModfg aug

G (A)E∼−→ Modfg aug

G (AE).

Proof. Obvious.

2.2. Smooth G-representations. In this subsection we give the basic definitions,and state the basic results, related to smooth, and admissible smooth, representa-tions of G over A. In the case when A is Artinian, our definitions will agree withthe standard ones. Otherwise, they may be slightly unorthodox, but will be themost useful ones for our later purposes.


2.2.1. Definition. Let V be a representation of G over A. We say that a vectorv ∈ V is smooth if:

(1) If v is fixed by some open subgroup of G.(2) v is fixed by some power mi of the maximal ideal of A.

We let Vsm denote the subset of V consisting of smooth vectors.

2.2.2. Remark. The following equivalent way of defining smoothness can be useful:a vector v ∈ V is smooth if and only if v is annihilated by an open ideal in A[[H]],for some (equivalently, any) open subgroup H of G.

2.2.3. Remark. If A is Artinian, then mi = 0 for sufficiently large i, and soautomatically miv = 0 for any element v of any A-module V . Thus condition (2)can be omitted from the definition of smoothness for such A, and our definitioncoincides with the usual one.

2.2.4. Lemma. The subset Vsm is an A[G]-submodule of V .

Proof. If vi ∈ Vsm (i = 1, 2) are fixed by open subgroups Hi ⊂ G, and annihilatedby mi1 and mi2 , then any A-linear combination of v1 and v2 is fixed byH1

⋂H2, and

annihilated by mmax i1,i2 , and hence also lies in Vsm. Thus Vsm is an A-submoduleof V . Also, if v ∈ Vsm is fixed by the open subgroup H ⊂ G, and annihilated bymi, then gv is fixed by gHg−1 ⊂ G for any g ∈ G, and again annihilated by mi.Thus Vsm is closed under the action of G.

2.2.5. Definition. We say that a G-representation V of G over A is smooth ifV = Vsm; that is, if every vector of V is smooth.

We let ModsmG (A) denote the full subcategory of ModG(A) consisting of smooth

G-representations. Clearly ModsmG (A) is in fact a Serre subcategory of ModG(A)

(i.e. it is closed under passing to subobjects, quotients, and extensions). In particu-lar, it is again an abelian category. The association V 7→ Vsm is a left-exact functorfrom ModG(A) to Modsm

G (A), which is right adjoint to the inclusion of ModsmG (A)

into ModG(A).The following lemma provides some useful ways for thinking about smooth rep-

resentations of G.

2.2.6. Lemma. Let V be a G-representation over A. The following conditions onV are equivalent.

(1) V is smooth.(2) The A-action on V and the G-action on V are both jointly continuous,

when V is given its discrete topology.(3) The A-action and the G-action on V ∗ are both jointly continuous when V ∗

is given its natural profinite topology.(4) For any compact open subgroup H of G, the H-action on V ∗ extends to a

continuous action of completed group ring A[[H]] on V ∗.(In both conditions (1) and (2), the ring A is understood to be equipped with itsm-adic topology.)

Proof. The straightforward (and well-known) proofs are left to the reader.

Lemma 2.2.6 implies that passing to Pontrjagin duals induces an anti-equivalence

(2) ModsmG (A) anti∼ // Modpro aug

G (A).

6 MATTHEW EMERTON

2.2.7. Definition. We say that a smooth G-representation V over A is admissibleif V H [mj ] (the mi-torsion part of the subspace of H-fixed vectors in V ) is finitelygenerated over A for every open subgroup H of G and every i ≥ 0.

2.2.8. Remark. If A is Artinian, so that mi = 0 for i sufficiently large, thenV H [mi] = V H for i sufficiently large, and so the preceding definition agrees withthe usual definition of an admissible smooth A-module.

We let ModadmG (A) denote the full subcategory of Modsm

G (A) consisting of ad-missible representations.

2.2.9. Lemma. A smooth G-representation V over A is admissible if and only ifV ∗ is finitely generated as an A[[H]]-module for some (equivalently any) compactopen subgroup H of G.

Proof. This is well-known.

Lemma 2.2.9 shows that the anti-equivalence (2) restricts to an anti-equivalence

(3) ModadmG (A)

anti∼ // Modfg augG (A).

2.2.10. Proposition. The category ModadmG (A) forms a Serre subcategory of the

abelian category ModG(A). In particular, it itself is an abelian category.

Proof. This follows directly from the anti-equivalence (3), together with Proposi-tion 2.1.6.

2.2.11. Remark. If V is any G-representation over A, and j ≥ 0, then V [mj ] isa successive extension of finitely many copies of V [m]. It thus follows from thepreceding result that if V is a smooth G-representation over A for which V [m] isan admissible G-representation over the field A/m (equivalently, if the condition ofDefinition 2.2.7 is assumed just in the case j = 1), then V is in fact an admissiblesmooth G-representation. In fact, it follows from Lemma 2.2.9, together with thetopological version of Nakayama’s lemma, that if H is an open pro-p subgroup of G(so that A[[H]] is a – typically non-commutative – local ring), then V is admissibleif and only if V H [m] is finite dimensional over A/m for one open pro-p subgroup ofG.

2.2.12. Definition. Let V be a G-representation over A.(1) We say that an element v ∈ V is locally admissible if v is smooth, and if

the smooth G-subrepresentation of V generated by v is admissible.(2) We let Vl.adm denote the subset of V consisting of locally admissible ele-

ments.

2.2.13. Lemma. The subset Vl.adm is an A[G]-submodule of V .

Proof. Let vi ∈ Vl.adm (i = 1, 2), and let Wi denote the G-subrepresentation of Vgenerated by vi. Since Modadm

G (A) is a Serre subcategory of ModsmG (A), we see

that the image of the map W1

⊕W2 → V induced by the inclusions Wi ⊂ V is

admissible. This image certainly contains any A-linear combination of the vi, andthus Vl.adm is an A-submodule of V . It is also clearly G-invariant, and thus is anA[G]-submodule of V .

2.2.14. Definition. We say that a G-representation V of G over A is locally ad-missible if V = Vl.adm; that is, if every vector of V is locally admissible.


We let Modl admG (A) denote the full subcategory of Modsm

G (A) consisting of locallyadmissible G-representations.

2.2.15. Proposition. The category Modl admG (A) is closed under passing to subrep-

resentations, quotients, and inductive limits in ModsmG (A). In particular, it itself is

an abelian category.

Proof. Since ModadmG (A) is closed under passing to subrepresentations and quo-

tients in ModsmG (A), the same is evidently true for Modadm

G (A). The closure underformation of inductive limits is also clear, since the property of being locally ad-missible is checked elementwise.

2.2.16. Remark. Clearly any finitely generated locally admissible representation isin fact admissible smooth. Since any representation is the inductive limit of finitelygenerated ones, we see that a representation is locally admissible if and only if itcan be written as an inductive limit of admissible representations.

The association V 7→ Vl.adm gives rise to a left-exact functor from ModG(A) toModl adm

G (A), which is right adjoint to the inclusion of Modl admG (A) into ModG(A).

The following diagram summarizes the relations between the various categoriesthat we have introduced:

ModG(A)

( )sm##

( )l.adm

++

ModsmG (A)?

OO

oo anti∼ // Modpro augG (A) // Modaug

G (A)

Modl admG (A)?

OO

ModadmG (A)?

OO

oo anti∼ // Modfg augG (A)

?

canonicaltopology

OO

(

LL

2.3. Some finiteness conditions. Let Z denote the centre of G.

2.3.1. Definition. Let V be an object of ModG(A).(1) We say that V is Z-finite if, writing I to denote the annihilator of V in

A[Z], the quotient A[Z]/I is a finite A-algebra.(2) We say that v ∈ V is locally Z-finite if the A[Z]-submodule of V generated

by v is Z-finite (equivalently, is finitely generated as an A-module). We letVZ−fin denote the subset of locally Z-finite elements of V .

(3) We say that V is locally Z-finite over Z if every v ∈ V is locally finiteover Z, i.e. if V = VZ−fin.

2.3.2. Lemma. (1) For any object V of ModG(A), the subset VZ−fin is anA[G]-submodule of V .

(2) If V1 and V2 are two (locally) Z-finite representations, then V1

⊕V2 is again

(locally) Z-finite.(3) Any A[G]-invariant submodule or quotient of a (locally) Z-finite represen-

tation is again (locally) Z-finite.

8 MATTHEW EMERTON

Proof. If V1 and V2 are A[G]-modules annihilated by the A-finite quotients A[Z]/I1and A[Z]/I2 of A[Z], respectively, then V1

⊕V2 is annihilated by A[Z]/(I1

⋂I2),

which is again A-finite (since it embeds into (A[Z]/I1)×(A[Z]/I2)). This proves (2)in the case of Z-finite representations.

Now suppose that v1 and v2 are two elements of VZ−fin, for some A[G]-module V .Let Ii denote the annihilator of vi in A[Z], and let Wi be the G-subrepresentationof V generated by vi. Since Z is the centre of G, we see that the Ii annihilates allof Wi. Thus each Wi is Z-finite, and thus so is W1

⊕W2. The image of the natural

map W1

⊕W2 → V contains all linear combinations of the vi, and is G-invariant.

Thus VZ−fin is an A[G]-submodule of V .Now let V1 and V2 be locally Z-finite A[G]-modules. Then (V1

⊕V2)Z−fin con-

tains V1 and V2, and so, since it is an A[G]-submodule of V1

⊕V2, equals all of

V1

⊕V2. This proves (2) in the case of locally Z-finite representations. Part (3) is

evident.

2.3.3. Lemma. If V is an object of ModG(A) which is finitely generated over A[G],then V is Z-finite if and only if V is locally Z-finite.

Proof. As we already observed in the proof of the preceding lemma, if an A[G]-module is generated by a single locally Z-finite vector, then it is Z-finite. If V isgenerated by finitely many such vectors, then it is a quotient of a direct sum offinitely many Z-finite representations, and so is again Z-finite (by parts (1) and (3)of the preceding lemma).

2.3.4. Lemma. If V is an object of ModG(A), then any locally admissible vectorin V is locally Z-finite.

Proof. Suppose that v ∈ V is locally admissible, let H ⊂ G be a compact opensubgroup that fixes v, and let mi annihilate v, for some i ≥ 0. If W ⊂ V denotesthe A[G]-submodule generated by v, then W is admissible smooth, by assumption,and is annihilated by mi, and so WH is finitely generated over A. Since WH isZ-invariant, we conclude that v is locally Z-finite, as claimed.

2.3.5. Lemma. If V is an object of ModadmG (A) which is finitely generated over

A[G], then V is Z-finite.

Proof. This follows directly from the preceding lemmas.

2.3.6. Lemma. Let V be an object of ModsmG (A), and consider the following con-

ditions on V .(1) V is of finite length (as an object of Modsm

G (A), or equivalently as an objectof ModG(A)), and is admissible.

(2) V is finitely generated as an A[G]-module, and is admissible.(3) V is of finite length (as an object of Modsm

G (A)), and is Z-finite.Then (1) implies (2) and (3).

Proof. The implication (1) ⇒ (2) is immediate. The implication (2) ⇒ (3) thenfollows from this together with the preceding lemma.

We make the following conjecture:

2.3.7. Conjecture. If G is a reductive p-adic group, then the three conditions ofLemma 2.3.6 are mutually equivalent.


2.3.8. Theorem. Conjecture 2.3.7 holds in the following two cases:

(1) G is a torus.(2) G = GL2(Qp).

Proof. Suppose first that G is a torus, so that G = Z. Let V be an object ofModsm

G (A). Taking into account Lemma 2.3.5, we see that each of conditions (2)and (3) of Lemma 2.3.6 implies that V is both finitely generated over A[Z] andZ-finite. These conditions, taken together, imply in their turn that V is finitelygenerated as an A-module. In particular, it is a finite length object of Modadm

G (A).Thus conditions (2) and (3) of Lemma 2.3.6 each imply condition (1).

Suppose now that G = GL2(Qp). The results of [1] and [2] imply that anyirreducible smooth G-representation that is Z-finite is admissible, and thus condi-tion (3) of Lemma 2.3.6 implies condition (1). It remains to show that the same istrue of condition (2) of Lemma 2.3.6. To this end, let V be an object of Modadm

G (A)that is finitely generated over A[G]. We must show that it is of finite length.

Write H = GL2(Zp)Z ⊂ G. Let S ⊂ V be a finite set that generates V overA[G], let W be the A[H]-submodule of V generated by S. Since V is smooth(by assumption), and Z-finite (by Lemma 2.3.5), we see that W is finitely gen-erated as an A-module, annihilated by some power of m. Since W contains thegenerating set S, the H-equivariant embedding W → V induces a G-equivariantsurjection c−IndG

H W → V. Thus it suffices to show that any admissible quotientV of c−IndG

H W is of finite length as an A[G]-module. Taking into account Propo-sition 2.2.10, we see in fact that it suffices to prove the analogous result with Wreplaced by one of its (finitely many) Jordan-Holder factors, which will be a fi-nite dimensional irreducible representation of H over k := A/m. Extending scalarsto a finite extension of k, if necessary, we may furthermore assume that W is anabsolutely irreducible representation of H over k, and this we do from now on.

Since V is a quotient of c−IndGH W , the evaluation map

(4) Homk[G](c−IndGH W,V )⊗k c−IndG

H W → V

is surjective. Lemma 2.3.9 below shows that Homk[G](c−IndGH W,V ) is a finite

dimensional k-vector space. It is also naturally a module over the endomorphismalgebra H := Endk[G](c−IndG

H W ). In [1] it is proved that H ∼= k[T ] (where Tdenotes a certain specified endomorphism whose precise definition need not concernus here). Thus Homk[G](c−IndG

H W,V ) is a finite dimensional k[T ]-module, andhence is a direct sum of k[T ]-modules of the form k[T ]/f(T )k[T ], for various non-zero polynomials f(T ) ∈ k[T ].

Taking into account the surjection (4), we see that it suffices to show that anyG-representation of the form (c−IndG

H W )/f(T )(c−IndGH W ) is of finite length. Fur-

ther extending scalars, if necessary, we may assume that f(T ) factors completelyin k[T ] as a product of linear factors, and hence are reduced to showing that forany scalar λ ∈ k, the quotient (c−IndG

H W )/(T − λ)(c−IndGH W ) is of finite length.

This, however, follows from the results of [1] and [2]. Thus the theorem is provedin the case of GL2(Qp).

2.3.9. Lemma. Let U and V be smooth representations of G over A. Suppose thatU is finitely generated over A[G], and that V is admissible. Then HomA[G](U, V )is a finitely generated A-module.

10 MATTHEW EMERTON

Proof. Let W ⊂ U be a finitely generated A-submodule that generates U over A[G],let H ⊂ G be a compact open subgroup of G that fixes W , and let i ≥ 0 be suchthat mi annihilates W . (Since U is a smooth representation of G over A, such Hand i exist.) Restricting homomorphisms to U induces an embedding

(5) HomA[G](U, V ) → HomA(W,V H [mi]).

Since V is an admissible smooth G-representation, its submodule V H is finitelygenerated over A, and so the target of (5) is finitely generated over A. The sameis thus true of the source, and the lemma follows.

2.4. $-adically admissible G-representations. In this subsection we introducea class of representations related to the considerations of [14].

2.4.1. Definition. We say that a G-representation V over A is $-adically contin-uous if:

(1) V is $-adically separated and complete.(2) The O-torsion subspace V [$∞] of V is of bounded exponent (i.e. is anni-

hilated by a sufficiently large power of $).(3) TheG-actionG×V → V is continuous, when V is given its$-adic topology.(4) The A-action A×V → V is continuous, when A is given its m-adic topology

and V is given its $-adic topology.

2.4.2. Remark. Conditions (3) and (4) of the definition can be expressed moresuccinctly as follows: for any i ≥ 0, the G-action on V/$iV is smooth, in the senseof 2.2.1.

We let Mod$−contG (A) denote the full subcategory of ModG(A) consisting of $-

adically continuous representations of G over A. Any object V of Mod$−contG (A)

sits in a canonical exact sequence

0→ V [$∞]→ V → Vfl → 0,

where Vfl is the maximal O-torsion free quotient of V . By assumption V [$∞] =V [$i] for i sufficiently large, and so for any j ≥ i, tensoring the above short exactsequence by O/$j over O induces a short exact sequence

0→ V [$∞] = V [$i]→ V/$jV → Vfl/$jVfl → 0,

while multiplication by $i induces an isomorphism Vfl∼−→ $iV, and hence embed-

dingsVfl/$

j−i → V/$jV.

Thus we see that both V [$∞] and Vfl are again objects of Mod$−contG (A), and the

study of of the category Mod$−contG (A) can largely be separated into the study of

O-torsion objects, and O-torsion free objects.

2.4.3. Proposition. Let 0 → V1 → V2 → V3 → 0 be a short exact sequence inModG(A).

(1) If V1 and V3 are objects of ModG(A), then so is V2.(2) Suppose furthermore that V2 is an object of Mod$−cont

G (A). Then the fol-lowing are equivalent:

(a) V1 is closed in the $-adic topology of V , and V3[$∞] has boundedexponent.

(b) V3 lies in Mod$−contG (A).


Furthermore, if these equivalent conditions hold, then V1 is also an object ofMod$−cont

G (A), and the $-adic topology on V2 induces the $-adic topologyon V1.

Proof. Suppose we are in the situation of (1). Since V2 is an extension of $-adically complete modules, it is again $-adically complete. If we choose i so largethat V1[$i] = V1[$∞] and V3[$i] = V3[$∞], then evidently V2[$2i] = V2[$∞],and so the O-torsion of V2 is of bounded exponent. Thus V2 lies in Mod$−cont

G (A).Suppose now that we are in the situation of (2). The surjection V3 → V2 induces

a surjection V2/$iV2 → V3/$

iV3 for each i ≥ 0. Thus V3/$iV3 is a smooth A[G]-

representation for each i ≥ 0, since this is true of V2/$iV2 by assumption. Since

V2 is assumed to be $-adically complete and separated, we see furthermore thatV1 is closed in V2 if and only if V3 (which is isomorphic to the quotient V2/V1) is$-adically complete and separated. From this we conclude the equivalence of (2)(a)and (2)(b).

It remains to show that, if these equivalent conditions hold, then V1 is an object ofMod$−cont

G (A), and the $-adic topology on V2 induces the $-adic topology on V1.Since V2[$∞] is of bounded exponent, by assumption, the same is true of V1[$∞].We have already shown in (1) that Mod$−cont

G (A) is closed under formation ofextensions, and so it suffices to show that (V1)fl is an object of Mod$−cont

G (A).Clearly V2/(V1[$∞]) lies in Mod$−cont

G (A) (since it is an extension of (V2)fl byV2[$∞]/V1[$∞]), and so replacing V1 by (V1)fl and V2 by V2/V1[$∞], we mayassume that V1 is O-torsion free.

Choose i ≥ 0 so that V3[$i] = V3[$∞]. For each j ≥ i, our original exactsequence yields an exact sequence

V3[$i] = V3[$j ]→ V1/$jV1 → V2/$

jV2.

Thus$i(V1

⋂$jV2) ⊂ $jV1,

and so$jV1 ⊂ V1

⋂$jV2 ⊂ $j−iV1,

since V1 is O-torsion free. Thus the $-adic topology on V2 induces the $-adictopology on V1. Since V1 is the kernel of the A[G]-linear (and hence O-linear) mapV2 → V3, and since V2 and V3 are $-adically complete and separated, we see thatV1 is complete and separated in the topology induced by the $-adic topology on V2,and so is in fact $-adically complete and separated. Thus V1 lies in Mod$−cont

G (A).

2.4.4. Corollary. The category Mod$−contG (A) is closed under the formation of

kernels, images, and extensions in the abelian category ModG(A).

Proof. The closure of Mod$−contG (A) under the formation of extensions is a restate-

ment of part (1) of the preceding proposition. Suppose now that 0→ V1 → V2 → V3

is an exact sequence of objects in ModG(A), with V2 and V3 lying in Mod$−contG (A).

Let V ′3 denote the image of V2 in V3. Since V3[$∞] has bounded exponent, by as-

sumption, so does V ′3 [$∞]. From part (2) of the preceding proposition, we conclude

that both V1 and V ′3 lies in Mod$−cont

G (A). This proves the closure of Mod$−contG (A)

under the formation of kernels and images.

12 MATTHEW EMERTON

2.4.5. Definition. We define an object V of Mod$−contG (A) to be admissible if the

induced G-representation on (V/$V )[m] (which is then a smooth G-representationover A/m, by Remark 2.4.2) is in fact an admissible smooth G-representation overA/m.

2.4.6. Remark. Since admissible smooth representations form a Serre subcategoryof ModG(A), we see that if V is a p-adically admissible G-representation over A,then in fact V/$iV is an admissible smooth representation for every i ≥ 0; cf.Remark 2.2.11.

We let Mod$−admG (A) denote the full subcategory of Mod$−cont

G (A) consistingof admissible representations.

2.4.7. Remark. When A is an object of Art(O), the categories Mod$−contG (A) and

ModsmG (A) coincide, as do the categories Mod$−adm

G (A) and ModadmG (A).

2.4.8. Lemma. If V is any object of Mod$−admG (A), then each of V [$∞] and Vfl

is also an object of Mod$−admG (A).

Proof. By assumption V [$∞] = V [$i] for i sufficiently large, and so for any j ≥ i,tensoring the above short exact sequence by O/$j over O induces a short exactsequence

0→ V [$∞] = V [$i]→ V/$jV → Vfl/$jVfl → 0,

while multiplication by $i induces an isomorphism Vfl∼−→ $iV, and hence embed-

dingsVfl/$

j−i → V/$jV.

Thus we see that both V [$∞] and Vfl are again objects of Mod$−admG (A).

2.4.9. Proposition. For any i ≥ 0, the functor V 7→ V ∗ induces an A-linear anti-equivalence of categories Mod$−cont

G (A)[$i] ∼−→ Modpro augG (A)[$i], which restricts

to an anti-equivalence of categories Mod$−admG (A)[$i] ∼−→ Modfg aug

G (A)[$i].

Proof. This follows by passing to$i-torsion in the anti-equivalences (2) and (3).

2.4.10. Corollary. The category Mod$−admG (A)[$∞] is a Serre subcategory of the

abelian category ModG(A). In particular, Mod$−admG (A)[$∞] is an abelian cate-

gory.

Proof. This follows directly from the proposition, together with the evident factthat Modfg aug

G (A)[$∞] is a Serre subcategory of Modpro augG (A).

2.4.11. Proposition. The functor V 7→ HomO(V,O) induces an A-linear anti-equivalence of categories Mod$−adm

G (A)fl ∼−→ Modfg augG (A)fl.

Proof. As is explained in the proof of [14, Thm. 1.2], the functor V 7→ HomO(V,O),the latter space being equipped with the weak topology (i.e. the topology of point-wise convergence), induces an equivalence of categories between the category of$-adically separated and complete flat O-modules and the category of compactlinear-topological flat O-modules. The reader may now easily check that this func-tor induces the required equivalence.

2.4.12. Proposition. The category Mod$−admG (A) is closed under the formation

of kernels, images, cokernels, and extensions in the abelian category ModG(A). Inparticular, Mod$−adm

G (A) is an abelian category.


Proof. Suppose first that 0 → V1 → V2 → V3 → 0 is a short exact sequence ofobjects in ModG(A), with V1 and V3 being objects of Mod$−adm

G (A). It followsfrom Corollary 2.4.4 that V2 is an object of Mod$−cont

G (A). For any i ≥ 0, wehave the exact sequence V1/$

iV2 → V2/$iV2 → V3/$

iV3 → 0. Since V1/$iV1 and

V3/$iV3 are assumed to be admissible smooth G-representations over A, the same

is true of V2/$iV3, by Proposition 2.2.10. Thus V2 is an object of Mod$−adm

G (A).Suppose now that 0 → V1 → V2 → V3 is an exact sequence of objects of

ModG(A), with V2 and V3 being objects of Mod$−admG (A). Let V ′

3 denote theimage of V2 in V3. It follows from Corollary 2.4.4 that V1 and V ′

3 are objects ofMod$−cont

G (A). Since V3[$∞] lies in Mod$−admG (A), so does V ′

3 [$∞], by Corol-lary 2.4.10. Choose i ≥ 0 so that V ′

3 [$i] = V ′3 [$∞]. For each j ≥ i, our given exact

sequence yields an exact sequence

V ′3 [$i] = V ′

3 [$j ]→ V1/$jV1 → V2/$

jV2 → V ′3/$

jV ′3 → 0.

Since the first and fourth terms of the exact sequence lie in ModadmG (A), we conclude

from Proposition 2.2.10 that the other two terms do also. We conclude that V1 andV ′

3 are both objects of Mod$−admG .

Suppose finally that V1 → V2 → V3 → 0 is an exact sequence of objects ofModG(A), with V1 and V2 being objects of Mod$−adm

G (A). We wish to show thatV3 is also an object of Mod$−adm

G (A). Applying HomO(– ,O) to our exact sequence,we then obtain the exact sequence

0→ HomO((V3)fl,O

)→ HomO

((V2)fl,O

)→ HomO

((V1)fl,O

)→ HomO(V3[$∞], E/O)→ HomO(V2[$∞], E/O)

Propositions 2.4.11 and 2.4.9 imply that, for any compact open subgroup H of G,each of HomO

((V1)fl,O

), HomO

((V2)fl,O

), and HomO(V2[$∞], E/O) is a finitely

generated A[[H]]-module. Since A[[H]] is Noetherian, the same is therefore trueof HomO

((V3)fl,O

)and HomO(V3[$∞], E/O). Thus (V3)fl and V3[$∞] are both

object of Mod$−admG (A) (by the same propositions), and thus V3 is also (being an

extension of the first by the second).

2.4.13. Proposition. If 0 → V1 → V2 → V3 → 0 is a short exact sequence inModG(A), and if V2 is an object of Mod$−adm

G (A), then the following are equivalent:(1) V1 is closed in the $-adic topology of V , and V3[$∞] has bounded exponent.(2) V3 lies in Mod$−cont

G (A).(3) V3 lies in Mod$−adm

G (A).

If these equivalent conditions hold, then V1 is also an object of Mod$−admG (A), and

the $-adic topology on V2 induces the $-adic topology on V1.

Proof. The equivalence of (1) and (2) follows from part (2) of Proposition 2.4.3, andclearly (2) implies (3). On the other hand, if (2) holds, then V3/$

iV3 is a quotient ofV2/$

iV2 for any i ≥ 0. The latter is an admissible A[G]-representation by assump-tion, and thus so is the former. Consequently V3 is an object of Mod$−adm

G (A),and so (2) implies (3). The remaining claims of the proposition follow from Propo-sition 2.4.3 (2) together with Proposition 2.4.12.

We have obvious equivalences of E-linearized categories Mod$−admG (A)flE

∼−→Mod$−adm

G (A)E and Modfg augG (A)flE

∼−→ Modfg augG (A)E , and so E-linearizing the

14 MATTHEW EMERTON

anti-equivalence of the preceding proposition yields an AE-linear anti-equivalenceof categories

(6) Mod$−admG (A)flE

∼−→ Modfg augG (A)flE .

We will reinterpret the source and target of (6) more concretely, after introducingsome further definitions and establishing some simple lemmas.

2.4.14. Definition. A unitary action of AE [G] on an E-Banach space V consists ofan AE [G]-module structure on V (extending its E-vector space structure) satisfyingthe following conditions:

(1) The G-action on V is unitary (i.e. V contains a bounded open G-invariantlattice) and continuous (i.e. the action map G× V → V is continuous).

(2) The induced A-action A × V → V is continuous when A is equipped withits m-adic topology.

2.4.15. Lemma. If V is an E-Banach space equipped with a unitary AE [G]-action,then the set of A[G]-invariant bounded open lattices in V is cofinal in the set of allbounded open lattices in V .

Proof. Condition (2) of Definition 2.4.14 together with the Banach-Steinhaus the-orem (see e.g. Prop. 6.15 together with Example (2) on p. 40 of [13]) implies thatthe action of A on V is equicontinuous. If we let V0 be any open bounded latticein V , then the A[G]-submodule V1 of V generated by V0 is again an open lattice,and is bounded, since the G-action is unitary and the A-action is equicontinuous.The A[G]-submodules $iV1 (for i ≥ 0) are then also A[G]-invariant bounded openlattices in V . As these form a neighbourhood basis of the zero element of V (sinceV is a Banach space) the lemma follows.

2.4.16. Lemma. Let V be an E-Banach space equipped with a unitary AE [G]-action.

(1) If V0 is any A[G]-invariant bounded open lattice of V , then the the inducedG-action on the A-module V0/$V0 is smooth.

(2) If furthermore the V0/$V0 is an admissible representation of A over G forone choice of V0, then it is so for any other choice.

Proof. If V0 is an A[G]-invariant bounded open lattice in V , then the actions of Gand A on V0/$V0 are continuous when the latter space is equipped with its discretetopology, which is to say, the A[G]-action on V0/$V0 is smooth. Suppose now thatV1 is a second such lattice. Scaling V1 if necessary, we may assume that V1 ⊂ V0.For some n > 0, we then also have the inclusion $nV0 ⊂ $V1, so that V1/$V1 isa subquotient of V0/$

nV0. If the A[G]-action on V0/$V0 is admissible, then, sinceV0/$

nV0 is a successive extension of finitely many copies of V0/$V0, it is again anobject of Modadm

G (A), by Proposition 2.2.10. It follows from the same propositionthat its subquotient V1/$V1 is an object of Modadm

G (A), as claimed.

2.4.17. Definition. Let V be an E-Banach space. An admissible unitary actionof AE [G] on V consists of a unitary AE [G]-action on V such that for some (orequivalently any) A[G]-invariant bounded bounded lattice V0 in V , the inducedG-action on (V0/$V0) (which is necessarily smooth) makes this representation anadmissible smooth G-representation over A. (The parenthetical claims follow fromLemma 2.4.16.)


2.4.18. Remark. It follows from Remark 2.2.11 that in the preceding definition,it is equivalent to require that V0/$V0)[m] be an admissible smooth representationof G over A/m.

2.4.19. Remark. Suppose that A = O. The notion of a unitary admissible actionof E[G] on an E-Banach space V , as given by the above definition, coincides withthe notion of a unitary representation of G on V which is admissible in the senseof [14].

Let ModuG(AE) denote the category of E-Banach spaces equipped with unitary

AE [G]-actions, with morphisms being continuous AE-linear G-equivariant maps.Let Modu adm

G (AE) denote the full subcategory of ModuG(AE) consisting of E-

Banach spaces equipped with unitary admissible AE [G]-actions.

2.4.20. Lemma. The functor V 7→ E ⊗O V induces an equivalence of categoriesMod$−adm

G (A)E∼−→ Modu adm

G (AE).

Proof. In the proof of [14, Thm. 1.2] it is observed that the functor V 7→ E ⊗O Vinduces an equivalence of categories between the E-linearization of the category of$-adically separated and complete flat O-modules (with morphisms being O-linearmaps) and the category of E-Banach spaces (with morphisms being continuous E-linear maps). The reader can easily check that this functor induces the equivalenceof the lemma.

2.4.21. Remark. The reader may easily check, in Definition 2.4.1, that condi-tion (2) is equivalent to the apparently weaker condition that the map G× V → Vbe left-continuous (i.e. that the map G → V defined by g 7→ gv is continuous foreach v ∈ V ) when V is equipped with its $-adic topology, and that condition (3)is equivalent to the apparently weaker condition that the map A× V → V be left-continuous when A is equipped with its m-adic topology and V is equipped withits $-adic topology. Similarly, using the Banach-Steinhaus theorem, the readermay check, in Definition 2.4.17, that condition (1) is equivalent to the apparentlyweaker condition that the G-action be separately continuous, and that condition (2)is equivalent to the apparently weaker condition that the A-action be separatelycontinuous (when A is equipped with is m-adic topology).

If V is an E-Banach space, we let V ′ denote the continuous dual space to V .

2.4.22. Proposition. The functor V 7→ V ′ induces an anti-equivalence of categoriesModu adm

G (AE) ∼−→ Modfg augG (AE).

Proof. This follows by E-linearizing the anti-equivalence of Proposition 2.4.11, andtaking into account the equivalences of Lemmas 2.4.20 and 2.1.9.

2.4.23. Corollary. The category Modu admG (AE) is closed under the formation of

closed subrepresentations, Hausdorff quotients, and strict extensions in the cate-gory Modu

G(AE), and is furthermore abelian. In particular, any morphism in thecategory has closed image.

Proof. Theorem 2.1.1 shows that A[[H]]E is Noetherian for any compact open sub-group of G, and thus that Modfg aug

G (AE) is closed under passing to subobjects, aswell as quotients and extensions, in the category of all augmented representationsof G over AE ; in particular, it is abelian. Proposition 2.4.22 then implies that the

16 MATTHEW EMERTON

analogous results hold for the category Modu admG (AE) (regarded as a full subcate-

gory of the exact category of all E-Banach spaces equipped with an AE [G]-actionsatisfying conditions (1) and (2) of Lemma 2.4.15).

2.4.24. Remark. In the case when A is finite over O, the preceding corollary isdue to Schneider and Teitelbaum [14].

3. Ordinary parts

In this section we take G to be a p-adic reductive group (by which we alwaysmean the group of Qp-points of a connected reductive algebraic group over Qp).We suppose that P is a parabolic subgroup of G, with unipotent radical N , thatM is a Levi factor of P (so that P = MN), and that P is an opposite parabolicto P , with unipotent radical N , chosen so that M = P

⋂P . (This condition

uniquely determines P .) Our goal in this section is to define the functor of ordinaryparts, which (for any object A of Comp(O)) is a functor OrdP : Modadm

G (A) →Modadm

M (A). In the following section we will show that it is right adjoint to parabolicinduction IndG

P: Modadm

M (A)→ ModadmG (A).

In fact, it is technically convenient to define OrdP in a more general context, asa functor OrdP : Modsm

P (A) → ModsmM (A), and this we do in Subsection 3.1. In

Subsection 3.2 we establish some elementary properties of OrdP . In Subsection 3.3,we prove that OrdP , when applied to admissible smooth G-representations, yieldsadmissible smooth M -representations. Finally, Subsection 3.4 extends the maindefinitions and results of the preceding subsections to the context of $-adicallycontinuous and $-adically admissible representations over A.

The functor OrdP is analogous to the locally analytic Jacquet functor JP studiedin [4] and [6], and many of the constructions and arguments of this section and thenext are suitable modifications of the constructions and arguments found in thosepapers. Typically, the arguments become simpler. (As a general rule, the theory ofOrdP is more elementary than the theory of the Jacquet functors JP , just as thetheory of p-adic modular forms is more elementary in the ordinary case than in themore general finite slope case.)

3.1. The definition of OrdP . We fix G, P , P , etc., as in the preceding discussion.We let A denote an object of Comp(O).

3.1.1. Definition. If V is a representation of N over A, and if N1 ⊂ N2 are compactopen subgroups of N , then let hN2,N1 : V N1 → V N2 denote the operator definedby hN2,N1(v) :=

∑n∈N2/N1

nv. (Here we are using n to denote both an element ofN2/N1, and a choice of coset representative for this element in N2. Since v ∈ V N1 ,the value nv is well-defined independently of the choice of coset representative.)

3.1.2. Lemma. If N1 ⊂ N2 ⊂ N3 are compact open subgroups of N , then

hN3,N2hN2,N1 = hN3,N1 .

Proof. This is immediate from the definition of the operators.

Fix a compact open subgroup P0 of P , and set M0 := M⋂P0, N0 := N

⋂P0,

and M+ := m ∈M |mN0m−1 ⊂ N0. Let ZM denote the centre of M , and write

Z+M := M+

⋂ZM . Note that each of M0 and N0 is a subgroup of G, while M+

and Z+M are submonoids of G.


3.1.3. Definition. If V is a representation of P over A, then for any m ∈M+, wedefine hN0,m : V N0 → V N0 via the formula hN0,m(v) := hN0,mN0m−1(mv).

3.1.4. Lemma. If m1,m2 ∈M+, then hN0,m1m2 = hN0,m1hN0,m2 .

Proof. We compute that

hN0,m1m2 = hN0,m1m2N0m−12 m−1

1m1m2

= hN0,m1N0m−1hm1N0m1,m1m2N0m−12 m−1

1m1m2

= hN0,m1N0m−1m1hN0,m2N0m−12m2 = hN0,m1hN0,m2 .

If m ∈M0, then (since in this case mN0m−1 = N0) the operator hN0,m coincides

with the given action of m on V N0 . In general, the preceding lemma shows thatthe operators hN0,m induce an action of the monoid M+ on V N0 , which we referto as the Hecke action of M+.

Regarding V N0 as a module over the monoid algebra A[M+], and so in particularits central subalgebra A[Z+

M ], via this action, we may consider the A[ZM ]-moduleHomA[Z+

M ](A[ZM ], V N0), as well as its submodule HomA[Z+M ](A[ZM ], V N0)ZM−fin.

3.1.5. Lemma. Let W be an A[Z+M ]-module. If φ ∈ HomA[Z+

M ](A[ZM ],W )ZM−fin,then the image im(φ) of φ is a finitely generated A-submodule of W , which isinvariant under the action of Z+

M .

Proof. Let ev1 : HomA[Z+M ](A[ZM ],W )ZM−fin → W denote the map given by

evaluation at 1 ∈ Z+M . For any φ ∈ HomA[Z+

M ](A[ZM ],W )ZM−fin, if U denotesthe A[ZM ]-submodule of HomA[Z+

M ](A[ZM ],W )ZM−fin generated by φ, then clearlyev1(U) coincides with the image im(φ) of φ. Since φ is locally ZM -finite, we seethat U is finitely generated over A, and hence the same is true of im(φ). Sincez1φ(z2) = φ(z1z2) for z1 ∈ Z+

M and z2 ∈ ZM , we see that im(φ) is invariant underthe action of Z+

M .

3.1.6. Lemma. Let W be an A[M+]-module.(1) The A[ZM ]-module structure on HomA[Z+

M ](A[ZM ],W ) extends naturallyto an A[M ]-module structure, characterized by the property that for anyz ∈ ZM , the map HomA[Z+

M ](A[ZM ],W ) → W induced by evaluation at zis M+-equivariant.

(2) If the M0-action on W is furthermore smooth, then the induced M -actionon HomA[Z+

M ](A[ZM ],W )ZM−fin is smooth.

Proof. From [4, Prop. 3.3.6], we see that the restriction map induces an isomorphism

(7) HomA[M+](A[M ],W ) ∼−→ HomA[Z+M ](A[ZM ],W ).

The source of this isomorphism has a natural A[M ]-module structure, which wemay then transport to the target. This A[M ]-module structure clearly satisfies thecondition of (1).

Suppose now that W is M0-smooth, and let φ ∈ HomA[Z+M ](A[ZM ],W )ZM−fin.

Since im(φ) is a finitely generated A-submodule of W , by Lemma 3.1.5, we may findan open ideal I in A[[M0]] which annihilates im(φ), and hence which annihilates φ

18 MATTHEW EMERTON

itself. (See Remark 2.2.2.) Thus the M -action on HomA[Z+M ](A[ZM ],W )ZM−fin is

smooth, proving (2).

We now give the main definition of this section.

3.1.7. Definition. If V is a smooth representation of P over A, then we write

OrdP (V ) := HomA[Z+M ](A[ZM ], V N0)ZM−fin,

and refer to OrdP (V ) as the P -ordinary part of V (or just the ordinary part of V ,if P is understood).

Lemma 3.1.6 shows that HomA[Z+M ](A[ZM ], V N0), and hence OrdP (V ), is nat-

urally an M -representation over A. Since the Hecke M0-action on V N0 coincideswith the given M0-action, it is smooth, and so the same lemma shows that theM -action on OrdP (V ) is smooth. Thus the formation of OrdP (V ) yields a functorfrom Modsm

P (A) to ModsmM (A).

3.1.8. Definition. We define the canonical lifting

(8) OrdP (V )→ V N0

to be the composite of the inclusion OrdP (V ) ⊂ HomA[Z+M ](A[ZM ], V N0) with the

map HomA[Z+M ](A[ZM ], V N0)→ V N0 given by evaluation at the element 1 ∈ A[ZM ].

Our choice of terminology is motivated by that of [3, §4].

Although the definition of the functor OrdP depends on the choice of compactopen subgroup P0 of P , we now show that it is independent of this choice, up tonatural isomorphism. To this end, suppose that P ′0 is an open subgroup of P0. WriteM ′

0 := M⋂P ′0, N

′0 := N

⋂P ′0, (M+)′ := m ∈ M |mN ′

0n−1 ⊂ N0, (Z+

M )′ :=(M+)′

⋂ZM .

We define the Hecke action of (M+)′ on V N ′0 in analogy to the Hecke action ofM+ on V N0 .

3.1.9. Proposition. Composition with the operator hN0,N ′0: V N ′0 → V N0 induces

an isomorphism HomA[(Z+M )′](A[ZM ], V N ′0) ∼−→ HomA[Z+

M ](A[ZM ], V N0).

Proof. We write Y ′ := M+⋂

(M+)′, and Y := Z+M

⋂(Z+

M )′ = Y ′⋂ZM . (The

notation is chosen to be compatible with that of [4, Prop. 3.3.2].) Since Y gener-ates ZM as a group [4, Prop. 3.3.2 (i)], the inclusions HomA[(Z+

M )′](A[ZM ], V N ′0) ⊂HomA[Y ](A[ZM ], V N ′0) and HomA[Z+

M ](A[ZM ], V N0) ⊂ HomA[Y ](A[ZM ], V N0) arein fact equalities. Thus composition with hN0,N ′0

does induce a map

φ : HomA[(Z+M )′](A[ZM ], V N ′0)→ HomA[Z+

M ](A[ZM ], V N0).

If m ∈ Y ′, then we compute:

(9) hN0,mhN0,N ′0= hN0,mN0m−1mhN0,N ′0

= hN0,mN0m−1hmN0m−1,mN ′0m−1m

= hN0,mN ′0m−1m = hN0,N ′0hN ′0,mN ′0m−1m = hN0,N ′0

hN ′0,m.

Thus hN0,N ′0: V N0 → V N ′0 intertwines the Hecke action of Y ′ on each of its source

and target. Since Y ′ and ZM together generate M [4, Prop. 3.3.2 (ii)], we see thatthe map φ is M -equivariant.


If z ∈ Y is such that zN0z−1 ⊂ N ′

0, then we define a map

(10) V N0 → V N ′0

via the formula v 7→ hN ′0,zN0z−1(zv). A computation similar to that of (9) (takinginto account the fact that z is central in M) shows that (10) also intertwines theHecke action of Y ′ on its source and target. Thus (10) induces an M -equivariantmap

ψ : HomA[Z+M ](A[ZM ], V N0)→ HomA[(Z+

M )′](A[ZM ], V N ′0).

One furthermore computes that

hN0,N ′0hN ′0,zN0z−1z = hN0,z, while hN ′0,zN0z−1zhN0,N ′0

= hN ′0,z.

Thus the composite φψ (resp. the composite φψ) coincides with the automorphismof HomA[Z+

M ](A[ZM ], V N0) (resp. HomA[(Z+M )′](A[ZM ], V N ′0)) induced by the action

of z, and in particular both φψ and ψφ are isomorphisms. This implies that φ isan isomorphism, as required.

The preceding result shows that OrdP is well-defined, up to canonical isomor-phism, independently of the choice of the open subgroup P0 of P . It is similarlyindependent of the choice of the Levi factor M of P . Indeed, if M ′ is another Levifactor of P , then we may canonically identify M and M ′ via the canonical isomor-phisms M ∼←− P/N

∼−→M ′. Furthermore, there is a uniquely determined elementn ∈ N such that nMn−1 = M ′, and the isomorphism induced by conjugation byn yields the same identification of M and M ′. Via this canonical identification, wemay and do regard any M ′ representation equally well as an M -representation.

Given a choice of compact open subgroup P0 ⊂ P , write P ′0 := nP0n−1, M ′

0 :=nM0n

−1 = nMn−1⋂P ′0, N

′0 := nN0n

−1 = N⋂P ′0, (M ′)+ := nM+n−1 = m′ ∈

M ′ |m′N ′0(m

′)−1 ⊂ N ′0, and Z+

M ′ := nZ+Mn−1 = (M ′)+

⋂ZM ′ . We define the

Hecke action of (M ′)+ on V N ′0 in analogy to the Hecke action of M+ on V N0 , anddenote by

Ord′P : ModG(A)→ ModM (A)

the functorV 7→ HomA[Z+

M′ ](A[ZM ′ ], V N ′0)ZM−fin.

(The analogue of Lemma 3.1.6 for P ′0 and M ′ shows that HomA[Z+M′ ]

(A[ZM ′ ], V N ′0),

and so also HomA[Z+M′ ]

(A[ZM ′ ], V N ′0)ZM−fin, is naturally an M ′-representation, and

hence an M -representation.) Thus Ord′P is the analogue of the functor OrdP ,but computed using the Levi factor M ′ of P rather than M. (Note that we havealready seen that this functor is well-defined independently of the particular choiceof compact open subgroup P ′0 used to compute it.)

Multiplication by n provides an isomorphism V N0∼−→ V N ′0 , which intertwines

the operator hm on V N0 with the operator hnmn−1 on V N ′0 , for each m ∈ M+.Consequently, multiplication by n induces an M -equivariant isomorphism

HomA[Z+M ](A[ZM ], V N0) ∼−→ HomA[Z+

M′ ](A[ZM ′ ], V N ′0),

and hence an M -equivariant isomorphism OrdP (V ) ∼−→ Ord′P (V ). This shows thatthe functor OrdP is well-defined, up to canonical isomorphism, independently ofthe choice of Levi factor of P .

20 MATTHEW EMERTON

3.2. Some elementary properties of OrdP . In this subsection, we record somesimple properties of the functor OrdP (V ). We maintain the notation of the pre-ceding subsection (A, P , M , N , P0, M0, N0, M+, ZM , Z+

M , etc.).

3.2.1. Lemma. If Wii∈I is an inductive system of smooth Z+M -modules2, then

the natural map of A[M ]-modules

(11) lim−→

i

HomA[Z+M ](A[ZM ],Wi)ZM−fin → HomA[Z+

M ](A[ZM ], lim−→

i

Wi)ZM−fin

is an isomorphism.

Proof. Let φ ∈ HomA[Z+M ](A[ZM ],Wj)ZM−fin for some index j in the direct set I,

and suppose that the image of φ under (11) vanishes, or equivalently, that the im-age of im(φ) under the map Wj → lim

−→i

Wi vanishes. Since Lemma 3.1.5 shows that

im(φ) is a finitely generated A-submodule ofWi, we may find an index k lying over jsuch that the image of im(φ) under the transition mapWj →Wk vanishes, or equiv-alently, such that the image of φ under the map HomA[Z+

M ](A[ZM ],Wj)ZM−fin →HomA[Z+

M ](A[ZM ],Wk)ZM−fin vanishes. This proves the injectivity of (11).Now let φ′ ∈ HomA[Z+

M ](A[ZM ], lim−→

i

Wi)ZM−fin. Lemma 3.1.5 shows that im(φ′)

is a finitely generated and Z+M -invariant A-submodule of lim

−→i

Wi. Thus we may

find a finitely generated A-module U ⊂ Wj that maps isomorphically onto im(φ′),for some sufficiently large index j. Let Z0 ⊂ Z+

M be a compact open subgroupthat fixes U elementwise, so that the Z+

M -action on U factors through Z+M/Z0.

Since A[Z+M/Z0] is a Noetherian ring, if we choose k over j large enough, then the

the image of the finitely generated A[Z+M/Z0]-module A[Z+

M/Z0]U in Wk will mapisomorphically onto its image in lim

−→i

Wi, which is to say, onto im(φ′). Consequently,

if we choose j large enough, then we may choose U to be an A[Z+M ]-submodule of

Wj that maps isomorphically onto im(φ′). We may then lift φ′ to an element φ ∈HomA[Z+

M ](A[ZM ], U)ZM−fin ⊂ HomA[Z+M ](A[ZM ],Wi)ZM−fin. This shows that (11)

is surjective.

3.2.2. Proposition. OrdP : ModP (A)→ ModM (A) is left-exact and additive, andcommutes with inductive limits.

Proof. The functors V 7→ V N0 and W 7→ HomA[Z+M ](A[ZM ],W )ZM−fin (on the

categories of P -representations and M+-modules respectively) are left exact andadditive, and (taking into account Lemma 3.2.1) commute with inductive limits.Thus OrdP also has these properties.

3.3. Preservation of admissibility. If V is an object of ModsmG (A), then we

may also regard it an an object of ModsmP (A), and so define OrdP (V ), an object of

ModsmM (A). The main result of this subsection is Theorem 3.3.3 below, which shows

that if V is an admissible smooth G-representation, then OrdP (V ) is an admissiblesmooth M -representation.

We begin by choosing a cofinal sequence Gii≥0 of compact open subgroups ofG, with each Gi normal in G0, and such that each Gi admits an Iwahori decompo-sition with respect to P and P . Recall that this latter condition means that, if we

2I.e., smooth under the action of a compact open subgroup of Z+M .


write Mi := Gi

⋂M, Ni := Gi

⋂N, and N i := Gi

⋂N , then the product map

N i ×Mi ×Ni → Gi

is a bijection.For each i ≥ j ≥ 0, we write Hi,j := N iMjN0. Each Hi,j is a compact open

subgroup of G0 (since we may rewrite it as Hi,j = GiMjN0, noting that MjN0 isa subgroup of M0N0, and thus of G0, and so normalizes Gi). The bijection

N i ×Mj ×N0∼−→ Hi,j

provides an Iwahori decomposition of Hi,j .

3.3.1. Lemma. If z0 ∈ Z+M is such that N j ⊂ z0N iz

−10 , for some i ≥ j ≥ 0, then

hN0,z0(VHi,j ) ⊂ V Hj,j .

Proof. Since z0 centralizes M , we see that

z0Hi,jz−10 = z0N iz

−10 Mjz0N0z

−10 ⊃ N jMjz0N0z

−10 = z0Hi,jz

−10

⋂Hj,j ,

and hence that z0V Hi,j ⊂ V z0Hi,jz−10

THj,j . Thus to prove the lemma, it suffices to

show that hN0,z0N0z−10

(V z0Hi,jz−10

THj,j ) ⊂ V Hj,j . Since M0, and so Mj , normalizes

N0 and z0N0z−10 , we see that hN0,z0N0z−1

0(V z0Hi,jz−1

0T

Hj,j ) is invariant under Mj

and N0. Since Hj,j = N jMjN0, it suffices to show that it is furthermore invariantunder N j .

Let n ∈ N j , and v ∈ V z0Hi,jz−10

THj,j . Fix a set nl of coset representatives for

N0/z0N0z−10 . Then

nhN0,z0N0z−10v = n

∑l

nlv =∑

l

nnlv =∑

l

n′lplv =∑

l

n′lv,

where we have used the Iwahori decomposition of Hj,j to rewrite each product nnl

in the form n′lpl, with n′l ∈ N0 and pl ∈ N jMj . We claim that the n′l again form aset of coset representatives for N0/z0N0z

−10 . Given this, we find that

nhN0,z0N0z−10v = hN0,z0N0z−1

0v,

and thus that V z0Hi,jz−10

THj,j is fixed by N j , as required.

Suppose now that (n′l)−1n′l′ ∈ z0N0z

−10 for some l 6= l′. Then

n−1l nl′ = p−1

l (n′l)−1n′l′pl′ ,

and the right hand side of this equation is a product of elements all lying inHi,jz

−10

⋂Hj,j . Since the left hand side lies in N , we see from the Iwahori decom-

position of Hi,jz−10

⋂Hj,j that in fact n−1

l nl′ ∈ z0N0z−10 , contradicting the fact

that they are distinct coset representatives for N0/z0N0z−10 . Thus n′l is indeed a

set of distinct coset representatives for N0/z0N0z−10 , and the lemma is proved.

3.3.2. Lemma. If U is an A[Z+M ]-module that is finitely generated over A, then

evaluation at the element 1 ∈ ZM induces an embedding HomA[Z+M ](A[ZM ], U) → U.

In particular, HomA[Z+M ](A[ZM ], U) is finitely generated over A.

Proof. If B denotes the image of A[Z+M ] in EndA(U), then B is a commutative finite

A-algebra, and so is isomorphic to the product of its localizations at its finitelymany maximal ideals, i.e. B ∼−→

∏m maximalBm. This factorization of B induces a

22 MATTHEW EMERTON

corresponding factorization of the B-module U , namely U =∏

m maximal Um, and aconsequent factorization

HomA[Z+M ](A[ZM ], U) ∼−→

∏m maximal

HomA[Z+M ](A[ZM ], Umi

).

We will say that a maximal ideal m is ordinary (resp. non-ordinary) if and onlyif no (resp. some) element z ∈ Z+

M has its image in B lying in m. If m is ordinary,then the A[Z+

M ]-module structure on Um extends in a unique manner to an A[ZM ]-module structure, and evaluation at 1 ∈ ZM induces an isomorphism

HomA[Z+M ](A[ZM ], Um) ∼−→ Um.

On the other hand, if m is non-ordinary, then it is easily seen that

HomA[Z+M ](A[ZM ], Um) = 0.

Thus evaluation at 1 induces an embedding

HomA[Z+M ](A[ZM ], U) ∼−→

∏m ordinary

Um ⊂ U,

as claimed.

3.3.3. Theorem. If V is an admissible smooth representation of G over A, thenOrdP (V ) is an admissible smooth representation of T , and the canonical lifting (8)is an embedding.

Proof. Since V is smooth by assumption, it is the direct limit of its submoduleV [mi] (i ≥ 0). Taking into account Proposition 3.2.2, we thus see that it suffices toprove the theorem with V replaced by V [mi] for some i ≥ 0. It is then evident thatOrdP (V ) := OrdP (V )[mi], and it remains to show that, for each j ≥ 0, OrdP (V )Mj

is finitely generated over A, and that the canonical lifting induces an embeddingOrdP (V )Mj → V N0 . We will compute OrdP (V ) using the compact open subgroupP0 := M0N0 of P . Observe that V MjN0 is invariant under the Hecke Z+

M -action onV N0 (since Z+

M is central in M), and thus

OrdP (V )Mj = HomA[Z+M ](A[ZM ], V N0)Mj

ZM−fin = HomA[Z+M ](A[ZM ], V MjN0)ZM−fin.

Let φ ∈ HomA[Z+M ](A[ZM ], V MjN0)ZM−fin. Lemma 3.1.5 shows that im(φ) is a

finitely generated Z+M -invariant A-module, and thus is contained in V Hi,j for some

sufficiently large i ≥ j. Choose z0 ∈ Z+M as in the statement of Lemma 3.3.1. If

z ∈ ZM , then φ(z) = φ(z0z−10 z) = hN0,z0φ(z−1

0 z) ∈ hN0,z0(VHi,j ) ⊂ V Hj,j (the

final inclusion holding by Lemma 3.3.1). Thus if we let U denote the maximalA-submodule of V Hj,j that is invariant under the Hecke Z+

M -action, then we seethat in fact im(φ) ⊂ U . Consequently, OrdP (V )Mj ⊂ HomA[Z+

M ](A[ZM ], U), andso is finitely generated over A, by Lemma 3.3.2. The same lemma shows that thecanonical lifting induces an embedding OrdP (V )Mj → U ⊂ V N0 .

3.4. Extension to the case of $-adically complete representations.

3.4.1. Definition. If V is an object of Mod$−contP (A), then we define

OrdP (V ) := lim←−

i

OrdP

((V/$iV )

).

(Note that each of the quotients V/$iV is an object of ModsmP (A/mj), by Re-

mark 2.4.6, and hence its P -ordinary part is defined.)


The space OrdP (V ) has a natural projective limit topology, if we endow eachterm OrdP (V/$iV ) appearing in the projective limit with its discrete topology.It is also $-adically complete (since each term OrdP (V/$iV ) appearing in theprojective limit is$-adically complete, and a projective limit of$-adically completeO-modules is itself $-adically complete). We will strengthen these observations,after proving a general lemma.

3.4.2. Lemma. Let F be any O-linear, O-module valued, left exact functor onModsm

G (A)[$∞], and extend F to a functor on Mod$−contG (A) via the formula

F (V ) := lim←−

i

F (V/$iV ), for any object V of Mod$−contG (A). Then for any such

V , and for sufficiently large j, the natural map F (V )/$jF (V ) → F (V/$j) is anembedding. In particular, F (V ) is $-adically complete, and F (V )[$∞] has boundedexponent.

Proof. Suppose first that V is O-torsion free. Then for any i, j ≥ 0, the short exactsequence

0 −→ V/$iV$j

−→ V/$i+jV −→ V/$jV −→ 0

gives rise to an exact sequence

0 −→ F (V/$iV ) $j

−→ F (V/$i+jV ) −→ F (V/$jV ).

Passing to the projective limit in i, we obtain an exact sequence

0 −→ F (V ) $j

−→ F (V ) −→ F (V/$jV ),

which in turn induces an embedding

F (V )/$jF (V ) → F (V/$j).

Since j ≥ 0 was arbitrary, this proves the assertion of the lemma for O-torsionfree V .

Let us turn to the general case. Fix i ≥ 0 so that V [$i] = V [$∞]. For any j ≥ i,we have the short exact sequence

0→ V [$i]→ V/$jV → Vfl/$jVfl → 0,

giving rise to the exact sequence

(12) 0→ F (V [$i])→ F (V/$jV )→ F (Vfl/$jVfl).

Passing to the direct limit over j, we obtain an exact sequence 0 → F (V [$i]) →F (V ) → F (Vfl). The result of the first paragraph shows that F (Vfl) is O-torsionfree, and so we see that F (V )[$∞] has bounded exponent. For j ≥ i, this exactsequence induces the exact sequence

0→ F (V [$i])→ F (V )/$jF (V )→ F (Vfl)/$jF (Vfl),

which in turn forms the top row of the following diagram, whose bottom row isgiven by (12):

0 // F (V [$i]) // F (V )/$jF (V ) //

F (Vfl)/$jF (Vfl)

0 // F (V [$i]) // F (V/$jV ) // F (Vfl/$

jVfl).

24 MATTHEW EMERTON

Since the outer two vertical arrows are embeddings, so is the middle vertical arrow.This completes the proof of the lemma in general.

3.4.3. Proposition. Taking ordinary parts is a functor OrdP : Mod$−contP (A) →

Mod$−contM (A). Furthermore, for any object V of Mod$−cont

G (A), and for any suf-ficiently large j, the natural map OrdP (V )/$jOrdP (V ) → OrdP (V/$jV ) is anembedding. In particular, the projective limit topology on OrdP (V ) coincides withthe $-adic topology on OrdP (V ),

Proof. This follows directly from the preceding lemma, since OrdP (V ) is left exact.

For each i ≥ 0, we have the canonical lifting OrdP (V/$iV ) → (V/$iV )N0 .These are compatible in an evident way as we change i, and so passing to theprojective limit, we obtain an M+-equivariant map

(13) OrdP (V )→ V N0 ,

which we again refer to as the canonical lifting.

3.4.4. Theorem. The passage to ordinary parts gives rise to a functor OrdP :Mod$−adm

G (A) → ModadmM (A). Furthermore, for any object V of Mod$−adm

G (A),the canonical lifting (13) is a closed embedding, when source and target are giventhere $-adic topologies.

Proof. Proposition 3.4.3 shows that OrdP (V ) is a $-adically continuous A[G]-module, and that for sufficiently large j, the quotient OrdP (V )/$jOrdP (V ) em-beds as a subrepresentation of OrdP (V/$jV ). The latter A[M ]-module lies inModadm

M (A), by Theorem 3.3.3. Thus so does the former, by Proposition 2.2.10.Consequently, OrdP (V ) is an object of Mod$−adm

G (A), as claimed. Finally, The-orem 3.3.3 shows that the canonical lifting OrdP (V/$jV ) → (V/$jV )N0 is anembedding, for any j ≥ 0. Consider the commutative diagram

OrdP (V )/$jOrdP (V ) //

V N0/$jV N0

OrdP (V/$jV ) // (V/$jV )N0 ,

in which the horizontal arrows arise from canonical lifting, and the vertical ar-rows are the natural ones. We have already noted that the left hand arrow is anembedding, for sufficiently large j, and that bottom horizontal arrow is also anembedding. Since V 7→ V N0 is left exact, and commutes with projective limits, wesee by Lemma 3.4.2 that the right hand vertical arrow is also an embedding forsufficiently large j. Thus the upper horizontal arrow is an injection for sufficientlylarge j, and so the canonical lift induces a closed embedding OrdP (V ) → V N0 asclaimed.

4. Adjunction formulas

4.1. Parabolic induction. Let A denote an object of Comp(O), with maximalideal m. If U is an object of Modsm

M (A), then we regard U as a P -representation,


via the projection P → P/N∼−→M , and define:

IndGPU := f : G→ U | f locally constant ,

f(pg) = pf(g) for all p ∈ P , g ∈ G .

The right regular action of G on functions induces a natural G-action on IndGPU ,

making it an A[G]-module.If U is an object of Mod$−cont

M (A), then again regarding U as a P -representation,we define:

IndGPU := f : G→ U | f continuous when U is given its $-adic topology ,

f(pg) = pf(g) for all p ∈ P , g ∈ G .

Again, the right regular action of G on functions induces a natural G-action onIndG

PU , making it an A[G]-module.

In order to establish some basic properties of the functor IndGP

, it will be con-venient to choose a continuous section σ : P\G → G to the projection G → P\G.(Such a section exists, since G is a locally trivial P -bundle over P\G, and hence wemay find a finite cover of P\G by disjoint open and closed sets over which this bun-dle is trivialized.) If we write Csm(P\G,U) (resp. C(P\G,U)) to denote the space oflocally constant (resp. continuous) U -valued functions on P\G, for an object U ofModsm

G (A) (resp. an object U of Mod$−contG (A), equipped with its$-adic topology),

then pulling back functions along σ induces natural A-linear isomorphisms

(14) IndGPU

∼−→ Csm(P\G,U), for any object U of ModsmM (A),

and

(15) IndGPU

∼−→ C(P\G,U), for any object U of Mod$−contM (A).

4.1.1. Lemma. If U is an object of Mod$−contM (A), then for each i ≥ 0, there is a

natural isomorphism

(IndGPU)/$i(IndG

PU) ∼−→ IndG

P(U/$iU).

Proof. Taking into account the isomorphism (15), it suffices to show that

C(X,U)/$i C(X,U) ∼−→ C(X,U/$iU)

for any profinite space X. For any j ≥ 0, we have the pair of short exact sequences

0 −→ ($j−iU + U [$i])/$jU −→ U/$jU$i

−→ $iU/$jU −→ 0

and0→ $iU/$jU → U/$jU → U/$iU → 0.

Note that the projective systems ($j−iU +U [$i])/$jUj≥i and $iU/$jUj≥i

both satisfy the Mittag-Leffler condition (in fact, in the latter the transition mapsare even surjective). Upon applying the (manifestly exact) functor Csm(X, –), weobtain the short exact sequences

0 −→ Csm(X, ($j−iU + U [$i])/$jU

)−→ Csm(X,U/$jU)

$i

−→ Csm(X,$iU/$jU) −→ 0

26 MATTHEW EMERTON

and

0→ Csm(X,$iU/$jU)→ Csm(X,U/$jU)→ Csm(X,U/$iU)→ 0.

Furthermore, the two projective systems C(X, ($j−iU + U [$i])/$jU

)j≥i and

C(X,$iU/$jU)j≥i again satisfy the Mittag-Leffler condition. Thus, passing tothe projective limit in j, we obtain short exact sequences

0 −→ C(X,U [$i]) −→ C(X,U) $i

−→ C(X,$iU) −→ 0

and0→ C(X,$iU)→ C(X,U)→ C(X,U/$iU)→ 0.

Putting these together yields the exact sequence

0 −→ C(X,U [$i]) −→ C(X,U) $i

−→ C(X,U) −→ C(X,U/$iU) −→ 0,

proving the lemma.

4.1.2. Lemma. For objects U of ModsmM (A), the formation of IndG

PU , commutes

with the formation of inductive limits.

Proof. Taking into account the isomorphism (14), it suffices to show that the forma-tion of Csm(X,U) commutes with the formation of inductive limits for any profinitespace X. However, this is clear, as any element of Csm(X,U) assumes only finitelymany different values in U (since X is compact).

4.1.3. Proposition. Parabolic induction U 7→ IndGPU gives rise to exact functors

ModsmM (A)→ Modsm

G (A) and Mod$−contM (A)→ Mod$−cont

G (A).

Proof. We first consider the case when U is smooth. As already noted in the proofof Lemma 4.1.1, the formation of locally constant functions is manifestly an exactfunctor. It follows from the isomorphism (14) that the formation of IndG

PU is exact

in U . We must show that if furthermore yields an object in ModsmG (A). Although

the section σ that we introduced above cannot be chosen to be G-equivariant, it canbe chosen to be H-equivariant, for some sufficiently small compact open subgroupH of G. The isomorphism (14) is then also H-equivariant. Since X is compact,it is a union of finitely many H-orbits. Thus it suffices to note that if U is anobject of Modsm

M (A), then Csm(H,U) is an object of ModsmG (A), since any element

of Csm(H,U) is uniformly locally constant (since H is compact) and assumes onlyfinitely many values in U .

We now consider the case of $-adically continuous representations. If 0→ U1 →U2 → U3 → 0 is a short exact sequence in Mod$−cont

G (A), then for any i ≥ 0 weobtain the short exact sequence 0 → U1/(U1

⋂$iU2) → U2/$

iU2 → U3/$iU3 →

0, which in turn yields a short exact sequence

0→ Csm(P\G,U1/(U1

⋂$iU2)

)→ Csm

(P\G,U2/$

iU2

)→ Csm

(P\G,U3/$

iU3

)→ 0.

Since the projective system U1/(U1

⋂$iU2)i≥0 has surjective transition maps,

so does the projective system C(P\G,U1/(U1

⋂$iU2)

)i≥0. Thus, passing to the

projective limit in i, we obtain a short exact sequence

0→ C(P\G,U1)→ C(P\G,U2)→ C(P\G,U3)→ 0.


(Here we have used Proposition 2.4.3 (2), which shows that the topology on U1

induced by the decreasing sequence of O-submodules U1

⋂$iU2 coincides with the

$-adic topology on U1.) Taking into account the isomorphism (15), we see thatIndG

PU induces an exact functor on Mod$−cont

M (A). It remains to show that it takesvalues in Mod$−cont

G (A).For any object U of Mod$−cont

M (A), clearly (IndGPU)[$∞] ∼−→ IndG

P(U [$∞]).

Thus (IndGPU)[$∞] is of bounded exponent, since this is true of U [$∞], by as-

sumption. The space IndGPU is also $-adically complete and separated, since the

isomorphism (15) identifies it with a space of continuous functions with valuesin the $-adically complete and separated space U . Finally, the isomorphism ofLemma 4.1.1, together with the result proved above in the case of smooth represen-tations, shows that the G-action on (IndG

PU)/$i(IndG

PU) is smooth. Thus IndG

PU

is indeed an object of Mod$−contG (A).

4.1.4. Proposition. If U is an admissible smooth (resp. locally admissible smooth,resp. $-adically admissible ) A[M ]-module, then IndG

PU is an admissible smooth

(resp. locally admissible smooth, resp. $-adically admissible ) A[G]-module.

Proof. In light of Proposition 4.1.3 and Lemmas 4.1.1 and 4.1.2, it suffices to con-sider the case when U is an object of Modadm

G (A). Let H be a compact opensubgroup of G, and let i ≥ 0. Write P\G as a finite union of H-orbits, sayP\G =

∐lj=1 PgjH. We then see that

(IndGPU)H [mi] ∼−→

l⊕j=1

f ∈ Csm(H,U [mi]) | f(g−1j pgjh) = pf(h)

for all p ∈ P⋂gjHg

−1j .

In particular, if we write M0 := M ∩⋂l

j=1 gjHg−1j , then we see that

(IndGPU)H [mi] → (UM [mi])l.

Since M0 is an open subgroup of M , and since U is assumed admissible, we seethat the target of this embedding is finitely generated over A, and thus so is thesource. This proves the proposition.

Suppose now that U is an object of ModsmM (G), so that elements of IndG

PU are

locally constant. If f is such an element, then its support, which is an open andclosed subset of G, is invariant under left translation by P , and so corresponds toa compact open subset of the quotient P\G. We will thus frequently speak of thesupport as f as a subset of P\G, rather than of G.

Composing the closed embedding N → G with the quotient map G→ P\G, weobtain an open immersion N → P\G, via which we regard N as an open subset ofP\G. Let (IndG

PU)(N) denote the A-submodule of IndG

PU consisting of elements

whose support (thought of as a compact open subset of P\G) is contained in N .Restricting such a function to N , we obtain a locally constant compactly supportedfunction N → U . Thus, if we denote Csmc (N,U) the A-module of all such functions,then we have a map of A-modules

(16) (IndGPU)(N)→ Csmc (N,U).

28 MATTHEW EMERTON

4.1.5. Lemma. The map (16) is an isomorphism.

Proof. Clearly this map is injective. On the other hand, given a function f ∈Csmc (N,U), we may extend it to a function on PN via the formula f(pn) = pf(n),and then extend it by zero to a function on G. Since f is compactly supported,this extension is locally constant, and yields an element of (IndG

PU)(N) that maps

to f under (16). Thus (16) is also surjective.

The translation action of P on P\G leaves its open subset N invariant. Con-cretely, we see that if mn ∈MN = P, and n′ ∈ N , then

n′ ·mn = m−1n′mn

(where the left side denotes the translation action of mn on n′, thought of as anelement of P\G). Thus (IndG

PU)(N) is P -invariant. We may transport this P -

action via the isomorphism of Lemma 4.1.5 to a P -action on Csmc (N,U). Of course,this P -action also admits a direct description, as follows:

(mnf)(n′) = mf(m−1n′mn), for m ∈M, n, n′ ∈ N, and f ∈ Csmc (N,A).

There is a natural isomorphism of A[P ]-modules

Csmc (N,A)⊗A U∼−→ Csmc (N,U),

if we equip the left hand side with the tensor product P -action, where P acts onCsmc (N,A) via the above description (taking the M -action on the coefficients A tobe trivial), and on U through its quotient M .

4.2. A simple adjunction formula. Fix a compact open subgroup P0 of P , anddefine M0, N0,M

+, etc., as in Subsection 3.1. We regard Csmc (N,A) as an A[P ]-module via the formula of the preceding subsection (taking the M -action on A tobe trivial). For any compact open subset Ω ⊂ N , we denote by 1Ω ∈ Csmc (N,A)the characteristic function of Ω (i.e. 1Ω is identically equal to 1 on Ω, and vanisheselsewhere). If V is an object of Modsm

P (A), then (since M normalizes N), the spaceHomA[N ](Csmc (N,A), V ) has a natural M -action.

4.2.1. Lemma. The map ev1N0: HomA[N ](Csmc (N,A), V )→ V N0 , induced by eval-

uation at the function 1N0 ∈ Csmc (N,A), is M+-equivariant, if we equip the targetwith its Hecke M+-action.

Proof. This is a simple calculation.

Since the source of ev1N0is an A[M ]-module, it induces an M -equivariant map

(17) HomA[N ](Csmc (N,A), V )→ HomA[M+](A[M ], V N0)( ∼−→ HomA[Z+M ](A[ZM ], V N0)

),

where the isomorphism, which was observed in the proof of Lemma 3.1.6 (1), followsfrom [4, Prop. 3.3.6].

4.2.2. Proposition. The map (17) is an isomorphism.

Proof. Given φ ∈ HomA[N ](Csmc (N,A), V ), let φ ∈ HomA[M+](A[M ], V N0) denotethe image of φ under (17). If m ∈M+, then

(18) φ(m) = (mφ)(1) = (mφ)(1N0) = φ(m1N0) = φ(1mN0m−1)


(where 1mN0m−1 denotes the characteristic function ofmN0m−1). Since any element

of Csmc (N,A) may be written in the form∑

i ni1mN0m−1 for some finite collectionni ⊂ N and some m ∈M+, we see that φ is completely determined by the func-tion φ, and thus (17) is injective. Conversely, given any φ ∈ HomA[M+](A[M ], V N0),we can define φ ∈ HomA[N ](Csmc (N,A), V ) via the formula (18), i.e.

φ(f) :=∑

i

niφ(m),

for any function f =∑

i niφ(m) ∈ Csmc (N,A); the M+-equivariance of φ ensuresthat φ(f) is well-defined, independently of the choice of representation of f as asum of this form. Thus (17) is also surjective.

4.2.3. Remark. The isomorphism (17) yields another proof of Proposition 3.1.9.

4.2.4. Proposition. If U and V are objects of ModsmM (A) and Modsm

P (A), respec-tively, then there is a natural isomorphism

HomA[P ]

(Csmc (N,U), V

) ∼−→ HomA[M ]

(U,HomA[Z+

M ](A[ZM ], V N0)).

Proof. We compute

HomA[P ]

(Csmc (N,U), V

) ∼−→ HomA[P ]

(Csmc (N,A)⊗A U, V

)∼−→ HomA[P ]

(U,HomA(Csmc (N,A), V )

)∼−→ HomA[M ]

(U,HomA[N ](Csmc (N,A), V )

)∼−→ HomA[M ]

(U,HomA[Z+

M ](A[ZM ], V N0)),

where the third isomorphism arises from the fact that the P -action on M factorsthrough M = P/N , and the final isomorphism is provided by Proposition 4.2.2.

4.2.5. Proposition. If U is a locally ZM -finite smooth representation of M over Athen there is a natural M -equivariant isomorphism U

∼−→ OrdP (Csmc (N,U)), char-acterized by the fact that its composite with the canonical lifting OrdP (Csmc (N,U))→Csmc (N,U)N0 is given by mapping an element u ∈ U to the constant function u sup-ported on N0.

Proof. If we write U =⋃

i∈I Ui as the union of finitely generated, ZM -invariant,A-modules, then we may write

Csmc (N,U)N0 ∼−→ lim−→

i∈I,z∈Z+M

Csm(z−1N0z, Ui)N0) ∼−→ lim−→

i∈I,z∈Z+M

A[z−1N0z/N0]⊗A Ui,

where, in forming the inductive limits, we direct Z+M via the relation of divisibility.

Lemma 3.2.1 yields an isomorphism

lim−→i,z

HomA[Z+M ](A[ZM ], A[z−1N0z/N0]⊗A Ui)ZM−fin

∼−→ Ordp(Csmc (N,U)).

It is easily checked that evaluation at 1 induces an isomorphism

HomA[Z+M ](A[ZM ], A[z−1N0z/N0]⊗A Ui)ZM−fin

∼−→ Ui,

where Ui is embedded into A[z−1N0z/N0]⊗A Ui via u 7→ 1⊗ u. Thus we find that

U∼−→ lim

−→i

Ui∼−→ Ordp(Csmc (N,U))

30 MATTHEW EMERTON

in the manner claimed. We leave it to the reader to verify that this isomorphismis M+-equivariant.

4.2.6. Corollary. If U is a locally ZM -finite smooth representation of M over A,and V is a smooth representation of P over A, then passing to ordinary partsinduces a natural isomorphism

HomA[P ]

(Csmc (N,U), V

) ∼−→ HomA[M ]

(U,OrdP (V )

).

Proof. This follows from the preceding propositions, together with the fact that,since U is locally ZM -finite by assumption, the inclusion

HomA[M ]

(U,OrdP (V )

)⊂ HomA[M ]

(U,HomA[Z+

M ](A[ZM ], V N0))

is in fact an equality.

4.3. The main adjunction formula. The main result of this subsection is The-orem 4.3.2, showing that IndG

Pand OrdP are adjoint functors.

Let U be a smooth, admissible representation of M over A. and let V be asmooth representation of G over A. We regard Csmc (N,U) as a P -invariant subspaceof IndG

PU , via the isomorphism of Lemma 4.1.5. In particular, if f ∈ Csmc (N,U)

and g ∈ G, then gf is a well-defined element of IndGPU .

4.3.1. Lemma. Let φ ∈ HomA[P ]

(Csmc (N,U), V

). If f ∈ Csmc (N,U) and g ∈ G are

chosen so that gf also lies in Csmc (N,U), then φ(gf) = gφ(f).

Proof. Let Ω denote the support of f in N . As in Subsection 3.3, choose a cofi-nal sequence Gii≥0 of compact open subgroup of G, each admitting an Iwahoridecomposition N i ×Mi × Ni

∼−→ Gi. Since Ω is compact, we may choose i andi′ sufficiently large so that Ω is a union of left cosets of Ni, i.e. so that Nin ⊂ Ωfor every n ∈ Ω, and such that f is Ni-invariant, and takes values in UMi [mi′ ]. Ifn ∈ Ω, we then see that (nf) |Ni

is a constant UMi [mi′ ]-valued function on Ni. IfX denotes the subspace of Csmc (N,U) consisting of all such functions (i.e. constantUMi [mi′ ]-valued functions on Ni), then X is elementwise-fixed by Gi (as one seesfrom the Iwahori decomposition of Gi), and is finite dimensional (since U is as-sumed to be admissible smooth). Thus φ(X) is a finite dimensional subspace of V ,and we may choose j ≥ i so that Hj fixes φ(X) elementwise.

The support of gf is equal to the translate Ωg−1 of Ω (computed in P\G). Byassumption, this again lies in N . Let us momentarily regard N as a closed subsetof G (rather than as an open subset of P\G), and compute the translate Ωg−1

inside G. Since its projection to P\G lies in N , we see that Ωg−1 is a compactsubset of PN . Let Ω denote the projection of Ωg−1 onto N (i.e. the projectiononto the first factor in the source of the isomorphism N×M×N ∼−→ PN). Choosean element z ∈ ZM so that zΩz−1 ⊂ N j , while z−1Niz ⊂ Ni.

Fix an n ∈ Ω. Since Ωg−1 ⊂ ΩP, we may write gn−1 = pn for some n ∈ Ω−1

and p ∈ P . By virtue of our choice of z, we have znz−1 ∈ N−1

j = N j . We nowcompute:

(19) g(f |z−1Nizn) = gn−1z−1(znf) |Ni= pnz−1(znf) |Ni

= pz−1znz−1(znf) |Ni= pz−1(znf) |Ni

.


All but the final equality are trivial manipulations. The final equality holds because(znf) |Ni

lies in X, and so if fixed by the element znz−1 ∈ N j ⊂ N i. Indeed,since (nf)Ni

is a constant UMi-valued function on Ni, we see that (znf) |zNiz−1 =z((nf) |Ni

)is a constant UMi-valued function on zNiz

−1 ⊃ Ni. Thus (znf) |Niis

again a constant UMi-valued on Ni, and so does lie in X.Applying φ to (19), we compute that

(20) φ(g(f |z−1Niz)

)= φ

(pz−1(znf) |Ni

)= pz−1φ

((znf) |Ni

)= pz−1(znz−1)φ

((znf) |Ni

)= gn−1z−1φ

((znf) |Ni

)= gφ

(n−1z−1(znf) |Ni

)= gφ(f |z−1Nizn).

Here the first equality comes from (19), the second and fifth from the P -equivarianceof φ, the third from the fact that φ

((znf) |Ni

)∈ φ(X) (by the discussion of the

preceding paragraph), and so is fixed by the element znz−1 ∈ N j . The fourth andthe final equality are both trivial manipulations.

Since z−1Niz ⊂ Ni, we may write Ω as a finite disjoint union of left z−1Nizcosets, say Ω =

∐l z−1Niznl. Accordingly, we may write f =

∑l f |z−1Niznl

, andthen compute (taking into account (20) that

φ(gf) =∑

l

φ(gf |z−1Niznl) =

∑l

gφ(f |z−1Niznl) = gφ(f).

This proves the lemma.

We now prove our main result.

4.3.2. Theorem. Let A be an object of Comp(O). If U is an object of ModadmM (A)

(resp. Mod$−admM (A)) and V is an object of Modadm

G (A) (resp. Mod$−contG (A) )

then passage to ordinary parts induces an isomorphism

HomA[G]

(IndG

PU, V

) ∼−→ HomA[M ]

(U,OrdP (V )

).

Consequently, the functors on ModsmG (A) and Mod$−cont

G (A) are right adjoint tothe functor U 7→ IndG

PU .

Proof. We first consider the case when U and V are smooth. Restricting maps fromIndG

PU to Csmc (N,U) induces a map

(21) HomA[G]

(IndG

PU, V

)→ HomA[P ]

(Csmc (N,U), V

)∼−→ HomA[G]

(A[G]⊗A[P ] Csmc (N,U), V

).

Since the natural map

A[G]⊗A[P ] Csmc (N,U)→ IndGPU

is surjective (clearly Csmc (N,U) generates IndGPU over G, since the G-translates

of N cover P\G), we see that (21) is injective. We claim that it is in fact anisomorphism. To prove this, we have to show that any map of A[G]-modules

φ : A[G]⊗A[P ] Csmc (N,U)→ V

necessarily factors through the quotient IndGPU of the source. In other words, we

have to show that if g1, . . . , gl is a finite sequence of elements of G, and f1, . . . , fl

32 MATTHEW EMERTON

is a finite sequence of elements of Csmc (N,U), such that

(22)l∑

i=1

gifi = 0 in IndGPU,

then

(23)l∑

i=1

giφ(fi)?= 0 in V.

It suffices to show that each point x of P\G has a compact open neighbourhoodΩx such that for any neighbourhood Ω′x ⊂ Ωx of x we have

(24)l∑

i=1

giφ(fi|Ω′xgi) ?= 0.

Indeed, we can then partition P\G into a finite disjoint union of such neighbour-hoods, say P\G =

∐sj=1 Ω′xj

, and writing

gifi =s∑

j=1

(gifi)|Ω′xj=

s∑j=1

gifi|Ω′xjgi,

so that

fi =s∑

j=1

fi|Ω′xjgi,

we conclude that if (24) holds, then so does (23).If x = Pg for some g ∈ G, then replacing (g1, . . . , gl) by (gg1, . . . , ggl), we see

that it suffices to treat the case when x equals the identity coset, which we identifywith the identity e of N under the open immersion N → P\G. Let Ωe be a compactopen neighbourhood of e in N , chosen so that Ωegi ⊂ N (as open subsets of P\G)for all i for which gi ∈ PN , and so that Ωegi (regarded as an open subset of P\G)is disjoint from the support of fi for all other i. It follows from (22) that

l∑i=1

(gifi)|Ωe=

l∑i=1

gifi|Ωegi= 0 in Csmc (N,U).

Applying the map φ to the second of these equalities (which is an equation involvingelements of Csmc (N,U), since f|Ωegi

= 0 if gi 6∈ N , by virtue of our choice of Ωe),and taking into account Lemma 4.3.1, we deduce that indeed

∑li=1 giφ(fi|Ωegi

) = 0,and thus we have shown that (21) is an isomorphism. Now composing it with theisomorphism of Proposition 4.2.4 completes the proof of the theorem in the casewhen A is Artinian.

We now consider the case when U and V are $-adically continuous. There is acommutative diagram

HomA[G]

(IndG

PU, V

) //

lim←−n

Hom(A/$n)[G]

(IndG

P(U/$nU), V/$nV

)

HomA[M ]

(U,OrdP (V )

)// lim←−n

HomA[M ]

(U/$nU,OrdP (V/$nV )

),


in which the vertical arrows are given by passing to ordinary parts, and the horizon-tal arrows arise from the isomorphism of Lemma 4.1.1 together with the definitionOrdP (V ) := lim

←−n

OrdP (V/$n). By what we have already proved, the right hand

vertical arrow is also an isomorphism. Thus so is the left hand vertical arrow.

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Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston,IL 60208

E-mail address: [email protected]

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