Second adjointness for representations of reductive p-adic groups

Joseph Bernstein

Tel Aviv University

§0. Introduction

0.1. In this paper, which was written in 1987, I continue the investigation of induced representations of reductive p-adic groups, started in [BZ]. The main tools of the investigation are induction functors i_{GM} and Jacquet functors r_{MG}. More precisely, let G be a reductive p-adic group and Alg G the category of algebraic (in other terminology, smooth) representations of G. For any parabolic subgroup P < G with Levi component M we define the induction functor i_{GM} : Alg M → Alg G and Jacquet functor r_{MG} : Alg G → Alg M as in [BZ].

Frobenius reciprocity implies that functor r_{MG} is left adjoint to i_{GM}. Recently, I have discovered to my great surprise, that these functors are also adjoint in the opposite direction. More precisely, let \overline{P} be the parabolic subgroup opposite to P with Levi component M. Then we can define functors \overline{i}_{G,M} : Alg M → Alg G and \overline{r}_{MG} : Alg G → Alg M in the same way as i and r, but using \overline{P} instead of P.

Main theorem. Functor i_{GM} is left adjoint to \overline{r}_{MG}, and \overline{i}_{GM} is left adjoint to r_{MG}.

This innocent-looking statement is in fact very powerful. For instance, it implicitly contains the strong admissibility theorem (indeed, it implies that functors r_{MG} commute with direct product and hence products of quasicuspidal representations are quasicuspidal. But this means that for a given open subgroup K ⊂ G there exists a uniform bound on supports of all K-invariant matrix coefficients of all cuspidal representations of G, i.e. all these supports lie in some subset S ⊂ G, compact modulo center).

The aim of this paper is to prove the main theorem and to show how it implies many important results about induced representation: description of the center of category Alg G, matrix Paley-Wiener theorem, cohomological duality in Alg G.

More precise versions of the theorem are formulated in §. They allow to prove Zelevinsky’s conjecture, that duality, which he defined on the Grothendieck group of representations of GL(n), actually carries irreducible representations into irreducible ones (see [Z]). I should add, that this way of proving Zelevinsky’s conjecture was suggested to me by V. Drinfeld many years ago. He explained to me that for the group G = SL(2), Ext^{1} (trivial representation) = Steinberg representation.

0.2. Contragredient properties of functor r_{HG}.

Another, essentially equivalent, form of the main theorem describes how to compute contragredient representations of r_{MG}(π). For induction functor we have the Frobenius reciprocity (i_{GM} (ρ))^{}˜ = i_{GM} (ρ̃), where ˜ denotes the contragredient representation (see [B2]).

Theorem. There is a functorial isomorphism

(r_{MG}(π))^{}˜ ≈ \overline{r}_{MG}(π̃), π ∈ Alg G .

0.3. Matrix Paley-Wiener theorem.

Let ρ be an irreducible cuspidal representation of M. Consider the family of induced representations π_{χ} = i_{GM} (χ · ρ), parametrized by unramified characters χ of M, with underlying family of vector spaces E_{χ}.

Let H = H(G) be the algebra of compactly supported locally constant measures on G. Any element h ∈ H(G) induces the family of operators h_{χ} = π_{χ}(h) : E_{χ} → E_{χ}.

This family has the following properties:

(PW1) h_{χ} is a regular function of parameter χ (unramified characters of M form a group isomorphic to (ℂ^{∗})^{ℓ} and function h_{χ} is algebraic on (ℂ^{∗})^{ℓ}).

(PW2) There exists an open subgroup K ⊂ G such that operators h_{χ} are left and right invariant with respect to π(K).

(PW3) For any intertwaining operator A : E_{χ} → E_{χ′} one has h_{χ′} ∘ A = A ∘ h_{χ}.

Theorem. Let a_{χ} : E_{χ} → E_{χ} be a family of operators, satisfying (PW1)- (PW3). Then a_{χ} = h_{χ} for some h ∈ H(G).

Remark. It is clear, that it is sufficient to check property (PW3) only on Zariski dense subsets of parameters χ and χ′.

This theorem follows easily from the following corollary of the main theorem: functor i_{GM} carries projective generators into projective generators.

0.4. Cohomological duality theorem.

Let us denote by γ_{L} and γ_{R} left and right actions of G on H(G). For any π ∈ Alg G, we can consider spaces Ext^{i}(π) = Ext_{A(G)}^{i}(π, (γ_{L}, H(G))) as G-modules, using right action γ_{R}.

Theorem. If π is irreducible then for exactly one index i Ext^{i}(π) ≠ 0. Moreover, representation Ext^{i}(π) is irreducible and π ↦ Ext^{i}(π) defines a duality on the set of equivalence classes of irreducible algebraic representations of G.

§1. Generalities from Algebra and Category Theory

1.1. Idempotented Algebras and Nondegenerate Modules.

We consider a class of rings slightly more general than rings with identity.

Definition. An associative ring H is called an idempotented ring if for each finite subset {x_{i}} ∈ H there exists an idempotent e ∈ H such that ex_{i} = x_{i} = x_{i}e for all i.

Example. Each ring with identity is an idempotented ring. More generally, let H_{α}, α ∈ I, be a direct system of rings and H = lim_{\overrightarrow{α∈I}} H_{α}. Suppose that the

ordered set I is filtered (i.e. for each α, β ∈ I there exists γ ∈ I such that α < γ, β < γ) and all H_{α} are rings with identities (but ring homomorphisms H_{α} ⟶ H_{β} for α < β are not supposed to map identities into identities). Then H is an idempotented ring.

In fact, any idempotented ring can be presented in such a way. Namely, consider the set I = IdemH of idempotents in H with partial order e ≤ f if eHe ⊂ fHf. Then H = lim_{\overrightarrow{e∈I}} eHe, where eHe is the ring with identity e.

Usually we consider H to be an algebra over some field k and call H an idempotented algebra.

A (left) module M over an idempotented ring H is called nondegenerate if HM = M or equivalently lim_{\overrightarrow{e∈IdemH}} eM = M. If H is a ring with identity, this is

just the usual condition that 1 acts on M as identity.

The category of nondegenerate H-modules we denote M(H). Each Hmodule M contains the maximal nondegenerate submodule HM, which we call the nondegenerate part of M. It is easy to see that M(H) is an abelian category with direct limits and filtered direct limits in M(H) are exact. Category M(H) also has arbitrary direct products (and, hence, inverse limits). Namely, for a family {M_{α} ∈ M(H)} the product ∏_{α} M_{α} in M(H) is equal to the nondegenerate part of the set theoretic direct product,

∏_{α} M_{α} = H (∏_{α}^{set} M_{α}) = lim_{\overrightarrow{e∈IdemH}} (∏_{α}(eM_{α})) .

1.2. Projective and Injective H-Modules. For each idempotent e ∈ H the functor M → eM is exact on M(H). Since eM = Hom_{H}(He, M), it shows, that He is a finitely generated projective object in M(H). The family of modules He for e ∈ IdemH form a system of projective generators for category M(H). In particular, M(H) has enough projective objects, i.e. each module M ∈ M(H) is a quotient of a projective one.

Similarly, one can see that M(H) has enough injective objects. Namely for each e ∈ IdemH and each injective ℤ-module U denote by I(e, U) the nondegenerate part of H-module Hom_{ℤ}(eH, U). Then the functor M → Hom_{H} (M, I(e, U)) = Hom_{ℤ}(eM, U) is exact on M(H), i.e. I(e, U) is an injective object, and {I(e, U)} form a system of injective cogenerators in M(H).

We will denote by M^{R}(H) the category of nondegenerate right H-modules, which we identify with category M(H^{∘}), where H^{∘} is the opposite algebra. We define in a usual way the tensor product M′ ⨂_{H} M of nondegenerate right and

left H-modules. It is easy to see that all the usual properties of ⨂_{H} hold in this case; we will use them freely. Note, that formula M′ ⨂_{H}(He) = M′e shows, that

He is a flat H-module, which implies that all projective H-modules are flat.

Let H be an idempotented algebra over a field k. For each H-module M ∈ M(H) we define the contragredient module M̃ ∈ M^{R}(H) as a nondegenerate part of the dual space M^{∗} = Hom_{k}(M, k), i.e. M̃ = lim_{\overrightarrow{e∈IdemH}}(M^{e})^{∗}.

Similarly we define the functor ∼: M^{R}(H) → M(H). It is easy to check that ∼ is an exact contravariant functor, with duality property Hom_{H}(M̃, N) = Hom_{H}(Ñ, M), M ∈ M(H), N ∈ M^{R}(H). In particular, ∼ maps projective objects into injective ones.

1.3. Hecke Algebras.

Let G be an ℓ-group, i.e. a Hausdorf topological group, which has a basis of neighbourhoods of e ∈ G, consisting of open compact subgroups (see [BZ1]). Let H = H(G) be the Hecke algebra of locally constant distributions (or complex valued measures) on G with compact support. Then H is an idempotented algebra (over ℂ) and category M(H(G)) is naturally identified with category M(G) of G-modules (see..., [BZ1]).

Let K ⊂ G be an open compact subgroup, e_{K} ⊂ H(G) be the normalized Haar measure on K. Then e_{K}H(G)e_{K} is the subalgebra H_{K}(G) of K-bi-invariant measures. The system of idempotents {e_{K}} is cofinal in Idem H(G),

i.e. H(G) = lim_{\overrightarrow{K}} H_{K}(G).

The involution ι : g ↦ g^{−1} on G defines the natural antiautomorphism ι : H(G) ⟶ H(G). Using this antiautomorphism we will usually identify M(H) with M^{R}(H), though sometimes it is move convenient to separate them.

1.4. Jordan-Hölder Content of a Module.

We want to describe some general properties of the category M(H). It is convenient to do it in a more general setting.

Let M be an abelian category with (arbitrary) direct sums (and, hence, direct limits). We will assume that M satisfies some axioms.

(A1) Filtered direct limits in M are exact.

In [Gr] this axiom is called AB... It is equivalent (see [ ]) to

(A1′) Let M_{α} ⊂ M be a filtered system of submodules, N ⊂ M. Then

N ∩ (∑_{α} M_{α}) = ∑_{α} (N ∩ M_{α}) .

An object M ∈ M is called finitely generated if for any filtered system of proper subobjects M_{α} ⊂ M the subobject ∑_{α} M_{α} ⊂ M is proper. For a finitely generated object M the functor Hom(M, ∗) : M → Ab preserves direct sums.

An object M ∈ M is called noetherian, if every of its subobjects is finitely generated or, equivalently, if each ascending chain of subobjects M_{1} ⊂ M_{2} ⊂ ... of M is stable.

Category M is called locally noetherian if each finitely generated object of M is noetherian.

(A2) Every object M ∈ M is a union of finitely generated subobjects.

In order to avoid set-theoretical troubles we also add

(A3) Isomorphism classes of finitely generated objects in M form a set.

We denote by IrrM the set of isomorphism classes of irreducible (i.e. simple) objects in M. For every E ∈ M we denote by JH(E) ⊂ IrrM the subset of irreducible subquotients of E.

For each idempotented ring H the category M = M(H) satisfies axioms A1 − A3. We will denote Irr(M(H)) by IrrH and IrrM(G) by IrrG (see 1.3).

Lemma. (i) Let E′ ⊂ E. Then JH(E) = JH(E′) ∪ JH(E/E′).

(ii) JH(E) = ∅ iff E = 0

(iii) If E_{α} ⊂ E, then JH (∑_{α} E_{α}) = ⋃_{α} JH(E_{α}).

Proof:

(i) is clear.

(ii) Let E ≠ 0. By A2 E has a nonzero finitely generated submodule E′. By Zorn’s lemma E′ has an irreducible quotient, i.e. JH(E) ≠ ∅.

(iii) Let I = {α} be the indexing set of E_{α}. If I is finite, the statement follows from (i) by induction. Hence, replacing system {E_{α}} by a system, consisting of finite sums of E_{α} we can assume, that {E_{α}} is a filtered direct system. Let Q = E′/E′′ be a simple subquotient of ∑_{α} E_{α}, i.e. E′′ ⫋ E′ ⊂ ∑_{α} E_{α}. Suppose that for all α Q ∉ JH(E_{α}). Then for every α E′∩(E′′ + E_{α}) = E′′. By A1′ E′∩∑_{α}(E′′ +E_{α}) = ∑ E′∩(E′′ +E_{α}) =

E′′, which contradicts the inclusion E′ ⊂ ∑_{α} E_{α}, since E′′ ≠ E′.

1.5. Decomposition of Categories. Suppose that the category M is split into a product of two subcategories M = M′ × M′′. This splitting induces a disjoint union decomposition IrrM = IrrM′ ∪ IrrM′′. We want to show that this decomposition completely describes the splitting.

For each subset S ⊂ IrrM denote by M(S) the full subcategory of M defined by M(S) = {E ∈ M | JH(E) ⊂ S}. Lemma 1.4 shows that M(S) is an abelian subcategory, closed with respect to subquotients, extensions and direct limits. For every E ∈ M we denote by E_{S} the union of all submodules E′ ⊂ E, which lie in M(S). Then E_{S} also lies in M(S). Let S′ ⊂ IrrM be another subset, which does not intersect S. Then for each E ∈ M(S) ∩ M(S′) we have JH(E) = ∅, i.e. E = 0. This implies that the categories M(S), M(S′) are orthogonal, i.e. Hom_{M}(E, E′) = 0 for E ∈ M(S), E′ ∈ M(S′). Also for every E ∈ M E_{S} ∩ E_{S′} = 0, i.e. E ⊃ E_{S} ⊕ E_{S′}.

Definition. We say that a subset S ∈ IrrM splits an object E ∈ M if E = E_{S} ⊕ E\overline{S}, where \overline{S} = IrrM\S. We say that S splits M if it splits all objects in M.

More generally, suppose we have a disjoint union decomposition IrrM = ⋃_{α∈A} S_{α}. We say that this decomposition {S_{α}} splits E if E = ⨁_{α∈A} E_{S_{α}}. We

say that the decomposition {S_{α}} splits M if it splits all objects in M. In this case M is equivalent to the category ∏_{α∈A} M(S_{α}).

Lemma. Let IrrM = ⋃_{α∈A} S_{α} be a disjoint union decomposition. Suppose it splits an object E ∈ M. Then it splits all subquotients of E.

Proof: Let E = ⨁_{α∈A} E_{α}, E_{α} ∈ M(S_{α}). It is sufficient to check that for every subobject L ⊂ EL = ∑_{α} (L ∩ E_{α}). Put C = L/ ∑_{α} (L ∩ E_{α}). Then for α

JH(C) ⊂ JH (L/L ∩ E_{α}) ⊂ JH (E/E_{α}) ⊂ ⋃_{β≠α} JH (E_{β}) ⊂ \overline{S_{α}} .

This implies, that JH(C) ⊂ ⋂_{α} (\overline{S_{α}}) = ∅, i.e. C = 0.

Remark. Let H be an idempotented ring. Suppose category M = M(H) has a decomposition M = ∏_{α} M_{α}. Applying this decomposition to the H-module

H we see that H = ⨁_{α} H_{α}. Since right multiplications in H are morphisms in

M(H) all H_{α} are two-sided ideals. It is easy to see that M_{α} = M(H_{α}). Conversely, each decomposition H = ⨁_{α} H_{α} of H into a direct sum of two-

sided ideals leads to the decomposition M(H) = ∏_{α} M(H_{α}).

1.6. Realization of an Abelian Category as a Category of Modules. Let M be an abelian category, satisfying A1 − A3. Let P ∈ M be a finitely generated projective object, Λ = End_{M}(P)^{∘} (^{∘} denotes the opposite algebra).

We define the functor r = r_{P} : M → M(Λ) by r(E) = Hom_{M}(P, E). It is exact and commutes with direct sums. Functor r has a left adjoint functor i = i_{P} : M(Λ) → M. Indeed, every Λ-module M can be presented as a cokernel of a morphism ν_{M} of free Λ-modules ν_{M} : ⨁_{α} Λ ⟶ ⨁_{β} Λ, where ν_{M}

is given by a matrix {ν_{αβ} ∈ Λ}. We define i(M) as a cokernel of a morphism ν′ : ⨁_{α} P ⟶ ⨁_{β} P, where ν′ is given by the same matrix {ν_{αβ} ∈ Λ}. In case

when M = M(H) the functor i can be described as i(M) = P ⨂_{Λ} M.

Lemma. Suppose that P is a generator of the category M, i.e. the functor r is faithful, or, equivalently, Hom_{M}(P, Q) ≠ 0 for Q ∈ IrrM. Then functor r and i are inverse and define an equivalence of categories

M i ; ⟵ ; ⟶ ; r M(Λ).

Proof: See [ ...]

This lemma allows us to realize M as a category of modules over some algebra with identity. This realization is not unique, it depends on the choice of P. Let us describe the relation between two such realizations.

Let A be an algebra with identity, P ∈ M(A) a finitely generated projective generator, Λ = (End_{A}P)^{∘}. Then P is an A − Λ-bimodule. We define a dual Λ − A-bimodule P^{∗} by P^{∗} = Hom_{A}(P, A).

Proposition. P^{∗} is a finitely generated projective generator in M(Λ), End_{Λ}(P^{∗}) = A^{∘} and the functors i : M(Λ) → M(A), r : M(A) → M(Λ) are canonically isomorphic to r(E) = P^{∗} ⨂_{A} E, E ∈ M(A) and i(M) = Hom_{Λ}(P^{∗}, M), M ∈

M(Λ).

Proof:

Step 1. For any E ∈ M(A) the natural morphism P^{∗} ⨂_{A} E → Hom_{A}(P, E) =

r(E) is an isomorphism.

Indeed, this is true for P = A, hence for P = A^{n} and hence for P which is a direct summand of A^{n}.

Step 2. Since P is a generator of M(A), A is a direct summand of P^{n} for some natural n. Hence r(A) = P^{∗} is a direct summand of r(P)^{n} = Λ^{n}, i.e. P^{∗} is a finitely generated projective Λ-module.

Step 3. Since functors r and i are mutually inverse, we have

Hom_{A}(E, i(M)) = Hom_{Λ}(r(E), M) =

= Hom_{Λ} (P^{∗} ⨂_{A} E, M) = Hom_{A}(E, Hom_{Λ} (P^{∗}, M))

which implies that i(M) is canonically isomorphic to Hom_{Λ} (P^{∗}, M). Since the functor i is faithful, P^{∗} is a generator of M(Λ).

Step 4. We have r(P) = Hom_{A}(P, P) = Λ ∈ M(Λ), r(A) = P^{∗} ⨂_{A} A = P^{∗}

and hence i(Λ) = Hom_{Λ}(P^{∗}, Λ) = P, i(P^{∗}) = Hom_{Λ}(P^{∗}, P^{∗}) = A ∈ M(A). This implies that as an algebra End_{Λ}(P^{∗}) = A^{∘}.

Corollary. P is a right projective Λ-module and End_{Λ}(P) = A.

Indeed, since P = Hom_{Λ}(P^{∗}, Λ), it is a right projective Λ-module, dual to P^{∗}. Hence End_{Λ}(P) = End_{Λ}(P^{∗})^{∘} = A.

1.7. Realization of a Subcategory as a Category of Modules.

Let P ∈ M be a finitely generated projective object, which we do not suppose to be a generator. Consider subset S = S_{P} ⊂ IrrM of irreducible quotients of P. We say that P splits the category M if the subset S splits M, i.e. M = M(S) × M(\overline{S}). (see...).

Corollary. Suppose P splits M. Then functors r, i give equivalence of categories M(S) i ; ⟵ ; ⟶ ; r M(Λ). Moreover, M(\overline{S}) = \{E ∈ M(S) | Hom(P, E) = 0\} M(S) = {E ∈ M(S) | E is a quotient of ⨁_{α} P }.

This easily follows from 1.6.

Example. Let H be an idempotented ring M = M(H). Choose an idempotent e ∈ H and put P = He. Then P is a finitely generated projective object in M, Λ = (End_{M}P)^{∘} coincides with the subalgebra eHe ⊂ H and functors r : M(H) → M(Λ), i : M(Λ) → M(H) are given by r(E) = eE, i(M) = P ⨂_{Λ} M.

We say that idempotent e splits M if the subset S = S_{e} = {ω ∈ IrrM | eω ≠ 0} splits M. In this case functors r and i give equivalence of categories M(S) i ; ⟵ ; ⟶ ; r M(Λ) and M(S) = \{E ∈ M(H) | E is generated by eE\}, M(\overline{S}) = \{E ∈ M(H) | eE = 0}.

1.8. The Central Algebra of M.

Let M be an abelian category.

Definition. The central algebra Z(M) is defined as Z(M) = End(Id_{M}), where Id_{M} : M → M is the identity functor. In other words, an element z ∈ Z(M) is a collection of morphisms z_{M} : M → M for all M ∈ ObM, such that for each morphism

α : M → N z_{N} ∘ α = α ∘ z_{M} .

If M = M(H) or M(G) we will also use notations Z(H) or Z(G) instead of Z(M(H)) or Z(M(G)).

Lemma. Let H be an idempotented ring. Then the morphism z ↦ z_{H} identifies Z(H) with the algebra End_{H×H^{∘}}(H) of endomorphisms of H which commute with right and left multiplications. In particular, if H has an identity, Z(H) is isomorphic to the center of H.

Proof: is straightforward, see...

Corollary. Let P be a finitely generated projective generator in M, Λ = (End_{Λ}P)^{∘}. Then the natural morphism z ↦ z_{P} ∈ Λ gives an isomorphism of Z(M) with the center of Λ.

This follows from the lemma and 1.6.

§2. Decomposition theorem

2.0. Let G be a connected reductive p-adic group, Θ(G) the set of infinitesimal characters of G, Θ(G) = ∪Θ its decomposition into the union of connected components. For each Θ consider the subset S_{Θ} = inf ·ch^{−1}(Θ) ⊂ IrrG and denote by M(Θ) = M(G, S_{Θ}) the corresponding subcategory in M(G), M(Θ) = {E ∈ M(G)|JH(E) ⊂ S_{Θ}} (see 1 ). In this section we prove the following

Decomposition theorem. M(G) = ∏_{Θ} M(Θ), where Θ runs through all con-

nected components of Θ(G).

Our proof follows the proof in [ ] with slight modifications, which we will use later.

Generalization. Let B be a commutative algebra with identity. Put M(Θ; B) = {E ∈ M(G; B)|E ∈ M(Θ) is G-module} . Then decomposition theorem implies that M(G; B) = ∏_{Θ} M(Θ; B).

2.1. Separation of compactly supported G-modules.

Let G be an arbitrary ℓ-group as in 1. A G-module E is called compactly supported if for each open compact subgroup K ⊂ G and each ξ ∈ E the function g ↦ (e_{K} ge_{K})ξ has a compact support of G. This implies that E has compactly supported matrix coefficients. Using this fact and arguing exactly like in a case of compact groups, one can prove the following.

Proposition. (see [ ]). Let V be a finitely generated compactly supported Gmodule. Then V is admissible and has finite length. The finite subset S = JH(V) ⊂ IrrG splits the category M(G) and each module E ∈ M(G; S) is completely reducible.

2.2. Separation of cuspidal components.

Let G be a reductive p-adic group. If the center Z(G) of G is compact, cuspidal G-modules are compactly supported and we can use 2.1 to separate them. In general they are compactly supported modulo center Z(G). To study this case we will use the following property of G.

(*) G has an open normal subgroup G^{0} such that Z(G) ∩ G^{0} is compact, Z(G) · G^{0} has finite index in G and the group Λ = G/G^{0} is a lattice, i.e. is isomorphic to ℤ^{d}, d ∈ ℤ^{+}.

It is easy to see that such a subgroup G^{0} is unique. By definition the group Ψ(G) of unramified characters of G coincides with

Hom (Λ, ℂ^{∗}) = {ψ : G → ℂ^{∗} : ψ|_{G^{0}} = 1} .

Lemma. Let (ρ, V) be a simple G-module. Then

(i) ρ|_{G^{0}} is completely reducible of finite length. The subset S_{ρ} = JH(ρ|_{G^{0}}) ⊂ IrrG^{0} is finite and is a G-orbit of the natural action of G on IrrG^{0}.

(ii) The correspondence ρ ↦ S_{ρ} gives a bijection of the set of Ψ(G) - orbits in IrrG and G-orbits in IrrG^{0}, i.e. S_{ρ} = S_{ρ′} iff ρ′ ≈ ψρ for some ψ ∈ Ψ(G).

(iii) The stabilizer St(ρ, Ψ) of ρ in Ψ(G) is finite. If we choose for each ψ ∈ St (ρ, Ψ) a nonzero morphism α_{ψ} : (ρ, V) → (ψρ, V), then {α_{ψ}} is a ℂ-basis of End_{G^{0}}(V).

Proof: (i), (ii) are proven in [ ]. (iii) Put A = End_{G^{0}}(V) and define the action of G on A by g(a) = ρ(g)aρ(g)^{−1}. This action is trivial on G^{0}. Because of Schur’s lemma it is also trivial on Z(G), so it is an action of the finite abelian group G/G^{0} · Z(G). Using this we can decompose A = ⊕A_{ψ}, where A_{ψ} are eigenspaces of the action. But A_{ψ} = Hom_{G}(ρ, ψρ) = ℂ · a_{ψ} by Schur’s lemma,

i.e. A = ⨁_{ψ} ℂ · a_{ψ} with ψ ∈ St(ρ, Ψ) ⊂ Hom (G/G^{0} · Z(G), ℂ^{∗}).

Harish-Chandra theorem. (see [ ]) Let π be a quasicuspidal G-module, i.e. r_{MG}(π) = 0 for all subgroups M ⫋ G. Then it is compactly supported modulo center, i.e. π|_{G^{0}} is compactly supported.

Corollary. Let (ρ, V) be a cuspidal irreducible G-module. Then the cuspidal component D = Ψ(G) · ρ ⊂ IrrG splits the category M(G).

Proof: Put S = S_{ρ} = JH(ρ|_{G^{0}} ) ⊂ IrrG^{0}. By 2.1 every G-module E has a decomposition E = E_{S} ⊕ ES̄ with E ∈ M(G^{0}; S), ES̄ ∈ M(G^{0}; \overline{S}). Since this decomposition is canonical it is G-invariant, i.e. E_{S} and ES̄ are G-submodules. Lemma 2.2 implies that E_{S} ∈ M(G, D), E⊥ ; \overline{S} ∈ M(G, \overline{D}).

2.3. Functors i_{GM} and r_{MG}.

In order to deal with noncuspidal components we will use functors i_{GM} and r_{MG}. Let us recall some elementary properties of these functors. For simplicity we consider only the case when M is a standard Levi subgroup.

(i) Transitivity. Let M < N < G. Then i_{GM} = i_{GN} ∘ i_{NM}, r_{MG} = r_{MN} ∘ r_{NG} (canonical isomorphisms).

(ii) Functor r_{MG} is left adjoint to i_{GM} (canonical adjointness). See [ ].

(iii) Functors i_{GM} and r_{MG} are exact and preserve direct sums. See [ ]

(iv) There exists a functorial isomorphism i_{GM} (σ̃) = (i_{GM} (σ))^{}˜ , σ ∈ M(M) (canonical isomorphism). See [ ]

(v) Functor r_{GM} maps finitely generated G-modules into finitely generated M-modules. See [ ].

(vi) Composition of functors r and i.

We need some notations. For each w ∈ W_{G} we fix a representative \overline{w} ∈ Norm (M_{0}, G). For each subgroup H ⊂ G we put w(H) = w̄Hw̄^{−1} and denote by w the corresponding functor w : M(H) → M(w(H)).

Let M, N < G. Each double coset W_{N}\W_{G}/W_{M} has a unique representative of minimal length; we denote the set of these representatives by W_{G}^{NM}. For each w ∈ W_{G}^{NM} we put

M_{w} = M ∩ w^{−1}(N) < M, N_{w} = w(M_{w}) = w(M) ∩ N < N .

Composition theorem. Consider functors F, F_{w} : M(M) → M(N), for w ∈ W_{G}^{NM}, defined by F = r_{NG} ∘ i_{GM}, F_{w} = i_{NNw} ∘ w ∘ r_{MwM}. More precisely, choose any ordering {w_{1}, …, w_{r}} of W_{G}^{NM} such that w_{i} < w_{j} implies i ≥ j (here < is the standard partial order on W, see [ ]). Then F has a canonical filtration 0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{V} = F and F_{i}/F_{i−1} is canonically isomorphic to F_{w_{i}}.

See the proof in [ ]. Canonicity of isomorphisms in .... and .... is discussed in appendix ...

(vii) Let K ⊂ G be an open compact subgroup. We will use the following simple lemma, which describes K-invariant vectors in induced G-modules.

Lemma. Let (P, M) be a standard parabolic pair. Fix a system (g_{1}, …, g_{n}) of representatives of double cosets P \G/K and consider open compact subgroups Γ_{1}, …, Γ_{n} ⊂ M defined by Γ_{i} = pr_{P →M} (P ∩ gKg^{−1}). Also fix a Haar measure on the unipotent radical U ⊂ P. Then for every V ∈ M(M) and E = i_{GM} (V)

there exists a canonical functorial isomorphism E^{K} ≈ ⨁_{i=1}^{n} V^{Γ_{i}}.

Proof: is straightforward.

2.4. Functors i_{G,D} and r_{D,G} and their properties.

Let (M, D) be a standard cuspidal block (notation (M, D) < (G, Θ(G))). It means that M < G and D is a cuspidal component of Θ(M). The subset Θ = i_{GM} (D) ⊂ Θ(G) is a connected component. We say that the component Θ corresponds to the block (M, D) and use the notation (M, D) < (G, Θ). Another standard cuspidal block (N, D′) corresponds to the same component Θ if and only if there exists w ∈ W_{G} such that w(M, D) = (N, D′), i.e. N = w(M), D′ = w(D). In this case we say that (N, D′) is associate to (M, D) (notation (N, D′) ∼ (M, D)).

Standard cuspidal blocks will play a role similar to standard Levi subgroups. By 2.2. M(D) is a direct summand of M(M). We denote by in_{D} : M(D) → M(M) and pr_{D} : M(M) → M(D) the corresponding inclusion and projection functors.

Consider the functors

i_{GD} = i_{GM} ∘ in_{D} : M(D) → M(G)
r_{DG} = pr_{D} ∘ r_{MG} : M(G) → M(D) .

The following properties of these functors immediately follow from 2.3.

(i) r_{DG} is left adjoint to i_{GD}.

(ii) i_{GD} and r_{DG} are exact and preserve direct sums.

(iii)

Composition theorem. Let (M, D), (N, D′) be standard cuspidal blocks F : r_{D′G} ∘ i_{GD} : M(D) → M(D′). Then F = 0 unless (M, D) ∼ (N, D′). If they are associate, F is glued from the functors w : M(D) → M(D′), where w ∈ {w ∈ W_{G}^{NM}|w(M, D) = (N, D′)}.

Proof: By composition theorem F is glued from pr_{D′} ∘ i_{NNw} ∘ w ∘ r_{MwM} ∘ in_{D}. If M_{w} ≠ M, we have r_{MwM} ∘ in_{D} = 0. If N_{w} ≠ N, we have pr_{D′} ∘ i_{NNw} = 0 (as right adjoint to r_{NwN} ∘ in_{D′} = 0). This proves the theorem.

Proposition. (i) The system of functors r_{DG} for all (M, D,) < (G, Θ(G)) is faithful, i.e. r_{DG}(E) = 0 for all (M, D) implies that E = 0.

(ii) Fix a connected component Θ ⊂ Θ(G). Then the system of functors r_{DG} with (M, D) < (G, Θ) is faithful on M(Θ).

(iii) Let E be a G-module such that r_{D′G}(E) = 0 for all standard cuspidal blocks (N, D′) which do not correspond to the component Θ. Then E ∈ M(Θ).

(iv) Conversely, if E ∈ M(Θ), then r_{D′G}(E) = 0 for (N, D′) ≮ (G, Θ).

(v) If (M, D) < (G, Θ), then i_{GD}(M(D)) ⊂ M(Θ).

Lemma. Let w ∈ IrrG, θ = inf ·ch w ∈ Θ(G) and Θ be a connected component of θ. There exists a cuspidal block (M, D) < (G, Θ) such that r_{DG}(w) ≠ 0. For each cuspidal block (N, D′) which does not correspond to Θ r_{D′G}(w) = 0.

Proof: We can find a cuspidal pair (M, ρ) such that M < G and w ∈ i_{GM} (ρ). Let D ⊂ Θ(M) be a connected component of ρ. Then Hom_{G}(w, i_{GD}(ρ)) = Hom(r_{DG}(w), ρ) ≠ 0, i.e. r_{DG}(w) ≠ 0. If (N, D′) ≁ (M, D), then r_{D′G}(w) ⊂ r_{D′G} ∘ i_{GD}(ρ) = 0 by composition theorem 2.4 (iii). Proof of the proposition. Since functors r_{DG} are exact the lemma implies (i), (ii) and (iii). Since for (N, D′) ≁ (M, D) r_{D′G} ∘ i_{GD} = 0, (iii) implies (iv).

Let us prove (iv). Let (N, D′) be a standard Levi block such that the corresponding component Θ′ differs from Θ. Put V = r_{D′G}(E) ∈ M(D′). By (v) i_{GN} (v) ∈ M(Θ′) and hence Hom_{N}(V, V) =Hom_{N}(r_{D′G}(E), V) =Hom_{G}(E, i_{GD′} (V)) = 0, i.e. V = 0.

Corollary. Let N < G, Θ ⊂ Θ(G) be a connected component. Consider all components Θ_{N} ⊂ i_{GN}^{−1}(Θ) ⊂ Θ(N) and the corresponding product category

M′ = ∏_{Θ_{N}} M(Θ_{N}). Then

i_{GN} (M′) ⊂ M(Θ) and r_{NG}(M(Θ)) ⊂ M′ .

Proof: it easily follows from the composition theorem in 2.3 and the proposition.

2.5. Proof of decomposition theorem.

Step 1. For each standard cuspidal block (M, D) define a functor T_{D} = i_{GD} ∘ r_{DG} : M(G) → M(G). Since the functor r_{GD} is left adjoint to i_{GD}, for each G-module E we have a canonical functorial morphism α_{D} : E → T_{D}(E). If L ⊂ E, then the restriction α_{D}|_{L} : L → T_{D}(E) corresponds to the morphism r_{DG}(L) → r_{DG}(E). Since the functor r_{DG} is exact, this morphism is an inclusion. This proves that α_{D}(L) = 0 if and only if r_{DG}(L) = 0.

Step 2. Consider the product morphism

α = ∏_{(M,D)} α_{D} : E → ∏_{(M,D)} T_{D}(E)

where the product is over all standard cuspidal blocks (M, D). Then r_{DG}(Ker α) = 0 for all (M, D), and, since {r} is a faithful system of functors, Ker α = 0.

Step 3. We want to show that the decomposition IrrG = ⋃_{Θ} S_{Θ} splits a G-module

E. Since E ⊂ ∏_{(M,D)} T_{D}(E) it is sufficient to check that {S_{Θ}} splits this

product (see 1...). By proposition 2.4. (v) {S_{Θ}} splits ⨁_{(M,D)} T_{D}(E), hence

it would be sufficient to prove that ⨁_{(M,D)} T_{D}(E) ≈ ∏_{(M,D)} T_{D}(E). This

follows from the following general statement.

(*) Let V_{Θ} ∈ M(Θ) : Θ ⊂ Θ(G). Then ⨁_{Θ} V_{Θ} ≈ ∏_{Θ} V_{Θ}.

Step 4. As we saw in ... ∏_{Θ} V_{Θ} = lim_{\overrightarrow{K}}(∏_{Θ} V_{Θ}^{K}). Hence (*) follows from

(**) Let K ⊂ G be an open compact subgroup. Then V_{Θ}^{K} = 0 for all but a finite number of components Θ, so ⨁_{Θ} V_{Θ}^{K} = ∏_{Θ} V_{Θ}^{K}.

Put S_{K} = {L ∈ IrrG|L^{K} ≠ 0}, Θ_{K}(G) = inf ·ch · S_{K}. If V_{Θ}^{K} ≠ 0, then V_{Θ} has an irreducible subquotient in S_{K}. Hence (**) follows from

(***) Θ_{K}(G) is a union of a finite number of components.

Let Θ ⊂ Θ(G) be a connected component (M, D) < (G, Θ), (ρ, V) ∈ D. For every ψ put E_{ψ} = i_{GM} (ψρ) ∈ M(G). The lemma 2.3 ( ) shows that the space E_{ψ}^{K} does not depend on ψ and is equal to ⨁_{i} V^{Γ_{i}}. For a given

infinitesimal character θ = (M, ψρ) ∈ Θ the fiber inf ·ch^{−1}(Θ) ⊂ IrrG coincides with JH(E_{ψ}). This implies, that θ ∈ Θ_{K}(G) iff ⨁_{i} V^{Γ_{i}} ≠ 0.

Hence Θ either lies in Θ_{K}(G) or does not intersect it, i.e. Θ_{K}(G) is a union of components. Moreover, Θ ⊂ Θ_{K}(G) iff D ⊂ Θ_{Γ_{i}}(M) for some i. So, using induction in dim M, we should estimate only the number of cuspidal components. In other words (***) follows from

(***′) Θ_{K}(G) contains a finite number of cuspidal connected components.

Step 5. Using 2.2 we see that (***′) is equivalent to

(****) Irr_{K}G^{0} has a finite number of compactly supported G^{0}-modules.

This statement is deduced in [ ] from the following

Uniform admissibiliy theorem. Let K ⊂ G be an open compact subgroup. There exists an effective constant C = C(G, K) such that for each simple Gmodule L dimL^{K} ≤ C(G, K).

Remark. The proof in [ ] does not give an effective estimate for the number and type of cuspidal components in Θ(G). In .... we will give an effective estimate.

2.6. The faithfulness of the functor r_{DG}.

Fix a connected component Θ ⊂ Θ(G). As we saw in 2.4. the system of functors {r_{DG}|(M, D) < (G, Θ)} is faithful on M(Θ). In fact, each of these functors is faithful. This fact allows us to simplify notations in many proofs.

Proposition. Let (M, D) < (G, Θ). Then the functor r_{DG} is faithful on M(Θ). In particular, for every G-module E ∈ M(Θ) the morphism α_{D} : E → T_{D}E, described in 2.5 is an inclusion.

The proof is based on the following lemma, due to Casselman

Lemma. Let M < G be a maximal Levi subgroup, D ⊂ Θ(M) a cuspidal component, ρ ∈ D. Suppose that for some w ∈ W_{G}, wM < G and w(M, D) ≠ (M, D). Then the G-module π = i_{GM} (ρ) is irreducible.

Proof:

Step 1. Let R(G) be the Grothendieck group of G-modules of finite length. By Langlands theory R(G) is generated by i_{GN} (ψσ), where N < G, ψ ∈ Ψ(N), σ ∈ IrrN is a tempered N-module.

Consider the infinitesimal character θ, corresponding to (M, ρ) and a subgroup R(θ) ⊂ R(G), generated by G-modules with infinitesimal character θ. Let i_{GN} (ψσ) ∈ R(θ). If N ≠ G then, since M is maximal, (N, ψσ) is conjugate to (M, ρ) and hence i_{GN} (ψσ) ≈ π. Hence if exclude the possibility N = G, then R(θ) = ℤ · π, i.e. π is irreducible.

Suppose there exists a tempered G-module σ ∈ IrrG, and ψ ∈ Ψ(G) such that ψσ ∈ R(θ). Replacing ρ by ψ^{−1}ρ we can assume that ψ = 1, i.e. inf. ch. σ = θ. Replacing the cuspidal pair (M, ρ) by a conjugate one we can assume, that σ ⫋ π.

Step 2. Since M is a maximal Levi subgroup, there exist modulo W_{M}, only one nontrivial element w ∈ W_{G} such that wM < G (see [ ]).

Put N = wM, D′ = wD, π′ = i_{GN} (wρ). We have r_{DG}(π) = ρ, r_{D′G}(π) = wρ. Since the system of functors r_{DG}, r_{D′G} is faithful on M(θ) and r_{DG}(σ) ≠ 0, this implies that r_{DG}(σ) = ρ, r_{D′G}(π/σ) = wρ and hence r_{D′G}(σ) = 0. This shows that π has length 2. Similarly, π′ has length 2. Since σ ∈ JH(π′) = JH(π) and σ ⊄ π′, there exists a nontrivial morphism π′ → σ.

Step 3. For every G-module τ denote by τ^{+} the Hermitian contragredient Gmodule. Then σ^{+} ≈ σ, since σ is tempered and hence unitary. Also ρ^{+} lies on the same component D as ρ, since D contains some unitary M-modules. This implies that τ = (π′)^{+} has a form τ = i_{GN} (ρ′) with ρ′ ∈ D′.

Nontrivial morphism π′ → σ gives a nontrivial morphism σ = σ^{+} → τ. But Hom_{G}(σ, τ) =Hom_{G}(σ, i_{GN} (ρ′)) =Hom_{N}(r_{D′G}(σ), ρ′) i.e. r_{D′G}(σ) ≠ 0, which contradicts Step 2. This contradiction proves the lemma.

Proof of the proposition.

Let E ∈ M(Θ), E ≠ 0. We have to prove that r_{DG}(E) ≠ 0. By ... we can find a standard cuspidal block (N, D′), associate to (M, D) such that r_{D′G}(E) ≠ 0. Let (N, D′) = w(M, D), w ∈ W_{G}. We call the map w : M → N elementary if there exists a Levi subgroup L < G such that M < L, N < L, w ∈ W_{L} and M is a maximal Levi subgroup in L. It is shown in [ ] that any map w : M → N can be obtained as a composition of elementary maps. Hence we can assume that w : M → N is elementary.

Let Θ′ = i_{LM} (D) = i_{LN} (D′) ⊂ Θ(L), V = r_{LG}(E) ∈ M(L). Since r_{D′L}(V) = r_{D′G}(E) ≠ 0, V has a nontrivial D′-component. Hence replacing G by L and E by the Θ′-component of V we can assume that M < G is a maximal Levi subgroup. We can also assume that (M, D) ≠ (N, D′), otherwise r_{DG}(E) = r_{D′G}(E) ≠ 0. Choose an irreducible subquotient w ∈ E. Then w ∈ JH(i_{GM} (ρ)) for some ρ ∈ D. By the lemma, i_{GM} (ρ) is irreducible, i.e. w = i_{GM} (ρ). This implies that r_{DG}(w) ≠ 0 and hence r_{DG}(E) ≠ 0.

Thus we have proved that r_{DG} is faithful on M(Θ). The same arguments as in 2.5 show that α_{D} : E → T_{D}E is an inclusion.

§3. Decomposition of category M(G) with respect to a compact subgroup

3.1. Let K ⊂ G be an open compact subgroup H_{K} = H_{K}(G). Put S_{K} = {L ⊂ IrrG|L^{K} ≠ 0}. We say that the subgroup K splits M(G) if the subset S_{K} splits M(G), i.e. M(G) = M(S_{K}) × M(\overline{S}_{K}). As shown in ... in this case we have

M(S_{K}) = {E ∈ M(G)|Eis generated by E^{K}},
M(\overline{S}_{K}) = \{E ∈ M(G)|E^{K} = 0\}

and the functors

r : M(S_{K}) → M(H_{K}),
i : M(H_{K}) → M(S_{K})

given by r(E) = E^{K}, i(M) = H ⨂_{H_{K}} M are mutually inverse equivalences of

categories.

We want to show that there are a lot of subgroups K which split M(G). In order to do this we describe some geometrical sufficient conditions on K.

First of all, let us notice, that if S_{K} is a union of subsets S_{Θ} for some components Θ, then K splits M(G). In fact, one can prove that any splitting subset S ⊂ IrrG is a union of S_{Θ} (it follows, for instance, from the description of Z(M(G)) below). So we want to find conditions which imply that S_{K} is a union of S_{Θ}.

3.2. Let P ⊂ G be a parabolic subgroup M = P/U. For a compact open subgroup K ⊂ G put K_{P} = K ∩ P, K_{M} = pr_{P →M} (K_{P}). Let K ⊂ G, Γ ⊂ M be open compact subgroups. Consider the following conditions on K and Γ.

(I) For each g ∈ G the subgroup (^{g}K)_{M} ⊂ M contains a subgroup, conjugate to Γ.

(II) For any open subgroup N ⊂ G the subset (pr_{P →M})^{−1}(Γ) · N contains a subgroup conjugate to K.

Note that these conditions are invariant with respect to conjugation of P, K or Γ.

Lemma. (see.....).

(i) Suppose K, Γ satisfy I. Then for each M-module V, V^{Γ} = 0 ⇒ i_{GM} (V)^{K} = 0.

(ii) Suppose K, Γ satisfy II. Then for each G-module E, E^{K} = 0 ⇒ rP_{MG}(E)^{Γ} = 0.

Proof:

(i) Follows from Lemma ....

(ii) V is isomorphic to E_{v} as Γ-module (see ...). Denote by A : E → E_{V} the natural projection. Suppose that E_{V}^{r} ≠ 0 and choose ξ ∈ E such that v = Aξ ∈ E_{V}^{r} \0. Let N be the stabilizer of ξ in g. Then for each g ∈ pr_{P →M} (r)^{−1} · N we have

A(gξ) = A(γn)ξ = γA(nξ) = γAξ = γv = v .

Choose a subgroup K′ ⊂ pr^{−1}(Γ) · U, conjugate to K. Then A(e_{K′} ξ) = v ≠ 0, i.e. E^{K′} ≠ 0 and E^{K} ≠ 0.

Corollary. Let K ⊂ G be an open compact subgroup such that for each parabolic subgroup P the pair K, Γ = K_{M} ⊂ M satisfy both conditions I and II. Then S_{K} is a union of S_{Θ} and hence K splits M(G).

Proof: Let Θ ⊂ Θ(G) be a connected component (M, D) < (G, Θ) a corresponding standard cuspidal block. Let L ∈ S_{Θ}. Then by ... r_{DG}(L) ≠ 0, so for some ψ ∈ Ψ(M) there exists an epimorphism r_{DG}(L) → ψρ and an inclusion L → i_{GM} (ψρ). Hence

L^{K} = 0 ⇒ r_{DG}(L)^{Γ} = 0 ⇒ V^{Γ} = 0and
V^{Γ} = 0 ⇒ i_{GM} (ψρ) = 0 ⇒ L^{K} = 0 .

Thus the condition L^{K} = 0 does not depend on L ∈ S_{Θ}, i.e. either S_{Θ} ⊂ S_{K} or S_{Θ} ⊂ \overline{S}_{K}.

Remarks.

(i) It is sufficient to check condition (I) for (finite number of) representatives {g_{i}} of double cosets P \G/Norm K. In particular, if K is a congruence subgroup, which is normalized by the maximal compact subgroup K_{0}, then Iwasava decomposition G = P K_{0} implies that I holds for Γ = K_{M}.

(ii) Let (P, \overline{P}) be a parabolic pair. Suppose that K ⊂ U Γ \overline{U}, where Γ ⊂ M = P ∩ \overline{P}. Then condition II holds. Indeed, put C = pr_{UΓ\overline{U}→\overrightarrow{V}}(K). Then we can find a ∈ Z(M) for which ^{a}C is arbitrarily small, and hence lie in N, which implies ^{a}K ⊂ U Γ N.

Examples.

(1) A congruence subgroup K of a nonzero level is normalized by K_{0} and satisfies KU ∪ K_{M} \overline{U} for each standard parabolic pair (P, \overline{P}). Hence it splits M(G).

(2) Let I be an Iwahori subgroup (see ...). Then it is easy to see that I ⊂ UI_{M} \overline{U}. Choosing representatives w ∈ W = K_{0}/I in P \G/I it is easy to check that K, K_{M} satisfy condition I for each standard parabolic subgroup p. Thus I splits M(G). Another proof of the fact see in [ ]. In this case S_{I} consists of one component S_{Θ}.

(3) The maximal compact subgroup K_{0} does not split M(G) since trivial and Steinberg G-modules ℂ and St lie over the same component Θ ⊂ Θ(G), but ℂ^{K_{0}} ≠ 0 while St^{K_{0}} = 0.

§4. Noetherian properties of M(G)

4.1. Structure of category M(D) for a cuspidal component D.

Let D ⊂ Θ(G) be a cuspidal component. Fix (ρ, V) ∈ D. Denote by F the algebra of regular functions on algebraic variety Ψ(G). It coincides with the group algebra of the lattice L = G/G^{0} and hence has a natural structure of G − F -module. This module describes a universal ψ(G)-family of unramified characters of G since its specialization at a point ψ ∈ Ψ(G) is ℂ_{ψ}.

We denote by Π(ρ) the G − F -modules Π(ρ) = F ⨂_{ℂ} V. As G-module Π(ρ)

does not depend on the choice of a point ρ ∈ D (up to a noncanonical isomorphism). So we denote this G-module as Π(D).

For every ψ ∈ Stab (ρ, Ψ(G)) we choose an isomorphism α_{F} : (ρ, V) → (ψρ, V) and extend it to the automorphism of Π(D) by α_{ψ}(v, f) = α_{ψ}(v)⊗ψ(f), where ψ(f) is defined as ψ(f)(ψ_{1}) = f(ψ^{−1}ψ_{1}).

Proposition. Let D ⊂ Θ(G) be a cuspidal component, (ρ, V) ∈ D.

(i) Π(D) is a finitely generated projective generator in the category M(D).

(ii) End_{G} Π(D) = ⨁_{ψ} F · a_{ψ} where ψ ∈ Stab(ρ, Ψ(G)).

Proof:

(i) Since F = ind_{G^{0}}^{G}(ℂ), where ℂ is the trivial G^{0}-module, Π(D) = ind_{G^{0}}^{G}(ρ|_{G^{0}} ). Hence for every G-module E we have Hom_{G}(Π(D), E) = Hom_{G^{0}}(V, E). If E ∈ M(D) its restriction to G^{0} is completely reducible (see 2.1), i.e. the functor E ↦ Hom_{G}(Π(D), E) = Hom_{G^{0}}(V, E) is exact and faithful. Hence Π(D) is a projective generator of M(D). Since G^{0} is open in G, Π(D) is finitely generated.

(ii) Hom_{G}(Π(D), Π(D)) = Hom_{G^{0}}(V, F ⊗ V) = = F ⨂_{ℂ} Hom_{G^{0}}(V, V), so the

statement follows from 2.2.

Using ... we see that the category M(D) has a fairly simple description. Namely, put Λ = End_{G}(Π(D))^{0}. Then M(D) is equivalent to the category M(Λ). The algebra Λ is a free module over the subalgebra F with generators a_{ψ}, i.e. Λ = ⨁_{ψ} F · a_{ψ} with ψ ∈ Stab(ρ, Ψ(G)), and following relations

(a) a_{ψ}fa_{ψ}^{−1} = ψ(f), f ∈ F.

(b) a_{ψ}a_{χ} = c(ψ, χ)a_{ψχ}, where c(ψ, χ) ∈ ℂ are some constants, defining a projective representation of Stab(ρ, Ψ(G)) in V.

Corollary. (i) The center Z(M(D)) of the category M(D) is isomorphic to the algebra Z(D) ⊂ F ⊂ End(Π(D)) of regular functions on D.

(ii) Category M(D) is locally noetherian.

(iii) Every finitely generated G-module E ∈ M(D) is Z(D) admissible.

Proof:

(i) Relations (a) - (b) show that Z(D) coincides with the center of Λ. Using ... we see that it coincides with Z(M(D)).

(ii) Since D ≈ Ψ(G)/Stab(ρ, Ψ(G)), F, and hence Λ, is a finitely generated Z(D)-module. Since Z(D) is a noetherian algebra, the category M(Λ) ≈ M(D) is locally noetherian.

(iii) Since ρ|_{G^{0}} is admissible (see 2.1), Π(D) is F -admissible and hence Z(D)- admissible. Since any finitely generated G-module E ∈ M(D) is a quotient of Π(D)^{n}, n ∈ ℤ^{+}, it is also Z(D) admissible.

4.2. Noetherian properties of M(G).

Theorem. Category M(G) is locally noetherian. Functors r and i map finitely generated modules into finitely generated ones.

Proof:

Step 1. Functor r maps finitely generated modules into finitely generated ones. This easily follows from Iwasava decomposition (see [ ]).

Step 2. Let (M, D) be a standard cuspidal block, V ∈ M(D) be a finitely generated M-module. Then G-module E = i_{GD}(V) is noetherian.

Let Θ = i_{GM} (D) ⊂ Θ(G). Then E ∈ M(Θ) (see ...). Since the functor r_{DG} is faithful and exact on M(Θ) it is sufficient to check, that r_{DG}(E) is noetherian. But by 2.4 r_{DG}(E) = r_{DG} ∘ i_{GD}(V) is glued from M-modules wV, w ∈ W (D), each of which is noetherian by Proposition 4.1.

Step 3. Let E be a finitely generated G-module. Then it is noetherian. Indeed, by 2... E imbeds into ⨁_{(M,D)} T_{D}E. Since it is finitely generated, its image

lies in a finite sum. Using Steps 1,2 we see that each G-module T_{D}E = r_{GD} ∘ r_{DG}(E) is noetherian, and hence E is noetherian.

Step 4. Let N < G, V ∈ M(N) be a noetherian M-module. Then i_{GN} (V) is noetherian G-module.

Repeating arguments in Step 3 we see that V is contained in a finite sum ⨁_{(M,D)} i_{ND} ∘ r_{DN} (V). Hence i_{GN} (V) is contained in a finite sum ⊕i_{GD} ∘ r_{DN} (V),

which is noetherian by Steps 1,2.

Generalization. Let B be a commutative noetherian ℂ-algebra with identity. Then category M(G; B) is locally noetherian, and functors i, r map noetherian G − B-modules into noetherian ones.

Generalization. Let B be a commutative algebra with identity. Then M(D; B) ≈ M(Λ ⨂_{ℂ} B), Z(M(D, B)) = Z(D) ⨂_{ℂ} B. If B is noetherian, then

Z(D) ⨂_{ℂ} B is noetherian, since Z(D) is a finitely generated ℂ-algebra. This

implies that M(D, B) is noetherian.

§5. Stabilization Theorem

5.1. Let K ⊂ G be an open compact subgroup. For each g ∈ G we put h(g) = e_{k}ge_{k} ∈ H_{K}, where g stands for δ-distribution at g. In other words, h(g) is the unique normalized bi-K-invariant measure, supported on KgK.

In some cases we have equalities h(a^{i}) = h(a)^{i} for i ≥ 0 or h(ab) = h(a)h(b). (geometrically it means that K^{i}gK = (KgK)^{i} and KabK = KaKbk respectively). We want to describe some sufficient conditions for these equalities. Essentially these conditions mean that a, b are dominant with respect to some parabolic pair.

Definition. Let (P, \overline{P}) be a parabolic pair. We say that subgroup K is in a good position with respect to (P, \overline{P}) if

(∗) K = K_{−}ΓK_{+} , where K_{−} = K ∩ \overline{U}, Γ = K ∩ M, K_{+} = K ∩ U .

Suppose (P, \overline{P}) and K are in a good position. We call element a ∈ M dominant with respect to (P, \overline{P} , K) if

(∗∗)  a^{−1}K_{−}a ⊂ K_{−} , aΓa^{−1} = Γ , aK_{+}a^{−1} ⊂ K_{+} .

For each compact subgroup C ⊂ G we denote by e_{c} the distribution on G, which is the image of the normalized Haar measure on c. If K is in a good position with respect to (P, \overline{P}), we have

e_{K} = e_{K_{−}}e_{Γ}e_{K_{+}} = e_{K_{+}}e_{Γ}e_{K_{−}} .

If a, b are dominant with respect to (P, \overline{P} , K) we have h(ab) = h(a)h(b). Indeed,

KaKbK = KaK_{+}ΓK_{−}bK = K(aK_{+}a^{−1})(aΓa^{−1})ab(b^{−1}K_{−}b)K = KabK .

Example. Let A ⊂ Z(M_{0}) be the maximal split torus, Λ = Hom_{alg.gr.}(A, F^{∗}) its character lattice, Σ ⊂ Λ the root system of G and Σ^{+} ⊂ Σ the system of positive roots, corresponding to P_{0}. Put A^{+} = {a ∈ A | |α(a)| ≤ 1 for all α ∈ Σ^{+}}. Then there exist arbitrary small open compact subgroups K ⊂ G (congruence subgroups) such that (P_{0}, \overline{P}_{0}) and K are in a good position, and all elements a ∈ A^{+} are dominant with respect to (P_{0}, \overline{P} , K). In particular, H_{K} contains a very big commutative subalgebra A = span{h(a)|a ∈ A^{+}}.

In fact these congruence subgroups are in a good position with respect to each standard parabolic pair (P, \overline{P}) and all elements in A^{+}∩Z(M) are dominant with respect to (P, \overline{P} , K) (see [ ]).

5.2. To each element g ∈ G naturally corresponds a parabolic pair. Namely, put P_{g} = {x ∈ G| the sequence g^{i}xg^{−i}, i = 1, 2, …, is bounded in G}.

Statement. P_{g} is a parabolic subgroup of G, (P_{g}, P_{g^{−1}}) is a parabolic pair.

For regular semisimple g the statement is proved in [c]. It is enough for our purposes.

Definition. Let (P, \overline{P}) be a parabolic pair. We say that an element a ∈ M is strictly dominant with respect to (P, \overline{P}) if (P, \overline{P}) = (P_{a}, P_{a^{−1}}). Geometrically it means that operators Ad a|_{U} and Ad a^{−1}|\overline{U} are strictly contractable and the family of operators {Ad a^{i} | i ∈ ℤ} is uniformly bounded on M.

Let (P, \overline{P}) and K be in a good position. We say that an element a ∈ M is strictly dominant with respect to (P, \overline{P} , K) if it is dominant and strictly dominant with respect to (P, \overline{P}).

Lemma. (i) Let g ∈ G, (P, \overline{P}) = (P_{g}, P_{g^{−1}}). There exist arbitrary small open subgroups K ⊂ G in a good position with respect to (P, \overline{P}) such that g is strictly dominant with respect to (P, \overline{P} , K).

(ii) Let K be in a good position with respect to (P, \overline{P}). There exist an element a ∈ Z(M) strictly dominant with respect to (P, \overline{P} , K).

Proof: Statement (i) is proved in [ ], (ii) is straightforward.

Fix an element strictly dominant with respect to (P, \overline{P} , K) and consider increasing sequences of subgroups

U_{n} = a^{−n}K_{+}a^{n} ⊂ U , \overline{U}_{n} = a^{n}K_{−}a^{−n} ⊂ \overline{U} .

When n → ∞ these subgroups become arbitrary large, when n → −∞ they become arbitrary small.

Put h = h(a). Using formulae in 5.1, we get for n ≥ 0

h^{n} = e_{K}a^{n}e_{K}

e_{K}a^{n} = a^{n}e_{U_{n}}e_{Γ}e_{\overline{U}_{−n}}
h^{n} = e_{K}a^{n}e_{K} = a^{n}e_{U_{n}}e_{K}

and similarly h^{n} = e_{K}e_{\overline{U}_{n}}a^{n}.

Proposition. Let E be a G-module, E_{U} the space of U-coinvariants of E (see...) and A : E → E_{U} the natural M-equivariant projection. Denote by A_{k} the corresponding morphism A_{K} : E^{K} → E_{Γ}^{U} = (E_{U})^{Γ}. Then

(i) A_{K}h^{n} = a^{n}A_{K}.

(ii) For ξ ∈ e^{K} h^{n}ξ = 0 iff e_{U_{n}}ξ = 0

In particular

Ker A_{K} = ⋃_{n} Ker e_{U_{n}} |_{E^{K}} = {ξ ∈ E^{K} | h^{n}ξ = 0 for larger} .

(iii) If ξ ∈ E is \overline{U}_{−n}-invariant, then a^{n}e_{Γ}Aξ = Ae_{K}a^{n}ξ. In particular, for each η ∈ E_{U}^{Γ} a^{n}η ∈ Im A_{K} for large n, i.e. ⋃_{n} a^{−n} Im A_{K} = E_{U}^{Γ}.

Proof: Formula h^{n} = a^{n}e_{U_{n}}e_{K} implies (i). Since the operator a on E_{U}^{Γ} is invertible, it also implies (ii). Using formula a^{n}e_{U_{n}}e_{Γ}e_{\overline{U}_{−n}} = e_{K}a^{n} we see that

a^{n}e_{Γ}A (e_{\overline{U}_{−n}}ξ) = a^{n}Ae_{U_{n}}e_{Γ}e_{\overline{U}_{−n}}ξ = Aa^{n}e_{U_{n}}e_{Γ}e_{\overline{U}_{−n}}ξ = Ae_{K}a^{n}ξ which proves (iii).

This proposition means, that space E_{U}^{Γ} together with operator g is naturally isomorphic to the localization of E^{K} with respect to operator h.

5.3. Stabilization Theorem. Let (P, \overline{P}) be a parabolic pair, K ⊂ G an open compact subgroup, in a good position with respect to (P, \overline{P}). Denote by C = C_{K} a constant in uniform admissibility theorem (see.....), i.e. a bound for dim L^{K} for L ∈ IrrG.

Let a ∈ M be an element strictly dominant with respect to (P, \overline{P} , K). Put h = h(a) ∈ H_{K}. For each G-module E consider h as an endomorphism of E^{K}.

Stabilization theorem. (i) For each G-module E there exists a unique decomposition E^{K} = E_{0}^{K} ⊕E_{∗}^{K} into h-invariant subspaces such that h^{c}E_{0}^{K} = 0 and h is invertible on E_{∗}^{K}. Namely, E_{0}^{K} = Ker h^{n} , E_{∗}^{K} = Im h^{n} for any n ≥ C.

(ii) Let C ⊂ U, \overline{C} ⊂ \overline{U} be sufficiently large open compact subgroups. Then for each G-module E

E_{0}^{K} = E^{K} ∩ Ker e_{C} , E_{∗}^{K} = e_{K}e\overline{C}E .

In particular, E_{0}^{K}, E_{∗}^{K} do not depend on the choice of a.

(iii) Consider the natural morphism A_{K} : E^{K} → E_{U}^{Γ}. Then E_{0}^{K} = Ker A_{K} , A_{K} : E_{∗}^{K} → E_{U}^{Γ} is an isomorphism.

Proof: Using formulas h^{n} = a^{n}e_{U_{n}}e_{K} = e_{K}e_{\overline{U}_{n}}a^{n}, we see that (i) implies (ii) for subgroups C ⊃ U_{n} = a^{−n}K_{+}a^{n} , \overline{C} ⊃ \overline{U}_{n} = a^{n}K_{−}a^{−n}. Using proposition 5.2 we see that (i) implies (iii). Hence it is enough to prove (i).

Step 1. Let L be a ℂ[x]-module, i.e. a vector space with an endomorphism x. We say that L is x-stable if L has an x-invariant decomposition L = L_{0}⊕L_{∗} such that xL_{0} = 0 and x is invertible on L_{∗}. Clearly, L is x-stable ⟺ L = Ker x ⊕ Im x ⟺ Ker x^{2} = Ker x, Im x^{2} = Im x ⟺ x is invertible on L/ Ker x ≈ Im x.

It is easy to check that the direct sum of x-stable modules is x-stable and for each morphism α : L → L′ of x-stable ℂ[x]-modules Ker α and Coker α are x-stable ℂ[x]-modules.

Step 2. Denote by M′ ⊂ M(G) the subcategory of G-modules E such that E^{k} is h^{C}-stable. We have to show that M′ = M(G).

As follows from Step 1 direct sums of modules in M′ and kernels and cokernels of morphisms of modules in M′ lie in M′.

Also, M′ contains all irreducible G-modules. Indeed, for each irreducible Gmodule L dim L^{K} ≤ C, and hence the sequence of subspaces Im h^{i} is constant for i ≥ C, i.e. h is invertible on Im h^{C}.

Step 3. Let B be a commutative noetherian ℂ-algebra, E B-admissible σ − B-module. Suppose that r_{MG}(E) is B-admissible M − B-module. Then for some n > 0 E^{K} is h^{n}-stable.

Indeed, since E^{K} is noetherian B-module, the sequence of submodules Ker h^{n} is stable. By proposition....Ker A^{K} = ⋃_{n} Ker h^{n}, and hence Ker A_{K} = Ker h^{n} for

some n > 0.

By proposition..... E_{U}^{Γ} is a union of B-submodules a^{−n} Im A_{k}. Since E_{U}^{Γ} is finitely generated B-module it is equal to a^{−Γ} Im A_{K} for some Γ > 0. Since a is invertible on E_{U}^{r} we see that E_{U}^{Γ} = Im A_{k} = E^{k}/ Ker A_{K}.

Thus the operator h is invertible on E^{K}/ Ker A_{k} = E^{K}/ Ker h^{n}, which implies that E^{K} is h^{n} stable.

Step 4. Let (N, D) be a standard cuspidal block, (ρ, V) ∈ D, Π(D) = F ⊗ V be G−F -module described in... Put (Π, E) = i_{G_{M}} (Π(D)). Then for some n > 0 E^{K} is h^{n}-stable.

It is sufficient to check that E and r_{MG}(E) are F -admissible modules. By composition theorem r_{MG}(E) is glued from M-modules i_{MM_{w}} ∘w(Π(D)). Hence F-admissibility of E and r_{MG}(E) follows from the following.

Lemma. The functor i_{GM} : M(A, B) → M(G, B) maps B-admissible modules into B-admissible ones.

This lemma is an immediate consequence of lemma...

Step 5. Module (Π, E) is step 4 which lies in M′, i.e. E^{K} is h^{C}-stable. Indeed, it is sufficient to check that Ker h^{n} ⊂ Ker h^{C}. Let ξ ∈ ker h^{n}, ξ′ = h^{C}ξ. For each ψ ∈ Ψ(M) consider specialization morphism Π(D) → ψρ and the corresponding morphism α_{ψ} : E → E_{ψ} = i_{GM}(ψρ).

Lemma. (see [ ]) For generic ψ G-module E_{ψ} is irreducible.

This lemma implies that for generic ψ E_{ψ} ∈ M′. Since h^{n}α_{ψ}(ξ) = 0, this implies that α_{ψ}(ξ′) = h^{C}α_{ψ}(ξ) = 0 and hence ξ′ = 0.

Step 6. Let (N, D) be a standard cuspidal block. Then i_{GN}(M(D)) ⊂ M′.

Let σ ∈ M(D). Since Π(D) is a projective generator in M(D) we can represent σ as a cokernel of some morphism γ : ⊕_{α}Π(D) → ⊕_{β}Π(D). Then i_{GN}(σ) = Coker (⊕_{α}Π → ⊕_{β}Π) (since functor i_{G_{N}} is exact and preserves direct sums). Since Π ∈ M′ Step 2 implies that σ ∈ M′.

Step 7. Each G-module E lies in M′. Indeed, we can embed E into module E′ = ⊕_{(N,D)}i_{GD} ∘ r_{DG}(E) as in..... By Step 6 E′ ∈ M′. Similarly we embed E′/E into E′′ ∈ M′. Then E = ker(E′ → E′′) lies in M′ by step 2.

5.4. Corollaries and Remarks to the Stabilization Theorem.

Generalized Jacquet Lemma. Let K be in a good position with respect to (P, \overline{P}). Then for each G-module E the morphism A_{K} : E^{K} → E_{U}^{Γ} is an epimorphism. Moreover, it has a right inverse morphism B, functorial in E, i.e. E_{U}^{Γ} can be realized in a natural way as a direct summand of E^{K}.

Corollary. Functor r_{MG}^{P} maps B-admissible G − B-modules into B-admissible M − B-modules.

We will prove more a general result.

Let B be a commutative ℂ-algebra with identity. Fix a class of objects C ⊂ M(B) closed with respect to isomorphisms, finite direct sums and taking of direct summands (i.e. for x ⊕ y ≈ Z, Z ∈ C iff X, Y ∈ C). Examples: C is the class of finitely generated B-modules, or the class of projective B-modules, or the class of flat B-modules and so on. We say that G − B-module E is of C-type if for each open compact subgroup K ⊂ G B-module E^{K} lies in C.

Proposition. Fix a class C ⊂ M(B) as above. Then functors i_{GM}^{P} : M(M, B) → M(G, B) r_{MG}^{P} : M(G, B) → M(M, B) map C-type modules into C-type modules.

Proof: For functor i_{GM} this follows from lemma..... Let E be a G − Bmodule of type C and Γ_{0} ⊂ M an open compact subgroup. Choose an open compact subgroup K ⊂ G in a good position with respect to (P, \overline{P}) such that Γ = K ∩ M ⊂ Γ_{0}. Then E_{U}^{Γ_{0}} is a direct summand of E_{U}^{Γ}, which is a direct summand of E^{K}. Hence B-module E^{Γ_{0}}U lies in C, which proves the proposition for functor r_{MG}.

Remark. 1. Consider the decreasing sequence of right ideals J_{n} = h^{n}H_{K} ⊂ H_{K}. Applying stabilization theorem to G-module H(G)e_{K} we see that it is stable, namely

(∗)  J_{n} = J_{C} for n ≥ C .

In fact this statement is equivalent to the theorem. Indeed, it implies that Im h^{n} = Im h^{C} for each G-module E. Using the natural anti-involution of H(G), given by the antiautomorphism g ↦ g^{−1} on G, we can deduce from (*) that H_{K}h^{n} = H_{K}h^{C} for n ≥ C, which implies that Ker h^{n} = Ker h^{C}.

Note, that (*) is purely geometrical statement, which has nothing to do with the representation theory. It would be very interesting to find a direct geometrical proof of (∗). Such proof would probably give a reasonably precise estimate for constant C in (∗). I was able to find such proof for congruence subgroups in GL(Z), but not for higher rank. Another form of the statement (∗), which does not involve the choice of a, is (∗∗) For sufficiently large open compact subgroups C ⊂ U the ideal J_{C} = e_{K}e_{C}H(G)e_{K} does not depend on C. Namely, this is true for C ⊃ a^{−C}K_{+}a^{C}.

5.5. An Effective Bound of the Number of Cuspidal Components With a Given Conductor.

Fix an open compact subgroup K ⊂ G. We want to give an effective bound of the number of cuspidal components D ⊂ Θ_{K}(G).

Let E-be a G-module, ξ ∈ E^{K}, ξ̃ ∈ Ẽ^{K}. We denote by φ_{ξ̃,ξ} the matrix

coefficient φ_{ξ̃,ξ}(g) = ⟨ξ̃, gξ̃⟩.

Proposition. There exists a compact subset S ⊂ G^{∘}, which can be effectively described in terms of G and K, such that for each quasicuspidal G-module E, ξ ∈ E^{K}, ξ̃ ∈ Ẽ^{K} the matrix coefficient φ_{\overline{ξ},ξ} vanishes on G^{∘}\S.

This proposition gives a desired bound. Indeed, let D_{1}, …, D_{r} be different cuspidal components in Θ_{K}(G), V_{i} ∈ D_{i}, 0 ≠ ξ_{i} ∈ V_{i}^{K}, 0 ≠ ξ̃_{i} ∈ V_{i}^{K}, φ_{i} = φ_{\overline{ξ}_{i},ξ_{i}} for i = 1, …, r. By 2.. matrix coefficients φ_{i} are linearly independent on G^{∘}. Since they vanish on G^{∘}\S and are K-biinvariant, their number r is less or equal to #(K\S/K).

Proof of Proposition. Let A ⊂ Z(M_{0}) be the maximal split torus, L the lattice of coweights of A, which we will identify with the quotient L = A/A^{∘} of A by its maximal compact subgroup. Let L^{∘} = L ∩ G^{∘} be the semisimple part of L, L^{∘+} = L^{∘} ∩ A^{+}, where A^{+} is defined in example 5.1. In other words, L^{∘+} = {a ∈ L | (α, a) ≤ 0 for all α ∈ Σ^{+}} is the Weyl chamber, corresponding to P_{0}.

Let us fix a homomorphism L → A, inverse to the projection A → L, and using it identifies L with a subgroup of A. By Cartan decomposition there exists a compact subset Ω ⊂ G^{∘} such that G^{∘} = Ω^{−1}L^{∘+}Ω.

Choose a congruence subgroup K′, which lies in the open subset ⋂_{x∈Ω} xKx^{−1}

and denote by C = C_{K′} the constant in uniform admissibility theorem for K′. Put S^{∘} = L^{∘+}\ [L^{∘+} + c(L^{∘+}\0)] , S = Ω^{−1}S^{∘}Ω. We claim that S is a desired subset. First of all, since L^{∘+} is a strictly convex cone, set S^{∘} is finite, i.e.., S is compact. Let E be a quasicuspidal G-module, \overline{ξ} ∈ E^{K}, ξ ∈ E^{K}, g ∈ G^{∘}\S. We want to show that φ_{\overline{ξ},xi}(g) = 0. By definition g = x^{−1}a′y, where x, y ∈ Ω, a′ ∈ L^{∘+} is of the form a′ = b + ca, b ∈ L^{∘,+}, a ∈ L^{∘+}\0. Put h(a) = e_{K′}ae_{K′} and similarly for a′, b. Since a ∈ L^{∘+}\0 the corresponding parabolic subgroup P_{a} differs from G, i.e. r_{MG}^{P}(E) = 0. Hence for each vector η ∈ E h(a)^{n}η = 0 for large n and by the stabilization theorem, h(a)^{C}η = 0. Hence

φ_{ξ̃ξ}(g) = φ_{xξ̃,yξ}(a′) = (xξ̃, a′yξ) = (xξ̃, h(a′)yξ) = (xξ̃, h(b)h(a)^{C}yξ) = 0

Here we used that vectors xξ̃ and yξ are K′-invariant. Formula h(a′) = h(b)h(a)^{C} follows from 5.1. Note, that addition in L becomes multiplication, when L is considered as a subgroup of G.

Remark. All bounds we described are effective, but quite excessive. The most excessive is the estimate for the constant C = C_{K} in the proof of uniform admissibility theorem. It would be interesting to find more precise bounds.

§6. Main Theorems About Functors Randi

6.1. Pairing Between Ẽ\overline{U} and E_{U}.

Let (P, \overline{P}) be a parabolic pair. For each G-module E denote by Ẽ the contragredient G-module and consider M-modules Ẽ\overline{U} = (Ẽ)\overline{U} and E_{U}.

Theorem. There exists a unique pairing \{ \} : Ẽ\overline{U} × E_{U} → ℂ satisfying the following condition on the asymptotic of matrix coefficients.

(ASS) Let K ⊂ G be an open compact subgroup, a ∈ M be an element strictly dominant with respect to (P, \overline{P}). Then there exists n_{0}, depending only on a and K, such that for each ξ̃ ∈ Ẽ, ξ ∈ E, i > n_{0} (\overline{ξ}, a^{i}ξ) = \{\overline{A}ξ̃, A^{i}ξ\} (here \overline{A} : Ẽ → Ẽ\overline{U}, A : E → E_{U} are natural projections).

The pairing { } is M-equivariant, functorial in E and it gives an isomorphism of M-modules Ẽ\overline{U} \underrightarrow{∼}(E_{U})^{∼}.

Corollary. There exists a canonical functorial isomorphism r\overline{P} ; MG(Ẽ) ≈ (r_{MG}^{P}(E))^{∼}. In particular, for a standard Levi subgroup M < G \overline{r}_{MG}(Ẽ) = r_{MG}(E)^{∼}.

Proof: Indeed, by definition r\overline{P} ; MG(Ẽ) = Ẽ\overline{U} ⊗ Δ1/2 ; \overline{U}, r_{MG}^{P}(E) = E_{U} ⊗ Δ_{U}^{1/2}. Since Δ\overline{U} and Δ_{U} are canonically dual (see appendix.....), the theorem implies the corollary.

6.2. Proof of Theorem 6.1.

Step 1. Let K ⊂ G be an open compact subgroup in a good position with respect to (P, \overline{P}), Γ = K ∩ M. First let us define the pairing \{ \} : Ẽr ; \overline{U} × E_{U}^{r} → ℂ. By the stabilization theorem A_{K} : E_{∗}^{K} → E_{U}^{Γ} is an isomorphism, so we can identify E_{U}^{Γ} with a subspace E_{∗}^{K} ⊂ E^{K}. Applying the stabilization theorem to the parabolic pair (\overline{P} , P), subgroup K and G-module Ẽ we can identify ẼΓ ; \overline{U} with the subspace Ẽ_{∗}^{K} ⊂ Ẽ^{K}. Then the restriction of the pairing ( , ) : Ẽ^{K} × E^{K} → ℂ defines a pairing \{ \} : ẼΓ ; \overline{U} × E_{U}^{Γ} → ℂ.

Step 2. Choose an element a ∈ M strictly dominant with respect to P, \overline{P} , K (see 5.2) and put h = h(a), h^{∗} = h(a^{−1}). For each ξ̃ ∈ Ẽ^{K}, ξ ∈ E^{K} we have

(ξ̃, a^{n}ξ) = (ξ̃, h^{n}ξ) = (ξ̃, h^{n}ξ) = ((h^{∗})^{n}ξ̃, ξ) .

Using stability theorem, we see that for n > C_{K}(ξ̃, a^{a}ξ) depends only on projections of ξ̃ on Ẽ_{∗}^{K} and of ξ and E_{∗}^{K}. This shows that the pairing { } satisfies condition (ASS) for a and K. Since h is invertible on E_{∗}^{K}, { } is uniquely determined by condition (ASS).

Step 3. Let K′ ⊂ K be a smaller subgroup, such that a is strictly dominant with respect to P, \overline{P} , K′. Consider the corresponding pairing \{ \} :′ ẼΓ′ ; \overline{U}×E_{U}^{Γ′} → ℂ. It satisfies (ASS) and by uniqueness property of { } the restriction of { }′ to ẼΓ ; \overline{U} ×E_{U}^{Γ} coincides with \{ \}. Hence, choosing smaller and smaller subgroups K, we can define a pairing \{ \} : Ẽ\overline{U} ×E\overline{U} → ℂ satisfying (ASS), and this pairing is unique. By construction the pairing { } does not depend on a. This implies that it is M-equivariant.

Step 4. For each subgroup K the space Ẽ^{K} is dual to E^{K} and the operator h^{∗} in Ẽ^{K} is dual to the operator h in E^{K}. Hence Ẽ_{∗}^{K} is dual to E_{∗}^{K}. By definition of \{ \} ẼΓ ; \overline{U} ≈ Ẽ_{∗}^{K} is dual to E_{U}^{Γ} ≈ E_{∗}^{K}, which implies that \{ \} gives an isomorphism of Ẽ\overline{U} with module (E_{U})^{∼} contragredient to E_{U}.

6.3. Completion of σ-Modules. We want to describe the pairing { } in a more direct and visual way, using the notion of completion of G-modules.

Definition. Let E be a G-module. We define its completion E^{∧} in any of three equivalent ways

(i) E^{∧} = Hom_{G}(H(G), E).

(ii) E^{∧} = lim_{\overrightarrow{K}} E^{K}, where the inverse limit is over all open compact subgroups

K ⊂ G and for K′ ⊂ K the connecting morphism E^{K;} → E^{K} is given by ξ ↦ e_{K}ξ.

(iii) E^{∧} is the completion of E in the topology, generated by open subset Ker e_{K} for open compact subgroups K ⊂ G.

The algebra D_{C}(G) of compactly supported distributions on G acts on the completion E^{∧} by dξ^{∧}(h) = ξ^{∧}(h∗d). This action is continuous in the topology, described in (iii) and its restriction to E ⊂ E^{∧} coincides with the natural action of D_{C}(G) on E. In particular, G acts on E^{∧}, but this representation usually is not smooth. The smooth part of E^{∧} coincides with E = H(G)E^{∧}.

It is easy to check that the functor E ↦ E^{∧} is exact and faithful. Moreover, if E′ ⊂ E, then (E′)^{∧} = Closure E′ in E^{∧} = {ξ^{∧} ∈ E^{∧} | H(G)ξ^{∧} ⊂ E′ ⊂ E}.

It is easy to check that (L̃)^{∧} ≈ L^{∗} (the dual space). This gives the following realization of E^{∧}, convenient for computations:

Let us realize E as a submodule of L̃ for some G-module L and then E^{∧} can be described as

E^{∧} = {ξ^{∗} ∈ L^{∗} | H(G)ξ^{∗} ⊂ E ⊂ L̃} .

6.4.

Theorem. Let (P, \overline{P}) be a parabolic pair, E a G-module. Then there exists a canonical isomorphism

\overline{A} : (E^{∧})^{U} \underrightarrow{∼}(E\overline{U})^{∧}

where (E^{∧})^{U} is the space of U-invariants in E^{∧}. For each ξ^{∧} ⊂ (E^{∧})^{U} the vector η^{∧} = \overline{A}ξ^{∧} is uniquely characterized by the following property.

(*) For each subgroup K ⊂ G in a good position with respect to (P, \overline{P}) \overline{A}e_{K}ξ^{∧} = e_{r}η^{∧}.

This theorem allows us to give another description of the pairing { } in theorem 6.1. Namely, applying it to G-module Ẽ we see that ((Ẽ)^{∧})^{U} = (E^{∗})^{U} = (E_{U})^{∗} is canonically isomorphic to (Ẽ\overline{U})^{∧}. Hence Ẽ\overline{U} = smooth part of (Ẽ\overline{U})^{∧} = smooth part of (E_{U})^{∗} = (E_{U})^{∼}, which is the statement of theorem 6.1.

Proof of the Theorem.

Step 1. Let K′ ⊂ K ⊂ G be open compact subgroups in a good position with respect to (P, \overline{P}). Then for each ξ′ ∈ E_{∗}^{K′} e_{K}ξ′ ∈ E_{∗}^{K} and Ae_{K}ξ′ = e_{r}Aξ′. Indeed, let C ⊂ U be a very large open compact subgroup, L = e_{C}E. By stabilization theorem (applied to \overline{P} , P, K) E_{∗}^{K} = e_{K}L and E_{∗}^{K′} = e_{K′}L, which implies that E_{∗}^{K} = e_{K}E_{∗}^{K′}. Moreover, for each η ∈ L A(e_{K}η) = A(e_{K_{+}}e_{r}e_{K_{−}}η) = e_{r}A(e_{K_{−}}η) = e_{r}A(η) and similarly for K′. Hence if ξ′ = e_{K′}η, we have A(e_{K}ξ′) = e_{Γ}A(η) = e_{Γ}(e_{Γ′}A(η)) = e_{Γ}A(ξ′).

Step 2. Consider the inverse system {e_{K}} where K runs through all good subgroups (i.e. open compact subgroups in a good position with respect to (P, \overline{P})). Step 1 shows that \{E_{∗}^{K}\} form a subsystem in \{E_{K}\} and \overline{A} : E_{∗}^{K} \underrightarrow{∼}EΓ ; \overline{U} gives an isomorphism of this subsystem with the system \{EΓ ; \overline{U}\}. This allows us to identify (E\overline{U})^{∧} = lim_{\overrightarrow{K}} E^{Γ})\overline{U}, with the subspace E_{∗}^{∧} = lim_{\overrightarrow{K}}(E_{∗}^{K}) ⊂ lim_{\overrightarrow{K}}(E^{K}) =

E^{∧}. Clearly E_{∗}^{∧} = {ξ^{∧} ∈ E^{∧} | e_{K}ξ^{∧} ⊂ E_{∗}^{K} for all good K}.

Step 3. Let us prove that E_{∗}^{∧} = (E^{∧})^{U}. Indeed P ξ^{∧} ∈ E_{∗}^{∧} ⟺ for all good K e_{K}ξ^{∧} ∈ E_{∗}^{K}P ⟺ for all good K and all open compact subgroups C ⊂ U e_{K}ξ^{∧} ∈ e_{K}e_{c}Eℙ ⟺ for all C ⊂ V and all good K e_{K}ξ^{∧} ∈ e_{K}e_{C}EP ⟺ for all C ⊂ U, ξ^{∧} lies in the closure of e_{c}E ⟺ for all C ⊂ U, e_{C}ξ^{∧} = ξ^{∧}.

This last condition implies that ξ^{∧} is U-univariant. Conversely, suppose that ξ^{∧} is U-invariant and prove that for each C ⊂ U e_{C}ξ^{∧} = ξ^{∧}. Choose a small subgroup K ⊂ G normalized by C. Then the vector ξ = e_{K}ξ^{∧} is C-invariant which implies that e_{C}ξ = ξ. Hence e_{K}e_{C}ξ^{∧} = e_{C}e_{K}ξ^{∧} = e_{C}ξ = ξ = e_{K}ξ^{∧}. Since this is true for arbitrary small K, e_{C}ξ^{∧} = ξ^{∧}.

6.5. Second Adjointness of Functors i and r.

Theorem. Let (P, \overline{P}) be a parabolic pair, M = P ∩ \overline{P}. Then the functor i_{GM}^{P} : M(M) → M(G) is canonically left adjoint to the functor r\overline{P} ; MG : M(G) → M(M). In particular, for a standard Levi subgroup M < G the functor i_{GM} is left adjoint to \overline{r}_{MG}.

This theorem follows from Theorem 6.4 and the following form of Frobenius reciprocity.

Proposition. Let G be an ℓ-group (see...), H ⊂ G a closed subgroup. Define the induction functor ind : M(H) → M(G) as in ([ ]), i.e., for V ∈ M(H) we define G-module E = ind(G, H, V) as

E = {f : G → V | f(hg) = hf(g) for h ∈ H, support of f is compact modulo H and f is locally constant}.

Define the twisted induction functor ind^{Δ}(V)) = ind (V ⊗ Δ_{G}Δ_{H}^{−1}). Then for each V ∈ M(H), E ∈ M(G) there is a canonical functorial isomorphism

Hom_{G} (ind^{Δ}(V), E) = Hom_{G}(V, E^{∧}) .

In other words, the functor ind^{Δ} is left adjoint to the functor S, given by S(E) = H-smooth part of E^{∧}.

Proof of Proposition. Let S(G) be the space of locally constant compactly supported functions on G with left action of G. We have a canonical isomorphism H(G) = S(G) ⊗ Δ_{G}(f ⊗ M → f · M). We will identify S(G) with ind(G, 1, ℂ) (since G acts on ind(ℂ) from the right, this identification involves change g ↦ g^{−1}). By transitivity of induction we have ind(G, H, S(H)) = S(G).

This implies, that

ind^{Δ}(H(H)) = ind(S(H)·Δ_{G}·Δ_{H}^{−1}·Δ_{H}) = Δ_{G}·ind(S(H)) = Δ_{G}·S(G) = H(G) .

Since ind^{Δ} is an exact functor, preserving direct sums and H(H) is a projective generator of M(H), ind^{Δ}(V) = H(G) ⨂_{H(H)} V. This implies, that

Hom_{G}(ind^{Δ}(V), E) = Hom_{G} (H(G) ⨂_{H(H)} V, E) =

= Hom_{H} (V, Hom_{G}(H(G), E)) = Hom_{H}(V, E^{∧}) =
= Hom_{H}(V, S(E)) .

All isomorphisms above are canonical.

Remark. Let us describe explicitly morphism α : V → ind^{Δ}(V)^{∧}, corresponding to identity morphism of ind^{Δ}(V). For v ∈ V we define α(v) ∈ ind^{Δ}(V)^{∧} by condition, that for each open compact subgroup K ⊂ G the function f_{K} = e_{K}α(v) ∈ ind^{Δ}(V) has the following form and vanishes outside of HK and

f(hK) = he_{H∩K}v ⊗ M_{G} ⊗ M_{H}^{−1} (M_{G}(K)^{−1}M_{H}(H ∩ K)) .

where M_{G} ∈ Δ_{G}, M_{H} ∈ Δ_{H}.

Proof of the Theorem. Let V ⊂ M(M), E ∈ M(G). Using canonical isomorphisms Δ_{G}Δ_{P}^{−1} = Δ_{U}^{−1} and Δ_{U}^{−1} = Δ\overline{U} we have

Hom_{G} (i_{GM}^{P}(V), E) = Hom_{G} (ind^{Δ}(G, P, V ⊗ Δ_{U}^{1/2})E) =

= Hom_{P} (V ⊗ Δ_{U}^{1/2}, E^{∨}) = Hom_{M} (V ⊗ Δ_{U}^{1/2}, (E^{∨})^{U}) =

= Hom_{M} (V, E\overline{U} ⊗ Δ1/2 ; \overline{U}) = Hom_{M} (V, r\overline{P} ; MGE) .

Remark. Let us write explicitly morphism α : V → \overline{r}_{MG}i_{GM}V. Let v ∈ V. Choose a subgroup K, in a good position with respect to (P, \overline{P}), such that e_{r}v = v. Then α(v) is represented by M_{jf}^{1/2}, where f : G → V ⊗_{Δ} U^{−1/2} is

supported on P K and for k ∈ K f(K) = vM_{U}^{−1/2} M_{G}^{−1}(K)M_{P} (K ∩ P).

Here M_{U} ∈ D_{U}, M\overline{U} ∈ Δ\overline{U} are dual and M_{G} = M\overline{U} · M_{P}. In particular, M_{G}^{−1}(K) M_{P} (K ∩ P) = M−1 ; \overline{U}(K_{−}). Identifying M_{U}^{−1/2} with M1/2 ; \overline{U} we can write

∫_{\overline{U}} α(v) = _{(∫K_{−}v·M\overline{U} )} M−1 ; \overline{U}(K_{−}) = v .

This shows that α coincides with the morphism in the composition theorem, corresponding to the big cell P \overline{P} and the point w = 1 ∈ P \overline{P} (see.....).