Cohomological properties of the canonical globalizations of Harish-Chandra modules

Consequences of Theorems of Kashiwara-Schmid, Casselman, and Schneider-Stuhler

Ulrich Bunke^{∗}and Martin Olbrich^{†}

July 18, 1996

Abstract

In this note we draw consequences of theorems of Kashiwara-Schmid, Casselman, and Schneider-Stuhler. Canonical globalizations of Harish-Chandra modules can be considered as coefficient modules for cohomology groups with respect to cocompact discrete subgroups or nilpotent Lie algebras. We obtain finiteness and comparison theorems for these cohomology groups.

Contents

1 Introduction 1

2 The comparison theorem for n_{P} -cohomology 5

3 Γ-cohomology 7 4 Properties of the smooth globalization 8

∗Humboldt-Universität zu Berlin, Institut für Reine Mathematik (SFB288), Ziegelstr. 13a, Berlin 10099. E-mail:ubunke@mathematik.hu-berlin.de

†Humboldt-Universität zu Berlin, Institut für Reine Mathematik (SFB288), Ziegelstr. 13a, Berlin 10099. E-mail:olbrich@mathematik.hu-berlin.de

5 Cohomology of S-arithmetric groups 10

1 Introduction

This note is based on one simple idea which will be applied in several ways in order to deduce properties of cohomology groups with coefficients canonical globalizations of Harish-Chandra modules. Some of these result were demanded by the theory of Selberg zeta functions [6], [4].

In the present section we formulate the basic theorems of Kashiwara/Schmid and Casselman and state our main results. The proofs are contained in the subsequent sections. The (arithmetic) framework for the last section is quite different from the preceeding ones. Here we combine results of the previous sections with those of Schneider/Stuhler. In the present section we only state a special case of the results of the last section.

Let G_{C} be a connected, reductive algebraic group and G = G_{R} be a real form of G_{C}. Let K ⊂ G be a maximal compact subgroup. By Mod(g, K) we denote the abelian category of (g, K)-modules and by HC(g, K) the full subcategory of Harish-Chandra modules of (g, K). Let C^{∞}(G)_{K} ∈ Mod(g, K) be the space of smooth functions on G which are right K-finite. The left g- and G-module structures of C^{∞}(G)_{K} induce corresponding structures on Hom_{(g,K)}(V, C^{∞}(G)_{K}) for any V ∈ Mod(g, K). Let RHom_{(g,K)}(., C^{∞}(G)_{K}) be the derived functor of Hom_{(g,K)}(., C^{∞}(G)_{K}) from Mod(g, K) to complex vector spaces. The following theorem is due to Kashiwara-Schmid [10].

Theorem 1.1 If V_{K} ∈ HC(g, K), then

RHom_{(g,K)}(V_{K}, C^{∞}(G)_{K}) ≅ MG(Ṽ_{K}) ,  (1)

where we view the right-hand side of (1) as the complex which is zero except at the zeroth place where it is the maximal globalization MG(Ṽ_{K}) (defined in [12], [10]) of the dual Harish-Chandra module Ṽ_{K}. The isomorphism is compatible with the natural G- and g-module structures on both sides.

In fact Kashiwara-Schmid define a natural topology on the left-hand side such that (1) becomes an isomorphism in a certain derived category of topological vector spaces (see [10], Sec. 3).

Next we deduce a slight generalization of this theorem. We say that a complex of Fréchet representations of G has property MG, if its differentials have closed range and its cohomology groups are, as topological representations of G, isomorphic to the maximal globalizations of their underlying Harish-Chandra modules (compare with [10]).

In order to simplify the notation we introduce the abbreviation

F (V_{K}) := Hom_{(g,K)}(V_{K}, C^{∞}(G)_{K}) .

Let V_{K} ∈ Mod(g, K) be finitely generated (as U(g)-module). Then we can define a natural Fréchet topology on F (V_{K}) as follows. Let W ⊂ V_{K} be a finite-dimensional K-invariant generating subspace of V_{K}. Then we have a quotient

U(g) ⊗_{U(k)} W → V_{K} → 0

in Mod(g, K), and applying F we obtain the embedding

0 → F (V_{K}) → F (U(g) ⊗_{U(k)} W) = C^{∞}(G ×_{K} W̃) .

This embedding induces a Fréchet topology on F (V_{K}), which is independent of the choice of W (see [12]).

Lemma 1.2 Let

P : … → P_{−2} → P_{−1} → P_{0} → 0

be a complex of finitely generated, F -acyclic modules in Mod(g, K), such that H^{∗}(P) ∈ HC(g, K). Then F (P) has property MG and

H^{∗}(F (P)) = F (H^{∗}(P)) .  (2)

Proof. We decompose P into short exact sequences

0 → Z_{i} → P_{i} →^{d} B_{i+1} → 0  (3)
0 → B_{i} → Z_{i} → H_{i} → 0 .  (4)

Since U(g) is Noether the modules Z_{i} are finitely generated. Applying the functor F to these short exact sequences we obtain the long exact sequences

0 → F (B_{i+1}) →^{F (d)} F (P_{i}) → F (Z_{i}) → R^{1}F (B_{i+1}) …  (5)
0 → F (H_{i}) → F (Z_{i}) → F (B_{i}) → R^{1}F (H_{i}) ….  (6)

Note that P_{i} (by assumption) and H_{i} (by Theorem 1.1) are F -acyclic. Since Z_{0} = P_{0}, we conclude R^{q}F (B_{0}) = R^{q}F (Z_{0}) = 0 for q ≥ 1 using (6). Now by (5) we obtain R^{q}F (P_{−1}) = R^{q}F (Z_{−1}) = 0 and inductively

R^{q}F (Z_{i}) = R^{q}F (P_{i}) = 0

for all i and q ≥ 1. From (3) we obtain

0 → F (B_{i+1}) →^{F (d)} F (P_{i}) → F (Z_{i}) → 0  (7)
F (Z_{i}) = F (P_{i})/F (B_{i+1})  (8)

and from (4)

0 → F (H_{i}) → F (Z_{i}) → F (B_{i}) → 0  (9)
0 → F (H_{i}) → F (P_{i})/F (B_{i+1}) → F (B_{i}) → 0 .  (10)

Thus F (H^{i}) is the middle cohomology of the sequence

0 → F (B_{i+1}) →^{F (d)} F (P_{i}) →^{α} F (B_{i}) → 0 .  (11)

Since ker(α) = ker(F (d̃) : F (P_{i}) → F (P_{i−1}) (where d̃: P_{i−1} → P_{i})) we have identified H^{∗}(F (P)) with F (H^{∗}(P)).

The complex (7) is an exact complex of Fréchet spaces and thus (8) represents F (Z_{i}) as a topological quotient. Moreover, (9) and (10) are exact complexes of Fréchet spaces, and the differential F (d) of (11) has closed range. Since F (H_{i}) = MG(F (H_{i})_{K}), the complex (11) has property MG. We conclude that F (P) has property MG, too. □

In this note we derive the following two consequences of Theorem 1.1 and its strengthening Lemma 1.2.

Let P = M_{P} A_{P} N_{P} be a Langlands decomposition of a parabolic subgroup P ⊂ G such that k ⊥ a_{P}, and K_{P} := M_{P} ∩ K, where m_{P} , a_{P} , n_{P} denote the Lie algebras of M_{P} , A_{P} , N_{P}. By Hecht-Schmid [8], if V_{K} ∈ HC(g, K), then the Lie algebra cohomology with respect to n_{P} satisfies

H^{∗}(n_{P} , V_{K}) ∈ HC(m_{P} ⊕ a_{P} , K_{P}).

If V is a representation of n_{P} on a topological vector space, then we equip H^{∗}(n_{P} , V) with the topology induced from the standard n_{P} -cohomology complex. For V_{K} ∈ HC(g, K) let mg(V_{K}) denote the minimal globalization of V_{K}.

Theorem 1.3 For V_{K} ∈ HC(g, K) we have topological isomorphisms

MG(H̃_{∗}(n_{P} , Ṽ_{K})) = H^{∗}(n_{P} , MG(V_{K}))  (12)
mg(H^{∗}(n_{P} , V_{K})) = H^{∗}(n_{P} , mg(V_{K})) .  (13)

In particular, the differentials of the standard n_{P} -cohomology complexes of mg(V_{K}) and MG(V_{K}) have closed range.

This results was also announced by Hecht-Taylor and Bratten. These authors use the technique of analytic localization (see [9], [3]) which is quite different from the method employed here.

Let Γ ⊂ G be a discrete, torsion-free, and cocompact subgroup. Then

L^{2}(Γ\G) = ⨁_{π∈Ĝ_{u}}^{Hilbert} N_{Γ}(π)V_{π},

where Ĝ_{u} is the unitary dual of G and N_{Γ}(π) ∈ N_{0} is the multiplicity of π in L^{2}(Γ\G). Let V_{π,K} ∈ HC(g, K) be the underlying Harish-Chandra module of V_{π} and Ann(V_{π,K}) ⊂ Z(g) be the annihilator of V_{π,K} in the center Z(g) of the universal enveloping algebra U(g) of g.

Theorem 1.4 For V_{K} ∈ HC(g, K) and a discrete, cocompact, torsion-free subgroup Γ ⊂ G we have

H^{∗}(Γ, MG(V_{K})) = ⨁ N_{Γ}(π)Ext_{(g,K)}^{∗}(Ṽ_{K}, V_{π,K}) ,  (14)
dim H^{∗}(Γ, MG(V_{K})) < ∞ ,  (15)
H^{∗}(Γ, mg(V_{K}))^{∗} = H^{dim(G/K)−∗}(Γ, MG(V_{K})),  (16)

where the sum in (14) is over the finite set of π ∈ Ĝ_{u} with Ann(V_{K}) ⊂ Ann(V_{π,K}).

The proofs of Theorems 1.3 and 1.4 are based on similar ideas and are the contents of Sections 2 and 3.

Besides the maximal and minimal globalizations there are further canonical globalizations V_{±∞} of a Harish-Chandra module V_{K}. Here V_{∞} is the space of smooth vectors in any Banach globalization of V_{K} and V_{−∞} is the space of distribution vectors (see [15], Ch.11). As above let P ⊂ G be a parabolic subgroup. The following theorem was shown by Casselman.

Theorem 1.5 The differentials of the standard n_{P} -homology complex of V_{∞} have closed range and we have topological isomorphisms

H_{∗}(n_{P} , V_{K})_{∞} = H_{∗}(n_{P} , V_{∞}) .

This implies

H^{∗}(n_{P} , V_{K})_{∞} = H^{∗}(n_{P} , V_{∞})  (17)
(H̃_{∗}(n_{P} , Ṽ_{K}))_{−∞} = H^{∗}(n_{P} , V_{−∞}).  (18)

In the present paper we show the following consequences of Theorem 1.5.

2 THE COMPARISON THEOREM FOR N_{P} -COHOMOLOGY 5

Theorem 1.6 Let Γ ⊂ G be a discrete, torsion-free, cocompact subgroup and V_{K} ∈ HC(g, K). Then dim H^{∗}(Γ, V_{±∞}) < ∞ and Poincaré duality holds:

H^{∗}(Γ, V_{∞})^{∗} = H^{dim(G/K)−∗}(Γ, V_{−∞}) .  (19)

Moreover we have H^{∗}(Γ, V_{∞}) ≅ H^{∗}(Γ, mg(V_{K})) and H^{∗}(Γ, V_{−∞}) ≅ H^{∗}(Γ, MG(V_{K})).

Theorem 1.7 Let W_{K_{P}} ∈ HC(m_{P} ⊕ a_{P} , K_{P}) and W_{∞} its smooth globalization. Then there is a spectral sequence with second term

E_{2}^{p,q} = ⨁ N_{Γ}(π)H^{p}(m_{P} ⊕ a_{P} , K_{P} , H^{q}(n_{P} , V_{π,∞} ⊗ W_{∞}))

converging to H^{p+q}(Γ, Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞}). Here the sum is over the finite set of π ∈ Ĝ_{u} with γ^{−1}(Ann(W_{K_{P}})) ⊂ Ann(V_{π,K}), where γ : Z(g) → Z(m_{P} ⊕a_{P}) is the Harish-Chandra homomorphism, and ⊗ is the injective topological tensor product.

A similar spectral sequence exists in a hyperfunction setting and was employed in [4], [6].

Finally we decribe a special case of our results of section 5. We adopt a slight change of the notation. Let G be a reductive, connected algebraic group defined over Q. Moreover, let S be a finite set of places of Q including the infinite place ω. We combine the results of Kashiwara-Schmid und Schneider-Stuhler [13] in order to prove finiteness results for the cohomology of S-arithmetic groups with coefficients in infinite-dimensional representations. A special case is the following theorem.

Theorem 1.8 Assume that rank_{Q}(G) = 0. If Γ ⊂ G(Q) is S-arithmetic and V_{K} ∈ HC(g, K), then H^{∗}(Γ, MG(V_{K})) is finite-dimensional.

Acknowledgement : While working on this paper the authors were supported by the Sonderforschungsbereich 288 ”Differential Geometrie und Quantenphysik” founded bu the DFG. We W. Schmid and A. Deitmar for helpful comments on this paper. We also thank W. Casselman and T. Bratten for communicating their unpublished results.

2 The comparison theorem for n_{P}-cohomology

We use the notation introduced in Section 1. We consider C^{∞}(G)_{K} as a (n_{P} , (g, K))- bimodule. Then Hom_{(g,K)}(., C^{∞}(G)_{K}) is a functor from Mod(g, K) with values in the category n_{P} -mod of n_{P} -modules. If I is an injective (n_{P} , (g, K))-bimodule, then Hom_{(g,K)}(V_{K}, I) is an injective n_{P} -module for any V_{K} ∈ Mod(g, K). Similarly, the functor n_{P}−inv of taking n_{P} -invariants carries (injective) (n_{P} , (g, K))-bimodules to (injective) (g, K)-modules. Let Rn_{P}−inv be the right-derived functor of n_{P}−inv. Then we have a natural isomorphism

Rn_{P}−inv ∘ RHom_{(g,K)}(., C^{∞}(G)_{K})
= R(n_{P}−inv ∘ Hom_{(g,K)}(., C^{∞}(G)_{K})).  (20)

Since G → N_{P} \G is a fibration with fibres which are isomorphic to an Euclidean space we have Rn_{P}−inv(C^{∞}(G)_{K}) = C^{∞}(N_{P} \G)_{K}. Inserting this into (20) we obtain

Rn_{P}−inv ∘ RHom_{(g,K)}(., C^{∞}(G)_{K})
= RHom_{(g,K)}(., C^{∞}(N_{P} \G)_{K}).

Let now V_{K} ∈ HC(g, K). Then by Theorem 1.1 we have

Rn_{P}−inv(MG(V_{K})) = Rn_{P}−inv ∘ RHom_{(g,K)}(Ṽ_{K}, C^{∞}(G)_{K})
= RHom_{(g,K)}(Ṽ_{K}, C^{∞}(N_{P} \G)_{K})
= RHom_{(g,K)}(Ṽ_{K}, Ind_{p,K_{P}}^{g,K} (C^{∞}(M_{P} A_{P})_{K_{P}}))
= RHom_{(p,K_{P})}(F_{p,K_{P}}^{g,K} (Ṽ_{K}), C^{∞}(M_{P} A_{P})_{K_{P}}),  (21)

where F_{p,K_{P}}^{g,K} is the forgetful functor, p is the Lie algebra of P, and we applied Shapiro’s Lemma (see [11], Thm. 7.4.2), in order to obtain the last line. n_{P} ⊂ p acts trivially on C^{∞}(M_{P} A_{P})_{K_{P} −finite}. Let n_{P}−coinv(V_{K}) := V_{K}/n_{P} V_{K} be the functor taking n_{P} -coinvariants and Ln_{P}−coinv be its left-derived functor. Since n_{P}−coinv maps (finitely generated, projective) (p, K_{P})-modules to (finitely generated, projective) (m_{P} ⊕ a_{P})- modules we have for W_{K_{P}} ∈ Mod(p, K_{P})

RHom_{(p,K_{P})}(W_{K_{P}} , C^{∞}(M_{P} A_{P})_{K_{P}})  (22)
= RHom_{(m_{P} ⊕a_{P} ,K_{P})}(Ln_{P}−coinv(W_{K_{P}}), C^{∞}(M_{P} A_{P})_{K_{P}}).

Put W_{K} := F_{p,K_{P}}^{g,K} (Ṽ_{K}). There is a projective resolution

P : … → P_{−2} → P_{−1} → P_{0} → Ṽ_{K} → 0

such that P_{i} = U(g) ⊗_{U(k)} E_{i}, where E_{i} are finite-dimensional representations of K. Then Ln_{P}−coinv(Ṽ_{K}) = n_{P}−coinv(P). Since n_{P}−coinv(P_{i}) ∈ Mod(m_{P} ⊕ a_{P} , K_{P}) are projective and finitely generated, and H^{∗}(n_{P}−coinv(P)) = H_{−∗}(n_{P} , V_{K}) ∈ HC(m_{P} ⊕ a_{P} , K_{P}) the complex n_{P}−coinv(P) satisfies the assumption of Lemma 1.2 (for M_{P} A_{P} in the place of G). We conclude that

Rn_{P}−inv(MG(V_{K})) = RHom_{(m_{P} ⊕a_{P} ,K_{P})}(n_{P}−coinv(P), C^{∞}(M_{P} A_{P})_{K_{P}})

and the right-hand side has property MG. We also obtain from (2)

H^{∗}(n_{P} , MG(V_{K})) = H^{∗}RHom_{(m_{P} ⊕a_{P} ,K_{P})}(n_{P}−coinv(P), C^{∞}(M_{P} A_{P})_{K_{P}})
= Hom_{(m_{P} ⊕a_{P} ,K_{P})}(H^{−∗}(n_{P}−coinv(P)), C^{∞}(M_{P} A_{P})_{K_{P}})
= MG(H̃_{∗}(n_{P} , Ṽ_{K}))

and thus (12).

As in [4], Lemma 4.3, we conclude that the differentials of the standard n_{P} -cohomology complex of MG(Ṽ_{K})

0 → MG(Ṽ_{K}) → MG(Ṽ_{K}) ⊗ Λ^{1}n_{P}^{∗} → MG(Ṽ_{K}) ⊗ Λ^{2}n_{P}^{∗} →

have closed range, too. The topological dual of the latter complex (up to a shift) is isomorphic in Mod(M_{P} A_{P}) to the standard n_{P} -cohomology complex of mg(V_{K}) tensored with Λ^{d}n_{P}, d := dim(n_{P}). It follows that

H^{∗}(n_{P} , MG(Ṽ_{K}))^{∗} = H^{d−∗}(n_{P} , mg(V_{K})) ⊗ Λ^{d}n_{P} .

We obtain using (12) that

mg(H^{∗}(n_{P} , V_{K})) = MG(H̃^{∗}(n_{P} , V_{K}))^{∗}
= MG(H̃_{d−∗}(n_{P} , V_{K}) ⊗ Λ^{d}n_{P})^{∗}
= (H^{d−∗}(n_{P} , MG(Ṽ_{K})) ⊗ Λ^{d}n_{P})^{∗}
= H^{∗}(n, mg(V_{K})) .

This implies (13) and finishes the proof of Theorem 1.3.

3 Γ-cohomology

Let Γ ⊂ G be a discrete, torsion-free, and cocompact subgroup. We consider C^{∞}(G)_{K} as a (Γ, (g, K))-bimodule. If I is an injective (Γ, (g, K))-bimodule and V ∈ Mod(g, K), then Hom_{(g,K)}(V, I) is an injective Γ-module. Let Γ−inv be the functor of taking Γ-invariants. If I is an injective bimodule, then Γ−inv(I) is an injective (g, K)-module. Since Γ acts freely on G we have

RΓ−inv(C^{∞}(G)_{K}) = C^{∞}(Γ\G)_{K} ,

where RΓ−inv is the right-derived functor of Γ−inv (compare [4], Lemma 2.4). We hence have natural isomorphisms

RΓ−inv ∘ RHom_{(g,K)}(., C^{∞}(G)_{K})
= RHom_{(g,K)}(., RΓ−inv(C^{∞}(G)_{K}))  (23)
= RHom_{(g,K)}(., C^{∞}(Γ\G)_{K}) .

If V_{K} ∈ HC(g, K), then by Theorem 1.1 we have

RΓ−inv(MG(V_{K})) = RHom_{(g,K)}(Ṽ_{K}, C^{∞}(Γ\G)_{K}) .

We decompose

L^{2}(Γ\G) = ⨁_{π∈Ĝ_{u}}^{Hilbert} N_{Γ}(π)V_{π}  (24)
= ⨁_{Ann(V_{K})⊂Ann(V_{π,K})} N_{Γ}(π)V_{π} ⊕ R ,  (25)

where R is the Hilbert sum of all summands of (24) not occuring in the first summand of (25). The sum in (25) is finite. Now we take the smooth and K-finite vectors and obtain

C^{∞}(Γ\G)_{K} = ⨁_{Ann(V_{K})⊂Ann(V_{π,K})} N_{Γ}(π)V_{π,K} ⊕ R_{∞,K}.

As in the proof of Lemma 6.3 [4] we show that RHom_{(g,K)}(Ṽ_{K}, R_{∞,K}) = 0, and we conclude

RΓ−inv(MG(V_{K})) = ⨁_{Ann(V_{K})⊂Ann(V_{π,K})} N_{Γ}(π)RHom_{(g,K)}(Ṽ_{K}, V_{π,K}) .

Since

dim Ext_{(g,K)}^{∗}(W_{K}, U_{K}) < ∞, U_{K}, W_{K} ∈ HC(g, K)

we see that H^{∗}(Γ, MG(V_{K})) is finite-dimensional. This proves (14) and (15).

There is a projective resolution of Ṽ_{K}

P : … → P_{1} → P_{0} → Ṽ_{K} → 0

in Mod(g, K) with P_{i} = U(g) ⊗_{U(k)} E_{i}, where E_{i} are finite-dimensional representations of K (see [2]). Thus Q := Hom_{(g,K)}(P, C^{∞}(G)_{K}) is a complex of Fréchet spaces (spaces of smooth sections of vector bundles over G/K) with continuous differentials (invariant differential operators). The complex Q is a Γ-acyclic resolution (by Theorem 1.1) of MG(V_{K}). Let ^{Γ}Q be the complex of Γ-invariants. The spectral theory of locally invariant differential operators on bundles over the compact locally symmetric space Γ\G/K shows that the differentials of ^{Γ}Q have closed range. Now we can prove Poincaré duality as in [4], Sec. 5.2. This implies (16) of Theorem 1.4.

4 Properties of the smooth globalization

Let W_{K_{P}} ∈ HC(m_{P} ⊕ a_{P} , K_{P}) and W_{∞} be its smooth globalization. We consider the topological tensor product of P -modules U := C^{∞}(G) ⊗ W_{∞} as a (Γ, P)-bimodule, Since C^{∞}(G) is nuclear it is not nessecary to specify the tensor product but take the injective one for explicitness. The Γ-module U is acyclic with respect to the functor Γ−inv. In fact the proof of [4], Lemma 2.4, carries over to the present situation.

Note that U is a smooth P -module but in general not a (p, K_{P})-module since its vectors need not be K_{P} -finite. Nevertheless we want to consider the right-derived functor of (p, K_{P})−inv on modules like U. By abuse of notation let R(p, K_{P})−inv(U) denote the concrete relative Lie-algebra cohomology complex

0 → U^{K_{P}} → (Λ^{1}(p/k_{P})^{∗} ⊗ U)^{K_{P}} → (Λ^{2}(p/k_{P})^{∗} ⊗ U)^{K_{P}} → ….  (26)

The smooth P -module U is (p, K_{P})−inv-acyclic in the sense that (26) is exact. This can be seen by recognising in (26) the de Rham complexes along the contractible P -orbits in G/K_{P} (compare Deitmar, [6], Lemma 4.1).

Note that

Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞} = (p, K_{P})−inv(U)

(again note the abuse of notation on the right hand side) and hence Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞} = R(p, K_{P})−inv(U) in the derived category of Γ-modules. R(p, K_{P})−inv(U) is a complex of Γ−inv-acyclic Γ-modules. Hence

RΓ−inv(Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞}) = RΓ−inv ∘ R(p, K_{P})−inv(U)
= R(p, K_{P})−inv(RΓ−inv(U))
= R(p, K_{P})−inv(C^{∞}(Γ\G) ⊗ W_{∞}) ,  (27)

where ⊗ is again the injective topological tensor product. We insert the finite, continuous decomposition

C^{∞}(Γ\G) = ⨁_{γ^{−1}Ann(W_{K_{P}})⊂Ann(V_{π,K})} N_{Γ}(π)V_{π,∞} ⊕ R_{∞} .

The action of D ∈ Z(g) on H^{∗}(Γ, Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞}) is either induced by the action of D on C^{∞}(G) or of γ(D) on W_{∞}. Applying again the technique of the proof of Lemma 6.3 in [4] we show that the summand R_{∞} does not contribute to the Γ-cohomology, and we have

RΓ−inv(Ind_{p,K_{P}}^{g,K} (W_{K_{P}})_{∞}) = ⨁_{γ^{−1}Ann(W_{K_{P}})⊂Ann(V_{π,K})} N_{Γ}(π)R(p, K_{P})−inv(V_{π,∞} ⊗ W_{∞}) .

Using the extension

0 → n_{P} → p → m_{P} ⊕ a_{P} → 0

we can write

R(p, K_{P})−inv(V_{π,∞} ⊗ W_{∞}) = R(m_{P} ⊕ a_{P} , K_{P})−inv(Rn_{P}−inv(V_{π,∞} ⊗ W_{∞}))

viewed as an explicit double complex. The columns are standard n_{P} -cohomology complexes. Let E_{2}^{p,q} = H^{p}(m_{P} ⊕ a_{P} , K_{P} , H^{q}(n_{P} , V_{π,∞} ⊗ W_{∞})) be the second stage of the spectral sequence converging to H^{p+q}(p, K_{P} , V_{π,∞} ⊗ W_{∞}). Theorem 1.7 follows.

We now prove Theorem 1.6. It is another theorem of Casselman (see [14], 3.8.3) that any module V_{K} ∈ HC(g, K) is a submodule as well as a quotient of a principal series representation associated to the minimal parabolic subgroup P_{0} of G . Using the long exact Γ-cohomology sequences it is easy to reduce the proof of Theorem 1.6 to the case of principal series representations associated to P_{0}. Thus 1.6 is a consequence of 1.7. In fact in this case W_{K_{P}} is finite-dimensional and the spectral sequence reduces to

E_{2}^{p,q} = ⨁ N_{Γ}(π)H^{p}(m_{P} ⊕ a_{P} , K_{P} , H^{q}(n_{P} , V_{π,K}) ⊗ W_{K_{P}})

(P = P_{0}) and thus consists of finite-dimensional vector spaces. The considerations above also imply that the differentials of (27) have closed range. Now Poincaré duality can be proved as in [4], Sec. 5.2. This finishes the proof of Theorem 1.6.

5 Cohomology of S-arithmetric groups

In this section we will change the notation of the groups slightly. By G we denote a reductive, connected algebraic group defined over Q. Moreover, let S be a finite set of places of Q including the infinite place ω. Let S_{f} := S \ {ω} the set of finite places of S.

The goal of this section is to prove the finiteness of the cohomology of an S-arithmetic group Γ ⊂ G(Q) with coefficients in an element of the class of representations which we will now describe.

For p ∈ S_{f} let C_{p} be the connected center of G(Q_{p}), χ_{p} : C_{p} → C^{∗} be a continuous character and Alg_{χ_{p}}(G(Q_{p})) be the category of smooth representations of G(Q_{p}) with central character χ_{p}. Fix an admissible V_{p} ∈ Alg_{χ_{p}}(G(Q_{p})) for each p ∈ S_{f}. Moreover, let V_{K} ∈ HC(g, K) and V_{−ω} be its maximal globalization (in the notation of the preceding sections this was MG(V_{K})). Then there is a natural representation of

G(R) × ∏_{p∈S_{f}} G(Q_{p})

on

W := Hom_{C} ( ⨂_{p∈S_{f}} V_{p}, V_{−ω})

given by (gϕ)(⊗v_{p}) = g_{ω}ϕ(⊗g_{p}^{−1}v_{p}), where g = (g_{ω}, (g_{p}|p ∈ S_{f})) and v_{p} ∈ V_{p}. By the usual diagonal embedding

Γ ↪ G(R) × ∏_{p∈S_{f}} G(Q_{p})

W becomes a Γ-module.

Theorem 5.1 Suppose that rank_{Q}(G) = 0. Then H^{∗}(Γ, W) is finite-dimensional.

The proof of the theorem also gives a way for computing the cohomology groups in terms of automorphic forms. In the case that V_{K} is finite-dimensional and V_{p} is trivial for all p ∈ S_{f}, this result has been obtained by Borel-Serre [1] (without the condition on rank_{Q}(G)).

The method should also work in the case rank_{Q}(G) ≥ 0. In this case one has to replace the maximal globalization V_{−ω} by the smooth globalization V_{−∞}. Then H^{∗}(Γ, W) would be finite-dimensional without any condition on rank_{Q}(G). The proof of the latter result should follow from unpublished recent results of Kashiwara-Schmid which were announced to us by Schmid and the results of Franke [7]. Using these results we currently can prove 5.1 in the case that rank_{Q}(G) = 1. If rank_{R}(G) = 1 and G is semisimple, then one can alternatively use the results of [5]. We will carry out this program in a paper which is in preparation.

Since a proof of the possible generalizations involves techniques which are very different from those used in the present paper we are going to prove the theorem as stated. It heavily depends on the result of Kashiwara-Schmid [10], Thm. 1.1, and results obtained by Schneider-Stuhler [13].

Proof.

Let X_{p} be the Bruhat-Tits building of G(Q_{p}). Schneider-Stuhler [13] construct a system of compact open subgroups U_{F}^{(e)} ⊂ G(Q_{p}), F a facet of X_{p}, e ∈ N_{0}, satisfying

1. gU_{F}^{(e)}g^{−1} = U_{gF}^{(e)}, ∀g ∈ G(Q_{p}), F a facet of X_{p}.

2. {U_{F}^{(e)}}_{e∈N_{0}} is a fundamental system of neighbourhoods of 1 ∈ G(Q_{p}).

3. If F ⊂ F̄′, then U_{F}^{(e)} ⊂ U_{F′}^{(e)}.

An orientation of a q ≥ 1-dimensional facet F ⊂ X_{p} is a generator c ∈ H_{q}(X_{p}^{q}, X_{p}^{q} \ F) ≅ Z, where X_{p}^{q} is the q-skeleton of X_{p}. By X_{p}^{(q)} we denote the set of all oriented q-facets (F, c) if q ≥ 1, and X_{p}^{(0)} denotes the set of vertices of X_{p}. If F, F′ ⊂ X_{p} are facets, dim(F) = q + 1, dim(F′) = q, then there is a natural map

∂_{F′}^{F} : H_{q+1}(X_{p}^{q+1}, X_{p}^{q+1} \ F) → H_{q}(X_{p}^{q}, X_{p}^{q} \ F′)

which is an isomorphism if F′ ⊂ F̄ and zero otherwise.

For all p ∈ S_{f} we choose e_{p} ∈ N_{0} sufficiently large such that V_{p} is generated by V_{p}^{U_{F}^{(e)}} for some facet F ⊂ X_{p}. This is possible by property 2. since V_{p} is admissible. Let e := max_{p∈S_{f}} e_{p}.

Let J_{f} be the set of all tuples l_{f} = (l_{p}|p ∈ S_{f}) with l_{p} ∈ {0, …, dim(X_{p})}. For l_{f} ∈ J_{f} set Z^{l_{f}} := ∏_{p∈S_{f}} X_{p}^{(l_{p})}. If x ∈ Z^{l_{f}}, then the stabilizer Γ_{x} ⊂ Γ of x is an arithmetic group acting cocompactly (rank_{Q}(G) = 0!) on the Riemannian symmetric space X_{ω} := G(R)/K(R). Let d(x) be the diameter of Γ_{x}\X_{ω}. Since Γ\ ∪_{l_{f} ∈J_{f}} Z^{l_{f}} is finite and d(gx) = d(x), we have

d := max_{x∈∪_{l_{f} ∈J_{f}} Z^{l_{f}}} d(x) < ∞ .

Let U ⊂ X_{ω} be the ball of radius d + 1 centered at the origin. Let J be the set of all tuples l = (k, l_{ω}, l_{f}), where k, l_{ω} ∈ N_{0}, l_{f} ∈ J_{f}. For l ∈ J and x ∈ Z^{l_{f}} let Z_{x}^{l} be the set of tuples A = (g_{0}U, …, g_{l_{ω}}U) with |A| := g_{0}U ∩ … ∩ g_{l_{ω}}U ≠ ∅, where g_{i} ∈ Γ_{x}. Set Z^{l} = ∪_{x∈Z^{l_{f}}} Z_{x}^{l} and denote a generic point of Z^{l} by (A, x), x ∈ Z^{l_{f}}, A ∈ Z_{x}^{l}.

Let

P^{.} : … → P_{2} → P_{1} → P_{0} → Ṽ_{K} → 0  (28)

be a projective resolution of Ṽ_{K} in the category Mod(g, K) with P_{i} of the special form U(g) ⊗_{U(k)} L_{i}, where L_{i} is a finite-dimensional representation of K(R). Let V^{k} be the sheaf of smooth sections of the homogeneous bundle G(R) ×_{K(R)} L_{k}. Sheafification of Hom_{(g,K)}(P^{.}, C^{∞}(G)_{K}) leads to a complex of sheaves

0 → V^{0} →^{d_{SK}} V^{1} →^{d_{SK}} V^{2} →^{d_{SK}} …,  (29)

where d_{SK} are differential operators.

For any l ∈ J, l = (k, l_{ω}, l_{f}) we define C^{l} to be the vector space of all

ϕ ∈ ∏_{(A,x)∈Z^{l}} Hom ( ⨂_{p∈S_{f}} V_{p}^{U_{Fp}^{(e)}}, V^{k}(|A|))

which are odd with respect to the change of the orientations of the facets (F_{p}, c_{p}) (x = ((F_{p}, c_{p})|p ∈ S_{f})) and with respect to permutations of the components of A.

The differentials d_{SK} induce differentials

d_{SK} : C^{l} → C^{l′} ,

where l = (k, l_{ω}, l_{f}) and l′ = (k + 1, l_{ω}, l_{f}). For any x ∈ Z^{l_{f}} the cover {gU}_{g∈Γ_{x}} leads to Čech complexes for the sheaves V^{k}. The Čech differential induces a differential

d_{ω} : C^{l} → C^{l′} ,

where l = (k, l_{ω}, l_{f}) and l′ = (k, l_{ω} + 1, l_{f}). For any p ∈ S_{f} we define a differential

d_{p} : C^{l} → C^{l′} ,

where l = (k, l_{ω}, l_{f}) and l′ = (k, l_{ω}, l′_{f}), l_{f} = (l_{q}|q ∈ S_{f}), l′_{f} = (l′_{q}|q ∈ S_{f}), l_{q} = l′_{q}, ∀q ∈ S_{f} \{p} and l′_{p} = l_{p}+1 as follows: Let ϕ ∈ C^{l} and (A, x′) ∈ Z^{l′}, x′ = ((F′_{q}, c′_{q})|q ∈ S_{f}),

⊗v_{q} ∈ ⨂_{q∈S_{f}} V_{q}^{U_{F′_{q}}^{(e)}}. Then we set

(d_{p}ϕ)_{(A,x′)}(⊗v_{q}) := ∑_{(F_{p},c_{p})∈X_{p}^{(lp)},∂_{Fp}^{F′_{p}}c′_{p}=c_{p}} ϕ_{(A,x)}(⊗v_{q}) ,

where x = ((F_{p}, c_{p})|p ∈ S_{f}) with (F_{q}, c_{q}) = (F′_{q}, c′_{q}) and ∀q ∈ S_{f} \ {p} . The differential d_{p} is well defined by property 3. of the system {U_{F_{q}}^{(e)}} and satisfies d_{p}^{2} = 0 (see [13]).

The system of spaces C^{l}, l ∈ J, and differentials d_{SK}, d_{q}, q ∈ S, forms a multicomplex.

Lemma 5.2 The total complex associated to ((C^{l}|l ∈ J), (d_{SK}, (d_{q}|q ∈ S)))

0 → W → D^{0} → D^{1} → D^{2} → …  (30)

is a Γ-acyclic resolution of W.

Proof. The action of Γ on C^{l} is defined by

(gϕ)_{(A,x)}(⊗v_{q}) := g^{∗}(ϕ_{(g^{−1}A,g^{−1}x)}(⊗g^{−1}v_{q}), ϕ ∈ C^{l} .

Since g^{−1}A ∈ Z_{g^{−1}x}^{l}, g^{−1}v_{q} ∈ V_{q}^{U_{g^{−1}Fq}^{(e)}} (by property 1. of the system {U_{F_{q}}^{(e)}}),

ϕ_{(g^{−1}A,g^{−1}x)}(⊗g^{−1}v_{q}) ∈ V^{k}(g^{−1}|A|) ,

and the action C^{l} ∋ ϕ ↦ gϕ ∈ C^{l} is well defined.

Let Z̆^{l_{f}} be the finite set of representatives of Γ\Z^{l_{f}}. For any x ∈ Z̆^{l_{f}} let Z̆_{x}^{l} be the finite set of representatives of Γ_{x}\X_{x}^{l}. Then the Γ-module C^{l} has the description

C^{l} = ⨁_{x∈Z̆^{l_{f}}} Ind_{{1}}^{Γ} ⨁_{(A,x)∈Z̆_{x}^{l}} Hom ( ⨂_{q∈S_{f}} V_{q}^{U_{Fq}^{(e)}}, V^{k}(|A|))

(x = ((F_{p}, c_{p})|p ∈ S_{f})), and this module is Γ-acyclic by Shapiro’s Lemma.

In order to finish the proof of the lemma we must show exactness of (30). Let l_{i} := (k, i, l_{f}). Since V^{k} are soft sheaves, the Čech complex

0 → C^{l_{0}} →^{d_{ω}} C^{l_{1}} →^{d_{ω}} C^{l_{2}} →^{d_{ω}} …  (31)

is exact in degree ≥ 1 and its zeroth cohomology is

C^{k,l_{f}} := ∏_{x∈Z^{l_{f}}} Hom ( ⨂_{q∈S_{f}} V_{q}^{U_{Fq}^{(e)}}, V^{k}(X_{ω})) ,

where x = ((F_{p}, c_{p})|p ∈ S_{f}).

Note that ⨂_{q∈S_{f}} V_{q}^{U_{Fq}^{(e)}} is finite-dimensional. We can apply Theorem 1 in order to conclude that for each l_{f} ∈ J_{f} the complex

0 → C^{0,l_{f}} →^{d_{SK}} C^{1,l_{f}} →^{d_{SK}} C^{2,l_{f}} →^{d_{SK}} …

is acyclic in degree ≥ 1 and its zeroth cohomology is

C^{l_{f}} := ∏_{x∈Z^{l_{f}}} Hom ( ⨂_{q∈S_{f}} V_{q}^{U_{Fq}^{(e)}}, V_{−ω}) ,

where x = ((F_{p}, c_{p})|p ∈ S_{f}).

Standard homological algebra shows that the cohomology of the total complex associated to the multicomplex ((C^{l_{f}} |l_{f} ∈ J_{f}), (d_{p}|p ∈ S_{f})) is equal to the cohomology of the complex (30).

For q ∈ S_{f} let S_{f}^{q} := {p ∈ S_{f}|q ≥ p}, J_{f}^{q} := {l_{f}^{q} = (l_{p}^{q}|p ∈ S_{f}^{q}), l_{p}^{q} ∈ {0, …, dim(X_{p})}}. Assume that we have already shown that the cohomology of the total complex associated to the multicomplex ((C^{l_{f}^{q}} |l_{f}^{q} ∈ J_{f}^{q}), (d_{p}|p ∈ S_{f}^{q})) is equal to the cohomology of the complex (30). Let r ∈ S_{f} be the largest prime smaller than q. Then we define ((C^{l_{f}^{r}} |l_{f}^{r} ∈ J_{f}^{r}), (d_{p}|p ∈ S_{f}^{r})) as the zeroth cohomology of

0 → C^{(l_{f}^{r} ,0)} →^{d_{q}} C^{(l_{f}^{r} ,1)} →^{d_{q}} C^{(l_{f}^{r} ,2)} →^{d_{q}} ….  (32)

If we could prove that (32) is acyclic in degree ≥ 1, then we would know that the cohomology of the total complex associated to the multicomplex ((C^{l_{f}^{r}} |l_{f}^{r} ∈ J_{f}^{r}), (d_{p}|p ∈ S_{f}^{r})) is equal to the cohomology of the complex (30). We reduce this induction step to the result of Schneider-Stuhler.

Let E_{p}^{i} be the space of all functions ω : X_{p}^{(i)} → V_{p} satisfying • ω has finite support,

• ω((F, c)) ∈ V_{p}^{U_{F}^{(e)}},

• if i ≥ 1, then ω((F, c)) = −ω((F, −c)).

Define the differential ∂_{p} : E_{p}^{i+1} → E_{p}^{i} by

(∂_{p}ω)((F′, c′)) = ∑_{(F,c)∈X_{p}^{(i+1)},∂_{F′}^{F}(c)=c′} ω((F, c)) .

Then Schneider-Stuhler (Theorem 1. and Corollary 2.,[13]) have shown that

E_{p}^{.} : … →^{∂_{p}} E_{p}^{2} →^{∂_{p}} E_{p}^{1} →^{∂_{p}} E_{p}^{0} → 0

is acyclic in degree ≠ 0 and its zeroth cohomology is isomorphic to V_{p}. This isomorphism is induced by ϵ : E_{p}^{0} → V_{p} given by ϵ(ω) := ∑_{F ∈X_{p}^{(0)}} ω(F).

We claim that the complex (32) is isomorphic to

_{y=((F_{p},c_{p})|p∈S_{f}^{r})∈∏_{p∈S^{r}_{f}} X_{p}^{(l^{r}_{p})}}^{∏} Hom ( ⨂_{p∈S_{f} \S_{f}^{q}} V_{p} ⊗ E_{q}^{.} ⊗ ⨂_{p∈S_{f}^{r}} V_{p}^{U_{Fp}^{(e)}}, V_{−ω}) .

In fact this is true if q is the largest prime of S_{f}.

Now we argue by induction. By the exactness of the tensor product and the Homfunctor for complex vector spaces the complex (32) is exact at all degrees ≥ 1 and its zeroth cohomology is just

_{y=((F_{p},c_{p})|p∈S_{f}^{r})∈∏_{p∈S^{r}_{f}} X_{p}^{(l^{r}_{p})}}^{∏} Hom ( ⨂_{p∈S_{f} \S_{f}^{q}} V_{p} ⊗ V_{q} ⊗ ⨂_{p∈S_{f}^{r}} V_{p}^{U_{Fp}^{(e)}}, V_{−ω}) .

Eventually we obtain that (30) is exact. □

We now finish the proof of the Theorem 5.1. Take the complex of Γ-invariant vectors of (30)

0 → ^{Γ}D^{0} → ^{Γ}D^{1} → ^{Γ}D^{2} → ….  (33)

Since there are equivariant partitions of unity the partial complex

0 → ^{Γ}C^{l_{0}} →^{d_{ω}} ^{Γ}C^{l_{1}} →^{d_{ω}} ^{Γ}C^{l_{2}} →^{d_{ω}} …  (34)

is exact in degree ≥ 1 and its zeroth cohomology is

ΓC^{(k,l_{f})} := ∏_{x∈Z̆^{l_{f}}} Hom ( ⨂_{q∈S_{f}} V_{q}^{U_{Fq}^{(e)}}, V^{k}(Y_{x})) ,

where x = ((F_{p}, c_{p})|p ∈ S_{f}) and Y_{x} := Γ\X_{ω}. Here V^{k}(Y_{x}) denotes the space of smooth sections of the locally homogeneous vector bundle Γ_{x}\G(R) ×_{K(R)} L_{k}. Thus the cohomology of (33) is equal to the cohomology of the total complex associated to the multicomplex ((C^{(k,l_{f})}|k ∈ N_{0}, l_{f} ∈ J_{f}), (d_{SK}, (d_{p}|p ∈ S_{f}))).

We form the double complex ((C^{(k,h)}|k, h ∈ N_{0}), (d_{SK}, d)) by taking the total complex with respect to the index l_{f}. There is a spectral sequence converging to the cohomology of (33) with second term

E_{2}^{h,k} = H_{d}^{h}H_{d_{SK}}^{k}(^{Γ}C^{(.,.)}) .

By the proof of Theorem 1.4 we have

H_{d_{SK}}^{k}(^{Γ}C^{(.,h)}) = _{l_{f} ∈J_{f} ,∑_{p∈S_{f}} l_{p}=h}^{⨁} ⨁_{x∈Z̆^{l_{f}}} Hom ( ⨂_{p∈S_{f}} V_{p}^{U_{Fp}^{(e)}}, H^{k}(Γ_{x}, V_{−ω})) ,

(l_{f} = (l_{p}, p ∈ S_{f}). Thus E_{2}^{h,k} is finite-dimensional for all h, k ∈ N_{0}, and this implies the assertion of Theorem 5.1. □

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