Fuchsian groups of the second kind and representations carried by

the limit set

Dedicated to Rolf Sulanke on the occasion of his 65’th birthday.

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

Contents

1 Introduction 1 2 The case Re(λ) > 0 3 3 Intertwining and scattering operators 7 4 The case Re(λ) < 0, λ ≠ −1/2, −3/2, … 12 5 The case λ = −1/2, −3/2, … 15 6 The case Re(λ) = 0, λ ≠ 0 20 7 Meromorphic continuations 24 8 Summary 26

∗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

1 Introduction

In this paper we study the cohomology of a Fuchsian group of the second kind with coefficients in certain infinite-dimensional representations. More precisely, we consider the representation on hyperfunctions with support on the limit set which transform according to a given complex conformal weight.

Let Γ ⊂ SL(2, R) =: G be a non-trivial discrete subgroup acting freely on the real hyperbolic plane H^{2} = SL(2, R)/S^{1} =: X.

One model for X is the upper half-plane R_{+}^{2} := {(x, y) ∈ R^{2}|y > 0} with the Riemannian metric \frac{1}{y^{2}} (dx^{2} +dy^{2}). Another model for the hyperbolic plane X is the Poincaré disc \{|z| < 1\} ⊂

C with the Riemannian metric \frac{ds^{2}+dt^{2}}{(1−|z|^{2})^{2}} , where z = s + it. Here G acts via the identification SL(2, R) ≅ SU(1, 1). Using the Poincaré disc model one can identify the circle S^{1} with the boundary ∂X of X at infinity. The action of the group G on X extends continuously to ∂X = S^{1}. The set of accumulation points (in the Euclidean topology) of an arbitrary orbit Γx, x ∈ X, is called the limit set Λ ⊂ ∂X. The complement Ω := ∂X \ Λ is called the ordinary set.

We assume that Γ acts convex cocompactly on X, i.e., Γ acts cocompactly on the convex hull of the limit set Λ. This implies that Γ is geometrically finite. Moreover Γ acts freely and cocompactly on Ω with the compact quotient B. Since Γ is non-trivial by assumption Λ contains at least two points.

The quotient Y := Γ\X is a complete hyperbolic Riemann surface without cusps. If Ω ≠ 0, then Y is non-compact and has infinite volume. The manifold B is a finite union of circles and can be viewed as the boundary at infinity of Y.

In the present paper we assume that Ω ≠ ∅. The cocompact case Ω = ∅ is contained as a special case in the results of [2].

Let T → S^{1} be the complexified tangent bundle of S^{1}. Since T is the complexification of a G-homogeneous oriented R-bundle we can form complex powers T^{λ} → S^{1}, λ ∈ C. In fact all bundles T^{λ} are trivial. They are distinguished by the G-action.

Let ∂ be the fundamental vector field of the action of U(1) ⊂ G = SU(1, 1). It provides a natural trivialization of T. Frequently we will identify a section ϕ∂^{λ} ∈ Γ(T^{λ}) with the function ϕ. For example we define L^{2}(S^{1}, T^{λ}) := L^{2}(S^{1})∂^{λ}.

The complex number λ parametrizes the principal series representation H^{λ} of G on the Hilbert space L^{2}(S^{1}, T^{λ−1/2}). By H_{−ω}^{λ} we denote the space of its hyperfunction vectors. As a topological vector space H_{−ω}^{λ} can be identified with the space of hyperfunction sections C^{−ω}(S^{1}, T^{λ−1/2}). A hyperfunction section f ∈ H_{−ω}^{λ} is by definition a continuous functional on the dual Fréchet space H_{ω}^{−λ} := C^{ω}(S^{1}, T^{−λ−1/2}).

If A ⊂ S^{1} is a closed subset, then we define C^{ω}(A, T^{−λ−1/2}) to be the dual Fréchet space of germs of analytic sections of T^{−λ−1/2} on A. There is a natural continuous restriction r_{A} : C^{ω}(S^{1}, T^{−λ−1/2}) → C^{ω}(A, T^{−λ−1/2}). The topological dual C^{−ω}(A, T^{λ−1/2}) = C^{ω}(A, T^{−λ−1/2})′ is the space of hyperfunctions with support on A, which can be viewed as a subspace of C^{−ω}(S^{1}, T^{λ−1/2}) via the adjoint ^{t}r_{A}.

Definition 1.1 By H_{−ω,Λ}^{λ} we denote the subspace of H_{−ω}^{λ} of hyperfunction sections with support in the limit set Λ.

Since Λ is a Γ-invariant closed subset H_{−ω,Λ}^{λ} is a Γ-invariant subspace of H_{−ω}^{λ}.

The goal of the present paper is to study the cohomology groups H^{∗}(Γ, H_{−ω,Λ}^{λ}). This group cohomology of Γ with coefficients H_{−ω,Λ}^{λ} is a purely algebraic object (see e.g. [1]). Our interest in these cohomology groups is motivated by a conjecture of S. Patterson [21] (see also [2], [3]) relating their dimensions with the order of the singularities of the Selberg zeta function associated to Γ.

From the algebraic point of view it might be surprising that the investigation of this cohomology involves mainly analytic methods. The point is that instead of the usually finite-dimensional representation of Γ we take the in general very large representation H_{−ω,Λ}^{λ}. Nevertheless the cohomology groups turn out to be finite-dimensional. This already is a non-trivial result. Moreover we are able to compute the cohomology in the sense that we relate it with topological data of Y or spectral data of the Laplacian on Y like resonances and L^{2}-eigenvalues.

Since Γ is the fundamental group of a non-compact Riemann surface with finitely many cylindrical ends it is a finitely generated free group. Hence the cohomological dimension of Γ is 1, in particular H^{p}(Γ, V) = 0, ∀p ≥ 2, for any representation V.

In the present paper we employ methods mainly based on the spectral analysis of the Laplace-Beltrami operators of X and Y. The methods of the present paper extend to the spherical principal series representations of the groups SO(1, n), Spin(1, n), n ≥ 2, (and probably all other R-rank one groups) of non-integral infinitesimal character. Since our goal is to explain the method we restrict our considerations to the case of Fuchsian groups. Another reason is that a complete result for the principal series with integral infinitesimal characters (as obtained in Section 5 for the SL(2, R)-case) is still not available in higher dimensions.

In forthcoming papers [4], [3] we consider a generalization to convex cocompact groups acting on rank-one symmetric spaces which employs almost solely analysis on the boundary ∂X. In higher dimensions it is desirable to have results for non-spherical principal series representations. The direct generalization of the methods of the present paper would lead to the spectral analysis of Laplace-type operators on (locally) homogeneous bundles on X and Y. Compared with the function case this analysis for bundles is much less understood. Therefore we need the new point of view.

2 THE CASE RE(λ) > 0 3

In the study of the cohomology H^{∗}(Γ, H_{−ω,Λ}^{λ}) a major role is played by the scattering operator. The properties of the scattering operator and the Eisenstein series in our case are well known by the work of Patterson [17], [18], [19]. For the reason of presentation we give short derivations of some of these facts.

Acknowledgment : We thank S. Patterson for explaining us his view on the topic of the present paper. The results of Section 5 were obtained while the authors visited the SFB 170 in Göttingen. We are grateful to H. Schlichtkrull and T. Oshima for clarifying discussion concerning hyperfunction and distribution boundary values. The authors were supported by the SFB 288 of the Deutsche Forschungsgemeinschaft.

2 The case Re(λ) > 0

In this section we construct a Γ-acyclic resolution of H_{−ω,Λ}^{λ}, Re(λ) > 0, using subspaces of smooth functions on X. We then employ this resolution in order to compute H^{∗}(Γ, H_{−ω,Λ}^{λ}).

Let A_{λ} := Δ − 1/4 + λ^{2}, where Δ is the Laplace-Beltrami operator of X. Let E := C^{∞}(X) and E(A_{λ}) := ker(A_{λ}) ⊂ E.

It was shown by Helgason ([6], [7], Introduction,Thm. 4.3), that any eigenfunction f ∈ E(A_{λ}) can be obtained by a Poisson transform of its boundary value in the sense of hyperfunctions. In greater detail, there exists ϕ∂^{λ−1/2} ∈ H_{−ω}^{λ}, Re(λ) ≥ 0, such that

f(z) = P_{λ}ϕ(z) := \frac{1}{2π} ∫_{S^{1}}(\frac{1 − |z|^{2}}{|z − b|^{2}} )^{λ+1/2}ϕ(b)db ,

where we employ the Poincaré disc model and write the application of the analytic functional ϕ to the real analytic kernel of the Poisson transform as an integral.

If Re(λ) ≥ 0, then the Poisson transform P_{λ} : H_{−ω}^{λ} → E(A_{λ}) is an isomorphism of representations of G. The inverse of the Poisson transform (up to scalar factor) is the boundary value map β_{λ} : E(A_{λ}) → H_{−ω}^{λ}. For Re(λ) > 0 it is given by

⟨β_{λ}(f), ϕ⟩ = lim_{r→1}(1 − r^{2})^{λ−1/2} ∫_{S^{1}} f(re^{iα})ϕ(e^{iα})\frac{dα}{2π}, ∀ϕ ∈ H_{ω}^{−λ} .

We have

P_{λ} ∘ β_{λ} = \frac{1}{\sqrt{π}} \frac{Γ(λ)}{Γ(1/2 + λ)}id_{E(A_{λ})} .  (1)

Next we characterize the subspace P_{λ}(H_{−ω,Λ}^{λ}) ⊂ E(A_{λ}) as the subspace of functions on which a certain set of seminorms is finite. We call a subset W ⊂ X admissible if it can be covered by the union of a finite number of Γ-translates of a fundamental domain of Γ. If W ⊂ X is admissible, k ∈ N_{0} := N ∪ {0}, and f ∈ E, then we define

q_{W,k}(f) := ∥A_{λ}^{k}f∥_{L^{2}(W)} ∈ [0, ∞] .

By E_{Λ} we denote the Fréchet space of all functions f ∈ E with q_{W,k}(f) < ∞ for all admissible W and k ≥ 0. It is easy to see that the seminorms associated to a countable set of admissible W ’s are sufficient in order to define the topology of E_{Λ}. The space E_{Λ} is a Γ-submodule of E.

Set E_{Λ}(A_{λ}) := E_{Λ} ∩ ker(A_{λ}).

Lemma 2.1

P_{λ}(H_{−ω,Λ}^{λ}) = E_{Λ}(A_{λ}) .

Proof. First we show the inclusion P_{λ}(H_{−ω,Λ}^{λ}) ⊂ E_{Λ}(A_{λ}). In the upper half-plane model the Poisson transform P_{λ} is given by the kernel P_{λ}((x, y), b) := (\frac{y}{y^{2}+(x−b)^{2}} )^{λ+1/2}, where b ∈ R. We can assume without loss of generality that ∞ ∈ Λ.

If W is admissible then its closure with respect to the Euclidean topology is separated from Λ. It is easy to check that the kernel b → P_{λ}(., b) defines an real analytic function from a neighbourhood of Λ to L^{2}(W). To see this for b near ∞ we use another coordinate system with infinite point ∞′ ∈ Λ and in which ∞ becomes finite. This is possible since Λ consists of at least two points.

The Poisson kernel satisfies A_{λ}P_{λ}(., b) = 0. If we pair the Poisson kernel with a hyperfunction ϕ ∈ H_{−ω,Λ}^{λ}, then we obtain a smooth eigenfunction of A_{λ} which is in L^{2}(W). Thus q_{W,k}(P_{λ}ϕ) < ∞ for all k ∈ N_{0}. Since W was arbitrary this proves the inclusion

P_{λ}(H_{−ω,Λ}^{λ}) ⊂ E_{Λ}(A_{λ}) .

The inclusion

E_{Λ}(A_{λ}) ⊂ P_{λ}(H_{−ω,Λ}^{λ})

is equivalent to the inclusion

β_{λ}(E_{Λ}(A_{λ})) ⊂ H_{−ω,Λ}^{λ} .

For the following argument it is more convenient to employ the Poincaré disc model with coordinates (r, α). Let W = (a, 1) × M, a ∈ (0, 1), be admissible with M ⊂ Ω open such that M̄ ⊂ Ω.

The theory of differential equations with regular singularities [10] allows one to define the boundary value β_{λ}(f) locally. If λ is a positive half-integer, then there is a slight complication but one can still take the boundary value β_{λ}(f) corresponding to the leading characteristic exponent λ − 1/2 (see [9], pages 24ff).

Let f ∈ C^{∞}(M ×(a, 1)) satisfy A_{λ}f = 0. Then the boundary value β_{λ}(f) ∈ C^{−ω}(M, T^{λ−1/2}) is obtained from an extension f̃ which is a hyperfunction on M × (a, 1/a) with support in M × (a, 1] still satisfying the eigenvalue equation A_{λ}f̃ = 0.

Let now f ∈ E_{Λ}(A_{λ}). Since f_{|W} ∈ L^{2}(W), the usual Sobolev regularity theorem implies that f has moderate growth along M (see the definition on page 26 of [15]). Thus f̃ is a distribution on (a, 1/a) × M. In fact f̃ is simply given by the extension of f by zero. By Lemma 2.19 of [15] we conclude that β_{λ}(f) is a distribution on M. Since we can choose M arbitrary we see that β_{λ}(f) is a distribution on Ω. It remains to verify that this distribution vanishes Let ϕ ∈ C_{c}^{∞}(Ω). Then the pairing ⟨β_{λ}(f), ϕ⟩ is defined and given by

⟨β_{λ}(f), ϕ⟩ = lim_{r→1}(1 − r^{2})^{λ−1/2} ∫_{S^{1}} f(re^{iα})ϕ(e^{iα})\frac{dα}{2π} .

Now let [e^{ib_{0}}, e^{ib_{1}}] ⊂ Ω such that supp(ϕ) ⊂ (e^{ib_{0}}, e^{ib_{1}}). The L^{2}-condition satisfied by f along Ω allows us to restrict the integration to the complement of (e^{ib_{0}}, e^{ib_{1}}), i.e.,

⟨β_{λ}(f), ϕ⟩ = lim_{r→1}(1 − r^{2})^{λ−1/2} ∫_{S^{1}\(e^{ib_{0}},e^{ib_{1}})} f(re^{iα})ϕ(e^{iα})\frac{dα}{2π} .

It follows that ⟨β_{λ}(f), ϕ⟩ = 0. Since ϕ was arbitrary we conclude β_{λ}(f)_{|Ω} = 0. □

Recall that a representation V of Γ is called Γ-acyclic if H^{p}(Γ, V) = 0, ∀p ≥ 1. A Γ-acyclic resolution of a representation W of Γ is an exact sequence of Γ-modules

0 → W → V^{0} → V^{1} → V^{2} → …,

where V^{p}, p ≥ 0, are Γ-acyclic. Let ^{Γ}V ⊂ V denote the subspace of invariant vectors of V. It is a well-known fact in homological algebra that H^{∗}(Γ, W) is the cohomology of the complex

0 → ^{Γ}V^{0} → ^{Γ}V^{1} → ^{Γ}V^{2} → ….

Lemma 2.2 E_{Λ} is Γ-acyclic.

Proof. We reduce the argument to the proof of Lemma 2.4, [2]. There a partition of unity {ρ_{g}}_{g∈Γ} satisfying g_{1}^{∗}ρ_{g} = ρ_{g_{1}g} is employed. Since in the present case the support of ρ_{g} is non-compact we must add a growth condition for the derivatives. Let F ⊂ X be a finite-sided fundamental domain of Γ and W ⊂ X be an ϵ-neighbourhood of F (ϵ > 0) with respect to the Euclidean metric induced by the Poincaré disc model such that W is admissible. Then we can find a partition of unity {ρ_{g}}_{g∈Γ} satisfying the equivariance condition above, supp(ρ_{g}) ⊂ gW, and |[Δ_{X}, ρ_{g}]|(r, α) = O(1 − r) uniformly with respect to α ∈ S^{1}. This partition of unity is compatible with the space E_{Λ} and the proof of Lemma 2.4 of [2] works. □

The following lemma shows that

0 ⟶ H_{−ω,Λ}^{λ} ⟶^{P_{λ}} E_{Λ} ⟶^{A_{λ}} E_{Λ} ⟶ 0  (2)

is a Γ-acyclic resolution of H_{−ω,Λ}^{λ}.

Lemma 2.3 A_{λ} : E_{Λ} → E_{Λ} is surjective.

Proof. Let ^{t}A_{λ} : E′_{Λ} → E′_{Λ} be the adjoint operator, where E′_{Λ} denotes the strong dual of E_{Λ}. It is enough to show that ^{t}A_{λ} is injective and has closed range (see Treves [27], Ch.II, Prop. 37.2).

Using suitable cut-off functions it is not difficult to show that C_{c}^{∞}(X) is dense in E_{Λ}. Hence there is a canonical embedding of E′_{Λ} into the distributions on X.

Let B′ ⊂ E′_{Λ} be a bounded subset. Since E_{Λ} is Fréchet, by a theorem of Schwartz [25], p.90, there exist k ∈ N_{0}, ϵ > 0, and an admissible W ⊂ X such that the closure of every connected component of X \ W (in the Euclidean topology induced by the Poincaré disc model) intersects Λ non-trivially and |⟨f, ϕ⟩| ≤ 1 for all f ∈ B′, ϕ ∈ C_{c}^{∞}(X) with q_{W,k}(ϕ) ≤ ϵ. Hence the support of the distributions f ∈ B′ is contained in W.

If f ∈ E′_{Λ} and ^{t}A_{λ}f = 0, then f is real analytic. Moreover, f is contained in some bounded set B′. Let W associated to B′ as in the preceding paragraph. Then f vanishes identically on the set X \ W with non-empty interior. Hence f = 0. This shows the injectivity of ^{t}A_{λ}.

We now prove that im(^{t}A_{λ}) is closed. We extend Δ − 1/4 to a non-negative self-adjoint unbounded operator on L^{2}(X). Since either Im(λ^{2}) ≠ 0 or λ^{2} > 0 the operator Δ − 1/4 + λ^{2} has a bounded inverse on L^{2}(X). We use Δ − 1/4 + λ^{2} in order to define the scale of Sobolev spaces H^{l}(X), l ∈ 2Z. In detail, Δ−1/4+λ^{2} : H^{l}(X) → H^{l−2}(X) is an isometry and H^{0}(X) = L^{2}(X).

There is a continuous inclusion of Fréchet spaces ⋂_{l} H^{l}(X) ↪ E_{Λ}. Dually there is a continuous restriction E′_{Λ} → lim_{l→∞} H^{−l}(X).

The range of ^{t}A_{λ} is closed, if im(^{t}A_{λ}) ∩ B′ is closed in B′ for any bounded set B′ ⊂ E′_{Λ}. Fix B′ and choose W and k as above. Let ^{t}A_{λ}h_{i} = f_{i} ∈ B′ converge to f. We claim that supp(h_{i}) ⊂ W. In fact assume by contradiction that supp(h_{i}) ∩ X \ W ≠ ∅, then by unique continuation supp(h_{i}) would contain a full connected component of X \ W, and this is absurd.

The sequence f_{i} converges in the direct limit lim_{l→∞} H^{−l}(X). In fact it already converges in H^{−2k}(X). Hence h_{i} → h = (Δ−1/4+λ^{2})^{−1}(f). Since supp(h_{i}) ⊂ W we also have supp(h) ⊂ W. It follows that h ∈ B′ and ^{t}A_{λ}h = f. This shows that the range of ^{t}A_{λ} is closed. □

Let Y := Γ\X be the Riemann surface corresponding to Γ and A_{λ,Y} := Δ_{Y} −1/4+λ^{2}, where Δ_{Y} is the Laplacian of Y. Define E_{Y} := {f ∈ C^{∞}(Y) | A_{λ,Y}^{k} f ∈ L^{2}(Y) ∀k ∈ N_{0}}. It is easy to check that ^{Γ}E_{Λ} = E_{Y}.

In order to calculate the cohomology of Γ with coefficients in H_{−ω,Λ}^{λ} we take the subcomplex of Γ-invariants of the complex (2). Hence H^{∗}(Γ, H_{−ω,Λ}^{λ}) is the cohomology of the complex

0 ⟶ E_{Y} ⟶^{A_{λ,Y}} E_{Y} ⟶ 0.

The essential spectrum of Δ_{Y} − 1/4 is the set [0, ∞) (see [17], [11]). The number −λ^{2} is separated from the essential spectrum of Δ_{Y} − 1/4. Thus there is a decomposition E_{Y} = ker_{L^{2}}(A_{λ,Y}) ⊕ ker_{L^{2}}(A_{λ,Y})^{⊥} and A_{λ,Y} acts as an isomorphism on ker_{L^{2}}(A_{λ,Y})^{⊥}. We obtain

Proposition 2.4 If Re(λ) > 0, then H^{0}(Γ, H_{−ω,Λ}^{λ}) = H^{1}(Γ, H_{−ω,Λ}^{λ}) = ker_{L^{2}}(A_{λ,Y}). There is a finite set Σ ⊂ (0, 1/2) such that ker_{L^{2}}(A_{λ,Y}) is non-trivial iff λ ∈ Σ. Moreover for λ ∈ Σ we have dim ker_{L^{2}}(A_{λ}) < ∞.

3 Intertwining and scattering operators

The topic of this section is an exposition of the theory of the scattering operator and Eisenstein series.

We introduce the spaces H_{∞}^{λ}(Ω) of smooth sections of T^{λ−1/2} on Ω and H_{−ω,(Λ)}^{λ} of hyperfunction sections of T^{λ−1/2} on S^{1} which restrict to smooth sections on Ω.

Since the sheaf of hyperfunctions is flabby we have an exact sequence

0 ⟶ H_{−ω,Λ}^{λ} ⟶ H_{−ω,(Λ)}^{λ} ⟶^{res} H_{∞}^{λ}(Ω) ⟶ 0

of Γ-modules. The long exact cohomology sequence gives

0 ⟶ ^{Γ}H_{−ω,Λ}^{λ} ⟶ ^{Γ}H_{−ω,(Λ)}^{λ} ⟶^{res} ^{Γ}H_{∞}^{λ}(Ω) ⟶

⟶ H^{1}(Γ, H_{−ω,Λ}^{λ}) ⟶ ….  (3)

Recall the definition of the finite set Σ := {λ ∈ C |1/2 > Re(λ) > 0, H^{0}(Γ, H_{−ω,Λ}^{λ}) ≠ 0}. For all λ ∈ C \ Σ, Re(λ) > 0, by Proposition 2.4 an invariant section ϕ_{λ} ∈ ^{Γ}H_{∞}^{λ}(Ω) has a unique invariant extension as a hyperfunction Φ_{λ} ∈ ^{Γ}H_{−ω,(Λ)}^{λ}, i.e., res(Φ_{λ}) = ϕ_{λ}. We want to show that Φ_{λ} depends holomorphically on λ. For this we must consider T^{λ−1/2} as a holomorphic family of bundles.

We fix a smooth positive invariant section ϱ∂ ∈ ^{Γ}H_{∞}^{3/2}(Ω) (see Perry [24], Prop.2.2, for a proof that such sections exist). Multiplication by

ϱ^{s}∂^{s} : H_{∞}^{λ}(Ω) → H_{∞}^{λ+s}(Ω), s ∈ C ,

is an isomorphism of Γ-modules. Let ϕ ∈ ^{Γ}H_{∞}^{1/2}(Ω) and set ϕ_{λ} := ϱ^{λ−1/2}ϕ∂^{λ−1/2} ∈ ^{Γ}H_{∞}^{λ}(Ω). By definition ϕ_{λ} is a holomorphic family of smooth Γ-invariant sections of the family T^{λ−1/2} on Ω. For λ ∈ C \ Σ, Re(λ) > 0, let Φ_{λ} ∈ ^{Γ}H_{−ω,(Λ)}^{λ} be the unique extension of ϕ_{λ}.

Writing Φ_{λ} = Φ̃_{λ}∂^{λ−1/2} we say that Φ_{λ} depends holomorphically on λ, iff Φ̃_{λ} is a holomorphic family of hyperfunctions. Identifying hyperfunctions with analytic functionals the space of hyperfunctions on S^{1} becomes a Fréchet space. It is known that a continuous, weakly holomorphic family of hyperfunctions is holomorphic.

Lemma 3.1 For λ ∈ C \ Σ, Re(λ) > 0, the family Φ_{λ} is a holomorphic family.

Proof. Consider the Poincaré series

∑_{g∈Γ} g′(b)^{λ+1/2}

for some b ∈ Ω. Here g′(b_{0}) is the conformal dilatation at b = b_{0} of the map S^{1} ∋ b ↦ g(b) ∈ S^{1} with respect to the Euclidean metric on S^{1} = ∂X in the Poincaré disc model. Let δ_{Γ} ∈ R be the smallest number such that the Poincaré series converges for all λ ∈ C with Re(λ) > δ_{Γ}.

For Re(λ) > δ_{Γ} the convergence is independent of the choice of b ∈ Ω and uniform in compact subsets of Ω. It is known that δ_{Γ} ≤ 1/2, that δ_{Γ} = 1/2 corresponds to the cocompact case Ω = ∅, and that δ_{Γ} + 1/2 is the Hausdorff dimension of Λ (Patterson [20], Sullivan [26]).

We claim that for Re(λ) > δ_{Γ} the hyperfunction extension Φ_{λ} of ϕ_{λ} is given by integration against ϕ_{λ}. This extension is in fact a measure. Let f ∈ C(S^{1}) and F ⊂ Ω be a compact fundamental domain for Γ. We have

< Φ_{λ}, f∂^{−λ−1/2} > := ∫_{Ω} ϱ^{λ−1/2}(b)ϕ(b)f(b)db  (4)
= ∑_{g∈Γ} ∫_{g^{−1}F} ϱ^{λ−1/2}(b)ϕ(b)f(b)db
= ∑_{g∈Γ} ∫_{F} ϱ^{λ−1/2}(gb)ϕ(gb)f(gb)d(gb)
= ∑_{g∈Γ} ∫_{F} g′(b)^{λ+1/2}ϱ^{λ−1/2}(b)ϕ(b)f(gb)db

= ∫_{F} (∑_{g∈Γ} g′(b)^{λ+1/2}f(gb)) ϱ^{λ−1/2}(b)ϕ(b)db .

Since f is uniformly bounded on S^{1} the right-hand side converges for Re(λ) > δ_{Γ}. It is easy to see that Φ_{λ} is a holomorphic family of measures with respect to the parameter λ. Hence it is a holomorphic family of hyperfunctions for Re(λ) > δ_{Γ}.

It remains to show that Φ_{λ} is holomorphic in the strip 0 < Re(λ) < 1/2, λ ∉ Σ. This requires analysis on the symmetric space X and its quotient Y = Γ\X. Let T : (0, ϵ) × Ω → X, ϵ > 0 sufficiently small, be the coordinates of a Γ- invariant neighbourhood of Ω introduced by Patterson [16] (see also Perry [23]). These coordinates depend on the choice of ϱ and are defined as follows:

y(r, b) := \frac{rϱ(b)}{1 + \frac{r^{2}}{4} ϱ′(b)^{2}}
x(r, b) := b − \frac{1}{2} \frac{r^{2}ϱ(b)ϱ′(b)}{1 + \frac{r^{2}}{4} ϱ′(b)^{2}} .

Here (x, y), y > 0, are coordinates of the upper half-plane model of X.

The map T satisfies gT (r, b) = T (r, gb) for g ∈ Γ, r ∈ (0, ϵ), b ∈ Ω. Hence we obtain induced coordinates of a collar neighbourhood E ⊂ Y at infinity:

T_{Y} : (0, ϵ) × B → Y .  (5)

In these coordinates the Laplacian Δ_{Y} has the form

Δ_{Y} = −(r \frac{∂}{∂r})^{2} + r \frac{∂}{∂r} + r^{2}Δ_{B} + rP (r \frac{∂}{∂r} , r \frac{∂}{∂b}) ,  (6)

where Δ_{B} is the Laplacian on B with respect to the metric induced by ϱ^{−2}(db)^{2} and P (u, v) is a polynomial in two variables u, v of degree two with coefficients depending on r, b which are bounded up to r = 0. Let A_{λ,Y} := Δ_{Y} − 1/4 + λ^{2}.

The volume form of Y near infinity can be written as r^{−2}ω, where ω is a smooth volume form which is bounded as r → 0.

Fix a cut-off function χ(r) being zero near r = ϵ and one near r = 0. Form f_{λ}(r, b) = χ(r)r^{−λ+1/2}ϕ(b). One can check that g_{λ} := A_{λ,Y} f_{λ} is a holomorphic family in L^{2}(Y) for Re(λ) < 1. For Re(λ) > 0, λ ∉ Σ, we can solve

A_{λ,Y} h_{λ} = −g_{λ}

in L^{2}(Y) and obtain a holomorphic family of eigenfunctions F_{λ} := f_{λ} + h_{λ} ∈ ker(A_{λ,Y}). Lifting F_{λ} to X we find a holomorphic family of invariant eigenfunctions F̃_{λ}. The boundary value β_{λ}(F̃_{λ}) is a hyperfunction and the desired extension Φ_{λ} of ϕ_{λ} for Re(λ) > 0, λ ∉ Σ.

Let W ⊂ X be any admissible subset. Then on W we have F̃_{λ}(x, y) ∼ y^{−λ+1/2}ϱ^{λ−1/2}(x)ϕ(x) (mod L^{2}(W)). Thus F̃_{λ} has the boundary value ϕ_{λ} along Ω. It follows that res β_{λ}(F̃_{λ}) = ϕ_{λ}.

For δ_{Γ} < Re(λ) < 1 the extension just constructed coincides with the extension defined by (4) by uniqueness. Since for Re(λ) > 0 the boundary value map β_{λ} depends holomorphically on λ the lemma follows. □

Definition 3.2 We define the Eisenstein series associated to ϕ ∈ ^{Γ}H_{∞}^{1/2}(Ω) and λ ∈ C with Re(λ) > 0, λ ∉ Σ, by

E(ϕ, λ) = P_{λ}(Φ_{λ}) .

For Re(λ) > δ_{Γ} we have

E(ϕ, λ)(x) = ∫_{S^{1}} P_{λ}(x, b)Φ̃_{λ}(b)db ,  (7)

where Φ_{λ} = Φ̃_{λ}∂^{λ−1/2}. The term ’series’ stems from the fact that by (4) Equation (7) can be written as

E(ϕ, λ)(x) = ∑_{g∈Γ} ∫_{F} P_{λ}(gx, b)ϱ^{λ−1/2}(b)ϕ(b)db .

The latter formula is the usual definition of the Eisenstein series given in the literature [17], [13], [14], [24], [16].

We now discuss the singularity of the extension Φ_{λ} at λ_{0} ∈ Σ. Let {F_{i}}, i = 1, …, k, be an orthonormal base of ker_{L^{2}}(A_{λ_{0},Y}). Then near λ_{0} the family F_{λ} has the form

F_{λ} = \frac{1}{λ_{0}^{2} − λ^{2}} ∑_{i=1}^{k}⟨F_{i}, g_{λ}⟩_{L^{2}(Y)}F_{i} + F̂_{λ} ,

where F̂_{λ} is holomorphic at λ = λ_{0} and orthogonal to ker_{L^{2}}(A_{λ_{0},Y}). Applying the boundary value map β_{λ} we obtain

Φ_{λ} = \frac{1}{λ_{0}^{2} − λ^{2}} ∑_{i=1}^{k}⟨F_{i}, g_{λ}⟩_{L^{2}(Y)}β_{λ}(F̃_{i}) + Φ̂_{λ} ,  (8)

where Φ̂_{λ} is holomorphic at λ = λ_{0}.

We define the boundary value β_{Y} (F_{i}) ∈ C^{∞}(B) by β_{Y} (F_{i})(b) = lim_{r→0} r^{−λ−1/2}F_{i}(r, b). The existence of this limit in C^{∞}(B) is not difficult to prove. In fact F̃_{i} is the Poisson transform of a Γ-invariant distributional section of T^{λ−1/2} with support on the limit set. The asymptotic expansion of the Poisson transform of this distribution near Ω yields F_{i}(r, b) ∼^{r→0} r^{λ+1/2}β_{Y} (F_{i})(b) + o(r^{λ+1/2}). A detailed argument will appear in a forthcoming paper [4]. We have

ϱ^{−λ−1/2}β_{Y} (F_{i})∂^{−λ−1/2} = res ∘ β_{−λ}(F̃_{i}) .  (9)

Applying Green’s formula and (6) we obtain

⟨F_{i}, g_{λ_{0}}⟩_{L^{2}(Y)} = ⟨F_{i}, A_{λ_{0},Y} f_{λ_{0}}⟩_{L^{2}(Y)} − ⟨A_{λ_{0},Y} F_{i}, f_{λ_{0}}⟩_{L^{2}(Y)}
= lim_{r→0}⟨F_{i}, r \frac{d}{dr} f_{λ_{0}}⟩_{L^{2}(\{r\}×B)} − lim_{r→0}⟨r \frac{d}{dr} F_{i}, f_{λ_{0}}⟩_{L^{2}(\{r\}×B)}
= −2λ_{0}⟨β_{Y} (F_{i}), ϕ⟩_{L^{2}(B)} .

Taking the Poisson transform of (8) and using (1) we obtain

Proposition 3.3 The Eisenstein series E(ϕ, λ) is meromorphic on Re(λ) > 0 with singularities in Σ and residues

\frac{1}{\sqrt{π}} \frac{Γ(λ_{0})}{Γ(1/2 + λ_{0})} ∑_{i=1}^{k}⟨β_{Y} (F_{i}), ϕ⟩_{L^{2}(B)}F_{i}

at λ = λ_{0} ∈ Σ.

Next we discuss the scattering operator. In the case of the trivial group Γ the scattering operator is just the Knapp-Stein intertwining operator J_{λ} : H_{−ω}^{λ} → H_{−ω}^{−λ}.

Consider the upper half-plane model and let ∂_{b} be the fundamental vector field of the translation group. Then C_{c}^{∞}(R)∂_{b}^{λ−1/2} ↪ H_{−ω}^{λ}. For Re(λ) < 0 the Knapp-Stein intertwining operator is defined as the integral operator

(J_{λ}ϕ∂_{b}^{λ−1/2})(b) = ∫_{R} |b − b′|^{−1−2λ}ϕ(b′)db′∂_{b}^{−λ−1/2}, ϕ ∈ C_{c}^{∞}(R) .

For Re(λ) ≥ 0 it is defined by analytic continuation and satisfies the following functional equation:

J_{−λ} ∘ J_{λ} = −\frac{cot(πλ)}{2λ}id .  (10)

J_{λ} is a meromorphic family of elliptic pseudodifferential operators with singularities at λ ∈ N_{0}. The theory of intertwining operators shows that J_{λ} extends to hyperfunctions and depends meromorphically on λ. The restriction of J_{λ} to H_{−ω,(Λ)}^{λ} defines a meromorphic family of Γintertwining operators H_{−ω,(Λ)}^{λ} → H_{−ω,(Λ)}^{−λ}. More details will again appear in [4].

We now define the scattering operator for non-trivial groups Γ.

Definition 3.4 For Re(λ) > 0, λ ∉ Σ ∪ N_{0}, the scattering operator S_{λ} is the operator

S_{λ} : ^{Γ}H_{∞}^{λ}(Ω) → ^{Γ}H_{∞}^{−λ}(Ω)

given by

ΓH_{∞}^{λ}(Ω) ∋ ϕ_{λ} ↦ Φ_{λ} ∈ ^{Γ}H_{−ω,(Λ)}^{λ} ↦ res ∘ J_{λ}(Φ_{λ}) ∈ ^{Γ}H_{∞}^{−λ}(Ω) .

Here Φ_{λ} ∈ H_{−ω,(Λ)}^{λ} is the unique Γ-invariant extension of ϕ_{λ} ∈ ^{Γ}H_{∞}^{λ}(Ω).

For Re(λ) > δ_{Γ}, λ ∉ N_{0}, ϕ ∈ ^{Γ}C^{∞}(Ω), and ϕ_{λ} = ϱ^{λ−1/2}ϕ∂_{b}^{λ−1/2} we have

S_{λ}ϕ_{λ}(b) = ∫_{Ω} |b − b′|^{−1−2λ}ϱ^{λ−1/2}ϕ(b′)db′∂_{b}^{−λ−1/2}
= ∑_{g∈Γ} ∫_{g^{−1}F} |b − b′|^{−1−2λ}ϱ^{λ−1/2}(b′)ϕ(b′)db′∂_{b}^{−λ−1/2}
= ∑_{g∈Γ} ∫_{F} |b − gb′|^{−1−2λ}ϱ^{λ−1/2}(gb′)ϕ(gb′)d(gb′)∂_{b}^{−λ−1/2}
= ∑_{g∈Γ} ∫_{F} |b − gb′|^{−1−2λ}g′(b′)^{λ+1/2}ϱ^{λ−1/2}(b′)ϕ(b′)db′∂_{b}^{−λ−1/2} ,

where the singularity of the kernel near b = b′ has to be interpreted in the regularized sense. Thus for Re(λ) > δ_{Γ}, λ ∉ N, the scattering operator is given by the integral kernel

∑_{g∈Γ} |b − gb′|^{−1−2λ}g′(b′)^{λ+1/2}

and this coincides with other definitions in the literature ([17], [16], [24]).

The scattering operator is a meromorphic family of elliptic pseudodifferential operators on B. To be precise, define S(λ) : C^{∞}(B) → C^{∞}(B) by

ϱ^{−1/2−λ}[S(λ)ϕ]∂^{−λ−1/2} := S_{λ}ϱ^{λ−1/2}ϕ∂^{λ−1/2}, ϕ ∈ C^{∞}(B) .

Then we have

Proposition 3.5 The family S(λ) : C^{∞}(B) → C^{∞}(B) is meromorphic for Re(λ) > 0 with poles at Σ and residues

ϕ ↦ \frac{1}{\sqrt{π}} \frac{Γ(λ_{0})}{Γ(1/2 + λ_{0})} ∑_{i=1}⟨β_{Y} (F_{i}), ϕ⟩_{L^{2}(B)}β_{Y} (F_{i})  (11)

at λ = λ_{0} ∈ Σ. Moreover, it has poles at λ ∈ N_{0} induced by the singularities of the on-diagonal regularization of the intertwining operator.

Equation (11) is obtained from (8), (9) and

J_{λ} ∘ β_{λ} = \frac{1}{\sqrt{π}} \frac{Γ(λ)}{Γ(1/2 + λ)}β_{−λ} .

4 THE CASE RE(λ) < 0, λ ≠ −1/2, −3/2, … 12

If S(λ) is regular, then the principal symbol of S(λ) is homotopic to the principal symbol of the identity operator. Hence the index of the pseudodifferential operator S(λ) vanishes.

In Section 7 we construct the meromorphic continuation of the Eisenstein series and the scattering operator using results from the scattering theory of Δ_{Y}. The necessary notation will be introduced in Section 6. Therefore we postpone the discussion of the meromorphic continuation until the end of this paper.

4 The case Re(λ) < 0, λ ≠ −1/2, −3/2, …

For 2λ ∉ Z the intertwining operator J_{λ} : H_{−ω}^{λ} → H_{−ω}^{−λ} is an isomorphism of G-modules. By restriction we obtain an isomorphism of Γ-modules J_{λ} : H_{−ω,(Λ)}^{λ} → H_{−ω,(Λ)}^{−λ}. The operator J_{λ} has first order poles at λ = k ∈ N. At these points we consider its renormalization J̃_{k} := lim_{λ→k}(λ− k)J_{λ}. By (10) it satisfies J̃_{k} ∘J_{−k} = −\frac{π}{2k}id and provides an isomorphism J̃_{k} : H_{−ω,(Λ)}^{k} → H_{−ω,(Λ)}^{−k}.

Lemma 4.1 For Re(λ) ≠ 0, λ ≠ −1/2, −3/2, …, the module H_{−ω,(Λ)}^{λ} is Γ-acyclic.

Proof. Using J_{λ} or J̃_{λ}, respectively, we can reduce the assertion to the case Re(λ) > 0. Since Γ acts freely and properly on Ω the modules H_{∞}^{λ}(Ω) are Γ-acyclic by [2], Lemma 2.4, for all λ ∈ C. Consider the exact sequence of Γ-modules

0 ⟶ H_{−ω,Λ}^{λ} ⟶ H_{−ω,(Λ)}^{λ} ⟶^{res} H_{∞}^{λ}(Ω) ⟶ 0 .

The associated long exact sequence reads

0 ⟶ H^{0}(Γ, H_{−ω,Λ}^{λ}) ⟶ H^{0}(Γ, H_{−ω,(Λ)}^{λ}) ⟶^{res} H^{0}(Γ, H_{∞}^{λ}(Ω)) ⟶^{δ}

⟶ H^{1}(Γ, H_{−ω,Λ}^{λ}) ⟶ H^{1}(Γ, H_{−ω,(Λ)}^{λ}) ⟶ 0 .

In case that λ ∉ Σ from Proposition 2.4 we immediately deduce H^{1}(Γ, H_{−ω,(Λ)}^{λ}) = 0 and thus Γ-acyclicity of H_{−ω,(Λ)}^{λ}.

Now assume that λ ∈ Σ. We have to show that δ is surjective. Let d := dim^{Γ}H_{−ω,Λ}^{λ} = dim H^{1}(Γ, H_{−ω,Λ}^{λ}) which is finite by Proposition 2.4. It is sufficient to show that

dim coker(res : ^{Γ}H_{−ω,(Λ)}^{λ} → ^{Γ}H_{∞}^{λ}(Ω)) ≥ d .

Let W̃ ⊂ ^{Γ}H_{∞}^{−λ}(Ω) be the subspace spanned by the boundary values ϱ^{−λ−1/2}β_{Y} (F)∂^{−λ−1/2} for all F ∈ ker_{L^{2}}(A_{λ,Y}) and V ⊂ ^{Γ}H_{∞}^{λ}(Ω) be the orthogonal complement of W̃ with respect to ⟨., .⟩_{L^{2}(B)}. Then codim V = d. It is sufficient to show that res(^{Γ}H_{−ω,(Λ)}^{λ}) ⊂ V.

Let Φ ∈ ^{Γ}H_{−ω,(Λ)}^{λ}. We must verify that ⟨β_{Y} (F), ϱ^{−λ+1/2}res(Φ)∂^{λ+1/2}⟩_{L^{2}(B)} = 0 for all F ∈ ker_{L^{2}}(A_{λ,Y}). Let E = P_{λ}Φ ∈ C^{∞}(Y). Then by Green’s formula

2λ⟨β_{Y} (F), ϱ^{−λ+1/2}res(Φ)∂^{λ+1/2}⟩_{L^{2}(B)}

= lim_{r→0}⟨F, r \frac{d}{dr} E⟩_{L^{2}(\{r\}×B)} − lim_{r→0}⟨r \frac{d}{dr} F, E⟩_{L^{2}(\{r\}×B)}
= ⟨F, A_{λ,Y} E⟩_{L^{2}(Y)} − ⟨A_{λ,Y} F, E⟩_{L^{2}(Y)}
= 0 .

This finishes the proof of the lemma for λ ∈ Σ. □

For Re(λ) < 0, λ ∉ −\frac{1}{2}N_{0}, the functional equation (10) implies the exactness of the sequence

0 ⟶ H_{−ω,Λ}^{λ} ⟶^{J_{λ}} H_{−ω,(Λ)}^{−λ} ⟶^{res∘J_{−λ}} H_{∞}^{λ}(Ω) ⟶ 0 .

Hence by Lemma 4.1 it is a Γ-acyclic resolution of H_{−ω,Λ}^{λ}. Considering the cohomology of the subcomplex of Γ-invariant vectors and taking Definition 3.4 of the scattering operator into account we obtain

Proposition 4.2 For Re(λ) < 0, λ ∉ −\frac{1}{2}N_{0}, −λ ∉ Σ, we have

H^{0}(Γ, H_{−ω,Λ}^{λ}) = ker(S_{−λ}) ,
H^{1}(Γ, H_{−ω,Λ}^{λ}) = coker(S_{−λ}) .

Since S_{−λ} is elliptic and index(S_{−λ}) = 0 we have

dim H^{0}(Γ, H_{−ω,Λ}^{λ}) = dim H^{1}(Γ, H_{−ω,Λ}^{λ}) < ∞ .

Now we discuss the points λ, where the intertwining operator or the scattering operator have singularities. First we consider the integral points k ∈ −N.

Proposition 4.3 If k ∈ −N, then we have

H^{0}(Γ, H_{−ω,Λ}^{k}) = H^{1}(Γ, H_{−ω,Λ}^{k}) = 0 .

Proof. We have the Γ-acyclic resolution

0 ⟶ H_{−ω,Λ}^{k} ⟶^{J_{−k}} H_{−ω,(Λ)}^{−k} ⟶^{res∘J̃_{k}} H_{∞}^{k}(Ω) ⟶ 0 .

If we define the renormalized scattering operator S̃_{k} := lim_{λ→k}(λ − k)S_{λ}, then we obtain

H^{0}(Γ, H_{−ω,Λ}^{−k}) = ker(S̃_{k}) ,
H^{1}(Γ, H_{−ω,Λ}^{−k}) = coker(S̃_{k}) .

Since S̃_{−λ} is elliptic and index(S̃_{k}) = 0 we have

dim H^{0}(Γ, H_{−ω,Λ}^{−k}) = dim H^{1}(Γ, H_{−ω,Λ}^{−k}) .

J̃_{k} and thus S̃_{k} are explicitly known differential operators. We claim that ker(S̃_{k}) = 0.

Using upper half-plane coordinates we have

J̃_{k}(f∂_{b}^{k−1/2}) = c_{k} (\frac{d}{dt})^{2k}f ∂_{b}^{−k−1/2}, f ∈ C_{c}^{∞}(R) ,

where c_{k} ≠ 0 is a certain constant.

We choose upper half-plane coordinates such that (0, ∞) ⊂ R becomes a connected component of Ω. Let f∂_{b}^{k−1/2} ∈ ker(S̃_{k}), f ∈ C^{∞}(Ω). On (0, ∞) the function f satisfies

(\frac{d}{dt})^{2k} f = 0

and thus f is a polynomial. The normalizer in Γ of the component (0, ∞) is the cyclic subgroup generated by the element g ∈ Γ acting on R by g : t ↦ a^{2}t for some a > 1. Since f∂_{b}^{k−1/2} is Γ-invariant we have f(a^{2}t) = a^{2k−1}f(t). This equation can only be satisfied if f = 0. This proves the claim. □

We now consider the case −λ ∈ Σ. In the proof of Lemma 4.1 we have seen that

dim coker(res : ^{Γ}H_{−ω,(Λ)}^{−λ} → ^{Γ}H_{∞}^{−λ}(Ω)) = d .

In fact res(^{Γ}H_{−ω,(Λ)}^{−λ}) = V is the orthogonal complement of the space spanned by the boundary values of the elements of ker_{L^{2}}(A_{−λ,Y}) and thus V is closed.

We choose a complement W ⊂ ^{Γ}H_{∞}^{−λ}(Ω) to V. By Proposition 3.5 the singular part of the scattering operator S_{−λ} vanishes on V. We fix an identification i : W →^{∼} ^{Γ}H_{−ω,Λ}^{−λ} and define a regularized scattering operator by (S_{−λ}^{reg})_{|V} := S_{−λ} and (S_{−λ}^{reg})_{|W} := res ∘ J_{−λ} ∘ i. Then S_{−λ}^{reg} is still an elliptic pseudodifferential operator of index zero.

Proposition 4.4 For −λ ∈ Σ we have

H^{0}(Γ, H_{−ω,Λ}^{λ}) = ker(S_{−λ}^{reg}) ,
H^{1}(Γ, H_{−ω,Λ}^{λ}) = coker(S_{−λ}^{reg}) .

Moreover, dim H^{0}(Γ, H_{−ω,Λ}^{λ}) = dim H^{1}(Γ, H_{−ω,Λ}^{λ}) = dim ker (S_{−λ}^{reg})_{|V} < ∞.

Proof. We have to show that

dim ker(res ∘ J_{−λ} : ^{Γ}H_{−ω,(Λ)}^{−λ} → ^{Γ}H_{∞}^{λ}(Ω)) = dim ker(S_{−λ}^{reg})  (12)
dim coker(res ∘ J_{−λ} : ^{Γ}H_{−ω,(Λ)}^{−λ} → ^{Γ}H_{∞}^{λ}(Ω)) = dim coker(S_{−λ}^{reg}) .  (13)

5 THE CASE λ = −1/2, −3/2, … 15

For any ϕ ∈ V by (8) we can construct the extension Φ ∈ ^{Γ}H_{−ω,(Λ)}^{−λ} thus obtaining a map ext : V → ^{Γ}H_{−ω,(Λ)}^{−λ} which is right inverse to res. It induces a decomposition ^{Γ}H_{−ω,(Λ)}^{−λ} = im(ext) ⊕^{Γ}H_{−ω,Λ}^{−λ}. Then the map

r̃es = res_{|im(ext)} ⊕ i^{−1} : im(ext) ⊕ H_{−ω,Λ}^{−λ} →^{∼} V ⊕ W

satisfies S_{−λ}^{reg} ∘ r̃es = res ∘ J_{−λ}. Equality (13) follows.

We claim that S_{−λ}^{reg}(V) is transverse to res ∘ J_{−λ}(^{Γ}H_{−ω,Λ}^{−λ}). At this point we employ the meromorphic continuation of the scattering operator to all of C and its functional equation constructed in Section 7. The functional equation implies that res∘J_{−λ}(^{Γ}H_{−ω,Λ}^{−λ}) ⊂ ker(S_{λ}) and S_{λ} ∘ (S_{−λ}^{reg})_{|V} = −\frac{cot(πλ)}{2λ}id_{V}.

We claim that the composition res ∘ J_{−λ} ∘ i is injective. In fact, the Poisson transform of a nontrivial element in the kernel of this composition would produce a non-trivial L^{2}-eigenfunction F of Δ_{Y} with β_{Y} (F) = 0. This is impossible, and (12) follows. □

5 The case λ = −1/2, −3/2, …

In this section we compute the cohomology of H_{−ω,Λ}^{λ} at the so-called topological points λ = −1/2, −3/2, . . .. The term ’topological’ stems from the observation that the Selberg zeta function associated to Γ has singularities at these points and the order of these singularities is related to the topology of Y, in particular, to the Euler characteristic [22].

Another special feature of the half-integer points is that the principal series representation H_{−ω}^{λ} is a reducible G-module. This leads to the existence of additional G-intertwining operators which are differential operators. Moreover, the Knapp-Stein intertwining operators are not invertible. Therefore the method of Section 4 breaks down.

We will employ the additional intertwining operators in an essential way in order to compute the cohomology of H_{−ω,Λ}^{λ} in terms of topological data of Y, i.e., the genus and the number of ends. It is remarkable that the result does not depend on resonances.

Let t be the parameter of R and d := d/dt. Then d^{k} is a G-intertwining operator mapping sections of T^{k/2−1/2} to sections of T^{−k/2−1/2}. For k ∈ 2N_{0} + 1 we have the exact sequences

0 → F_{k} → H_{−ω}^{k/2} →^{d^{k}} H_{−ω}^{−k/2} → F_{k} → 0

of G-modules, where F_{k} is the finite-dimensional representation of G of dimension k.

We define the Γ-modules M_{k}, N_{k} by

0 → H_{−ω,Λ}^{k/2} →^{d^{k}} H_{−ω,Λ}^{−k/2} → M_{k} → 0

0 → N_{k} → H_{−ω}^{k/2}(Ω) →^{d^{k}} H_{−ω}^{−k/2}(Ω) → 0 ,

where H_{−ω}^{λ}(Ω) denotes the hyperfunction sections of T^{λ−1/2} on Ω.

The short exact sequence of complexes

0  0  0
↓  ↓  ↓
0 → H_{−ω,Λ}^{k/2} → H_{−ω}^{k/2} → H_{−ω}^{k/2}(Ω) → 0
d^{k} ↓ d^{k} ↓ d^{k} ↓
0 → H_{−ω,Λ}^{−k/2} → H_{−ω}^{−k/2} → H_{−ω}^{−k/2}(Ω) → 0
↓  ↓  ↓
0  0  0

induces a long exact cohomology sequence which specializes to

0 → F_{k} → N_{k} → M_{k} → F_{k} → 0 .  (14)

Since H_{−ω,Λ}^{k/2} has trivial Γ-cohomology by Proposition 2.4, we have

H^{∗}(Γ, M_{k}) ≅ H^{∗}(Γ, H_{−ω,Λ}^{−k/2})

leading to the problem of computing the cohomology of M_{k}.

Recall that by assumption Ω ≠ ∅ and Γ is non-trivial. It follows that Γ is a free group on 2g + t − 1 generators

Γ = ⟨a_{1}, …, a_{g}, b_{1}, …, b_{g}, σ_{1}, …, σ_{t−1}⟩ ,

where g is the genus of the surface Y and t is the number of boundary components, i.e., dim H^{0}(B). The generators σ_{i} correspond to the boundary circles. The missing circle σ_{t} can be expressed in terms of the other generators.

Let Σ_{i} ≅ Z be the group generated by σ_{i}.

Lemma 5.1 As a representation of Γ we have N_{k} = ⨁_{i=1}^{t} Ind_{Σ_{i}}^{Γ}F_{k|Σ_{i}}. Moreover, the embedding u : F_{k} → N_{k} is given by F_{k} ∋ v ↦ ⊕_{i=1}^{t}(γ ↦ γ^{−1}v).

Proof. As a vector space, N_{k} = ⨁_{components of Ω} F_{k}, where we identify F_{k} with the kernel of d^{k}, i.e., with the polynomials of order at most k − 1. The Γ-action is easy to check. □

The cohomology of N_{k} can be computed using the Shapiro Lemma [1]:

H^{∗}(Γ, N_{k}) = H^{∗}(Γ, ⨁_{i=1}^{t} Ind_{Σ_{i}}^{Γ}F_{k|Σ_{i}}) = ⨁_{i=1}^{t} H^{∗}(Σ_{i}, F_{k|Σ_{i}}) .

Now σ_{i} is hyperbolic and thus has a unique fixed line in F_{k}. It follows that H^{p}(Σ_{i}, F_{k|Σ_{i}}) = C for p = 0, 1 and H^{p}(Σ_{i}, F_{k|Σ_{i}}) = 0 for p ≥ 2. We finally obtain

H^{∗}(Γ, N_{k}) = H^{∗}(B) .

From the exact sequence (14) we see that H^{∗}(Γ, M_{k}) is finite-dimensional and χ(Γ, M_{k}) = χ(Γ, N_{k}) = 0.

We split the sequence (14) into two short exact sequences

0 → F_{k} → N_{k} → R_{k} → 0 ,
0 → R_{k} → M_{k} → F_{k} → 0 ,

where R_{k} is a certain Γ-module.

We obtain two long exact cohomology sequences

0 → H^{0}(Γ, F_{k}) → H^{0}(B) → H^{0}(Γ, R_{k}) → ,
→ H^{1}(Γ, F_{k}) → H^{1}(B) → H^{1}(Γ, R_{k}) → 0  (15)
0 → H^{0}(Γ, R_{k}) → H^{0}(Γ, M_{k}) → H^{0}(Γ, F_{k}) →
→ H^{1}(Γ, R_{k}) → H^{1}(Γ, M_{k}) → H^{1}(Γ, F_{k}) → 0 .  (16)

In the following discussion we will assume that Γ is non-elementary. Then for k ≥ 2 we have

H^{0}(Γ, F_{k}) = 0 and dim H^{1}(Γ, F_{k}) = (2g − 2 + t)k ,

while in case k = 1 we have

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

Let h^{1} := dim H^{1}(Γ, M_{k}) and q be the dimension of the image of u_{∗} : H^{1}(Γ, F_{k}) → H^{1}(B). Then we can read off from (15) that dim H^{1}(Γ, R_{k}) = t − q. In case k ≥ 2 from (16) we obtain

h^{1} = t − q + (2g − 2 + t)k .  (17)

We claim that (17) also holds in case k = 1. We must show that i : H^{0}(Γ, M_{1}) → H^{0}(Γ, F_{1}) vanishes. If [ω] ∈ H^{0}(Γ, M_{1}) is represented by ω ∈ ^{Γ}H_{−ω,Λ}^{−1/2}, then i([ω]) ∈ H^{0}(Γ, F_{1}) ≅ C is just given by the integral of the one-form ω over ∂X:

i(ω) = ∫_{∂X} ω .

The claim is a consequence of the following Lemma.

Lemma 5.2 Assume that there exists ω ∈ ^{Γ}H_{−ω,Λ}^{−1/2} with ∫_{∂X} ω = 1. Then g = 0 and t = 2, hence Γ is elementary.

Proof. Let Γ̃ = π_{1}(SȲ), where SȲ is the sphere bundle of the compact surface with boundary Ȳ = Γ\X ∪ Ω. Then Γ̃ is a central extension of Γ by Z:

0 → Z → Γ̃ → Γ → 0 .

The action of Γ on ∂X lifts to an action of Γ̃ on the universal cover ˜∂X → ∂X.

Let D = {|z| ≤ 1} = X ∪ ∂X in the Poincaré disc model and SD be the sphere bundle. The velocity vector field of the curve t ↦ e^{2πit} provides an embedding γ : ∂X ↪ SD. Let γ̃ : ˜∂X ↪ S̃D be a lift of this embedding to the universal cover S̃D → SD.

Note that SȲ = Γ\SD and S̃D → SȲ is the universal cover. By convention the velocity vector field of the projection to Ȳ of the curve [0, 1] ∋ t ↦ e^{2πit} ∈ D represents the generator 1 ∈ Z ≅ ker(Γ̃ → Γ).

If g ∈ Γ, then g maps γ to itself. Let g̃ ∈ Γ̃ be the covering transformation of S̃D → SD over g such that the induced map g̃_{|γ̃} : γ̃ → γ̃ has fixed points. This condition determines g̃ uniquely. The group Γ̃ is generated by g̃, g ∈ Γ, and the covering transformations k, k ∈ Z, over id_{SD}.

The presentation of Γ in terms of the generators a_{i}, b_{i}, i = 1, …, g, and σ_{j}, j = 1, …, t, corresponds to the choice of a polygonal fundamental domain F ⊂ D of Γ. The boundary faces of F correspond to curves representing a_{i}, b_{i}, σ_{j} and the closed curve ∂F represents

∏_{i=1}^{g}[a_{i}, b_{i}] ∏_{j=1}^{t} σ_{j} .

Let [∂F] ⊂ Ȳ be the image of ∂F under the projection D \ Λ → Ȳ. We choose a section V of SȲ_{|[∂F]} such that the restriction of V to the curve representing σ_{j} represents σ̃_{j}. The face of F representing σ_{j} contains a piece [a, b] ∋ t ↦ e^{2πit}, and we can choose V such that on this piece V is just the velocity vector field.

Viewed as a closed oriented curve in SȲ the field V represents an element [V] ∈ Z ≅ ker(Γ̃ → Γ). This element is given by

[V] = deg(V : ∂F → S^{1}) .

We claim that [V] = 2−2g−t. In fact glue discs D_{j}, j = 1, …, t, to the boundary components of Ȳ. Then we obtain a closed Riemann surface Ŷ of genus g. We extend V to a section V̂ of SŶ with non-degenerate singularities. The total index of V̂ is 2 − 2g. On each disc D_{j} we can choose V̂ such that it has a singularity of index 1. The index of V̂ on Ȳ is just deg(V) = [V]. Thus [V] + t = 2 − 2g and the claim follows.

Let â_{i}, b̂_{i}, i = 1, …, g denote the elements of Γ̃ which are represented by the restriction of V to the faces of F representing a_{i}, b_{i}.

Assume that ω ∈ ^{Γ}H_{−ω,Λ}^{−1/2} satisfies ∫_{∂X} ω = 1. We lift ω to ˜∂X obtaining ω̃. We also fix a point s ∈ ˜∂X in the lift Ω̃ of Ω. The following defines a group homomorphism

Γ̃ ∋ g̃ ↦ c(g) := ∫_{s}^{g̃s} ω̃ ∈ C .

If g̃ stabilizes a connected component U of Ω̃, then c(g̃) = 0. In fact let h̃ ∈ Γ̃ be such that h̃s ∈ U. Then g̃h̃s ∈ U and

c(g̃) = c(g̃h̃) − c(h̃) = ∫_{h̃s}^{g̃h̃s} ω̃ = 0

since ω_{|U} = 0. If k ∈ Z ≅ ker(Γ̃ → Γ), then c(k) = k.

By construction the lifts σ̃_{j} fix components of Ω̃ and thus c(σ̃_{j}) = 0, j = 1, …, t. We obtain

0 = c (∏_{i=1}^{a}[â_{i}, b̂_{i}] ∏_{j=1}^{t} σ̃_{j}) = c(2 − 2g − t) = 2 − 2g − t .

We conclude that g = 0 and t = 2 (If g = 1 and t = 0, then Y would be a torus and not hyperbolic). □

In order to compute h^{1} it remains to compute t − q.

Lemma 5.3 We have t − q = 1 for k = 1 and t − q = 0 for k ≥ 2.

Proof. We claim that the composition

r : H^{1}(Γ, F_{k}) ⟶^{u_{∗}} H^{1}(Γ, N_{k}) ⟶^{Shapiro} ⨁_{i=1}^{t} H^{1}(Σ_{i}, F_{k|Σ_{i}})

is induced by the embeddings Σ_{i} ↪ Γ, i = 1, …, t.

On the category of Γ-modules we consider the functor morphism v : id → ⨁_{i=1}^{t} Ind_{Σ_{i}}^{Γ}F_{i} given by

v_{W} : W ∋ w ↦ ⊕_{i=1}^{t}{γ ↦ γ^{−1}v} ∈ ⨁_{i=1}^{t} Ind_{Σ_{i}}^{Γ}F_{i}(W) ,

where F_{i} is the forgetful functor from Γ-modules to Σ_{i}-modules. In particular we have u = v_{F_{k}}. Since there is a canonical isomorphism ⨁_{i=1}^{t} ^{Γ}Ind_{Σ_{i}}^{Γ}F_{i}(W) = ⨁_{i=1}^{t} ^{Σ_{i}}W the functor morphism v induces a functor morphism ^{Γ}v : ^{Γ}(.) → ⨁_{i=1}^{t} ^{Σ_{i}}(.) which is the canonical embedding. Here G(.) denotes the functor which takes G-invariants. By Shapiro’s Lemma the derived functors R^{∗} ^{Σ_{i}}(.) ∘ F_{i} and R^{∗} ^{Σ_{i}}(.) on the category of Γ-modules coincide. This proves the claim.

The Γ-module F_{k} induces a local system F_{k} on Y ∪ B. Since the compact manifold Y ∪ B and the components of its boundary B are classifying spaces of Γ and Σ_{i}, respectively, we obtain the following commutative diagram:

\begin{matrix}H^{1}(Γ, F_{k}) & →^{r} & ⨁_{i=1}^{t} H^{1}(Σ_{i}, F_{k|Σ_{i}}) \\ ↓ ≀ & & ↓ ≀ \\ H^{1}(Y ∪ B, F_{k}) & → & H^{1}(B, F_{k}) \end{matrix} .

The assertion of the lemma follows from the long exact sequence

… → H^{1}(Y ∪ B, F_{k}) → H^{1}(B, F_{k}) → H^{2}(Y ∪ B, B, F_{k}) → 0

6 THE CASE RE(λ) = 0, λ ≠ 0 20

and

H^{2}(Y ∪ B, B, F_{k}) = \{ \begin{matrix}C , & k = 1 \\ 0 , & k ≠ 1 \end{matrix} .

□

Proposition 5.4 Let Γ be non-elementary. Then we have

dim H^{0}(Γ, H_{−ω,Λ}^{−k/2}) = dim H^{1}(Γ, H_{−ω,Λ}^{−k/2}) = \{ \begin{matrix}(2g − 2 + t)k , & k ≥ 3, odd \\ 2g + t − 1 , & k = 1 \end{matrix} .

If Γ is elementary, then the limit set consists of two points. H^{∗}(Γ, H_{−ω,Λ}^{λ}) is represented by derivatives of delta distributions located at the two points of Λ. The following result is an easy exercise.

Proposition 5.5 If Γ is non-trivial elementary, then

dim H^{0}(Γ, H_{−ω,Λ}^{−k/2}) = dim H^{1}(Γ, H_{−ω,Λ}^{−k/2}) = 2, k = 1, 3, 5, ….

6 The case Re(λ) = 0, λ ≠ 0

In this section we employ methods from scattering theory in order to show that the cohomology groups H^{∗}(Γ, H_{−ω,Λ}^{iμ}), μ ∈ R \ {0}, vanish. It is not surprising that the point μ = 0 provides additional difficulties. In fact one can show

Proposition 6.1 Let S̃_{0} := lim_{λ→0} λS_{λ}. Then

dim H^{0}(Γ, H_{−ω,Λ}^{0}) = dim H^{1}(Γ, H_{−ω,Λ}^{0}) = dim ker(S̃_{0} − \frac{1}{π} id) < ∞ .

The proof appears in [3] and is based on techniques which are quite different from the scattering theoretic methods of the present paper.

The idea used in this section is similar to that of Section 2. Let λ = iμ ≠ 0 and A_{iμ} := Δ − 1/4 − μ^{2}. By the theorem of Helgason the Poisson transform provides an embedding P_{iμ} : H_{−ω,Λ}^{iμ} ↪ E.

In Section 2 we have characterized the subspace P_{λ}(H_{−ω,Λ}^{λ}) ⊂ E, Re(λ) > 0, using L^{2}- conditions on admissible sets W. If Re(λ) = 0, then L^{2}-conditions are to strong.

In order to characterize P_{iμ}(H_{−ω,Λ}^{iμ}) we introduce the following seminorms indexed by k ≥ 1, i ≥ 0, compact sets K ⊂ X and admissible subsets W ⊂ X of the special form W = M ×(0, a) ⊂ R_{+}^{2}, a > 0, for compact M ⊂ Ω (written in the coordinates (x, y), y > 0, of the upper half-plane model):

q_{W,k}(f) := ∥A_{iμ}^{k}f∥_{B(W)}
p_{W} (f) := ∥f∥_{B^{∗}(W)}
s_{W} (f) := p_{W} (D_{μ}f)
r_{K,i}(f) := ∥A_{iμ}^{i}f∥_{L^{2}(K)} .

We explain the undefined notation. The operator D_{μ} is given in the upper half-plane coordinates by D_{μ} := y \frac{d}{dy} − (iμ + 1/2). The B- and B^{∗}-norms are defined by

∥f∥_{B(W)} := ∑_{j≥0} 2^{j/2}∥f∥_{L^{2}(W ∩Ω_{j})}
∥f∥_{B^{∗}(W)} := sup_{j≥0} 2^{−j/2}∥f∥_{L^{2}(W ∩Ω_{j})},

where Ω_{j} := {(x, y) ∈ R_{+}^{2}| − ln(y) ∈ [2^{j−1}, 2^{j})}, j ≥ 1, and Ω_{0} := {(x, y) ∈ R_{+}^{2}|y ≥ e}. By B(W) and B^{∗}(W) we denote the Banach spaces of functions on W with finite B- or B^{∗}-norm.

We also introduce the closed subspace

∘B^{∗}(W) = \{f ∈ B^{∗}(W)| lim sup_{R} \frac{1}{R}∥f∥_{L^{2}(M×(a,e^{−R}))}^{2} = 0\} .

These norms are natural in the framework of scattering theory [8], Ch.14, [23].

Let E_{Λ}^{μ} be the Frechet space of all f ∈ E such that for all W and K described above q_{W,k}(f) < ∞, ∀k ≥ 1, p_{W} (f) < ∞, s_{W} (f) < ∞, r_{K,i}(f) < ∞, ∀i ≥ 0 and D_{μ}f_{|W} ∈ ^{∘}B^{∗}(W). Note that we do not require q_{W,0}(f) < ∞. Again it is not difficult to see that a countable set of such seminorms is sufficient to define the topology of E_{Λ}^{μ}. The space E_{Λ}^{μ} is a Γ-submodule of E. Set E_{Λ}^{μ}(A_{iμ}) = E_{Λ}^{μ} ∩ ker(A_{iμ}).

Lemma 6.2

P_{iμ}(H_{−ω,Λ}^{iμ}) = E_{Λ}^{μ}(A_{iμ})

Proof. We first prove that P_{iμ}(H_{−ω,Λ}^{iμ}) ⊂ E_{Λ}^{μ}(A_{iμ}). Let P_{iμ}((x, y), b) = (\frac{y}{y^{2}+(x−b)^{2}} )^{iμ+1/2} be the kernel defining the Poisson transform (with respect to the upper half-plane coordinates). Then for any compact set K ⊂ X the function b ↦ P_{iμ}(., b) is real analytic from ∂X to L^{2}(K). Moreover A_{iμ}P_{iμ}(., b) = 0. Thus for any ϕ ∈ H_{−ω,Λ}^{iμ} we have r_{K,i}(P_{iμ}ϕ) < ∞ for all i ≥ 0. Moreover, for any admissible W = M × (0, a) we have q_{W,k}(P_{iμ}ϕ) = 0, k ≥ 1.

The map b ↦ P_{iμ}(., b) is real analytic from a neighbourhood of Λ to B^{∗}(W) and D_{μ}P_{iμ}(., b) ∈ ∘B^{∗}(W). For ϕ ∈ H_{−ω,Λ}^{iμ} it follows p_{W} (P_{iμ}ϕ) < ∞ and P_{iμ}ϕ ∈ ^{∘}B^{∗}(W). This proves P_{iμ}(H_{−ω,Λ}^{iμ}) ⊂ E_{Λ}^{μ}(A_{iμ}).

We now show the opposite inclusion E_{Λ}^{μ}(A_{iμ}) ⊂ P_{iμ}(H_{−ω,Λ}^{iμ}). As in the proof of Lemma 2.1 we employ the boundary value map.

Under certain growth conditions f ∈ E(A_{iμ}) has an asymptotic expansion

f(x, y) ∼^{y→0} y^{iμ+1/2}β_{−iμ}(f)(x) + y^{−iμ+1/2}β_{iμ}(f)(x) + O(y^{3/2}) .

Here β_{±iμ}(f) are smooth sections of T^{±iμ−1/2}. The boundary value map extends to arbitrary eigenfunctions f ∈ E(A_{iμ}), but in general β_{±iμ}(f) ∈ H_{−ω}^{±iμ}. Equation (1) still holds.

By the theory of differential equations with regular singularities one can define the boundary value locally. The condition f ∈ E_{Λ}^{μ}(A_{iμ}) again implies that f has moderate growth along Ω and that β_{iμ}(f)_{|Ω} is a distribution (see the corresponding argument in the proof of Lemma 2.1). We again have to show that this distribution vanishes.

The argument is similar to the corresponding argument in the proof of Lemma 2.1. We again employ the coordinates of the Poincaré disc model. Let ψ ∈ C_{c}^{∞}(Ω) and [e^{ib_{0}}, e^{ib_{1}}] ⊂ Ω be such that supp(ψ) ⊂ (e^{ib_{0}}, e^{ib_{1}}).

Then ⟨β_{iμ}(f), ψ⟩ is defined and given by

⟨β_{iμ}(f), ψ⟩ = lim_{R→∞} \frac{1}{R} ∫_{0}^{R} f_{ψ}(t)e^{t(1/2−iμ)}dt ,

where

f_{ψ}(t) = ∫_{S^{1}} f(t, α)ψ(e^{iα})\frac{dα}{2π} ,

and the t-coordinate is the hyperbolic distance from the origin. Using (\frac{d}{dt} + iμ + 1/2)f = −D_{μ}f + o(t^{−1}) and D_{μ}f ∈ ^{∘}B^{∗} along Ω we obtain

\frac{1}{R} ∫_{0}^{R} ∫_{(e^{ib_{0}},e^{ib_{1}})}(\frac{d}{dt} + iμ + 1/2)f(t, α)e^{t(∓iμ−1/2)}ψ(e^{iα})\frac{dα}{2π}e^{t}dt

≤ (R^{−1/2}∥D_{μ}f∥_{L^{2}((0,R)×(e^{ib_{0}},e^{ib_{1}}))} + o(R^{−3/2}))∥ψ∥_{L^{2}((e^{ib_{0}},e^{ib_{1}}))}
→^{R→∞} 0 .

By partial integration this implies on the one hand

lim_{R→∞} \frac{1}{R}e^{R(1/2+iμ)} ∫_{(e^{ib_{0}},e^{ib_{1}})} f(R, α)ψ(e^{iα})\frac{dα}{2π} = 0

and on the other hand

2iμ lim_{R→∞} \frac{1}{R} ∫_{0}^{R} ∫_{(e^{ib_{0}},e^{ib_{1}})} f(t, α)ψ(e^{iα})\frac{dα}{2π}e^{t(1/2−iμ)}dt

= − lim_{R→∞} \frac{1}{R}e^{R(1/2−iμ)} ∫_{(e^{ib_{0}},e^{ib_{1}})} f(R, α)ψ(e^{iα})\frac{dα}{2π} .

Thus

lim_{R→∞} \frac{1}{R} ∫_{0}^{R} ∫_{(e^{ib_{0}},e^{ib_{1}})} f(t, α)ψ(e^{iα})\frac{dα}{2π} e^{t(1/2−iμ)}dt = 0 .

We conclude that ⟨β_{iμ}(f), ψ⟩ = 0. Since ψ was arbitrary we have β_{iμ}(f)_{|Ω} = 0. □

Our next task is to extend the inclusion 0 → H_{−ω,Λ}^{iμ} → E_{Λ}^{μ} to a Γ-acyclic resolution of H_{−ω,Λ}^{iμ}. As in Section 2 we employ the operator A_{iμ}.

Let Ẽ_{Λ} be the Frechét space of functions f ∈ E such that q_{W,k}(f) < ∞ for all admissible W ⊂ X of the form M × (0, a) and k ≥ 0, and r_{K,i}(f) < ∞ for all compact K ⊂ X, i ≥ 0. One can check that a countable number of such seminorms is sufficient in order to define the topology of Ẽ_{Λ}. The space Ẽ_{Λ} is a Γ-submodule of E.

The next lemma is proved in analogy with Lemma 2.2.

Lemma 6.3 E_{Λ}^{μ}, Ẽ_{Λ} are Γ-acyclic.

The following lemma implies that

0 ⟶ H_{−ω,Λ}^{iμ} ⟶^{P} E_{Λ}^{μ} ⟶^{A_{iμ}} Ẽ_{Λ} ⟶ 0  (18)

is a Γ-acyclic resolution of H_{−ω,Λ}^{iμ}.

Lemma 6.4 A_{iμ} : E_{Λ}^{μ} → Ẽ_{Λ} is surjective.

Proof. We consider the adjoint operator ^{t}A_{iμ} : Ẽ′_{Λ} → (E_{Λ}^{μ})′. We must show that ^{t}A_{iμ} is injective and has closed range. One can check that C_{c}^{∞}(X) is dense in Ẽ_{Λ} (note that this is not true for E_{Λ}^{μ}). Therefore there is a canonical embedding of Ẽ′_{Λ} into the space of distributions on X.

In fact if f ∈ Ẽ′_{Λ}, then there exists an admissible W ⊂ X, such that f is a distribution of finite order and with support in W (see the proof of Lemma 2.3 for a similar argument).

If ^{t}A_{iμ}f = 0, then f is real analytic. Since it vanishes on a non-empty open subset of X, it vanishes identically. Thus ^{t}A_{iμ} : Ẽ′_{Λ} → (E_{Λ}^{μ})′ is injective.

We now turn to the problem of proving that ^{t}A_{iμ} has closed range. As in Lemma 2.3 we can restrict the consideration to a bounded set B′ ⊂ (E_{Λ}^{μ})′. Then there is a subset U ⊂ X being the union of a compact set K ⊂ X and an admissible set W of the special form W = (b, 1) × M, M ⊂ Ω compact, such that the closure of X \U (with respect to the Euclidean topology induced by the Poincaré disc model) intersects Λ non-trivially and supp(h) ⊂ U holds for all h ∈ B′. Though h can not be identified with a distribution, there is an obvious notion of support of h. So supp(h) ⊂ U means that ⟨h, f⟩ = 0 for all f ∈ E_{Λ}^{μ} with f_{|U} = 0.

Let f_{i} ∈ Ẽ′_{Λ} be a sequence such that ^{t}A_{iμ}f_{i} =: h_{i} ∈ B′ and h_{i} → h in (E_{Λ}^{μ})′. Then unique continuation implies supp(f_{i}) ⊂ U. We have to construct f ∈ Ẽ′_{Λ} satisfying ^{t}A_{iμ}f = h.

Let E_{∅}^{μ}, Ẽ_{∅} be defined like E_{Λ}^{μ}, Ẽ_{Λ} replacing Λ by the empty set and Ω by ∂X. Moreover let A_{iμ} : E_{∅}^{μ} → E_{∅} be given by Δ − 1/4 − μ^{2} and ^{t}A_{iμ} be its adjoint. Then f_{i} ∈ Ẽ′_{∅} and ^{t}A_{iμ}f_{i} → h holds in (E_{∅}^{μ})′.

It follows easily from Perry [23] (essentially from Lemma 7.1 below) that A_{iμ} : E_{∅}^{μ} → E_{∅} is a topological isomorphism. It is here where we need the assumption μ ≠ 0. Thus f_{i} → f = ^{t}A_{iμ}^{−1}h in (E_{∅})′. Since supp(f_{i}) ⊂ U we also have supp(f) ⊂ U and hence f ∈ Ẽ′_{Λ} and ^{t}A_{iμ}f = h. Thus the range of ^{t}A_{iμ} : Ẽ′_{Λ} → (E_{Λ}^{μ})′ is closed. This finishes the proof of surjectivity of A_{iμ} and of the lemma. □

Using the coordinates (5) of a collar neighbourhood E of infinity T_{Y} : (0, ϵ) × B → E ⊂ Y (see (5)) we define the spaces B(Y), B^{∗}(E) and ^{∘}B^{∗}(E) in a similar way as B(W), B^{∗}(W) and ∘B^{∗}(W) (see [23] for details). Let A_{iμ,Y} = Δ_{Y} − 1/4 − μ^{2}, where Δ_{Y} is the Laplacian on Y and D_{μ} = \frac{d}{dr} − iμ. Then it is easy to see that

ΓE_{Λ}^{μ} = E_{Y}^{μ}, ^{Γ}Ẽ_{Λ} = Ẽ_{Y}

with

E_{Y}^{μ} := {f ∈ C^{∞}(Y)|∥A_{iμ,Y}^{k} f∥_{B(Y)} < ∞, k ≥ 1, ∥f∥_{B^{∗}(E)} < ∞, D_{μ}f ∈ ^{∘}B^{∗}(E)}
Ẽ_{Y} := {f ∈ C^{∞}(Y)|∥A_{iμ,Y}^{k} f∥_{B(Y)} < ∞, k ≥ 0} .

Now H^{∗}(Γ, H_{−ω,Λ}^{λ}) is given by the cohomology groups of

0 ⟶ E_{Y}^{μ} ⟶^{A_{iμ,Y}} Ẽ_{Y} ⟶ 0 .

Again from [23] (see Lemma 7.1) we conclude that A_{iμ,Y} : E_{Y}^{μ} → Ẽ_{Y} is a topological isomorphism. Hence ker(A_{iμ,Y}) = coker(A_{iμ,Y}) = 0.

Proposition 6.5 If Re(λ) = 0, λ ≠ 0, then we have

H^{∗}(Γ, H_{−ω,Λ}^{λ}) = 0 .

7 Meromorphic continuations

In Section 3 we have constructed a meromorphic continuation of the scattering operator up to the imaginary axis. In the present section we first show that it is continuous at the imaginary axis. Then we employ the functional equation in order to provide the meromorphic continuation to all of C. We also obtain the meromorphic continuation of the Eisenstein series.

The meromorphic continuation of the scattering operator and the Eisenstein series for surfaces was first obtained by Patterson [17],[19],[19]. Generalizations to higher dimensions can be found in [16], [24], [13], [12], [4]. Therefore we can keep the arguments sketchy. In particular, we do not discuss at all the point λ = 0 .

Consider the operator A_{λ,Y} = Δ_{Y} − 1/4 + λ^{2}. For Re(λ) > 0, λ ∉ Σ, it is invertible on L^{2}(Y). If λ approaches the imaginary axis we consider its inverse defined on the slightly smaller space B(Y) ⊂ L^{2}(Y) with values in the slightly larger space L^{2}(Y) ⊂ B^{∗}(Y).

Lemma 7.1 (Perry, [23], Prop. 5.5) The inverse A_{λ,Y}^{−1} : B(Y) → B^{∗}(Y) is a weakly continuous family of bounded operators uniformly bounded on compact subsets of (C \ ({0} ∪ Σ)) ∩ {Re(λ) ≥ 0}.

We now employ the constructions and notations introduced in Section 3. Let ϕ ∈ C^{∞}(B) = ΓH_{∞}^{1/2}(Ω). and consider ϕ_{λ} := ϱ^{λ−1/2}ϕ∂^{λ−1/2} ∈ ^{Γ}H_{∞}^{λ}(Ω). We want to construct a hyperfunction extension Φ_{λ} ∈ H_{−ω,(Λ)}^{λ}. The sequence (3) together with Propositions 2.4 and 6.5 implies that Φ_{λ} exists for Re(λ) ≥ 0, λ ∉ Σ, λ ≠ 0. As we have seen in Section 3 the family Φ_{λ} is holomorphic near the imaginary axis.

We now show that Φ_{λ} is continuous up to Re(λ) = 0, λ ≠ 0. We construct the family of approximate eigenfunctions f_{λ} as in Section 3 and note that g_{λ} := A_{λ,Y} f_{λ} ∈ B(Y). Lemma 7.1 shows that F_{λ} = f_{λ} − A_{λ,Y}^{−1} g_{λ} extends continuously to Re(λ) = 0, λ ≠ 0, as a family of eigenfunctions in C^{∞}(Y) (weakly continuous in B^{∗}(Y)). Lifting F_{λ} to F̃_{λ} on the universal cover X and taking the boundary value Φ_{λ} = β_{λ}F_{λ} we see that the family of hyperfunctions Φ_{λ} is continuous up to the imaginary axis, too. Here we employ the fact that β_{λ} depends continuously on λ.

We define the right limit S_{λ+0} of the scattering matrix for λ ≠ 0, Re(λ) = 0, by

S_{λ+0}ϕ_{λ} := (res ∘ J_{λ})Φ_{λ+0} .

Then S(λ) is continuous in the strong topology as a family of operators on C^{∞}(B) for Re(λ) ≥ 0, λ ∉ Σ and λ ∉ N_{0}.

The Knapp-Stein intertwining operators satisfy the functional equation

J_{λ} ∘ J_{−λ} = −\frac{cot(πλ)}{2λ} .

It follows for Re(λ) = 0, λ ≠ 0, that

S_{λ+0} ∘ S_{−λ+0}ϕ_{−λ} = lim_{ϵ→0}(res ∘ J_{λ+ϵ})res^{−1}(res ∘ J_{−λ+ϵ}Φ_{−λ+ϵ})
= res ∘ J_{λ} ∘ J_{−λ}Φ_{−λ+0}
= −\frac{cot(πλ)}{2λ}ϕ_{−λ} .

We define S_{λ} for Re(λ) < 0 by

S_{λ} := −\frac{cot(πλ)}{2λ}S_{−λ}^{−1} .

A standard application of the meromorphic Fredholm theory for Fréchet spaces [5] as in [18] shows that S_{λ} is meromorphic with at most finite-dimensional singularities at λ ∉ \frac{1}{2}N_{0}. We only have to show continuity of S_{λ} at Re(λ) = 0. In fact we have

S_{λ+0} − S_{λ−0} = S_{λ+0} + \frac{cot(πλ)}{2λ}S_{−λ+0}^{−1} = 0 .

Lemma 7.2 S(λ) is a meromorphic family of elliptic pseudodifferential operators on C^{∞}(B). It has at most finite-dimensional poles at the spectral points λ ∈ Σ (of first order) and in the set of resonances (i.e., Re(λ) < 0 and H^{0}(Γ, H_{−ω,Λ}^{λ}) ≠ 0). There are further singularities at λ ∈ N_{0}. The scattering operator satisfies the functional equation S_{−λ} ∘ S_{λ} = −\frac{cot(πλ)}{2λ}.

Corollary 7.3 The set of all λ ∈ C with H^{∗}(Γ, H_{−ω,Λ}^{λ}) ≠ 0 is discrete.

We now can find an analytic continuation of Φ_{λ}. For Re(λ) < 0, λ not a resonance or in \frac{1}{2}Z, we define

Φ_{λ} := −2λtan(πλ)J_{−λ} ∘ res^{−1} ∘ S_{λ}ϕ_{λ} .

Then for Re(λ) = 0, λ ≠ 0, we have

Φ_{λ−0} = −2λtan(πλ)J_{−λ+0} ∘ res^{−1} ∘ S_{λ−0}ϕ_{λ−0}
= −2λtan(πλ)J_{−λ} ∘ J_{λ} ∘ res^{−1}ϕ_{λ}
= res^{−1}ϕ_{λ}
= Φ_{λ+0} .

It follows that Φ_{λ} extends as a meromorphic family of hyperfunctions. We obtain the Eisenstein series by applying the Poisson transform: E(ϕ, λ) = P_{λ}Φ_{λ}.

Proposition 7.4 The Eisenstein series has a meromorphic continuation to all of C with poles in Σ, the set of resonances, and the negative half-integers. The Eisenstein series satisfies the functional equation

E(ϕ, λ) = \sqrt{π} \frac{Γ(1/2 − λ)}{Γ(−λ)}E(S(λ)ϕ, −λ) .

The functional equation follows easily from the corresponding functional equation of the Poisson transform [9], Prop. 6.2.

8 Summary

Let Γ be a non-trivial, torsion-free Fuchsian group of the second kind without parabolic elements. By H_{−ω,Λ}^{λ} we denote the hyperfunction vectors of the principal series representation H_{−ω}^{λ} with support in the limit set of Γ. The cohomology groups H^{∗}(Γ, H_{−ω,Λ}^{λ}) are finite-dimensional and have vanishing Euler characteristic. In greater detail:

• If Re(λ) > 0, then we have

H^{0}(Γ, H_{−ω,Λ}^{λ}) = H^{1}(Γ, H_{−ω,Λ}^{λ}) = ker_{L^{2}}(Δ_{Y} − 1/4 + λ^{2}) ,

where Δ_{Y} is the Laplace-Beltrami operator on the hyperbolic surface Y := Γ\H^{2}. Let Σ be the set of all λ, Re(λ) > 0, with ker_{L^{2}}(Δ_{Y} − 1/4 + λ^{2}) non-trivial. It is known that Σ ⊂ (0, δ_{Γ}], and if δ_{Γ} > 0, then δ_{Γ} ∈ Σ.

• If Re(λ) = 0, λ ≠ 0, then we have

H^{∗}(Γ, H_{−ω,Λ}^{λ}) = 0 .

• If λ = 0, then

dim H^{0}(Γ, H_{−ω,Λ}^{0}) = dim H^{1}(Γ, H_{−ω,Λ}^{0}) = dim ker(S̃_{0} − \frac{1}{π} id) ,

where S̃_{0} is defined in Section 6.

• If Re(λ) < 0, λ ∉ −\frac{1}{2}N, −λ ∉ Σ, then we have

H^{0}(Γ, H_{−ω,Λ}^{λ}) = ker(S_{−λ})  (19)
H^{1}(Γ, H_{−ω,Λ}^{λ}) = coker(S_{−λ}) ,

where S_{λ} is the scattering operator, and dim H^{0}(Γ, H_{−ω,Λ}^{λ}) = dim H^{1}(Γ, H_{−ω,Λ}^{λ}).

• For Re(λ) < 0, −λ ∈ Σ, (19) holds with S_{−λ} replaced by S_{−λ}^{reg} defined in Section 4.

• For Re(λ) < 0, −λ ∈ N, we have

H^{∗}(Γ, H_{−ω,Λ}^{λ}) = 0 .

• If λ = −k/2, k odd, then we have for non-elementary Γ

dim H^{0}(Γ, H_{−ω,Λ}^{−k/2}) = dim H^{1}(Γ, H_{−ω,Λ}^{−k/2}) = \{ \begin{matrix}(2g − 2 + t)k , & k ≥ 3 \\ 2g + t − 1 , & k = 1 , \end{matrix}

and for elementary Γ

dim H^{0}(Γ, H_{−ω,Λ}^{−k/2}) = dim H^{1}(Γ, H_{−ω,Λ}^{−k/2}) = 2 .

Here g is the genus and t is the number of boundary components of Y.

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