Resolutions of distribution globalizations of Harish-Chandra modules and cohomology

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

Contents

1 Introduction 0 2 Surjectivity on weighted spaces 4 3 The standard resolution 12 4 𝔫-cohomology 15 5 Γ-acyclicity of S_{∞}E 17 6 The range of the Laplacian on Y 22 7 Γ-cohomology 25 8 Fuchsian groups of the first kind 27

1 Introduction

This paper represents a part of a longer project of the authors originated from a conjecture of S. J. Patterson on the divisor of the Selberg zeta function. The project is concerned with the investigation of the cohomology of discrete subgroups Γ of semisimple Lie groups G with coefficients in ”natural” infinite-dimensional representations of G.

The present paper can be considered as a continuation of [4]. There we have treated the case of cocompact subgroups Γ ⊂ G of rank-one Lie groups acting on the space of analytic or hyperfunction vectors V_{π,±ω} of Banach representations V_{π} of G. In the present paper we consider the cohomology of Γ with coefficients in the smooth and distribution vectors V_{π,±∞} of V_{π}, and we admit non-cocompact subgroups Γ of finite covolume vol(Γ\G) < ∞.

∗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

The main idea in [4] was to use Γ-acyclic resolutions of V_{π,−ω} given by complexes of Ginvariant differential operators acting on smooth sections E_{i} := C^{∞}(X, E_{i}) of homogeneous vector bundles E_{i} over the symmetric space X := G/K. Exactness of these complexes was a consequence of the surjectivity of Ω − μ : E_{i} → E_{i}, where Ω denotes the Casimir of G, and μ ∈ ℂ. Finiteness of the cohomology groups was implied by the theory of elliptic differential operators on compact manifolds.

Replacing the space E_{i} by the space of sections of moderate growth S_{∞}E_{i} or the space of sections of uniform moderate growth S_{umg}E_{i} one can hope to resolve the space of distribution vectors V_{π,−∞} of V_{π}. As already mentioned in [4], the main technical point is the surjectivity of Ω − μ : S_{∗}E_{i} → S_{∗}E_{i}, ∗ = ∞ or ∗ = umg. We prove this result in Section 2.

We obtain information about the cohomology groups H^{∗}(Γ, V_{π,−∞}) by investigation of the kernel and the cokernel of Ω − μ : ^{Γ}S_{∞}E_{i} → ^{Γ}S_{∞}E_{i}, where ^{Γ}S_{∞}E_{i} denotes the space of Γ-invariant vectors in S_{∞}E_{i}. In contrast to the cocompact case, where elliptic theory implies finiteness of these spaces, in the finite-covolume case the analysis becomes more complicated. In Section 6 we deal with this problem using Gelfand’s inequality.

In principle the results of Sections 2 and 6 could be obtained in a uniform way by using suitable Paley-Wiener theorems. Indeed, surjectivity of Ω − μ : S_{∞}E → S_{∞}E has essentially been shown this way in [21] for the trivial bundle E = X × ℂ. For arbitrary bundles E one would need a Paley-Wiener theorem for non-K-finite sections of E (such a theorem has been announced by Casselman). The corresponding Paley-Wiener theorem for the locally symmetric situation is folklore (we thank the referee for pointing this out). A special case was e.g. considered in [7]. We think that our approach in Section 6 is much more elementary.

In [4] it was quite easy to show that for any homogeneous vector bundle E → X the space of smooth sections E is Γ-acyclic, i.e., H^{p}(Γ, E) = 0, ∀p ≥ 1. Because of the growth conditions the corresponding problem for S_{∞}E is much more complicated. We prove Γ-acyclicity of S_{∞}E in Section 5. Note that we were not able to show that S_{umg}E is Γ-acyclic, too. This is the reason for working with S_{∞}E instead of the more common space S_{umg}E.

We now give the precise statement of our main result. Let G be a connected semisimple rank one Lie group with finite center. Let 𝔤 be the Lie algebra of G and K be its maximal compact subgroup. By HC(𝔤, K) we denote the category of Harish-Chandra modules of G. The Harish-Chandra module (π, V_{π,K}) ∈ HC(𝔤, K) has natural globalizations (analytic, smooth, distribution, and hyperfunction vectors)

V_{π,ω} ⊂ V_{π,∞} ⊂ V_{π,−∞} ⊂ V_{π,−ω} .

Here V_{π,∗} is a complete locally convex vector space admitting a continuous G-action such that the underlying (𝔤, K)-module is V_{π,K}.

Let Γ ⊂ G be a torsion-free discrete subgroup of finite covolume. The cohomology groups of Γ with coefficients in the different globalizations V_{π,∗} again become locally convex topological vector spaces. The results described above combined with our theory for the hyperfunction case [4] yield

Theorem 1.1 H^{∗}(Γ, V_{π,−∞}) is finite-dimensional and Hausdorff. Moreover, if Γ is cocompact, then for all p ≥ 0 there are topological isomorphisms

H^{p}(Γ, V_{π,ω}) = H^{p}(Γ, V_{π,∞})  (1)
H^{p}(Γ, V_{π,−ω}) = H^{p}(Γ, V_{π,−∞})  (2)
H^{p}(Γ, V_{π,∗})^{∗} = H^{dim(G/K)−p}(Γ, V_{π̃,−∗}), ∗ = ω, ∞ .  (3)

A detailed analysis of the cohomology of Fuchsian groups of the first kind with coefficients in principal series representations is given in Section 8. We express the cohomology in terms of automorphic and cusp forms. For the non-irreducible principal series representations we find explicit formulas for the dimensions of the cohomology groups in terms of the topology of Y = Γ\X.

It is worth noting that for cocompact Γ there is an alternative way to prove Theorem 1.1. Let P ⊂ G be a minimal parabolic subgroup with Langlands decomposition P = MAN which is compatible with K, and let 𝔫 be the Lie algebra of N. If (π, W_{π,K}) ∈ HC(𝔤, K), then the Lie algebra cohomology groups H^{∗}(𝔫, W_{π,K}) are known to be finitedimensional MA-modules. Using the decomposition of L^{2}(Γ\G) into a Hilbert sum of irreducible unitary representations of G and Casselman’s submodule theorem one can express the cohomology groups H^{∗}(Γ, V_{π,−∞}) via the 𝔫-cohomology groups H^{∗}(𝔫, W_{π,−∞}) of the distribution globalization of certain unitary Harish-Chandra modules (compare [4], Cor. 6.4., [6]). Hence Theorem 1.1 becomes a consequence of the following comparison theorem due to Casselman (unpublished) and its analog for the analytic globalization (proved e.g. in [3], [4], and [6]).

Theorem 1.2 There are isomorphisms of MA-modules

H^{p}(𝔫_{P} , V_{π,∞}) ≅ H^{p}(𝔫, V_{π,K}) .

The natural topology on H^{p}(𝔫_{P} , V_{π,∞}) is Hausdorff.

For special cases this result goes back to the thesis of Osborne [20].

The methods of the present paper also give an independent proof of Theorem 1.2 (see Section 4). However, according to Casselman, this theorem holds true for G of arbitrary rank. This in combination with a recent result of Kashiwara-Schmid [17], Thm. 6.13, can be used to generalize Theorem 1.1 to the case of cocompact lattices in arbitrary semisimple Lie groups (see [6]).

Our main motivation for considering the Γ-cohomology of globalizations of Harish-Chandra modules was to prove a conjecture of Patterson about the singularities of the Selberg zeta functions associated to Γ.

Let G be a semisimple Lie group of real rank one and P be a minimal parabolic subgroup of G. Choose a Langlands decomposition P = MAN such that 𝔞 ⊥ 𝔨 where 𝔞 is the Lie algebra of A. Let (σ, V_{σ}) ∈ M̂ be an irreducible representation of M, and Γ ⊂ G be a discrete cocompact torsion-free subgroup. Then the Selberg zeta function Z_{Γ}(s, σ), s ∈ 𝔞_{ℂ}^{∗}, is defined as the analytic continuation of the infinite product

Z_{Γ}(s, σ) = ∏_{[g]∈CΓ,n_{Γ}(g)=1} ∏_{k=0}^{∞} det (1 − a_{g}^{−ρ−s}S^{k}(Ad(m_{g}a_{g})_{𝔫}^{−1}) ⊗ σ(m_{g})) .

Here CΓ is the set of hyperbolic conjugacy classes in Γ, n_{Γ}(g) is the maximal number n ∈ ℕ such that g = h^{n} for some h ∈ Γ and m_{g} ∈ M, a_{g} ∈ A^{+} are such that g is conjugated in G to m_{g}a_{g}. S^{k}(Ad(m_{g}a_{g})_{𝔫}^{−1}) stands for the k’th symmetric power of Ad(m_{g}a_{g})^{−1} restricted to 𝔫 and ρ ∈ 𝔞^{∗} is defined by ρ(H) := \frac{1}{2}tr(ad(H)_{𝔫}). The infinite product converges for Re(s) > ρ. In this generality the Selberg zeta function was introduced by Fried [11]. He applied Ruelle’s techniques of hyperbolic dynamics and gave a meromorphic continuation of Z_{Γ}(s, σ).

The parameters (σ, λ) ∈ M̂ × 𝔞_{ℂ}^{∗} also define a principal series representation of G on the Hilbert space

H^{σ,λ} = {f : G → V_{σ} | f(gman) = a^{λ−ρ}σ(m)^{−1}f(g), ∀man ∈ MAN, f_{|K} ∈ L^{2}},

where G acts by the left regular representation. By H_{−∞}^{σ,λ} we denote the space of its distribution vectors.

S. Patterson [22] conjectured the relationship of the singularities of Selberg zeta functions with the cohomology of Γ with coefficients in the distribution vectors of principal series representations. In [4] we have shown an analog of Pattersons conjecture for the hyperfunction vectors of principal series representations. Pattersons original conjecture now follows from (2).

Theorem 1.3 The cohomology H^{p}(Γ, H_{−∞}^{σ,λ}) is finite-dimensional for all p ≥ 0,

χ(Γ, H_{−∞}^{σ,λ}) = ∑_{p=0}^{∞}(−1)^{p} dim H^{p}(Γ, H_{−∞}^{σ,λ}) = 0,  (4)

χ(Γ, Ĥ_{−∞}^{σ,λ}) = ∑_{p=0}^{∞}(−1)^{p} dim H^{p}(Γ, Ĥ_{−∞}^{σ,λ}) = 0,  (5)

and the order of Z_{Γ}(s, σ) at s ∈ 𝔞_{ℂ}^{∗} can be expressed in terms of the group cohomology of Γ with coefficients in H_{−∞}^{σ,λ} as follows :

ord_{s=λ≠0}Z_{Γ}(s, σ) = − ∑_{p=0}^{∞}(−1)^{p}p dim H^{p}(Γ, H_{−∞}^{σ,λ}),  (6)

ord_{s=0}Z_{Γ}(s, σ) = − ∑_{p=0}^{∞}(−1)^{p}p dim H^{p}(Γ, Ĥ_{−∞}^{σ,0}),  (7)

where Ĥ_{−∞}^{σ,λ} is a certain non-trivial extension of H_{−∞}^{σ,λ} with itself.

Our description of the cohomology for non-cocompact Γ of finite covolume implies that the analog of Pattersons conjecture is false in that case.

Acknowledgment : We thank W. Casselman for explaining us his proof of Theorem 1.2. We are also grateful to W. Schmid and J. Franke for discussing their results with us. The authors were supported by the SFB 288 of the Deutsche Forschungsgemeinschaft.

2 Surjectivity on weighted spaces

From now on let G be a connected semisimple Lie group with finite center of real rank one and K be its maximal compact subgroup. Let G ×_{K} V_{γ} = E be a finite-dimensional G-homogeneous vector bundle over the rank-one symmetric space X = G/K. Let E := C^{∞}(X, E) be the space of smooth sections of E. It can be identified with the space of smooth functions f on G with values in V_{γ} satisfying f(gk) = γ(k)^{−1}f(g), ∀k ∈ K. Let U(𝔤) be the universal enveloping algebra of the Lie algebra 𝔤 of G. For L ∈ 𝔤 and a smooth function f on G we set

f(Lg) := \frac{∂}{∂t}_{|t=0}f(exp(tL)g) and f(gL) := \frac{∂}{∂t}_{|t=0}f(g exp(tL)) .

These definitions extend canonically to L ∈ U(𝔤).

We define an increasing sequence of G-submodules S_{R}E ⊂ E, R ∈ ℝ, by

S_{R}E := {f ∈ E | p_{R,L}(f) < ∞ ∀L ∈ U(𝔤)},

where the seminorms p_{R,L} are defined by

p_{R,L}(f) := sup_{G} e^{−Rdist(O,gK)}|f(Lg)| ,

and O ∈ X is the class [K] ∈ G/K considered as the base point of X. S_{R}E is a Fréchet space and for R ≤ R′ we have a continuous inclusion S_{R}E ↪ S_{R′}E. Let S_{R}E′ := (S_{−R}E)^{∗} be the topological conjugate dual of S_{−R}E. The adjoint of the inclusion S_{−R′}E ↪ S_{−R}E is the restriction S_{R}E′ → S_{R′}E′, and we form

S_{∞}E′ := lim_{R^{⟶}} S_{R}E′ .

The space S_{∞}E′ is the topological conjugate dual of ⋂_{R} S_{R}E. Since C_{c}^{∞}(X, E) is dense in ⋂_{R} S_{R}E, we have an inclusion S_{∞}E′ ↪ C^{−∞}(X, E).

We also consider

S_{∞}E := {f ∈ E | ∀L ∈ U(𝔤) ∃R ∈ ℝ s.t. p_{R,L}(f) < ∞} .

The G-invariant Hermitian scalar product on E induces an inclusion S_{∞}E ⊂ S_{∞}E′. Note that the space S_{∞}E could also be defined using right instead of left derivatives. The symbol S_{∞}E is an abuse of notation since

S_{∞}E ≠ lim_{R∈ℝ^{⟶}}S_{R}E .

S_{∞}E is usually called the space of sections of E of moderate growth.

Let Ω be the Casimir operator of G, and for fixed μ ∈ ℂ set B := −Ω − μ. The aim of this section is to show

Theorem 2.1 The operator B : S_{∞}E → S_{∞}E is surjective.

We will proceed as follows. Inspired by the method used by Franke [10] we first establish that S_{∞}E′ is Fin_{μ}-acyclic (the functor Fin_{μ} is introduced below). In this main analytic step we employ Arthur’s Paley-Wiener theorem for the Harish-Chandra-Schwartz space. An algebraic argument based on results of Casselman-Wallach on the globalization functor then implies the surjectivity of B on S_{∞}E′. Eventually, the surjectivity on S_{∞}E follows by a regularity result.

Let ℂ[B] be the ring of all polynomials in B. We consider the functor Fin_{μ} on the category of ℂ[B]-modules defined by

Fin_{μ}(V) = {v ∈ V | ∃l ≥ 0 B^{l}v = 0} .

This functor is left exact and its higher derived functors are denoted by Fin_{μ}^{i}(V). Since the inductive limit is an exact functor on the category of ℂ[B]-modules and

Fin_{μ}(V) ≅ lim_{j^{⟶}} Hom_{ℂ[B]}(ℂ[B]/(B)^{j}, V)

we have

Fin_{μ}^{i}(V) ≅ lim_{j^{⟶}} Ext_{ℂ[B]}^{i}(ℂ[B]/(B)^{j}, V) .  (8)

Since ℂ[B] is a regular ring of dimension one we have Fin_{μ}^{i}(V) = 0 for i ≥ 2. Thus a ℂ[B]-module V is called Fin_{μ}-acyclic, iff Fin_{μ}^{1}(V) = 0. Fin_{μ}^{1}(V) can be computed using the Koszul complexes

0 → Ext_{ℂ[B]}^{0}(ℂ[B]/(B)^{k}, V) → V →^{B^{k}} V → Ext_{ℂ[B]}^{1}(ℂ[B]/(B)^{k}, V) → 0 , k ∈ ℕ .

In order to perform the direct limit in (8) note that the map

Ext_{ℂ[B]}^{1}(ℂ[B]/(B^{k}), V) → Ext_{ℂ[B]}^{1}(ℂ[B]/(B^{k+1}), V)

is induced by B : V → V. Hence we have

Fin_{μ}^{1}(V) = lim_{k^{⟶}} coker(B^{k} : V → V) .

Proposition 2.2 S_{∞}E′ is Fin_{μ}-acyclic.

We consider the Schwartz space

S_{−log}E := {f ∈ E | q_{L,k}(f) < ∞ ∀k ∈ ℕ, L ∈ U(𝔤)} ,

where

q_{L,k}(f)^{2} = ∫_{G} dist(O, gK)^{k}|f(Lg)|^{2}dg

and its topological conjugate dual space

S_{log}E′ = (S_{−log}E)^{∗} .

There is an inclusion S_{log}E′ ↪ S_{∞}E′ and for R ∈ ℝ sufficiently small we have a continuous map S_{R}E′ → S_{log}E′. As a first step we prove

Lemma 2.3 S_{log}E′ is Fin_{μ}-acyclic.

Proof. Fix a minimal parabolic subgroup P = MAN of G, where the Langlands decomposition is chosen such that 𝔞 ⊥ 𝔨. Using the generalization of the Helgason-Fourier transform to bundles (see [5], 1.4.3 , Branson-Olafsson-Schlichtkrull [2]) and a result of Arthur [1] we can identify the Schwartz space S_{−log}E with

[S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{γ})]^{W} ⊕ ⨁_{i=1}^{r} V_{i,∞} .

Here V_{i,∞} are the spaces of smooth vectors of certain irreducible discrete series representations, S(ı𝔞^{∗}) is the Schwartz space on ı𝔞^{∗} and W = ℤ_{2} is the Weyl group. ˆ⊗ denotes the nuclear tensor product. The operation of W on S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{γ}) is implemented by a family of unitary operators

A_{λ} : C^{∞}(K ×_{M} V_{γ}) → C^{∞}(K ×_{M} V_{γ}), λ ∈ ı𝔞^{∗} ,

which are closely related to Knapp-Stein intertwining operators of principal series representations. The non-trivial element w ∈ W acts by (wf)(−λ, k) = (A_{λ}f(λ, .))(k). Dually, we can identify S_{log}E′ with

[S′(ı𝔞^{∗}) ˆ⊗C^{−∞}(K ×_{M} V_{γ})]^{W} ⊕ ⨁_{i=1}^{r} V_{i,−∞} .

On the Fourier image the operator B acts as multiplication by the polynomial P (λ) := |λ|^{2} + |ρ|^{2} − γ(Ω_{M}) − μ, λ ∈ ı𝔞^{∗}, on the first summand, and as a scalar on V_{i,−∞}. Here ρ is as usual the half-sum of the positive restricted roots, and Ω_{M} is the Casimir operator of M.

It follows that Fin_{μ}^{1}(V_{i,−∞}) = 0. In order to see that

Fin_{μ}^{1}([S′(ı𝔞^{∗}) ˆ⊗C^{−∞}(K ×_{M} V_{γ})]^{W}) = 0

we show that the multiplication by P is surjective on [S′(ı𝔞^{∗}) ˆ⊗C^{−∞}(K ×_{M} V_{γ})]^{W}. We consider the dual situation and show that the multiplication by P on [S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{γ})]^{W} is injective and has closed range. Injectivity is easy since P (λ) is non-invertible at most on a set of codimension 1 in ı𝔞^{∗} × K/M. Note that

C^{∞}(K ×_{M} V_{γ}) = ⨁_{σ∈M̂} ⨁_{h=1}^{[γ:σ]} C^{∞}(K ×_{M} V_{σ})

and P (λ) acts by the scalar polynomial P_{σ}(λ) = |λ|^{2} +|ρ|^{2} −σ(Ω_{M})−μ on each summand S′(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{σ}). Let Z_{σ} be the set of zeros of P_{σ}(λ). Let n_{σ}(λ) be the order of the zero of P_{σ} at λ ∈ Z_{σ}. Then the range of the multiplication by P_{σ} on S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{σ}) consists of all sections f ∈ S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{σ}) such that f(λ, kM) has a zero of order n_{σ} for all λ ∈ Z_{σ} and for all k ∈ K. This vanishing condition is a closed condition. The range of P on [S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{γ})]^{W} consists of all f ∈ [S(ı𝔞^{∗}) ˆ⊗C^{∞}(K ×_{M} V_{γ})]^{W} such that f_{σ}(λ, kM) has a zero of order n_{σ}(λ) for all λ ∈ Z_{σ}, k ∈ K and σ. This condition is again closed.

This finishes the proof of the lemma. □

The next lemma provides the bridge between S_{∞}E′ and S_{log}E′.

Lemma 2.4 Let k ≥ 1. For any f ∈ S_{∞}E′ there exists a g ∈ S_{∞}E′ with f −B^{k}g ∈ S_{log}E′.

Proof. It is enough to show that for any f ∈ S_{R}E′ there exists g ∈ S_{R}E′ such that f − Bg is in the range of the restriction S_{R−1}E′ → S_{R}E′. In fact, it is sufficient that this holds for all weights R in a dense subset of ℝ. Then the assertion of the lemma follows by a finite iteration with respect to k and R.

We retain the notations of the proof of Lemma 2.3 and consider the polar coordinates ℝ^{+} × K/M of X \ {O}: (t, kM) ↦ k exp(tH)K, where H ∈ 𝔞^{+} has length 1. Then we have

E_{|X\{O}} ≅ ⨁_{σ∈M̂} E(σ) ,  (9)

where E(σ) = ℝ^{+} × K ×_{M} V_{γ}(σ). Here V_{γ}(σ) is the σ-isotypic component of V_{γ}. In these coordinates we define

B_{rad} := −(\frac{d^{2}}{dt^{2}} + 2|ρ|\frac{d}{dt} + γ(Ω_{M})) − μ .  (10)

In fact, B_{rad} is the constant term along ℝ^{+} of B: B − B_{rad} = e^{−t}Q, where Q has bounded coefficients up to infinity (see, e.g., [27], Prop. 9.1.2.11). B_{rad} leaves the splitting (9) invariant. Let P_{σ} : E_{|X\{O}} → E(σ) be the projection. Then B_{rad} = ∑_{σ} B_{rad}(σ)P_{σ}, and there exist x_{σ}, y_{σ} ∈ ℂ such that B_{rad}(σ) = −(\frac{d}{dt} − x̄_{σ})(\frac{d}{dt} − ȳ_{σ}). We exclude the weights R = Re(x_{σ}), Re(y_{σ}) for all σ from the following consideration.

Let S_{−R}E_{0} ⊂ S_{−R}E be the subspace of all sections vanishing in a neighbourhood of the origin O ∈ X. Then S_{−R}E_{0} is a direct limit of Fréchet spaces. Let S_{R}E′_{0} be the topological conjugate dual of S_{−R}E_{0}. Then S_{R}E′ ⊂ S_{R}E′_{0}. Choose a cut-off function χ ∈ C^{∞}(ℝ^{+}) such that χ(t) = 0 for t < 1 and χ(t) = 1 for t > 2. Then the multiplication by χ defines a continuous operator S_{R}E′_{0} → S_{R}E′.

Below we will construct a continuous solution operator H : S_{−R}E_{0} → S_{−R}E_{0} such that HB_{rad}^{∗} − id : S_{−R}E_{0} → S_{−R}E_{0} is continuous, and if supp(f) ⊂ [2, ∞) × K/M, then (HB_{rad}^{∗} − id)f = 0. Here B_{rad}^{∗} is the formal adjoint of B_{rad}. The adjoint H^{∗} : S_{R}E′_{0} → S_{R}E′_{0} then has the property that B_{rad}H^{∗}ϕ − ϕ ∈ E′ for ϕ ∈ S_{R}E′_{0}. Here E′ is the space of distributions with compact support. The composition χ ∘ H_{|S_{R}E′}^{∗} : S_{R}E′ → S_{R}E′ satisfies B_{rad}χ∘H^{∗}ϕ−ϕ ∈ E′, ϕ ∈ S_{R}E′. Hence BχH^{∗}ϕ−ϕ = e^{−t}QχH^{∗}ϕ(modE′). But e^{−t}QχH^{∗}ϕ as well as E′ are contained in the range of the restriction S_{R−1}E′ → S_{R}E′. This proves the induction step assuming that we have already constructed H.

Set S_{−R}E_{0}(σ) := P_{σ}(S_{−R}E_{0}). It is enough to construct solution operators H_{x_{σ}}, H_{y_{σ}} : S_{−R}E_{0}(σ) → S_{−R}E_{0}(σ) such that H_{x_{σ}}(\frac{d}{dt} + x_{σ})f = χf, ∀f ∈ S_{−R}E_{0}(σ), and similarly for H_{y_{σ}}. Then H := − ∑_{σ} H_{x_{σ}}H_{y_{σ}}P_{σ} has the required properties.

For R > Re(x_{σ}) we set (simplifying the notation by omitting the k-variable)

(H_{x_{σ}}f)(t) := −χ(t)e^{−x_{σ}t} ∫_{t}^{∞} e^{x_{σ}s}f(s)ds .

Since f ∈ S_{−R}E_{0}(σ) we have for some C < ∞ and all s ∈ ℝ^{+} that |f(s)| ≤ Ce^{−Rs}. Hence the integral converges. If R < Re(x_{σ}), then we set

(H_{x_{σ}}f)(t) := χ(t)e^{−x_{σ}t} ∫_{0}^{t} e^{x_{σ}s}f(s)ds .

It is easy to check that H_{x_{σ}}(\frac{d}{dt} + x_{σ})f = χf. We must show that

H_{x_{σ}}(S_{−R}E_{0}(σ)) ⊂ S_{−R}E_{0}(σ)  (11)

and that H_{x_{σ}} is continuous.

To define the topology of S_{−R}E_{0}(σ) it is sufficient to consider a set of seminorms {p̃_{−R,L}} with L = L_{1} ⊗ L_{2} ∈ U(𝔞) ⊗ U(𝔨). Set (Lf)(t, k) = f(L_{1}t, L_{2}k). Then

p̃_{−R,L} := sup_{t∈ℝ^{+},k∈K} e^{Rt}|(Lf)(t, k)| .

It is clear that LH_{x}f = H_{x}Lf + W_{L}f, where W_{L} : S_{−R}E_{0}(σ) → C_{c}^{∞}(X \ O, E(σ)) is continuous. Thus in order to show that H_{x_{σ}} satisfies (11) and is continuous it is enough to verify the estimate

p̃_{−R,1}(H_{x_{σ}}f) ≤ Cp̃_{−R,1}(f) .

We employ that |f(t, k)| ≤ p̃_{−R,1}(f)e^{−Rt}. If R > Re(x_{σ}), then

p̃_{−R,1}(H_{x_{σ}}f) = sup_{t∈ℝ^{+},k∈K} e^{Rt}|(H_{x_{σ}}f)(t, k|
≤ sup_{t∈ℝ^{+},k∈K} e^{Rt}p̃_{−R,1}(f)e^{−Re(x_{σ})t} ∫_{t}^{∞} e^{(Re(x_{σ})−R)s}ds
≤ Cp̃_{−R,1}(f) .

If R < Re(x_{σ}), then

p̃_{−R,1}(H_{x_{σ}}f) = sup_{t∈ℝ^{+},k∈K} e^{Rt}|(H_{x_{σ}}f)(t, k)|
≤ sup_{t∈ℝ^{+},k∈K} e^{Rt}p̃_{−R,1}(f)e^{−Re(x_{σ})t} ∫_{1}^{t} e^{(Re(x_{σ})−R)s}ds
≤ Cp̃_{−R,1}(f) .

This finishes the construction of H and hence the proof of the lemma. □

Now we finish the proof of Proposition 2.2. Because of Lemma 2.3 it is enough to show that the natural map Fin_{μ}^{1}(S_{log}E′) → Fin_{μ}^{1}(S_{∞}E′) is surjective. Let f ∈ S_{∞}E′ represent an element in coker(B^{k} : S_{∞}E′ → S_{∞}E′) and thus in Fin_{μ}^{1}(S_{∞}E′). According to Lemma 2.4 we find g ∈ S_{∞}E′ such that f − B^{k}g ∈ S_{log}E′. But f − B^{k}g defines an element in Fin_{μ}^{1}(S_{log}E′) which is mapped to the class of f in Fin_{μ}^{1}(S_{∞}E′). □

Proposition 2.5 The operator B : S_{∞}E′ → S_{∞}E′ is surjective.

Proof. It follows from the proof of Lemma 2.3 that if B does not annihilate one of the discrete series representations V_{i}, then B is surjective on S_{log}E′. In this case Lemma 2.4 implies the proposition. The reason for working with the functor Fin_{μ} is that B in general is not surjective on S_{log}E′.

Let S_{∞}E′(B^{k}) := ker(B^{k} : S_{∞}E′ → S_{∞}E′) and C be defined by the following exact sequence:

0 → S_{∞}E′(B) → S_{∞}E′ →^{B} S_{∞}E′ → C → 0 .  (12)

We want to show that C = 0.

Since BC = 0 and BS_{∞}E′(B) = 0 by Proposition 2.2 the sequence (12) consists of Fin_{μ}-acyclic spaces. Thus applying Fin_{μ} to (12) we obtain the exact sequence

0 → S_{∞}E′(B) → Fin_{μ}(S_{∞}E′) →^{B} Fin_{μ}(S_{∞}E′) → C → 0 .

By definition we have

Fin_{μ}(S_{∞}E′) = lim_{k^{⟶}} S_{∞}E′(B^{k}) ,

and in order to verify that C = 0 it is enough to show that

lim_{k^{⟶}} coker(B : S_{∞}E′(B^{k}) → S_{∞}E′(B^{k})) = 0 .  (13)

The space of K-finite vectors S_{∞}E′(B^{k})_{K} is a Harish-Chandra module and S_{∞}E′(B^{k}) is its canonical distribution vector globalization ([26], Ch. 11). We will explain these facts in greater detail in Section 3. Since the globalization functor is exact (Wallach [26], Ch. 11, Casselman [9]) to show (13) is suffices to verify that

lim_{k^{⟶}} coker(B : S_{∞}E′(B^{k})_{K} → S_{∞}E′(B^{k})_{K}) = 0 .  (14)

Taking power series at O we obtain the following isomorphism (see, e.g., [4], Lemma 3.5)

S_{∞}E′(B^{k})_{K} ≅ Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}(B^{k}) .  (15)

Since the multiplication by B on (U(𝔤) ⊗_{U(𝔨)} Vγ̃)_{K} is injective we obtain by K-type wise algebraic dualization

coker (B : Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K} → Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}) = 0 .

Thus Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K} is Fin_{μ}-acyclic. Applying Fin_{μ} to the exact sequence of Fin_{μ}- acyclic spaces

0 → Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}(B) → Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K} →^{B} Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K} → 0

we obtain the exact sequence

0 → Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}(B) → Fin_{μ}(Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}) →^{B}

Fin_{μ}(Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}) → 0 .

The exactness at the last place is equivalent to

lim_{k^{⟶}} coker(B : Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}(B^{k}) → Hom_{U(𝔨)}(U(𝔤), V_{γ})_{K}(B^{k})) = 0 .

The claim (14) now follows from (15). □

In order to prove Theorem 2.1 it remains to show the following regularity result.

Lemma 2.6 If f ∈ S_{∞}E′, Bf ∈ S_{∞}E, then f ∈ S_{∞}E.

Proof. Let W̃ be a G-invariant fundamental solution of B, i.e., BW̃ = W̃ B = id on C_{c}^{∞}(X, E). Its existence is ensured by the surjectivity of B on C^{−∞}(X, E). Consider X × X with the diagonal G-action. Let χ ∈ ^{G}C^{∞}(X × X) be a G-invariant cut-off function defined by χ(x, y) = θ(dist(x, y)), where θ ∈ C^{∞}(ℝ), θ(r) ∈ [0, 1], θ(r) = 1 for r < 1 and θ(r) = 0 for r > 2.

Multiplying the distributional kernel of W̃ by χ we obtain a G-invariant parametrix W of B. It has finite propagation (the support of W ϕ is contained in a 1-neighbourhood of the support of ϕ for all ϕ ∈ C_{c}^{∞}(X, E)) and is applicable to arbitrary distributions in C^{−∞}(X, E). Let f ∈ S_{∞}E′ and Bf = F ∈ S_{∞}E. Then f = W F + Sf, where S is a G-invariant smoothing operator on E. We must show that W F, Sf ∈ S_{∞}E.

Let L ∈ U(𝔤). Then

(Sf)(Lg) = ⟨f, s(LgK, .)⟩ ,

where s is the integral kernel of S and s(LgK, .) ∈ C_{c}^{∞}(X, E). For any L_{1} ∈ U(𝔤) there is a P ∈ ℝ depending on L_{1}, L such that for any R ∈ ℝ

p_{−R,L_{1}}(s(LgK, .)) = sup_{h∈G} e^{Rdist(O,hK)}|s(LgK, L_{1}hK)| ≤ Ce^{(P +R)dist(O,gK)} ,

where C may depend on L, L_{1}, R, P but not on g. In fact, for all A, B ∈ U(𝔤) we have sup_{g,h∈G} |s(gA, hB)| < ∞ since S is G-invariant. If f ∈ S_{R}E′ for some R, then there is a finite set {L_{1}, …, L_{r}} ⊂ U(𝔤) such that for all L ∈ U(𝔤) there exists P ∈ ℝ with

p_{R+P,L}(Sf) = sup_{g∈G} e^{−(R+P)dist(O,gK)}|⟨f, s(LgK, .)⟩|
≤ C(f) sup_{g∈G,i=1,...,r} e^{−(R+P)dist(O,gK)}p_{−R,L_{i}}(s(LgK, .))
< ∞ .

This proves Sf ∈ S_{∞}E.

Let L ∈ U(𝔤). We now show that there exists R′ ∈ ℝ such that p_{R′,L}(W F) < ∞. Let w be the distributional kernel of W. w(O, .) is a distribution supported in the ball of radius 2 with center in O. Using the G-invariance of W we can write

|(W F)(LgK)| = |⟨w(O, .), F (Lg .)⟩|
≤ C sup_{i=1,...,r, dist(O,x)≤2} |F (LgL_{i}x)| ,

where {L_{1}, …, L_{r}} ⊂ U(𝔤) is a certain finite subset. There is another finite set {L′_{1}, …, L′_{r′}} ⊂ U(𝔤) and an exponent Q ∈ ℝ such that

sup_{i=1,...,r} |F (LgL_{i}x)| ≤ C′e^{Qdist(O,gK)} sup_{j=1,...,r′} |F (LL′_{j}gx)| .

Hence for R large enough we have

p_{R+Q,L}(W F) ≤ C′′ sup_{j=1,...,r′} p_{R,LL′_{j}}(F) < ∞ .

□

In the theory of automorphic forms it is convenient to work with the space of sections of uniform moderate growth S_{umg}E ⊂ S_{∞}E defined by

S_{umg}E := {f ∈ E | ∃R ∈ ℝ ∀L ∈ U(𝔤) s.t. sup_{g∈G} e^{−Rdist(O,gK)}|f(gL)| < ∞} .

The proof of the above regularity result also works if one replaces S_{∞}E by S_{umg}E. Thus Theorem 2.1 holds with this replacement, too. The reason for considering S_{∞}E here is that we can prove Γ-acyclicity for this space only (see Section 5).

3 The standard resolution

In this section we use the spaces S_{∞}E for certain bundles E in order to construct resolutions of the distribution globalization of a Harish-Chandra module. The construction parallels completely the one given in [4] for the maximal globalization. The only new ingredient is Theorem 2.1. For the convenience of the reader we repeat the arguments from [4].

Let (π, V_{π,K}) ∈ HC(𝔤, K) be a Harish-Chandra module. Then V_{π,K} decomposes into a direct sum of joint generalized eigenspaces of Z(𝔤). Since the summands can be treated separately, without loss of generality we can assume that there exist μ ∈ ℂ and k ∈ N such that B := (−Ω − μ)^{k} ∈ Ann(V_{π,K}), i.e., BV_{π,K} = 0.

Let W be a finite-dimensional K-stable subspace of the dual V_{π̃,K} of V_{π,K} in the category HC(𝔤, K), which generates V_{π̃,K} as a U(𝔤)-module. Let E_{0} → X be the homogeneous vector bundle G×_{K} W̃ and E_{0} be the space of its smooth sections. Using any globalization V_{π} of V_{π,K} (i.e. a representation of G such that (V_{π})_{K} = V_{π,K}) we define an embedding

i : V_{π,K} ↪ E_{0} ≅ [C^{∞}(G) ⊗ W̃]^{K}

that is characterized by

⟨i(v)(g), w⟩ := ⟨w, π(g^{−1})v⟩,  v ∈ V_{π,K}, w ∈ W, g ∈ G.  (16)

In fact i maps into S_{∞}E_{0} and the closure of i(V_{π,K}) in S_{∞}E_{0} is contained in S_{∞}E_{0}(B) and constitutes the distribution vector globalization V_{π,−∞} of V_{π,K} (Wallach [26], Ch. 11, Casselman [9]).

We will also consider the spaces E_{0}^{for} := Hom_{U(𝔨)}(U(𝔤), W̃) of formal power series sections of E_{0} and V_{π,for} := V_{π̃,K}^{∗} of formal power series vectors of V_{π,K}. There is an exact functor from HC(𝔤, K) to the category of (not necessarily K-finite) (𝔤, K)-modules which sends V_{π,K} to V_{π,for}. Note that V_{π,for} = ∏_{γ∈K̂} V_{π,K}(γ), V_{π,K}(γ) denoting the γ-isotypic component.

For homogeneous vector bundles E and F on X we denote by D(E, F) the set of G-invariant differential operators D : E → F.

Proposition 3.1 There exist homogeneous vector bundles E_{1}, E_{2}, … on X and Ginvariant differential operators D_{i} ∈ D(E_{i}, E_{i+1}), i = 0, 1, …, such that the embedding i : V_{π,−ω} ↪ E_{0}(B) can be extended to a (possibly infinite) exact sequence

0 → V_{π,−∞} →^{i} S_{∞}E_{0}(B) →^{D_{0}} S_{∞}E_{1}(B) →^{D_{1}} S_{∞}E_{2}(B) →^{D_{2}} ….  (17)

This sequence remains to be exact on the level of formal power series :

0 → V_{π,for} →^{i} E_{0}^{for}(B) →^{D_{0}} E_{1}^{for}(B) →^{D_{1}} E_{2}^{for}(B) →^{D_{2}} ….  (18)

Proof. Let Z(E) be the image of Z(𝔤) in D(E, E). For any vector bundle E → X the ℂ[B]-module Z(E) is finitely generated ([4], Lemma 2.3). In order to make the exposition transparent we include a proof of the following well-known result.

Lemma 3.2 For any vector bundle E → X we have E(B)_{K} ∈ HC(𝔤, K).

Proof. Let (γ, V_{γ}) be the finite-dimensional representation of K corresponding to E and (γ̃, Vγ̃) be its dual. We consider the K-equivariant embedding

i : Vγ̃ ↪ { \atop E(B)_{K}}

defined by

i(ṽ)(f) := ⟨ṽ, f(e)⟩ ,

where we identify the fibre of E at e = [K] with V_{γ}. Let T := U(𝔤)(i(Vγ̃)). For any t ∈ T the dimension of Z(𝔤)t can be estimated by the dimension of a generating subspace of the ℂ[B]-module Z(E). Thus T is a locally Z(𝔤)-finite and finitely generated U(𝔤)- module. By a theorem of Harish-Chandra ([25], 3.4.7), T ∈ HC(𝔤, K). The canonical map E(B)_{K} → T̃ is injective by the analyticity of solutions of the equation Bf = 0. In fact, an element in the kernel of this map would have a vanishing Taylor series at e. We obtain that T ↪ { \atop E(B)_{K}} is surjective. Thus T = { \atop E(B)_{K}} and E(B)_{K} ∈ HC(𝔤, K) since the dual of a Harish-Chandra module is a Harish-Chandra module, too ([25], 4.3.2). □

Lemma 3.3 Let V_{π,K} be a Harish-Chandra submodule of E(B)_{K}. Then there exist a homogeneous vector bundle F and an operator D ∈ D(E, F) such that ker D ∩ E(B)_{K} = V_{π,K}. We also have ker D ∩ S_{∞}E(B) = V_{π,−∞}.

Proof. According to the proof of Lemma 3.2 there is a surjection

U(𝔤) ⊗_{U(𝔨)} Vγ̃ → { \atop E(B)_{K}} .

Let W be a finite-dimensional K-stable generating subspace of the Harish-Chandra module V_{π,K}^{⊥} ⊂ { \atop E(B)_{K}}. Then we choose a K-equivariant map α such that the following diagram

W ⟶^{α} U(𝔤) ⊗_{U(𝔨)} Vγ̃
↓  ↓
V_{π,K}^{⊥} ⟶  { \atop E(B)_{K}}

commutes. This is possible since U(𝔤) ⊗_{U(𝔨)} Vγ̃ is K-semisimple.

We set F := G ×_{K} W̃. The map α can be considered as an element of

[U(𝔤) ⊗_{U(𝔨)} Vγ̃ ⊗ W̃]^{K} ≅ [U(𝔤) ⊗_{U(𝔨)} Hom(V_{γ}, W̃)]^{K} .

The latter space is canonically isomorphic to D(E, F) via the right regular representation R of U(𝔤) on C^{∞}(G)⊗V_{γ}. Thus α defines an element D ∈ D(E, F). If α(w) = ∑ X_{i} ⊗v_{i}, then

⟨w, Df⟩ = ∑⟨v_{i}, R_{X_{i}}f⟩ ∈ C^{∞}(G), w ∈ W, v_{i} ∈ Vγ̃, X_{i} ∈ U(𝔤) .

Let f ∈ S_{∞}E(B), X ∈ U(𝔤) and w ∈ W. Then we have

⟨w, L_{X}Df(1)⟩ = ⟨w, DL_{X}f(1)⟩
= ∑⟨v_{i}, R_{X_{i}}L_{X}f(1)⟩  (19)
= ⟨w, L_{X}f⟩
= ⟨L_{X^{op}}w, f⟩ ,

where X → X^{op} is the anti-automorphism of U(𝔤) induced by the multiplication with −1 on 𝔤. Since f is real analytic we have Df = 0 iff the left hand side of (19) vanishes for all X ∈ U(𝔤) and w ∈ W, while by construction of W we have f ∈ V_{π,−∞} iff the right hand side does. The lemma follows. □

In order to construct the bundles E_{i} and operators D_{i} of Proposition 3.1 we iterate Lemma 3.3. D_{i}(E_{i}(B)_{K}) is a Harish-Chandra submodule of E_{i+1}(B)_{K}. Therefore we find a bundle E_{i+2} and an operator D_{i+1} ∈ D(E_{i+1}, E_{i+2}) such that ker D_{i+1} ∩ E_{i+1}(B)_{K} = D_{i}(E_{i}(B)_{K}). We obtain an exact sequence of Harish-Chandra modules

0 → V_{π,K} →^{i} E_{0}(B)_{K} →^{D_{0}} E_{1}(B)_{K} →^{D_{1}} E_{2}(B)_{K} →^{D_{2}} ….

Applying the distribution vector globalization functor (which is exact) we end up with (17). Analogously, we want to obtain (18) by taking formal power series vectors. This is possible since for any homogeneous vector bundle E we have

(E(B)_{K})_{for} = E^{for}(B)

([4], Lemma 3.5). The proof of Proposition 3.1 is now complete. □

The Proposition 3.1 provides a resolution of V_{π,−∞} by spaces S_{∞}E_{i}(B). We now employ a Koszul complex construction in order to replace the eigenspaces S_{∞}E_{i}(B) by section spaces S_{∞}E_{i}. We recall the following fact from [4].

Lemma 3.4 Let E, F be homogeneous vector bundles on X and A ∈ D(E, F) such that AE(B) = 0. Then A = HB for some H ∈ D(E, F).

Let V_{π,K}, E_{i}, D_{i} be as in Proposition 3.1.

Proposition 3.5 There exist H_{i} ∈ D(E_{i}, E_{i+2}), i ≥ 0, making the following into an exact complex:

0 → V_{π,−∞} → S_{∞}E_{0} \underrightarrow{( \begin{matrix}D_{0} \\ B \end{matrix} )} \begin{matrix}S_{∞}E_{1} \\ ⊕ \\ S_{∞}E_{0} \end{matrix} \underrightarrow{( \begin{matrix}D_{1} & H_{0} \\ −B & D_{0} \end{matrix} )} \begin{matrix}S_{∞}E_{2} \\ ⊕ \\ S_{∞}E_{1} \end{matrix} \underrightarrow{( \begin{matrix}D_{2} & H_{1} \\ B & D_{1} \end{matrix} )} ….  (20)

4 𝔑-COHOMOLOGY 15

We shall call (20) a standard resolution of V_{π,−∞}. Proof. In order to construct the operators H_{i} we apply Lemma 3.4 to A = D_{i+1}D_{i}. The exactness of (20) is easily reduced to the exactness of (17) and the surjectivity of B : S_{∞}E_{i} → S_{∞}E_{i} that was proved in Theorem 2.1. □

4 𝔫-cohomology

Fix a minimal parabolic subgroup P = MAN, the Langlands decomposition chosen compatible with 𝔨, and let 𝔫 be the Lie algebra of N. Let B = (−Ω − λ)^{l} for some λ ∈ ℂ, l ∈ ℕ, and S_{∞}E(B) = {f ∈ S_{∞}E | Bf = 0}.

Lemma 4.1 We have

H^{p}(𝔫, S_{∞}E) = H^{p}(𝔫, S_{∞}E(B)) = 0, ∀p ≥ 1.

Proof. We consider the standard 𝔫-cohomology complex (Λ^{∗}𝔫^{∗} ⊗ S_{∞}E, d_{𝔫}). We have to show that it is exact in all non-zero degrees. The idea is to use the contraction of the N-orbits in X in order to define a zero-homotopy

I : Λ^{p}𝔫^{∗} ⊗ S_{∞}E → Λ^{p−1}𝔫^{∗} ⊗ S_{∞}E , p ≥ 1 .

Using the Iwasawa decomposition G = NAK we can identify

Λ^{p}𝔫^{∗} ⊗ S_{∞}E = {f ∈ C^{∞}(NA, Λ^{p}𝔫^{∗} ⊗ V_{γ})

| ∀L_{1} ∈ U(𝔫), L_{2} ∈ U(𝔞) ∃R ∈ ℝ

s.t. sup_{na∈NA} e^{−Rdist(O,naK)}|f(L_{1}nL_{2}a)| < ∞} .

We further identify C^{∞}(NA, Λ^{p}𝔫^{∗} ⊗ V_{γ}) with the partial de Rham complex C^{∞}(NA, Λ^{p}T^{∗}N ⊗ V_{γ}) using the trivialization of T N by left fundamental vector fields.

For t ∈ [0, 1] we define the contraction Ψ_{t} : NA → NA by Ψ_{t}(exp(X)a) := exp(tX)a, X ∈ 𝔫, a ∈ A. Let T_{t} := \frac{d}{ds}_{|s=0}Ψ_{t+s} ∈ C^{∞}(NA, Ψ_{t}^{∗}(T N)).

The homotopy I is now defined by

Iω := ∫_{0}^{1} Ψ_{t}^{∗}(T_{t} ⌋ω)dt , ω ∈ C^{∞}(NA, Λ^{p}T^{∗}N ⊗ V_{γ}) .  (21)

As in the Poincaré Lemma for the usual de Rham complex one shows that Id_{𝔫}+d_{𝔫}I = id. Since 𝔫 is nilpotent we have

If(exp(X)a) = ∫_{0}^{1} Φ(t, X)f(exp(tX)a)dt , X ∈ 𝔫, a ∈ A ,

where f ∈ C^{∞}(NA, Λ^{p}𝔫^{∗} ⊗ V_{γ}), and Φ : [0, 1] × 𝔫 → Hom(Λ^{p}𝔫^{∗}, Λ^{p−1}𝔫^{∗}) is a polynomial function. It remains to verify that if f ∈ Λ^{p}𝔫^{∗} ⊗ S_{∞}E, then If ∈ Λ^{p−1}𝔫^{∗} ⊗ S_{∞}E.

Let f ∈ Λ^{p}𝔫^{∗} ⊗ S_{∞}E, L_{1} ∈ U(𝔫), L_{2} ∈ U(𝔞). Let Δ : U(𝔫) → U(𝔫) ⊗ U(𝔫) be the coproduct given for X ∈ 𝔫 by Δ(X) := X ⊗ 1 + 1 ⊗ X. Write Δ(L_{1}) = ∑_{α} A_{α} ⊗ B_{α}. Due to the nilpotency of 𝔫 for any L ∈ U(𝔫) there are polynomial functions p_{L} : [0, 1] × 𝔫 → Hom(Λ^{p}𝔫^{∗}, Λ^{p−1}𝔫^{∗}) and q_{L} : [0, 1] × 𝔫 → U(𝔫) and R_{α} ∈ ℝ such that

|If(L_{1} exp(X)L_{2}a)| = | ∑_{α} ∫_{0}^{1} t^{p−1}p_{A_{α}}(t, X)f(q_{B_{α}}(t, X) exp(tX)L_{2}a)dt|

≤ ∑_{α} C_{α}(1 + |X|)^{n_{α}} ∫_{0}^{1} e^{R_{α}dist(O,exp(tX)aK)}dt .

Now using that dist(O, exp(tX)aK) ≤ dist(O, exp(X)aK) for t ∈ [0, 1] and (1 + |X|)^{n_{α}} ≤ C′e^{Qdist(O,exp(X)aK)} for some C′ < ∞, Q > 0 and all a ∈ A, X ∈ 𝔫 and α, we see that for R := max({0} ∪ {R_{α}})

|If(L_{1} exp(X)L_{2}a)| ≤ Ce^{(R+Q)dist(O,exp(X)aK)} .

We conclude that If ∈ Λ^{p−1}𝔫^{∗} ⊗ S_{∞}E. The first assertion of the lemma

H^{p}(𝔫, S_{∞}E) = 0 , ∀p ≥ 1

follows.

This together with Theorem 2.1 shows that

0 → S_{∞}E(B) → S_{∞}E →^{B} S_{∞}E → 0

is an 𝔫-acyclic resolution of S_{∞}E(B). Taking 𝔫-invariants we obtain the complex

0 → ^{𝔫}S_{∞}E(B) → S_{∞}C^{∞}(A) ⊗ V_{γ} →^{^{𝔫}B} S_{∞}C^{∞}(A) ⊗ V_{γ} → 0 .  (22)

Here ^{𝔫}B is the restriction of B to the subspace of 𝔫-invariant vectors and S_{∞}C^{∞}(A) is the space of all smooth functions on A with derivatives growing not faster than exponentially. As a second order translation-invariant differential operator^{𝔫}B is surjective on S_{∞}C^{∞}(A). Thus the complex (22) is again exact, and the remaining assertion of the lemma follows. □

Let (π, V_{π,K}) ∈ HC(𝔤, K). Recall that H^{∗}(𝔫, V_{π,−∞}) carries a natural MA-module structure.

Theorem 4.2 The inclusion V_{π,−∞} ↪ V_{π,for} induces an isomorphism

H^{p}(𝔫, V_{π,−∞}) ⟶^{∼} H^{p}(𝔫, V_{π,for}) .

5 Γ-ACYCLICITY OF S_{∞}E 17

Moreover H^{p}(𝔫, V_{π,−∞}) = H^{p}(𝔫, V_{π,−ω}) = H^{p}(𝔫, V_{π,for}), and all spaces are finite-dimen-sional. The 𝔫-cohomology of V_{π,−∞} satisfies Poincaré duality

H^{p}(𝔫, V_{π,−∞})^{∗} ≅ H^{m−p}(𝔫, V_{π̃,∞}) ⊗ Λ^{m}𝔫,  (23)

where m = dim(𝔫). We also have

H^{p}(𝔫, V_{π,∞}) ≅ H^{p}(𝔫, V_{π,ω}) ≅ H^{p}(𝔫, V_{π,K}) .

Proof. The spaces S_{∞}E(B) and E^{for}(B) are 𝔫-acyclic (see Lemma 4.1 and [4], Lemma 2.3). Thus the cohomology H^{p}(𝔫, V_{π,∗}) for ∗ = −∞, for is isomorphic to the cohomology of the subcomplex of 𝔫-invariants of (17), (18), respectively. The following lemma implies the first assertion of the theorem.

Lemma 4.3 For any homogeneous vector bundle E → X we have

𝔫S_{∞}E(B) = ^{𝔫}E(B) = ^{𝔫}E^{for}(B) .

Proof. The U(𝔞)-module

𝔫E^{for}(B) ≅ ({ \atop E(B)_{K}}/𝔫({ \atop E(B)_{K}}))^{∗}

is finite-dimensional (see [25], Ch.4). Therefore it splits into generalized weight spaces 𝔫E^{for}(B)_{μ}, μ ∈ 𝔞_{ℂ}^{∗}. f ∈ ^{𝔫}E^{for}(B)_{μ}, considered as a formal power series on 𝔞, satisfies the differential equation

(H + μ(H))^{k}f = 0 ∀H ∈ 𝔞  (24)

for a certain k ∈ ℕ. The solutions of (24) have the form

P (H)e^{−μ(H)}, P ∈ S(𝔞^{∗}) ⊗ V_{γ} .

They extend to smooth 𝔫-invariant sections in ^{𝔫}S_{∞}E(B). □

The proof of the remaining assertions of the theorem is parallel to the proofs of the corresponding facts in [4]. □

5 Γ-acyclicity of S_{∞}E

Let Γ ⊂ G be a discrete, torsion-free subgroup of finite covolume and Y := Γ\X be the corresponding locally symmetric space. Let E → X be a G-homogeneous vector bundle and S_{∞}E be the space of its sections of moderate growth. In order to use the standard resolution (20) for the computation of H^{∗}(Γ, V_{π,−∞}), V_{π,K} ∈ HC(𝔤, K), we have to know that S_{∞}E is Γ-acyclic, i.e., H^{p}(Γ, S_{∞}E) = 0 for all p ≥ 1.

Compared with the corresponding result for E the proof of the Γ-acyclicity of S_{∞}E is surprisingly involved. The idea is to show the Γ-acyclicity of the space S_{∞}C^{∞}(G) of functions of moderate growth on G by looking at the de Rham complex on Y twisted with the flat vector bundle associated to the Γ-module S_{∞}C^{∞}(G). One would like to replace this de Rham complex by the subcomplex consisting of forms of moderate growth (w.r.t. the y ∈ Y -variable) without changing the cohomology. This is possible by Lemma 5.2 and Corollary 5.3. One observes that the cochain spaces of the latter complex are isomorphic to the spaces of forms of moderate growth on X with values in the space of functions of moderate growth on Γ\G. The resulting complex can be shown to be acyclic using the contraction of X as in the proof of the Poincaré Lemma. The Γ-acyclicity of S_{∞}C^{∞}(G) easily implies that of S_{∞}E.

We consider the space of functions of moderate growth S_{∞}C^{∞}(G) on G defined by

S_{∞}C^{∞}(G) := {f ∈ C^{∞}(G) | ∀L ∈ U(𝔤) ∃R ∈ ℝ s.t. sup_{g∈G} e^{−Rdist(O,gK)}|f(Lg)| < ∞} .

Let π : X × G → X be the projection. We also consider the following de Rham type complexes of Γ-modules

• (C^{∗}, d), where

C^{p} := {ω ∈ C^{∞}(X × G, π^{∗}Λ^{p}T^{∗}X) | ∀L_{1}, L_{2} ∈ U(𝔤), ∀W ⊂^{⊂}X

∃R ∈ ℝ s.t. sup_{g∈G,x∈W} e^{−Rdist(O,gK)}|ω(L_{1}x, L_{2}g)| < ∞}

≅ {f ∈ C^{∞}(G × G, Λ^{p}T_{O}^{∗}X) | f(gk, h) = k^{−1}f(g, h) ∀k ∈ K, ∀L_{1}, L_{2} ∈ U(𝔤),

∀W ⊂^{⊂}G ∃R ∈ ℝ s.t. sup_{h∈G,g∈W} e^{−Rdist(O,hK)}|f(g, Lh)| < ∞}

with the Γ-action γf(g, h) := f(γ^{−1}g, γ^{−1}h), and the de Rham differential differentiates w.r.t. the first variable;

• (S^{∗}, d) ⊂ (C^{∗}, d), where

S^{p} := {f ∈ C^{p} | ∀L_{1}, L_{2} ∈ U(𝔤) ∃R, Q ∈ ℝ s.t.

sup_{g,h∈G} e^{−Rdist(O,gK)}e^{−Qdist(O,hK)}|f(L_{1}g, L_{2}h)| < ∞} ;

• (S_{+}^{∗}, d) coincides with (S^{∗}, d) as a complex of vector spaces, but has a different Γ-module structure: γf(g, h) := f(g, γ^{−1}h).

The complex of Γ-invariants (^{Γ}C^{∗}, d) can be considered as the de Rham complex on Y twisted with the flat bundle associated to the Γ-module S_{∞}C^{∞}(G).

Lemma 5.1 H^{∗}(Γ, S_{∞}C^{∞}(G)) is equal to the cohomology of the complex (^{Γ}C^{∗}, d).

Proof. The contraction of X along radial rays induces a zero-homotopy of the de Rham complex (C^{∗}, d). We leave to the reader to verify that the zero homotopy is compatible with the growth conditions with respect to the second variable. We will discuss a similar problem in the proof of Lemma 5.5. Thus the complex (C^{∗}, d) is exact in all non-zero degrees. Its zeroth cohomology is the Γ-module S_{∞}C^{∞}(G). It can be shown along the lines of [4], Lemma 2.4. that (C^{∗}, d) consists of Γ-acyclic modules. Thus (C^{∗}, d) constitutes a Γ-acyclic resolution of S_{∞}C^{∞}(G). □

The manifold Y has finitely many cusps each diffeomorphic to B_{i} × [0, ∞), where B_{i} ≅ Γ ∩ N_{i}\N_{i}, N_{i} being the nil-radical of the stabilizer P_{i} of the corresponding cusp. B_{i} is compact. Attaching copies of the cusp bases B_{i} as a boundary at infinity we obtain the Borel-Serre compactification of Y which is a manifold with boundary Ȳ.

There is a natural sheafification (\underline{^{Γ}S^{∗}}, d) of (^{Γ}S^{∗}, d) on Ȳ: for an open set U ⊂ Ȳ the vector space \underline{^{Γ}S^{p}}(U) consists of those forms ω ∈ C^{∞}((U ∩ Y) × G, π^{∗}T^{∗}Y) which satisfy

sup_{ΓgK∈U∩Y,h∈G} |ω(L_{1}g, L_{2}h)|e^{−Rdist_{Y} (O_{Y} ,ΓgK)}e^{−Qdist(O,hK)} < ∞

for any L_{1}, L_{2} ∈ U(𝔤) and appropriate Q, R ∈ ℝ which may depend on L_{1}, L_{2}. Analogously, sheafifying (^{Γ}C^{∗}, d) we obtain the complex of sheaves (\underline{^{Γ}C^{∗}}, d) on Ȳ.

A complex is called acyclic if it is exact in all non-zero degrees.

Lemma 5.2 The complex of sheaves (\underline{^{Γ}S^{∗}}, d) is acyclic.

Proof. Let y ∈ Ȳ. Then we have to show that the complex of germs (\underline{^{Γ}S^{∗}}_{y}, d) is acyclic. If y ∈ Y, then \underline{^{Γ}S^{∗}}_{y} = \underline{^{Γ}C^{∗}}_{y}, and we employ the standard homotopy formula associated to the radial contraction of small balls in Y with center at y in the same way as in the proof of Lemma 5.1.

It remains to consider y ∈ ∂Y. Set N := N_{i} and write y = (Γ ∩ N)n_{0} × {∞}, where n_{0} ∈ N. Let G_{j} ⊂ 𝔫, j ∈ ℕ, be a fundamental system of balls around 0, U_{j} := (Γ ∩ N)n_{0} exp(G_{j}) ⊂ B_{i}. Then W_{j} := U_{j} × (j, ∞] ⊂ B_{i} × (0, ∞] is a fundamental system of neighbourhoods of y in Ȳ. We decompose forms ω on W_{j} as ω = ω_{1} ⊕ ω_{2} ∧ dt, such that \frac{∂}{∂t} ⌋ ω_{j} = 0, j = 1, 2. One can check that

\underline{^{Γ}S^{p}}(W_{j}) = V^{p}(W_{j}) ⊕ V^{p−1}(W_{j}) ∧ dt ,

where

V^{p}(W_{j}) := {ω ∈ C^{∞}(U_{j} × (j, ∞) × G, π_{1}^{∗}Λ^{p}T^{∗}U_{j})|

∀L_{1} ∈ U(𝔫), L_{2} ∈ ℂ[\frac{∂}{∂t}], L_{3} ∈ U(𝔤) ∃R, Q ∈ ℝ

s.t. sup_{t∈(j,∞),h∈G} e^{−tR}e^{−Qdist(O,hK)}|ω(L_{1}n, L_{2}t, L_{3}h)| < ∞} ,

and π_{1} : U_{j} × (j, ∞) × G → U_{j} is the projection onto the first factor. The complex (\underline{^{Γ}S^{∗}}(W_{j}), d) is the total complex of a double complex. The latter consists of two rows equal (V^{∗}(W_{j}), d_{𝔫}), where d_{𝔫} is the differential of the de Rham complex of U_{j}. The vertical differential of the double complex is the differentiation along the t-direction given by ±\frac{∂}{∂t}. The balls U_{j} can be contracted radially. The associated zero homotopy of the de Rham complex of U_{j} extends to V^{∗}(W_{j}). Again we leave the verification to the reader (see also the proof of Lemma 4.1). Thus the rows (V^{∗}(W_{j}), d_{𝔫}) of the double complex are acyclic. The zeroth horizontal cohomology is equal to ^{𝔫}V^{0}(W_{j}), the functions in V^{0}(W_{j}) that do not depend on n ∈ U_{j}. To finish the proof of the Lemma we must show that \frac{∂}{∂t} : ^{𝔫}V^{0}(W_{j}) → ^{𝔫}V^{0}(W_{j}) is surjective. In fact, given g ∈ ^{𝔫}V^{0}(W_{j})

f(n, t, h) := ∫_{j}^{t} g(n, s, h)ds

solves the equation \frac{∂}{∂t}f = g such that f ∈ ^{𝔫}V^{0}(W_{j}). □

Corollary 5.3 The inclusion (^{Γ}S^{∗}, d) ↪ (^{Γ}C^{∗}, d) induces an isomorphism in cohomology.

Proof. To any finite open covering of Ȳ there is an associated partition of unity which is compatible with the sheafs \underline{^{Γ}S^{∗}} and \underline{^{Γ}C^{∗}}. Thus \underline{^{Γ}S^{∗}} and \underline{^{Γ}C^{∗}} are acyclic with respect to the global section functor.

The complex of sheaves corresponding to (^{Γ}C^{∗}, d) is locally acyclic by the standard Poincaré Lemma. Moreover the inclusion of sheaves (\underline{^{Γ}S^{∗}}, d) ↪ (\underline{^{Γ}C^{∗}}, d) induces an isomorphism of the zeroth cohomology sheaves. By Lemma 5.2 (\underline{^{Γ}S^{∗}}, d) is locally acyclic. Thus the inclusion (\underline{^{Γ}S^{∗}}, d) ↪ (\underline{^{Γ}C^{∗}}, d) is a quasi-isomorphism. Since the sheaves \underline{^{Γ}S^{∗}}, \underline{^{Γ}C^{∗}} are acyclic with respect to the global section functor, the induced map of the complexes of global sections (^{Γ}S^{∗}, d) ↪ (^{Γ}C^{∗}, d) is a quasi-isomorphism, too. □

Lemma 5.4 The map T : X × G → X × G given by (x, g) ↦ (gx, g) induces an isomorphism of complexes of Γ-modules

T^{∗} : (S^{∗}, d) ⟶^{∼} (S_{+}^{∗}, d) .

Proof. It is clear that T^{∗} commutes with d and intertwines the Γ-actions. We show that T^{∗}(S^{p}) ⊂ S^{p}. In a similar manner one can prove the opposite inclusion (T^{−1})^{∗}(S^{p}) ⊂ S^{p}, where T^{−1}(x, g) = (g^{−1}x, g). Both inclusions together imply the lemma.

Let f ∈ S^{p}, L_{1}, L_{2} ∈ U(𝔤). Let r = deg(L_{1}) + deg(L_{2}) and {X_{i}} be a basis of U_{r}(𝔤), the elements of U(𝔤) of order ≤ r. Then there exist Q, R ∈ ℝ such that

sup_{g,h∈G,i,j} e^{−Rdist(O,gK)}e^{−Qdist(O,hK)}|f(X_{i}g, X_{j}h)| < ∞ .

Let Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) be the coproduct induced by X ↦ X ⊗ 1 + 1 ⊗ X, X ∈ 𝔤, and write Δ(L_{2}) = ∑ A_{α} ⊗ B_{α}. We have

T^{∗}f(L_{1}g, L_{2}h) = ∑_{α} f(A_{α}hL_{1}g, B_{α}h) .

Interchanging h and L_{1} we obtain for some C < ∞, P ∈ ℝ

|T^{∗}f(L_{1}g, L_{2}h)| ≤ Ce^{P dist(O,hK)}e^{Rdist(O,hgK)}e^{Qdist(O,hK)}
≤ Ce^{Rdist(O,gK)}e^{(P +Q+R)dist(O,hK)} , ∀g, h ∈ G .

In order to deduce the second inequality we have employed that dist(O, hgK) ≤ dist(O, hK) +dist(O, gK). These estimates show that T^{∗}f ∈ S^{p}. □

Lemma 5.5 The de Rham complex (^{Γ}S_{+}^{∗}, d) is acyclic.

Proof. In contrast to (^{Γ}S^{∗}, d) the complex (^{Γ}S_{+}^{∗}, d) is a twisted de Rham complex on X. So we could try to employ the contraction of X along radial rays in order to prove the lemma. In view of the work we have already done it is more convenient to contract first the N-orbits in X, where N comes from a fixed Iwasawa decomposition G = NAK.

As in the proof of Lemma 5.2 the de Rham complex (^{Γ}S_{+}^{∗}, d) is the total complex of a double complex consisting of two rows each equal to the 𝔫-cohomology complex (Λ^{∗}𝔫 ⊗^{Γ}S_{+}^{0}, d_{𝔫}). We identify Λ^{p}𝔫 ⊗^{Γ}S_{+}^{0} with

{f ∈ C^{∞}(N × A × Γ\G, Λ^{p}𝔫)

| ∀L_{1} ∈ U(𝔫), L_{2} ∈ U(𝔞), L_{3} ∈ U(𝔤) ∃R, Q ∈ ℝ

s.t. sup_{na∈NA,g∈G} e^{−Rdist(O,naK)}e^{−Qdist_{Y} (O_{Y} ,ΓgK)}|f(L_{1}n, L_{2}a, L_{3}g)| < ∞} .

We show that the complex (Λ^{∗}𝔫 ⊗^{Γ}S_{+}^{∗}, d_{𝔫}) is acyclic. Let I be the zero homotopy (21) which we extend trivially to the g-variable.

We claim that I : Λ^{p}𝔫 ⊗^{Γ}S_{+}^{0} → Λ^{p−1}𝔫 ⊗^{Γ}S_{+}^{0}. The same discussion as in the proof of Lemma 4.1 leads to the following estimate. Fix L_{1} ∈ U(𝔫), L_{2} ∈ U(𝔞), L_{3} ∈ U(𝔤). Let r := deg(L_{1}) and {X_{i}} be a basis of U_{r}(𝔫). For all f with

sup_{i,na∈NA,g∈G} e^{−P dist_{Y} (O_{Y} ,ΓgK)}e^{−Rdist(O,naK)}|f(X_{i}n, L_{2}a, L_{3}g)| < ∞

we have for some Q ∈ ℝ

sup_{na∈NA,g∈G} e^{−P dist_{Y} (O_{Y} ,ΓgK)}e^{−(R+Q)dist(O,naK)}|(If)(L_{1}n, L_{2}a, L_{3}g)| < ∞ .

This implies the claim.

6 THE RANGE OF THE LAPLACIAN ON Y 22

Hence the rows of the double complex are acyclic and their zeroth cohomology is equal to

V := {f ∈ C^{∞}(A × Γ\G) | ∀L_{2} ∈ U(𝔞), L_{3} ∈ U(𝔤) ∃R, Q ∈ ℝ

s.t. sup_{a∈A,g∈G} e^{−R| log(a)|}e^{−Qdist_{Y} (O_{Y} ,ΓgK)}|f(L_{2}a, L_{3}g)| < ∞} .

The vertical differential given by H : V → V, H ∈ 𝔞^{+}, |H| = 1, is surjective. In fact, let f ∈ V and set

F (a, g) := ∫_{1}^{a} f(b, g)db .

Then HF = f and F ∈ V. This proves the lemma. □

Theorem 5.6 The Γ-modules S_{∞}C^{∞}(G) and S_{∞}E are acyclic.

Proof. Lemma 5.1 combined with Corollary 5.3 and Lemma 5.4 shows that H^{∗}(Γ, S_{∞}C^{∞}(G)) is equal to the cohomology of the complex (^{Γ}S_{+}^{∗}, d) which is acyclic by Lemma 5.5. The Γ-acyclicity of S_{∞}C^{∞}(G) follows.

S_{∞}C^{∞}(G) carries a right K-module structure which commutes with the left Γ-module structure. This induces a K-action on (^{Γ}C^{∗}, d). H^{∗}(Γ, S_{∞}E) is the cohomology of the complex ([^{Γ}C^{∗} ⊗ V_{γ}]^{K}, d). Let [z] ∈ H^{p}(Γ, S_{∞}E), p ≥ 1, be represented by a K-invariant cocycle z ∈ [^{Γ}C^{p} ⊗ V_{γ}]^{K}. Then by the Γ-acyclicity of S_{∞}C^{∞}(G) we have z = db for some possibly non-invariant (p − 1)-cochain b ∈ ^{Γ}C^{p−1} ⊗ V_{γ}. Let b̄ the average of b with respect to K. Then also db̄ = z and hence 0 = [z] ∈ H^{p}(Γ, S_{∞}E). This finishes the proof of the theorem. □

6 The range of the Laplacian on Y

In order to determine the cohomology of the subcomplex of Γ-invariants of the standard resolution (20) we need to know the range of the operator B on the sections of moderate growth on Y. It will turn out that the range of B consists of all sections orthogonal to the space of cusp forms annihilated by B. In particular, B has closed range and a finite-dimensional cokernel.

First we introduce a scale of weighted spaces on Y. Let E_{Y} := Γ\E be the locally homogeneous bundle over Y. A section f ∈ C^{∞}(Y, E_{Y}) =: E_{Y} can be viewed as a function on f : G → V_{γ} satisfying f(hgk) = γ^{−1}(k)f(g) for all h ∈ Γ, k ∈ K. We define an increasing sequence of Fréchet spaces S_{R}E_{Y}, R ∈ ℝ, by

S_{R}E_{Y} := {f ∈ E_{Y} | p_{Y;R,L}(f) < ∞ ∀L ∈ U(𝔤)} ,

where the seminorms are defined by

p_{Y;R,L}(f) := sup_{g∈G} e^{−Rdist(O_{Y} ,ΓgK)}|f(gL)| .

Note that (in contrast to the definition of S_{R}E) we employ the right action of U(𝔤) to define p_{Y;R,L}(f). In fact, to define S_{R}E_{Y} it is sufficient to consider the seminorms p_{Y;R,Ω^{k}}(f), Ω being the Casimir operator of G, k ∈ ℕ_{0}. Let

S_{−∞}E_{Y} := ⋂_{R∈ℝ}S_{R}E_{Y}

with the natural topology of the intersection. Elements in S_{−∞}E_{Y} which are (sums of) eigensections of Ω are called cusp forms. Let S_{∞}E′_{Y} be the topological conjugate dual of S_{−∞}E_{Y}.

Our main interest is in the space of sections of moderate growth S_{∞}E_{Y} := ^{Γ}S_{∞}E. It has the alternative description

S_{∞}E_{Y} = {f ∈ E_{Y} | ∀L ∈ U(𝔤) ∃R ∈ ℝ s.t. sup_{g∈G} e^{−Rdist_{Y} (O_{Y} ,ΓgK)}|f(gL)| < ∞} .

The Hermitian scalar product of E_{Y} induces an inclusion S_{∞}E_{Y} ↪ S_{∞}E′_{Y}. Fix μ ∈ ℂ and set B := −Ω − μ. We consider B as an operator on S_{∞}E′_{Y}.

Lemma 6.1 The operator B^{∗} : S_{−∞}E_{Y} → S_{−∞}E_{Y} has closed range.

Proof. Let {f_{i}} be a sequence in S_{−∞}E_{Y} such that B^{∗}f_{i} =: h_{i} → h ∈ S_{−∞}E_{Y}. We have to find f ∈ S_{−∞}E_{Y} with B^{∗}f = h. If ker B^{∗} ≠ 0, then we can project f_{i} to the L^{2}-orthogonal complement of ker B^{∗}. Thus we can assume that f_{i} ⊥ ker B^{∗}.

We can assume that for some R ∈ ℝ the sequence p_{Y;R,1}(f_{i}) is bounded. If not, dividing f_{i} by p_{Y;R,1}(f_{i}) we obtain a sequence f̃_{i} with p_{Y;R,1}(f̃_{i}) = 1 and B^{∗}f̃_{i} → 0. We show below that f̃_{i} has a subsequence converging to F ∈ S_{−∞}E_{Y}. Now B^{∗}F = 0 and F ⊥ ker B^{∗}. Hence F = 0 contradicting p_{Y;R,1}(F) = 1.

Since for all k ∈ ℕ_{0} the sequence p_{Y;R,Ω^{k}}(h_{i}) is bounded we conclude that p_{Y;R,Ω^{k}}(f_{i}) is bounded, too.

Consider a cusp c_{l} of Y associated to the minimal parabolic subgroup P = MAN ⊂ G. Let 𝔞^{+} be the positive half-line in 𝔞 determined by N and H ∈ 𝔞^{+} be the unit vector. For T ∈ ℝ set A_{T} := {exp(tH) | t > T }. Then a neighbourhood U_{l} of infinity of this cusp can be identified with Γ ∩ N\N × A_{d_{l}}, d_{l} ∈ ℝ large. For f ∈ E_{Y}, the constant term F_{P} ∈ C^{∞}(U_{l}, E_{Y |U_{l}}) is defined by

F_{P} (na) = \frac{1}{vol(Γ ∩ N\N)} ∫_{Γ∩N\N} F (n′na)dn′ .

Let χ ∈ C^{∞}(Y) be a cut-off function being one on Γ ∩ N\N × A_{d_{l}+1} and zero outside of U_{l}. Then F_{c_{l}} = χF_{P} ∈ E_{Y}. If Y has the cusps c_{l}, l = 1, …, r, then we set F_{c} := ∑_{l=1}^{r} F_{c_{l}}.

Applying this construction to our sequence f_{i} we obtain a decomposition f_{i} = f_{i,c}+f_{i,r} with f_{i,r} := f_{i} − f_{i,c}. We want to show that for a suitable chosen subsequence both parts converge.

Since f_{i} is bounded in S_{R}E_{Y}, by Gelfand’s inequality in the form of [10], Thm. 5, the sequence f_{i,r} is bounded in S_{−∞}E_{Y}. Since for R < R′ the embedding S_{R}E_{Y} ↪ S_{R′}E_{Y} is compact, any bounded sequence in S_{−∞}E_{Y} has a converging subsequence. Thus by taking a subsequence we can assume that f_{i,r} converges in S_{−∞}E_{Y}.

For y ∈ ⋃_{l=1}^{r} U_{l} and dist(y, ∂U_{l}) > 1, for l = 1, …, r, we have (B^{∗}f)_{c}(y) = B^{∗}f_{c}(y). Consider again the cusp c_{l} and the coordinates Γ ∩ N\N × A_{d_{l}}. Up to the sign in front of the first order term the N-radial part of B coincides with B_{rad} considered in the proof of Lemma 2.4, (10). Hence there are commuting x, y ∈ End(V_{γ}) such that for t > d_{l} + 1

(B^{∗}F_{c_{l}})(n exp(tH)) = −((\frac{d}{dt} + x)(\frac{d}{dt} + y)F_{c_{l}})(n exp(tH)) .

Assume that R < − max(∥x∥, ∥y∥). We set for F ∈ S_{R}E_{Y}

(H_{c_{l}}F)(na) := −χ(na)e^{−y log(a)} ∫_{log(a)}^{∞} e^{(y−x)s} ∫_{s}^{∞} e^{xt}F_{c_{l}}(n exp(tH))dtds .

Again, if Y has cusps c_{l}, l = 1, …, r, then we set H_{c} := ∑_{l=1}^{r} H_{c_{l}}. Then H_{c} : S_{R}E_{Y} → S_{R}E_{Y} is continuous, and supp(H_{c}B^{∗}F − F_{c}) ⊂ V, where V ⊂ Y is compact and independent of F. The proof of continuity is similar to the corresponding argument in the proof of Lemma 2.4. Notice that Ω^{k}H_{c} − H_{c}Ω^{k} = W is a continuous operator W : S_{R}E_{Y} → C_{c}^{∞}(Y, E_{Y}) and supp(W f) ⊂ V′, where V′ ⊂ Y is compact and independent of f.

Thus F_{i} := H_{c}h_{i} − f_{i,c} is a bounded sequence in S_{−∞}E_{Y}. Hence F_{i} has a subsequence converging in S_{−∞}E_{Y}. Since H_{c}h_{i} converges in S_{−∞}E_{Y} by taking a subsequence we can assume that f_{i,c} converges in S_{−∞}E_{Y}, too. Let f be the limit of f_{i} = f_{i,c} + f_{i,r} for this subsequence. Then B^{∗}f = h. This finishes the proof of the lemma. □

Let E_{Y} (B)_{cusp} the space of cusp forms in the kernel of B.

Corollary 6.2 We have

B(S_{∞}E′_{Y}) = (E_{Y} (B)_{cusp})^{⊥} ⊂ S_{∞}E′_{Y} .

Proof. S_{−∞}E_{Y} is a Fréchet space. Thus Lemma 6.1 implies that imB ⊂ S_{∞}E′_{Y} is weak^{∗} closed (see, e.g., [24]). It follows that imB = (ker B^{∗})^{⊥}. If ker B^{∗} ≠ 0, then μ is real and therefore ker B^{∗} = E_{Y} (B)_{cusp}. □

The final result of this section is Theorem 6.3 Let B : S_{∞}E_{Y} → S_{∞}E_{Y} given by B := (−Ω − μ)^{k} for some k ∈ ℕ. Then

im(B) = (E_{Y} (B)_{cusp})^{⊥} ∩ S_{∞}E_{Y} .

Proof. Obviously, it is enough to prove the theorem for k = 1. In order to apply the regularity result Lemma 2.6 we need a pull back of distributions π^{∗} : S_{∞}E′_{Y} → S_{∞}E′, where π : X → Y is the projection.

We define a continuous map π_{∗} : S_{−∞}E → S_{−∞}E_{Y}. For f ∈ S_{−∞}E let (π_{∗}f)(Γg) := ∑_{γ∈Γ} f(γg). We show that the sum converges and π_{∗} is continuous (compare also [8], Prop. 1.11).

There is a Q ∈ ℝ such that sup_{g∈G} ∑_{γ∈Γ} e^{−Qdist(γgK,O)} =: C < ∞. For k ∈ ℕ_{0} we have

p_{Y;−R,Ω^{k}}(π_{∗}f) = sup_{g∈G} e^{Rdist_{Y} (O_{Y} ,ΓgK)}| ∑_{γ∈Γ} f(γgΩ^{k}K)|
≤ sup_{g∈G} e^{Rdist_{Y} (O_{Y} ,ΓgK)} ∑_{γ∈Γ} |f(Ω^{k}γgK)|
≤ sup_{g∈G} e^{Rdist_{Y} (O_{Y} ,ΓgK)} ∑_{γ∈Γ} e^{−(R+Q)dist(O,γgK)}p_{−(R+Q),Ω^{k}}(f)
≤ sup_{g∈G} p_{−(R+Q),Ω^{k}}(f) ∑_{γ∈Γ} e^{Qdist(γgK,O)}
≤ Cp_{−(R+Q),Ω^{k}}(f) .

This estimate shows the continuity of π_{∗}.

Let π^{∗} : S_{∞}E′_{Y} → S_{∞}E′ be the adjoint of π_{∗}. We claim that π^{∗} is injective. Let E_{c,Y} be the space of smooth sections of E_{Y} with compact support. Then a standard partition of unity argument shows that E_{c,Y} ⊂ im(π_{∗}). Thus π_{∗} has dense range, and the claim follows. (In fact more is true: Casselman ([8], Thm. 2.2) has shown that π_{∗} is surjective, at least for arithmetic Γ.)

To prove the theorem it suffices to show that if f ∈ S_{∞}E′_{Y} with F = Bf ∈ S_{∞}E_{Y}, then f ∈ S_{∞}E_{Y}. Let W be the parametrix of B constructed in the proof of Lemma 2.6. Then we have π^{∗}f = W F + Sπ^{∗}f (viewing F ∈ ^{Γ}S_{∞}E). We have already shown that W F, Sπ^{∗}f ∈ S_{∞}E. Since W, S are G-equivariant we obtain π^{∗}f ∈ ^{Γ}S_{∞}E. It follows from the injectivity of π^{∗} that f ∈ S_{∞}E_{Y} as desired. □

As at the end of Section 2 we see that Theorem 6.3 remains true if one replaces S_{∞}E_{Y} by the space of sections of uniform moderate growth S_{umg}E_{Y} := ^{Γ}S_{umg}E.

7 Γ-cohomology

In this section we discuss properties of the Γ-cohomology of distribution vector globalizations of admissible representations of G.

Let (π, V_{π,K}) ∈ HC(𝔤, K) and Γ ⊂ G be a discrete torsion-free subgroup of finite covolume. Let B = (−Ω − μ)^{k} such that BV_{π,K} = 0.

Proposition 7.1 We have

dim H^{p}(Γ, V_{π,−∞}) < ∞, ∀p ≥ 0.

Proof. Let

0 → V_{π,−∞} → S_{∞}E_{0} \underrightarrow{( \begin{matrix}D_{0} \\ B \end{matrix} )} \begin{matrix}S_{∞}E_{1} \\ ⊕ \\ S_{∞}E_{0} \end{matrix} \underrightarrow{( \begin{matrix}D_{1} & H_{0} \\ −B & D_{0} \end{matrix} )} \begin{matrix}S_{∞}E_{2} \\ ⊕ \\ S_{∞}E_{1} \end{matrix} \underrightarrow{( \begin{matrix}D_{2} & H_{1} \\ B & D_{1} \end{matrix} )} …  (25)

be a standard resolution (see Proposition 3.5) of V_{π,−∞}. It is a Γ-acyclic resolution of V_{π,−∞} by Theorem 5.6. The cohomology of the subcomplex of Γ-invariant vectors is isomorphic to H^{∗}(Γ, V_{π,−∞}).

For any locally homogeneous vector bundle E_{Y} → Y we denote by E_{Y} (B)_{cusp} the space of cusp-forms in E_{Y} (B) and by S_{∞}E_{Y} (B) the kernel of B in S_{∞}E_{Y}. We consider the subcomplex of the complex of Γ-invariants of (25)

0 → S_{∞}E_{0,Y} (B) \underrightarrow{( \begin{matrix}D_{0} \\ 0 \end{matrix} )} \begin{matrix}S_{∞}E_{1,Y} (B) \\ ⊕ \\ E_{0,Y} (B)_{cusp} \end{matrix} \underrightarrow{( \begin{matrix}D_{1} & H_{0} \\ 0 & D_{0} \end{matrix} )} \begin{matrix}S_{∞}E_{2,Y} (B) \\ ⊕ \\ E_{1,Y} (B)_{cusp} \end{matrix} \underrightarrow{( \begin{matrix}D_{2} & H_{1} \\ 0 & D_{1} \end{matrix} )} … (26)

and claim that its cohomology is H^{∗}(Γ, V_{π,−∞}). In fact, let (f_{i}, f_{i−1}) ∈ S_{∞}E_{i,Y} ⊕ S_{∞}E_{i−1,Y} be a cochain in the complex of Γ-invariants of (25). By the results of Theorem 6.3 there is a unique decomposition f_{i−1} = f_{i−1}^{im} + f_{i−1}^{cusp}, where f_{i−1}^{im} ∈ BS_{∞}E_{i−1,Y} and f_{i−1}^{cusp} ∈ E_{i−1,Y} (B)_{cusp}. Thus modulo a coboundary the cochain (f_{i}, f_{i−1}) is equivalent to (f̃_{i}, f_{i−1}^{cusp}). If in addition (f_{i}, f_{i−1}) and hence (f̃_{i}, f_{i−1}^{cusp}) is a cocycle, then Bf̃_{i} = 0, since D_{i−1} : E_{i−1,Y} (B)_{cusp} → E_{i,Y} (B)_{cusp} and the range of B : S_{∞}E_{i,Y} → S_{∞}E_{i,Y} is transverse to E_{i,Y} (B)_{cusp}. We conclude that the cohomology of (26) surjects onto H^{∗}(Γ, V_{π,−∞}).

Assume now that f_{i} = H_{i−2}g_{i−2} + D_{i−1}g_{i−1}, f_{i−1} = (−1)^{i−1}Bg_{i−1} + D_{i−2}g_{i−2}, and f_{i−1} ∈ E_{i−1,Y} (B)_{cusp}, g_{i−2} ∈ E_{i−2,Y} (B)_{cusp}. It follows that Bg_{i−1} = 0 and thus (g_{i−1}, g_{i−2}) is an (i − 1)-cochain of (26). This shows that the cohomology of (26) maps injectively to H^{∗}(Γ, V_{π,−∞}) proving the claim.

The proposition now follows since (26) is a complex of finite-dimensional vector spaces (see e.g. [12], Thm.1). □

If Γ is cocompact, then we can prove a Poincaré duality theorem as in [4], Proposition 5.2. It relates the Γ-cohomology of the distribution vector globalization of a Harish-Chandra module of G with the Γ-cohomology of the smooth globalization of its dual.

Proposition 7.2 Let Γ be cocompact. The Γ-cohomology of V_{π,±∞} satisfies Poincaré duality

H^{p}(Γ, V_{π,−∞})^{∗} ≅ H^{n−p}(Γ, V_{π̃,∞}),

where n = dim(X).

Let V_{π,±ω} be the minimal and maximal globalizations of V_{π,K}, respectively. If Γ is cocompact, then E_{Y} (B)_{cusp} = E_{Y} (B) = S_{∞}E_{Y} (B). Combining [4], Proposition 5.1 with Proposition 7.1 we obtain

Corollary 7.3 Let Γ be cocompact. Then for (π, V_{π,K}) ∈ HC(𝔤, K) we have

H^{∗}(Γ, V_{π,ω}) = H^{∗}(Γ, V_{π,∞})
H^{∗}(Γ, V_{π,−∞}) = H^{∗}(Γ, V_{π,−ω}) ,

where the identifications are induced by the natural inclusions of the globalizations.

If Γ has torsion, then there exists a cofinite torsion free normal subgroup Γ′ ⊂ Γ. Employing the spectral sequence for the group cohomology associated to the extension

0 → Γ′ → Γ → F → 0 ,

where F is some finite group, one easily obtains a generalization of the results of the present section to Γ. The spectral sequence degenerates at the second term since the higher cohomology of a finite group with coefficients in a vector spaces over ℂ vanishes. It follows H^{∗}(Γ, V) = H^{∗}(Γ′, V)^{F} for any Γ-module V over ℂ. In particular, in Proposition 7.1 we can relax the condition that Γ is torsion-free.

8 Fuchsian groups of the first kind

In this section we give a detailed discussion of the Γ-cohomology of a Fuchsian group of the first kind with coefficients in the distribution vector globalization of principal series representations. In this case we know explicit standard resolutions. As a by-product we also obtain the cohomology groups corresponding to the discrete series, and thus gain in this case a rather complete knowledge of the Γ-cohomology with coefficients in the distribution globalization of Harish-Chandra modules.

Let Γ ⊂ P SL(2, ℝ) =: G be a discrete torsion-free subgroup of finite covolume. Such a Γ is called a Fuchsian group of the first kind and it acts freely on the hyperbolic plane X = H^{2}. The quotient Y = Γ\X is a complete Riemann surface of finite volume.

The group G acts on the circle S^{1} which can be identified with the boundary ∂X of X using the Poincaré disc model. Let T → S^{1} be the complexified tangent bundle of S^{1}. It is G-homogeneous and we can form complex powers T^{λ} → S^{1}, λ ∈ ℂ. The number λ ∈ ℂ parametrizes a principal series representation (π^{λ}, H^{λ}) of G on the Hilbert space L^{2}(S^{1}, T^{λ−1/2}). By H_{−∞}^{λ} we denote the space of its distribution vectors.

For λ ≠ −1/2, −3/2, −5/2, … combining a theorem of Helgason ([15], [16] Introduction, Thm. 4.3) with the characterization of the distribution vector globalization via matrix coefficients given at the beginning of Chapter 3 we see that the Poisson transform P_{λ} is an G-equivariant isomorphism

P_{λ} : H_{−∞}^{λ} →^{∼} S_{∞}E(B) ,

where B = −Ω − 1/4 + λ^{2} and E = X × ℂ is the trivial bundle (in this special situation this fact was first obtained by Lewis [18]). Thus a standard resolution of the principal series representation H_{−∞}^{λ} for λ ≠ −1/2, −3/2, −5/2, … is simply

0 → H_{−∞}^{λ} →^{P_{λ}} S_{∞}E →^{B} S_{∞}E → 0 .

The complex (26) reduces to

0 → S_{∞}E_{Y} (B) →^{0} E_{Y} (B)_{cusp} → 0 .  (27)

Proposition 8.1 For λ ≠ −1/2, −3/2, −5/2, … we have

H^{0}(Γ, H_{−∞}^{λ}) = S_{∞}E_{Y} (B)
H^{1}(Γ, H_{−∞}^{λ}) = E_{Y} (B)_{cusp}
H^{2}(Γ, H_{−∞}^{λ}) = 0.

Moreover, χ(Γ, H_{−∞}^{λ}) = dim H^{0}(Γ, H_{−∞}^{λ}) − dim H^{1}(Γ, H_{−∞}^{λ}) = r, where r is the number

of cusps of Y. If H^{1}(Γ, H_{−∞}^{λ}) ≠ 0, then λ ∈ \{ \begin{matrix}ıℝ ∪ (−1/2, 1/2) & Y non-compact \\ ıℝ ∪ (−1/2, 1/2] & Y compact \end{matrix} \}.

Proof. The first part of the proposition follows immediately from (27). Since E_{Y} (B)_{cusp} ⊂ ker_{L^{2}}(B : E_{Y} → E_{Y}) and spec_{L^{2}}Δ_{Y} ⊂ [0, ∞) we see that H^{1}(Γ, H_{−∞}^{λ}) = 0 for λ ∉ ıℝ ∪ (−1/2, 1/2]. If λ = 1/2, then ker_{L^{2}}(B) is generated by the constant function which is a cusp form exactly when Y is compact.

Let p : S_{∞}E_{Y} (B) → ℂ^{2r} be the linear map taking the constant term. For any cusp the constant term has two components (the incoming and the outcoming). It is known that dim im(p) = r. In fact, the range of p is generated by the constant terms of regular Eisenstein series and their residues. The scattering matrix fixes the relation between the two components of the constant term. Since ker(p) = E_{Y} (B)_{cusp}, the assertion about the Euler characteristic follows. □

Note that Proposition 8.1 generalizes immediately (with the same proof) to spherical principal series representations of any rank-one group G.

Now we discuss the points λ = −k/2, k = 1, 3, 5 … taking the structure of H_{−∞}^{±k/2} as a G-module into account. We first exploit the exact sequence

0 → F_{k} → H_{−∞}^{k/2} → D_{k}^{+} ⊕ D_{k}^{−} → 0 ,  (28)

where F_{k} is the finite-dimensional representation of G of dimension k and D_{k}^{±} are the distribution vectors of holomorphic and anti-holomorphic discrete series representations. Let r denote the number of cusps of Y and g denote the genus.

We exclude for a moment the case k = 1, r = 0 from the considerations. Then we have H^{i}(Γ, H_{−∞}^{k/2}) = 0, i ≥ 1, and dim H^{0}(Γ, H_{−∞}^{k/2}) = dim S_{∞}E_{Y} (B) = r by Proposition 8.1. The long exact cohomology sequence associated to (28) gives

0 → H^{0}(Γ, F_{k}) → H^{0}(Γ, H_{−∞}^{k/2}) → H^{0}(Γ, D_{k}^{+} ⊕ D_{k}^{−}) →^{δ} H^{1}(Γ, F_{k}) → 0

H^{1}(Γ, D_{k}^{+} ⊕ D_{k}^{−}) = 0 .  (29)

It follows that

dim H^{0}(Γ, D_{k}^{+} ⊕ D_{k}^{−}) = r + dim H^{1}(Γ, F_{k}) − dim H^{0}(Γ, F_{k}) .  (30)

An investigation of the long exact sequence associated to

0 → D_{k}^{+} ⊕ D_{k}^{−} → H_{−∞}^{−k/2} → F_{k} → 0

leads to

0 → H^{0}(Γ, D_{k}^{+} ⊕ D_{k}^{−}) → H^{0}(Γ, H_{−∞}^{−k/2}) → H^{0}(Γ, F_{k}) → 0  (31)
H^{1}(Γ, H_{−∞}^{−k/2}) = H^{1}(Γ, F_{k}) .

Proposition 8.2 If k > 1 or r > 0, then we have

dim H^{0}(Γ, H_{−∞}^{−k/2}) = r + dim H^{1}(Γ, F_{k}) = \{ \begin{matrix}2g − 1 + 2r & k = 1 \\ k(2g − 2) + (k + 1)r & k > 1 \end{matrix}
dim H^{1}(Γ, H_{−∞}^{−k/2}) = dim H^{1}(Γ, F_{k}) = \{ \begin{matrix}2g − 1 + r & k = 1 \\ k(2g − 2 + r) & k > 1 \end{matrix}
dim H^{2}(Γ, H_{−∞}^{−k/2}) = 0 ,

while for r = 0 the following holds:

dim H^{0}(Γ, H_{−∞}^{−1/2}) = dim H^{0}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) = 2g
dim H^{1}(Γ, H_{−∞}^{−1/2}) = dim H^{1}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) + dim H^{1}(Γ, F_{1}) − H^{0}(Γ, F_{1}) = 2g + 1
dim H^{2}(Γ, H_{−∞}^{−1/2}) = dim H^{2}(Γ, F_{1}) = 1 .

In any case we obtain χ(Γ, H_{−∞}^{−k/2}) = r.

Proof.In the first case the result is obtained from (30) and (31) by the following well-known formulas:

dim H^{0}(Γ, F_{k}) = \{ \begin{matrix}1 & k = 1 \\ 0 & k > 1 \end{matrix}
dim H^{1}(Γ, F_{k}) = \{ \begin{matrix}2g − 1 + r & k = 1 \\ k(2g − 2 + r) & k > 1 \end{matrix}
dim H^{2}(Γ, F_{k}) = 0 .

In the case r = 0, k = 1 a discussion of the long exact sequence associated to (28) leads to

dim H^{0}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) = dim H^{1}(Γ, F_{1}) = 2g , dim H^{1}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) = 2 .

The analog of (31) in the present case reads as

0 → H^{0}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) →^{∼} H^{0}(Γ, H_{−∞}^{−1/2}) →^{0} H^{0}(Γ, F_{1}) →^{δ} H^{1}(Γ, D_{1}^{+} ⊕ D_{1}^{−}) →  (32)
→ H^{1}(Γ, H_{−∞}^{−1/2}) → H^{1}(Γ, F_{1}) → 0
H^{2}(Γ, H_{−∞}^{−1/2}) = H^{2}(Γ, F_{1}) .

To see the first isomorphism in (32) we employ the following fact: If

0 → U_{K} → V_{K} → W_{K} → 0

is a non-split exact sequence of Harish-Chandra modules and Γ is cocompact, then ΓU_{−∞} →^{∼} ^{Γ}V_{−∞}. Now the result for r = 0 and k = 1 follows. □

The aim of the following discussion is to understand Proposition 8.2 in the framework of standard resolutions.

Let K be the canonical bundle of X (viewing X as a complex manifold) and K^{i} be its i’th power.

Lemma 8.3 There is a Poisson transform P such that

0 ⟶ H_{−∞}^{−k/2} ⟶^{P} S_{∞}K^{(k+1)/2} ⊕ S_{∞}K^{−(k+1)/2} ⟶

\underrightarrow{( \begin{matrix}¯∂^{(k+1)/2} ⊕ ∂^{(k+1)/2} \\ −Ω − \frac{k^{2}−1}{4} \end{matrix} )} \begin{matrix}S_{∞}K^{0} \\ ⊕ \\ S_{∞}K^{(k+1)/2} ⊕ S_{∞}K^{−(k+1)/2} \end{matrix} ⟶^{(Ω+ \frac{k^{2}−1}{4} , ¯∂^{(k+1)/2}⊕∂^{(k+1)/2})} S_{∞}K^{0} ⟶ 0 .

is a standard resolution of H_{−∞}^{−k/2}.

Here, in abuse of notation, ¯∂ : K^{(l+1)/2} → K^{(l−1)/2}, l > 0 odd, is the contraction of the Dolbeaut operator with the Kähler form and similarly ∂ : K^{−(l+1)/2} → K^{−(l−1)/2} is the contraction of the holomorphic part of the canonical connection with the Kähler form. We write ¯∂^{(k+1)/2}, ∂^{(k+1)/2} for the composition of the corresponding number of ¯∂’s, ∂’s, respectively.

Proof. Define B := −Ω − \frac{k^{2}−1}{4}. In view of the results of Section 3 it is enough to show that there is a Poisson transform P such that

0 ⟶ H_{−∞}^{−k/2} ⟶^{P} S_{∞}K^{(k+1)/2}(B) ⊕ S_{∞}K^{−(k+1)/2}(B) ⟶^{¯∂^{(k+1)/2}⊕∂^{(k+1)/2})} S_{∞}K^{0}(B) ⟶ 0

is an exact sequence of the type (17).

It can be seen from (28) that W := H^{k/2}(γ_{k+1}) ⊕ H^{k/2}(γ_{−k−1}), where γ_{±(k+1)} is the

minimal K-type of D_{k}^{±}, generates the Harish-Chandra module H_{K}^{k/2} ≅ { \atop H_{K}^{−k/2}}. The twoparameter family of identifications of W̃ with the fibre of K^{(k+1)/2} ⊕ K^{−(k+1)/2} over O yields a family P_{a,b}, a, b ∈ ℂ^{∗}, of injective Poisson transforms defined by (16). As G-modules the eigenspaces S_{∞}K^{±(k+1)/2}(B) have three-step composition series’ with composition factors D_{k}^{±}, F_{k}, D_{k}^{∓} (see [19]). The operators ¯∂^{(k+1)/2}, ∂^{(k+1)/2}, respectively, annihilate exactly the first composition factor consisting of (anti-)holomorphic forms which is contained in the image of P_{a,b} for any a, b. Above we have shown that S_{∞}K^{0}(B) is isomorphic to H_{−∞}^{k/2} having composition factors D_{k}^{+} ⊕ D_{k}^{−}, F_{k}. Thus ¯∂^{(k+1)/2} ⊕ ∂^{(k+1)/2} is surjective. Now it is not difficult to see that one can adjust the parameters a, b such that imP_{a,b} = ker(¯∂^{(k+1)/2} ⊕ ∂^{(k+1)/2}). □

Now we take the Γ-invariant vectors in the standard resolution. The complex (26) reduces to

0 ⟶ S_{∞}K_{Y}^{(k+1)/2}(B) ⊕ S_{∞}K_{Y}^{−(k+1)/2}(B) ⟶
\underrightarrow{( \begin{matrix}¯∂^{(k+1)/2} ⊕ ∂^{(k+1)/2} \\ 0 \end{matrix} )} \begin{matrix}S_{∞}K_{Y}^{0} (B) \\ ⊕ \\ K_{Y}^{(k+1)/2}(B)_{cusp} ⊕ K_{Y}^{−(k+1)/2}(B)_{cusp} \end{matrix} ⟶ (33)

⟶^{(0, ¯∂^{(k+1)/2}⊕∂^{(k+1)/2})} K_{Y}^{0} (B)_{cusp} ⟶ 0 .

In order to make further reductions we assume that Y is not compact (the case of compact Y is an easy exercise and left to the reader). This assumption implies that K_{Y}^{0} (B)_{cusp} = 0 (see Proposition 8.1).

We claim that for k > 1 the eigenspaces S_{∞}K_{Y}^{±(k+1)/2}(B) consist entirely of (anti-)holo-morphic forms. In each cusp neighbourhood we introduce upper half-plane coordinates (x, y), y >> 0, z = x + ıy, and trivialize K_{Y}^{(k+1)/2} using the section (\frac{dz}{y})^{(k+1)/2}. Then the operator B has the form y^{2}Δ+ı(k +1)y \frac{∂}{∂x} − \frac{k^{2}−1}{4}. If ϕ ∈ S_{∞}K_{Y}^{(k+1)/2}(B), then its constant term is a linear combination of y^{−k}dz^{(k+1)/2} and dz^{(k+1)/2}. It follows that ¯∂ϕ ∈ L^{2}. But ker_{L^{2}}(B) = 0 on K_{Y}^{(k−1)/2}, k > 1, and thus ¯∂ϕ = 0. The claim for K_{Y}^{−(k−1)/2} follows by complex conjugation. Thus the differentials of (33) vanish for k > 1.

The spaces

A_{k+1} := ker(¯∂: S_{∞}K_{Y}^{(k+1)/2}(B) → S_{∞}K_{Y}^{(k−1)/2}(B))
Ā_{k+1} := ker(∂ : S_{∞}K_{Y}^{−(k+1)/2}(B) → S_{∞}K_{Y}^{−(k−1)/2}(B))

are called the holomorphic and anti-holomorphic automorphic forms of weight k + 1. The subspaces

S_{k+1} := K_{Y}^{(k+1)/2}(B)_{cusp} ⊂ A_{k+1}
S̄_{k+1} := K_{Y}^{−(k+1)/2}(B)_{cusp} ⊂ Ā_{k+1}

are the holomorphic and anti-holomorphic cusp-forms of weight k + 1. Since for k > 1 the space S_{∞}K_{Y}^{0} (B) (note that B depends on k) is generated by Eisenstein series its dimension is equal to the number of cusps r. For k = 3, 5, … we have

dim H^{0}(Γ, H_{−∞}^{−k/2}) = dim(A_{k+1} ⊕ Ā_{k+1})
dim H^{1}(Γ, H_{−∞}^{−k/2}) = dim(S_{k+1} ⊕ S̄_{k+1}) + r .

Employing dim A_{k+1} = dim Ā_{k+1} = k(g − 1) + r(k + 1)/2 and dim S_{k+1} = dim S̄_{k+1} = k(g − 1) + r(k − 1)/2 (see [23], Thm. 2.23) we recover the results of Proposition 8.2.

Finally, we consider the case k = 1. In this case both

¯∂: S_{∞}K_{Y}^{1} (B) → S_{∞}K_{Y}^{0} (B) ,
∂ : S_{∞}K_{Y}^{−1}(B) → S_{∞}K_{Y}^{0} (B)

have a one-dimensional range spanned by the constant function. Note the r-dimensional space S_{∞}K_{Y}^{0} (B) is generated by the regular Eisenstein series and the constant function. We obtain for k = 1

dim H^{0}(Γ, H_{−∞}^{−1/2}) = dim(A_{2} ⊕ Ā_{2}) + 1
dim H^{1}(Γ, H_{−∞}^{−1/2}) = dim(S_{2} ⊕ S̄_{2}) + r − 1 .

Using dim A_{2} = dim Ā_{2} = g − 1 + r, dim S_{1} = dim S̄_{2} = g (see [23], Thm. 2.23) we again recover the results of Proposition 8.2.

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