Cohomology of S-arithmetic groups in globalizations of Harish-Chandra modules and automorphic forms

Ulrich Bunke, Werner Hoffman and Martin Olbrich

September 20, 1996

Contents

1 The result 0

2 First reduction 2

3 Analysis on the symmetric space 5

4 A spectral sequence 7

1 The result

Let G denote a semisimple simply connected linear algebraic group defined over a number field F. Let g denote the Lie algebra of G(F_{∞}) and K_{∞} be a maximal compact subgroup of G(F_{∞}), where F_{∞} := ∏_{ν|∞} F_{ν}, ν running over all archimedean places of F. By HC(g, K_{∞}) we denote the category of Harish-Chandra modules, i.e, of finitely generated, admissible (g, K_{∞})-modules. If V ∈ HC(g, K_{∞}), then let V_{−∞} denote the distribution vector globalization of V defined as follows. If Ṽ_{B} is any Banach representation of G(F_{∞}) with underlying (g, K_{∞})-module Ṽ (where Ṽ denotes the dual Harish-Chandra module of V), then let Ṽ_{∞} be the Fréchet space of smooth vectors in Ṽ_{B}. Now V_{−∞} is by definition the topological dual of Ṽ_{∞}. This definition is independent of the choice of Ṽ_{B}.

Let S be an non-empty finite set of non-archimedean places of F and put

G(A^{S}(F)) := \begin{matrix}∏ & ′ \\ ν∉S,ν|/∞ \end{matrix} G(F_{ν}) ,

where ∏′ stands for the restricted product. Let K_{f}^{S} be an open compact subgroup of G(A^{S}(F)). Then G(F) ∩ K_{f}^{S} is called an S-arithmetic subgroup of G(F).

The goal of the present note is to study, for V ∈ HC(g, K_{∞}), the cohomology groups

H^{∗}(G(F) ∩ K_{f}^{S}, V_{−∞}) .  (1)

In case that V is finite-dimensional these cohomology groups were considered in [1]. Here we extend those results to infinite-dimensional V ∈ HC(g, K_{∞}). The result, and in parts the arguments, will be parallel to the finite-dimensional case. We consider the direct limit of the spaces (1) over all compact open subgroups K_{f}^{S} ⊂ G(A^{S}(F))

H^{∗}(G, S, V) := lim_{K_{f}^{S}^{⟶}} H^{∗}(G(F) ∩ K_{f}^{S}, V_{−∞}) .

It carries a G(A^{S}(F))-action, and we recover the cohomology groups (1) by

H^{∗}(G(F) ∩ K_{f}^{S}, V_{−∞}) = H^{∗}(G, S, V)^{K_{f}^{S}} .

In order to state our result concerning H^{∗}(G, S, V) we need some terminology. Let Π(G)_{cusp} denote the set of irreducible unitary cuspidal automorphic representations of G(A(F)). We will use the same letter for such a representation and the corresponding (g, K_{∞}; G(A_{f}(F)))- module obtained by taking K_{∞}-finite G(A_{f}(F))-smooth vectors, where A_{f}(F) := A^{∅}(F). Each π ∈ Π(G)_{cusp} has a restricted tensor product decomposition

π = ⨂_{ν} π_{ν} ,

where π_{ν} is an irreducible unitarizable smooth admissible representation of G(F_{ν}) for finite ν, and an irreducible unitarizable Harish-Chandra module for infinite ν. The finite product ⨂_{ν|∞} π_{ν} is a Harish-Chandra module π_{∞} ∈ HC(g, K_{∞}).

For a non-archimedean place ν let ST_{ν} denote the Steinberg representation of G(F_{ν}). By Π(G)_{cusp}^{S} we denote the subset of π ∈ Π(G)_{cusp} with π_{ν} ≅ ST_{ν} for all ν ∈ S. If π ∈ Π(G)_{cusp}, then we define

π_{f}^{S} := ⨂_{ν∉S,ν|/∞} π_{ν} .

Our main result is the following extension of [1], Thm. 1, to infinite-dimensional V ∈ HC(g, K_{∞}).

Theorem 1.1 Let G be a semisimple simply connected linear algebraic group defined over a number field F, S be a finite set of non-archimedean places of F, and assume that G(F_{ν}) is non-compact for some ν ∈ S. Let V ∈ HC(g, K_{∞}). Then the G(A^{S}(F))-module H^{∗}(G, S, V) is the direct sum of

H_{ST}^{∗} (G, S, V) := ⨁_{π∈Π(G)_{cusp}^{S}} π_{f}^{S} ⊗ Ext_{g,K_{∞}}^{∗}(Ṽ , π_{∞})[−D] ,

where D = ∑_{ν∈S} rank_{F_{ν}} G, and of the constant G(A^{S}(F))-module

H_{const}^{∗}(G, S, V) = Ext_{g,K_{∞}}^{∗}(Ṽ , C) .

\underline{Remark :} Our proof of the theorem depends on a result of Kashiwara and Schmid (announced to us by Schmid) which is stated here as Theorem 2.2. If rank_{R}(G) = 1, then alternatively one can use the corresponding result of [4].

It turns out that the cohomology of S-arithmetic groups behaves much simpler than that of arithmetic groups (S = ∅). In Section 4 we provide a spectral sequence converging to the cohomology of arithmetic groups with coefficients in the distribution globalization of a Harish-Chandra module. It implies at least:

Theorem 1.2 Let G be as in 1.1. For any compact open subgroup K_{f} ⊂ G(A_{f}(F)) we have

dim(H^{∗}(G, ∅, V)^{K_{f}}) < ∞ ,

i.e., the cohomology groups of arithmetic subgroups of G(F) with coefficients in distribution vector globalizations of Harish-Chandra modules are finite-dimensional.

2 First reduction

Let Γ_{G(F)}^{G(A^{S}(F))} denote the right-adjoint functor of the restriction functor from the category of smooth representations of G(A^{S}(F)) to the category of G(F)-modules [1], (2.1), and let RΓ_{G(F)}^{G(A^{S}(F))} denote its right-derived functor. We define the object

H(G, S, V) := RΓ_{G(F)}^{G(A^{S}(F))}(V_{−∞})

in the derived category of smooth G(A^{S}(F))-modules. It was shown in [1], Ch. 2, that its cohomology equals H^{∗}(G, S, V) and that

H(G, S, V) = RHom_{G(F_{S})}(C, H(G, ∅, V)) ,  (2)

where F_{S} := ∏_{ν∈S} F_{ν}. While in [1] a de Rham type resolution is used to represent H(G, ∅, V), for infinite-dimensional V we have to employ a different, more complicated resolution.

There are finite-dimensional representations W_{i}, i = 0, 1, …, of K_{∞} such that we can construct a projective resolution of (g, K_{∞})-modules

P : … → U(g) ⊗_{U(k)} W_{1} → U(g) ⊗_{U(k)} W_{0} → Ṽ → 0 ,

where k is the Lie-algebra of K_{∞}, U(g), U(k) denote the universal enveloping algebras, and Ṽ is the (g, K_{∞})-module dual of V.

Let ∥ . ∥ : G(A(F)) → R_{+} be a height function. Choose a compact open subgroup K_{f} ⊂ G(A_{f}(F)) and set K := K_{∞}K_{f}. By C_{mg}^{∞}(G(A(F))) we denote the space of smooth functions on G(A(F)) which are right K-finite and such that for any R ∈ U(g) there is a Q ∈ R with

sup_{g∈G(A(F))} ∥g∥^{−Q}|f(gR)| < ∞ .

By C_{mg}^{∞}(G(F)\G(A(F))) (“mg” stands for “moderate growth”) we denote the subspace of G(F)- invariants of C_{mg}^{∞}(G(A(F))). In a similar way we define C_{mg}^{∞}(G(F_{∞})) as the space of K_{∞}-finite smooth functions of moderate growth. Via right translations, we consider C_{mg}^{∞}(G(F)\G(A(F)))

as a (g, K_{∞}; G(A^{S}(F)))-module. It is clear from [1], (2.1), that Γ_{G(F)}^{G(A_{f} (F))}C_{mg}^{∞}(G(F_{∞})) = C_{mg}^{∞}(G(F)\G(A(F))), where G(F_{∞}) acts on C_{mg}^{∞}(G(F_{∞})) by left translations.

Proposition 2.1 In the derived category of smooth G(A^{S}(F))-modules we have the following ismorphisms:

H(G, ∅, V) ≅ Hom_{g,K_{∞}}(P, C_{mg}^{∞}(G(F)\G(A(F))))
≅ RHom_{g,K_{∞}}(Ṽ , C_{mg}^{∞}(G(F)\G(A(F)))) .

Proof. We need the following deep theorem which was announced to us by W. Schmid.

Theorem 2.2 (Kashiwara-Schmid) If V ∈ HC(g, K_{∞}), then

Ext_{g,K_{∞}}^{p}(Ṽ , C_{mg}^{∞}(G(F_{∞}))) = \{ \begin{matrix}V_{−∞} & p = 0 \\ 0 & p ≥ 1 \end{matrix}

as G(F_{∞})-modules.

In case that rank_{R}(G) = 1 an alternative proof of that theorem was given in [4].

Theorem 2.2 shows that

Hom_{g,K_{∞}}(P, C_{mg}^{∞}(G(F_{∞})))

represents V_{−∞} in the derived category of G(F)-modules. This complex consists of spaces C_{mg}^{∞}(G(F_{∞}) ×_{K_{∞}} W) for certain finite-dimensional K_{∞}-modules W. To prove the proposition it

is sufficient to show that C_{mg}^{∞}(G(F_{∞}) ×_{K_{∞}} W) is Γ_{G(F)}^{G(A_{f} (F))}-acyclic. As argued in [1], Ch. 3, it is enough to show that C_{mg}^{∞}(G(F_{∞}) ×_{K_{∞}} W) is acyclic with respect to the functors which take the invariants with respect to arithmetic subgroups G(F) ∩ K_{f}, where K_{f} ⊂ G(A_{f}(F)) is compact open. But this follows easily (see [4]) from the following theorem.

Theorem 2.3 For any arithmetic subgroup G(F) ∩ K_{f} we have

H^{p}(G(F) ∩ K_{f}, C_{mg}^{∞}(G(F_{∞}))) = 0, ∀p ≥ 1 .

This theorem will be proved in Section 3. Assuming Theorem 2.3 we have finished the proof of Proposition 2.1. □

In order to follow further the lines of arguments of [1] we need a reduction from moderate growth to uniform moderate growth. A function f ∈ C_{mg}^{∞}(G(F)\G(A(F))) belongs to the subspace C_{umg}^{∞}(G(F)\G(A(F))) if there is a Q ∈ R such that for any R ∈ U(g)

sup_{g∈G(A(F))} ∥g∥^{−Q}|f(gR)| < ∞ .

Proposition 2.4 The inclusion C_{umg}^{∞}(G(F)\G(A(F))) ↪ C_{mg}^{∞}(G(F)\G(A(F))) induces an isomorphism in the derived category of smooth G(A^{S}(F))-modules

Hom_{g,K_{∞}}(P, C_{umg}^{∞}(G(F)\G(A(F)))) →^{∼} Hom_{g,K_{∞}}(P, C_{mg}^{∞}(G(F)\G(A(F)))) .  (3)

Proof. Let J ⊂ Z(g) (where Z(g) denotes the center of U(g)) be the annihilator of Ṽ, and let Fin_{J} denote the functor on the category of (g, K_{∞})-modules which takes in each (g, K_{∞})-module the subspace J-finite vectors (see [6], Ch. 4). By Fin_{J}^{i}, i ≥ 0, we denote the corresponding higher derived functors. It was shown by Franke [6], Ch. 7.4, and Ch. 8, Prop. 1, that for all i > 0

Fin_{J}^{i} C_{mg}^{∞}(G(F)\G(A(F))) = 0 ,
Fin_{J}^{i} C_{umg}^{∞}(G(F)\G(A(F))) = 0 ,

and that

Fin_{J} C_{mg}^{∞}(G(F)\G(A(F))) = Fin_{J} C_{umg}^{∞}(G(F)\G(A(F))) .  (4)

Since J annihilates the cohomology of the complexes on both sides of (3) the natural morphism of functors Fin_{J} → id induces quasi-isomorphisms

Hom_{g,K_{∞}}(P, Fin_{J}C_{umg}^{∞}(G(F)\G(A(F)))) →^{∼} Hom_{g,K_{∞}}(P, C_{umg}^{∞}(G(F)\G(A(F))))
Hom_{g,K_{∞}}(P, Fin_{J}C_{mg}^{∞}(G(F)\G(A(F)))) →^{∼} Hom_{g,K_{∞}}(P, C_{mg}^{∞}(G(F)\G(A(F)))) .

The proposition follows from (4). □

Now we argue further as in [1], Ch. 3. Let L_{V}^{2} (G(F)\G(A(F))) denote the space of functions f ∈ C_{umg}^{∞}(G(F)\G(A(F))) which are square-integrable and annihilated by some power of J. Based on the analysis of [6], in [1] it was shown that the inclusion

L_{V}^{2} (G(F)\G(A(F))) ↪ C_{umg}^{∞}(G(F)\G(A(F)))

induces an isomorphism

RHom_{G(F_{S})}(C, Hom_{g,K_{∞}}(P, L_{V}^{2} (G(F)\G(A(F)))))
→^{∼} RHom_{G(F_{S})}(C, Hom_{g,K_{∞}}(P, C_{umg}^{∞}(G(F)\G(A(F))))) .

Decomposing L_{V}^{2} (G(F)\G(A(F))) = ⨁_{π} π into irreducible discrete summands we end up with

RHom_{G(F_{S})}(C, H(G, ∅, F)) ≅ ⨁_{π} π_{f}^{S} ⊗ ⨂_{ν∈S} RHom_{G(F_{ν})}(C, π_{ν}) ⊗ Hom_{g,K_{∞}}(P, π_{∞}) .

Since P was a projective resolution of Ṽ, we can replace Hom_{g,K_{∞}}(P, π_{∞}) by RHom_{g,K}(Ṽ , π_{∞}). Evaluating RHom_{G(F_{ν})}(C, π_{ν}) we finish the proof of Theorem 1.1 as in [1], Ch. 4. □

3 Analysis on the symmetric space

Let G be a connected semisimple algebraic group defined over Q. Fix a maximal compact subgroup K_{∞} ⊂ G(R). By X we denote the symmetric space G(R)/K_{∞}. Let X̄ be its Borel-Serre ‘compactification’ (which, of course, is not compact, but admits an cocompact action of arithmetic subgroups of G) [3].

We define the sheaf version (C^{∗}, d) of the de Rham complex of X̄ twisted with C_{mg}^{∞}(G(R)). Let Ū ⊂ X̄ be open. Let U ⊂ G(R) be K_{∞}-invariant such that U/K_{∞} = X ∩ Ū. Then

C^{∗}(Ū) = {f ∈ C^{∞}(U × G, Λ^{∗}(g/k))^{K_{∞}} |

∀L = (L_{1}, L_{2}) ∈ U(g) ⊗ U(g), V ⊂ U ∃R ∈ R s.t.

sup_{h∈V, g∈G} ∥g∥^{R}|f(hL_{1}, gL_{2})| < ∞} .

Here V runs over all K_{∞}-invariant open subsets of U such that clo(V) ⊂ U.

By S^{∗} we denote the subsheaf of C^{∗} given by

S^{∗}(Ū) = {f ∈ C^{∞}(U × G, Λ^{∗}(g/k))^{K_{∞}} |

∀L = (L_{1}, L_{2}) ∈ U(g) ⊗ U(g), V ⊂ U ∃R ∈ R s.t.

sup_{h∈V, g∈G} ∥g∥^{R}∥h∥^{R}|f(hL_{1}, gL_{2})| < ∞} ,

where V runs over all K_{∞}-invariant open sets with V/K_{∞} = V̄ ∩ X, and V̄ is an open subset of Ū such that clo(V̄) ⊂ Ū. Then (S^{∗}, d) is a subcomplex of sheaves of (C^{∗}, d).

Since X̄ is locally contractible, the complex (C^{∗}, d) is acyclic. The corresponding result for (S^{∗}, d) requires some argument.

Lemma 3.1 The complex (S^{∗}, d) is acyclic.

Proof. If x ∈ X̄, then x has a fundamental system of contractible neighbourhoods. We will show that the associated zero homotopy of (C_{x}^{∗}, d) is compatible with the subcomplex of germs (S_{x}^{∗}, d).

Let P ⊂ G be a parabolic subgroup defined over Q and P (R) = M_{P} (R)^{1}A_{P} (R)^{0}N_{P} (R) be a Langlands decomposition. Then P (R) acts transitively on X. The geodesic action of A_{P} (R)^{0} on X is defined by γ(a, pK_{∞}) = paK_{∞}, where p ∈ P (R), a ∈ A_{P} (R)^{0}. We put e(P) := A_{P} (R)^{0}\X. Then X̄ is the disjoint union of faces e(P), where P runs over all parabolic subgroups of G defined over Q.

We now describe a fundamental system of neighbourhoods of x ∈ X̄ given by Siegel sets. Let x ∈ e(P) for some P. The map A_{P} (R)^{0} ∋ exp(a) ↦ (e^{−⟨α,a⟩})_{α∈Δ_{P}} defines an isomorphism A_{P} (R)^{0} ≅ ∏_{α∈Δ_{P}} R_{+}. Using this identification we define the compactification Ā_{P} (R)^{0} ≅ ∏_{α∈Δ_{P}} (R_{+} ∪ {0}). Then X(P) = Ā_{P} (R)^{0} ×_{A_{P} (R)^{0}} X = ∪_{P ⊂Q}e(Q) is an open neighbourhood of x. In this picture x = {0} × [y] for some y ∈ M_{P} (R)^{1}N_{P} (R). Let

Ā_{P,T} =: {a ∈ Ā_{P} (R)^{0}, ⟨α, log(a)⟩ > T ∀α ∈ Δ_{P} } .

If Ω ⊂ M_{P} (R)^{1}N_{P} (R) is a precompact neighbourhood of y, then we define the Siegel set

S_{T,Ω} := ΩĀ_{P,T} K_{∞} .

Let [Ω] ⊂ e(P) be the image of Ω under the canonical projection M_{P} (R)^{1}N_{P} (R) → e(P). Choosing a sequence of sets Ω such that [Ω] forms a fundamental sequence of contractible neighbourhoods of y and letting T tend to infinity, the corresponding sequence of Siegel sets S_{Ω,T} forms a fundamental sequence of contractible neighbourhoods of x. In the remainder of this proof we fix a particular element of this sequence. Let

A_{P,T} := Ā_{P,T} ∩ A_{P} (R)^{0} .

For any R ∈ R there exist c > 0, C < ∞, Q, S ∈ R such that

c∥ωa∥^{Q} ≤ ∏_{α∈Δ_{P}} a^{Rα} ≤ C∥ωa∥^{S}

for all ω ∈ Ω, a ∈ A_{P,T}. The space S^{∗}(S_{Ω,T}) is characterized by the following conditions: We have f ∈ S^{∗}(S_{Ω,T}), iff for all W ⊂ Ω such that W̄ ⊂ Ω, and for all L = (L_{1}, L_{2}, L_{3}), L_{2} ∈ U(a_{P}), L_{3} ∈ U(g), L_{1} a left-invariant differential operator on M_{P} (R)^{1}N_{P} (R), there exists R ∈ R such that

sup_{ω∈W,a∈A_{P,T} ,g∈G(R)} ∥g∥^{R} ∏_{α∈Δ_{P}} a^{Rα}|f(ωL_{1}aL_{2}, gL_{3})| < ∞ .

Let ǎ_{P} denote the dual of the Lie algebra of A_{P} (R)^{0}. We define the space

E_{P,T}^{∗} = {f ∈ C^{∞}(A_{P,T} × G, Λ^{∗}ǎ_{P}) |∀L_{2} ∈ U(a_{P}), L_{3} ∈ U(g) ∃R ∈ R s.t.

sup_{a∈A_{P,T} ,g∈G(R)} ∥g∥^{R} ∏_{α∈Δ_{P}} a^{Rα}|f(aL_{2}, gL_{3})| < ∞} .

Then we have an inclusion (E_{P,T}^{∗} , d) ↪ (S^{∗}(S_{Ω,T}), d). Using the chain homotopy induced by the contraction of [Ω] (which is usually employed in the proof of the Poincaré lemma) and the characterization of S^{∗}(S_{Ω,T}) given above it is easy to see that this inclusion is a quasi isomorphism.

If Q is a parabolic subgroup contaning P then using the projection A_{P,T} → A_{Q,T} we can view (E_{Q,T}^{∗} , d) as a subcomplex of (E_{P,T}^{∗} , d). The contraction of the fibres of this projection induces a chain homotopy between this inclusion and the indentity of (E_{Q,T}^{∗} , d). By induction we see that (E_{P,T}^{∗} , d) is quasi isomorphic to the complex (C_{mg}^{∞}(G(R)), 0) consisting of one non-trivial space. The lemma follows. □

Lemma 3.2 The sheaves S^{∗} and C^{∗} are fine sheaves.

Proof. It follows from [5], Lemma 2.3 (a) (or [2], Sec. 6.4), that for any finite open covering of Γ\X̄ there exists a subordinated partition of unity in C_{mg}^{∞}(Γ\G(R)/K_{∞}). Furthermore, multiplication by f ∈ C_{mg}^{∞}(Γ\G(R)/K_{∞}) defines sheaf homomorphisms S^{∗} → S^{∗}, C^{∗} → C^{∗}. This implies the lemma. □

We now prove Theorem 2.3. By restriction of scalars we can assume that F = Q. Let Γ ⊂ G(R) be a torsion-free arithmetic subgroup. The complex of global sections (C(X̄), d) is a Γ-acyclic resolution of C_{mg}^{∞}(G(R)). The complex of Γ-invariants is the complex of global sections of the quotient sheaf (Γ\C^{∗}, d) over Γ\X̄. We must show that H^{j}(Γ\C^{∗}(Γ\X̄), d) = 0 for all j ≥ 1. Since Γ acts freely and properly on X̄, by Lemma 3.1 the inclusion

(Γ\S^{∗}, d) ↪ (Γ\C^{∗}, d)

is a quasi isomorphism of sheaves. By Lemma 3.2 we conclude

H^{∗}(Γ\S^{∗}(Γ\X̄), d) →^{∼} H^{∗}(Γ\C^{∗}(Γ\X̄), d) .

It was shown in [4] that H^{j}(Γ\S^{∗}(Γ\X̄), d) = 0 for all j ≥ 1. (The proof in [4] was given in the R-rank one case, but it immediately generalizes to the current situation.) This proves Theorem 2.3. □

4 A spectral sequence

In this section we discuss a spectral sequence converging to the cohomology of an arithmetic group with coefficients in the distribution vector globalization of a Harish-Chandra module. In particular, we show that the cohomology groups are finite-dimensional (Theorem 1.2). The spectral sequence is essentially based on the filtration of the space of automorphic forms constructed in [6]. We refer to this reference for details and notation.

By restriction of scalars we assume without loss of generality that G is defined over Q. Let {P } be a class of associated parabolic subgroups over Q. Then in [6] a groupoid M_{J,{P },∞}^{k,T,i} is defined. Here k ∈ N_{0} and J ⊂ Z(g) is a cofinite ideal. The index i ∈ N indicates the filtration step. For T we refer to [6]. The objects of this groupoid are triples t = (R, Λ, χ), where:

• R = M_{R}A_{R}N_{R} is a standard parabolic subgroup containing an element of {P } and satisfying rank_{G}(P) = k.

• Λ : A_{R}(Q)A_{R}(R)^{0}\A_{R}(A) → C^{∗} is a continuous character (here A = A(Q)). The differential λ_{t} of the archimedean component is required to satisfy a condition depending on T and i.

• χ : Z(m) → C is a character. λ and χ also have to satisfy a condition which is connected with J.

The morphisms of the groupoid are given by certain finite Weyl sets. To a triple t = (R, Λ, χ) we associate the space V (t) consisting of square integrable K ∩ R(A)-finite functions f on R(Q)N_{R}(A)A_{R}(R)^{0}\R(A) satisfying:

• For any standard parabolic subgroup Q ⊂ R not belonging to {P } the constant term f_{Q} is orthogonal to the space of cusp forms on M_{Q}(Q)\M_{Q}(A).

• f(ga) = Λ(a)f(g), ∀a ∈ A_{R}^{1}(A).

• f is a χ-eigenvector.

By D_{t} we denote the symmetric algebra S(a_{R}^{G})_{C} which is an R(A)-module as explained in [6]. It is shown in there that one can view the map t = (R, Λ, χ) ↦ M(t) = Ind_{R(A)}^{G(A)}V (t) ⊗ D_{t} as a

functor from M_{J,{P },∞}^{k,T,i} to (g, K_{∞}, G(A_{f}))-modules.

Let now V be a Harish-Chandra module and J = Ann_{Z(g)}(Ṽ). Then the complex H(G, ∅, V) can be decomposed into a direct sum ⨁_{{P }} H(G, ∅, V)_{{P }} over the classes of associated parabolic

subgroups. The piece H(G, ∅, V)_{{P }} has a further decomposition ⨁_{k=0}^{rank_{G}(P)} H(G, ∅, V)_{{P }}^{k}. As in [6], 7.4, there is a spectral sequence with E_{1}^{p,q}-term

colim_{t∈M_{J,{P },∞}^{k,T,p}}Ext_{g,K_{∞}}^{p+q}(Ṽ , M(t))

converging to H^{∗}(H(G, ∅, V)_{{P }}^{k}). In order to prove Theorem 1.2 it is sufficient to show that the E_{1}^{p,q} are admissible G(A_{f})-modules.

There is a further spectral sequence with second term

F_{2}^{i,j} = Ind_{R(A_{f})}^{G(A_{f})}Ext_{l_{R},K∩R(R)}^{i}(H_{j}(n_{R}, Ṽ), V (t) ⊗ D_{t})

converging to Ext_{g,K_{∞}}^{p+q}(Ṽ , M(t)) by Shapiro’s Lemma. We decompose l_{R} = m_{R} ⊕ a_{R} and H_{∗}(n_{R}, Ṽ) with respect to generalized a_{R}-characters. Hence as (a_{R} ⊕ m_{R}, K_{∞} ∩ M_{R}(R))- module we can decompose H_{∗}(n_{R}, Ṽ) into a finite sum of terms of the type U_{θ} ⊗ W_{θ}, where U_{θ} is a finite-dimensional vector space on which a_{R} acts by a generalized character θ, and W_{θ} is an admissible (m_{R}, M_{R}(R) ∩ K_{∞})-module. We know that Ext_{a_{R}}^{∗}(U_{θ}, D_{t}) is finite-dimensional. If U, W are admissible (m_{R}, M_{R}(R) ∩ K_{∞})-modules, then Ext_{m_{R},M_{R}(R)∩K_{∞}}^{∗}(U, W) is finitedimensional. This and the definition of V (t) implies that Ext_{l_{R},K∩R(R)}^{i}(H_{j}(n_{R}, Ṽ), V (t) ⊗ D_{t}) is an admissible R(A_{f})-module for all i, j. We conclude that F_{2}^{i,j} is an admissible G(A_{f})-module for all i, j. This finishes the proof of Theorem 1.2. □

Unfortunately the further simplifications of the spectral sequence above obtained in [6] for finite-dimensional V do not apply for general V.

References

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[3] A. Borel and J.-P. Serre. Corners and arithmetic groups. Comm.Math.Helv., 48:244–297, 1974.

[4] U. Bunke and M. Olbrich. Cohomological properties of the smooth globalization of a Harish-Hhandra module. Preprint, SFB288, No.177, 1995.

[5] W. Casselman. Canonical extensions of harish-chandra modules. Canadian J. Math., 41:385– 438, 1989.

[6] J. Franke. Harmonic analysis in weighted L_{2}-spaces. preprint, 1995.