The Spectrum of Kleinian Manifolds

Ulrich Bunke^{∗}and Martin Olbrich^{†} September 14, 1999

Contents

1 Introduction 1 2 Geometric preparations 7 3 Analytic preparations 10 4 Push-down and extension 12 5 Meromorphic continuation of ext 19 6 Invariant distributions on the limit set 31 7 Consequences of unitarity 42 8 Abstract harmonic analysis on Γ\G 47 9 Tempered invariant distribution vectors 52 10 Eisenstein series, wave packets, and scalar products 60 11 The Plancherel theorem and spectral decompositions 68

1 Introduction

Let G be a real simple linear connected Lie group of real rank one, and let Γ ⊂ G be a convex-cocompact, non-cocompact, torsion-free, discrete subgroup. This paper is devoted to the decomposition of the right regular representation of G on L^{2}(Γ\G) into irreducibles, the so called Plancherel decomposition. We also allow twists by finite-dimensional unitary representations (φ, V_{φ}) of Γ, i.e., we investigate the right regular representation of G on the space

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@uni-math.gwdg.de

†Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:olbrich@uni-math.gwdg.de

L^{2}(Γ\G, φ) := {f : G → V_{φ} | f(gx) = φ(g)f(x) ∀g ∈ Γ, x ∈ G, ∫_{Γ\G} |f(x)|^{2} dx < ∞} .

Let K ⊂ G be a maximal compact subgroup. Then X := G/K is a Riemannian symmetric space of negative curvature which is the universal covering of the locally symmetric space Y := Γ\X. Our assumption on Γ implies that Y has infinite volume and no cusps. We call such a locally symmetric space a Kleinian manifold. Let (γ, V_{γ}) be a finite-dimensional unitary representation of K. Then we form the homogeneous vector bundle V (γ) := G ×_{K} V_{γ} over X and the locally homogeneous vector bundle V_{Y} (γ, φ) := Γ\(V (γ) ⊗ V_{φ}) over Y. Let 𝔤 denote the Lie algebra of G, U(𝔤) the universal enveloping algebra of 𝔤 and Z its center. Via the left regular action of U(𝔤) on C^{∞}(X, V (γ)) any A ∈ Z gives rise to a G-invariant differential operator. This operator descends to C^{∞}(Y, V_{Y} (γ, φ)). Here the Casimir operator Ω_{G} ∈ Z, which defines an essentially selfadjoint elliptic operator of second order acting on L^{2}(Y, V_{Y} (γ, φ)), is of particular interest.

Our initial motivation was to obtain the spectral decomposition of the space of sections L^{2}(Y, V_{Y} (γ, φ)) of the bundle V_{Y} (γ, φ) over Y with respect to the Casimir operator and other locally invariant differential operators. However, the isomorphism

L^{2}(Y, V_{Y} (γ, φ)) ≅ [L^{2}(Γ\G, φ) ⊗ V_{γ}]^{K}

implies that the Plancherel decomposition of L^{2}(Γ\G, φ) is more or less equivalent to the desired spectral decompositions for all the bundles at once (for details see Section 11). Our main results are the Plancherel theorem Theorem 11.1 and its consequences for spectral decompositions obtained in Theorem 11.2.

The structure of the Plancherel decomposition depends on the critical exponent δ_{Γ} of Γ (see Definition 2.2). For technical reasons we have to exclude discrete subgroups of the isometry group of X = 𝕆H^{2} with δ_{Γ} ≥ 0 (because our method of the meromorphic continuation of Eisenstein series involves the ”embedding trick” (see below) which does not work in this case). Then Theorem 11.1 provides a decomposition

L^{2}(Γ\G, φ) = L^{2}(Γ\G, φ)_{ac} ⊕ L^{2}(Γ\G, φ)_{cusp} ⊕ L^{2}(Γ\G, φ)_{res} ⊕ L^{2}(Γ\G, φ)_{U} .

Here L^{2}(Γ\G, φ)_{ac} decomposes further into a sum of direct integrals corresponding to the unitary principal series representations of G, each occuring with infinite multiplicity, L^{2}(Γ\G, φ)_{cusp} decomposes into discrete series representations of G, each discrete series representation of G occurs with infinite multiplicity. These two parts, which are in a sense the main contribution to L^{2}(Γ\G, φ), have essentially the same structure as in the case of the trivial group, i.e., in the Plancherel theorem for L^{2}(G) due to Okamoto [41], Hirai [30], and Harish-Chandra [25], [26]. The remaining two parts L^{2}(Γ\G, φ)_{res} and L^{2}(Γ\G, φ)_{U} can only be non-trivial if δ_{Γ} ≥ 0. They consist of a direct sum of non-discrete series representations of G with real infinitesimal character occuring with finite multiplicity. The subsbscript res stands for residual spectrum. Indeed, the space L^{2}(Γ\G, φ)_{res} is generated by residues of Eisenstein series. The ”stable” subspace L^{2}(Γ\G, φ)_{U} is of similar nature but is orthogonal to the residues of Eisenstein series.

It contains representations of integral infinitesimal character only. The understanding of its significance deserves further study.

Concerning the spectral decomposition of the operator −Ω_{G} acting on L^{2}(Y, V_{Y} (γ, φ)) it follows that the absolute continuous spectrum consists of finitely many branches [c_{i}, ∞] of infinite multiplicity, where the constants c_{i} ∈ ℝ are computable in terms of γ, that there is no singular continuous spectrum, and that the discrete spectrum is finite. The set of eigenvalues with infinite multiplicity coincides with the discrete spectrum of −Ω_{G} on L^{2}(X, V (γ)). Eigenvalues of finite multiplicity can only occur if δ_{Γ} ≥ 0. The corresponding eigenspaces split into a residual and a ”stable” part, where the residual part is generated by residues of Eisenstein series. The ”stable” part can occur only if γ is non-trivial. Note that the Plancherel theorem also provides a finer decomposition of the generalized eigenspaces of the Casimir operator with respect to the algebra of locally invariant diffrential operators D(G, γ). For instance, the Casimir operator may have embedded eigenvalues which are isolated with respect to D(G, γ). For more information see Section 11.

Spectral decompositions of L^{2}(Y, V_{Y} (γ, φ)) (respectively partial results; sometimes also cusps are allowed) were previously obtained for

• trivial γ and surfaces by Patterson [44], compare also [23] and [20]

• higher dimensional real hyperbolic manifolds and trivial γ by Lax-Phillips [33], [34], [35], Perry [48], Mazzeo-Melrose [39], Mandouvalos [37]

• differential forms on real hyperbolic manifolds by Mazzeo-Phillips [40]

• differential forms on complex-hyperbolic manifolds by Epstein-Melrose-Mendoza [22], Epstein-Melrose [21].

As in most of these papers the crucial step towards a spectral decomposition is the construction and meromorphic continuation of Eisenstein series and the scattering matrix (at least up to the critical axis). Besides the papers just cited this problem for trivial γ and real hyperbolic manifolds Y is treated e.g. in [45], [46], [43], [16], [36], [1], [38], [49]. Our approach differs philosophically, if not technically, from these papers. After two sections of preparatory character Sections 4-7 contain the development of our geometric version of ”scattering theory”. The emphasis is on analysis on the sphere at infinity, i.e., the geodesic boundary ∂X of X. The advantage of this approach becomes manifest if one goes beyond the case of the trivial representation γ. Indeed, as experience shows, ”meromorphic objects” (Eisenstein series, scattering matrices, Selberg zeta functions etc.) correspond to families of bundles on the boundary rather than to bundles on the (locally) symmetric space itself. The key notions which we are going to discuss here are the extension map ext and invariant distributions supported on the limit set. Very similar ideas appear in the work of van den Ban-Schlichtkrull and Delorme on the Plancherel formula for reductive symmetric spaces (see e.g. [5], [9], [14], [7]).

Let P ⊂ G be a minimal parabolic subgroup. Then ∂X can be viewed as the homogeneous space G/P. Set M := K ∩ P. Let σ ∈ M̂ be an irreducible representation of M. Finitedimensional irreducible representations of P then come in families {σ_{λ}}_{λ∈𝔞_{ℂ}^{∗}} parametrized by a one-dimensional complex vector space 𝔞_{ℂ}^{∗}. They determine families of homogeneous vector bundles {V (σ_{λ})}_{λ∈𝔞_{ℂ}^{∗}} over ∂X. Sections of these bundles carry representations of G, the socalled principal series representations. We are interested in the space of Γ-invariant distribution sections ^{Γ}C^{−∞}(∂X, V (σ_{λ})) as well as in its twisted version ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)), where φ is a finite-dimensional representation of Γ. It turns out to be useful to allow non-unitary twists φ, too. ∂X is the union of the domain of discontinuity Ω of Γ and the limit set Λ. The most interesting invariant distributions are those which are supported on the limit set. By a slight abuse of notation we denote the space of such distributions by ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). A prominent example of an invariant distribution supported on the limit set is given by the Patterson-Sullivan measure which is an element in ^{Γ}C^{−∞}(Λ, V (1_{δ_{Γ}})). Here 1 stands for the trivial representation of M, and the critical exponent δ_{Γ} is viewed as an element of 𝔞_{ℂ}^{∗} in a natural way. Eventually, it will turn out that for unitary φ the representations appearing in L^{2}(Γ\G, φ)_{res} ⊕ L^{2}(Γ\G, φ)_{U} can be parametrized by the set {(σ, λ) ∈ M̂ × 𝔞_{ℂ}^{∗} | Re(λ) ≥ 0,^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0}.

In order to construct elements of ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) we proceed as follows. We consider the boundary at infinity of Y, the compact manifold B := Γ\Ω. The bundle V (σ_{λ}, φ) on ∂X induces a corresponding bundle V_{B}(σ_{λ}, φ) on B such that

C^{−∞}(B, V_{B}(σ_{λ}, φ)) ≅ ^{Γ}C^{−∞}(Ω, V (σ_{λ}, φ)) .

Using this isomorphism we want to construct the extension map

ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

which extends an invariant distribution on Ω across the limit set Λ. This is possible if λ belongs to some right half-plane depending on the critical exponent δ_{Γ}. In fact, we define ext to be the adjoint of an operator π_{∗} which associates to each smooth section of V (σ̃_{−λ}, φ̃) the Γ-average of its restriction to Ω. Because of the convergence of the Poincare series this average exists for Re(λ) sufficiently large and depends holomorphically on λ in that region. The first task (Section 5) is to obtain a meromorphic continuation of ext to all of 𝔞_{ℂ}^{∗}.

Classically, for trivial γ, Eisenstein series are obtained by averaging the Poisson kernel P_{λ}(x, b), x ∈ X, b ∈ Ω ⊂ ∂X, over Γ. Then the pairing of this Eisenstein series with a distribution ϕ on B yields the eigenfunction E(λ, ϕ) on Y, also called Eisenstein series. Using the extension map we can rewrite

E(λ, ϕ) = P_{λ} ∘ ext(ϕ) ,  (1)

where P_{λ} is the Poisson transform. It is Equation (1) which we will use to define Eisenstein series for general bundles. Thus, in our approach the extension map ext is the primary object. Once ext is understood, the Eisenstein series will not cause any essential additional difficulties.

There is a second important object closely related to ext, the scattering matrix S_{λ} which we define as follows

S_{λ} := res ∘ J_{λ} ∘ ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{−λ}, φ)) ,  (2)

where

res : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{λ}, φ))

is induced by restriction from ∂X to Ω, and

J_{λ} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X, V (σ_{−λ}, φ))

is the scattering matrix for the trivial group Γ known in representation theory as Knapp-Stein intertwining operator.

Initially, ext and S_{λ} are defined on some right half-plane. Their meromorphic continuation proceeds in three surprisingly simple steps which we are going to sketch now.

For step one assume that ext is defined on a half-plane {Re(λ) > −ϵ} for some ϵ > 0. Then we show the functional equations

S_{λ} = S_{−λ}^{−1}  (3)
ext = J_{−λ} ∘ ext ∘ S_{λ}  (4)

for |Re(λ)| < ϵ. Under the additional hypotheses σ = 1 we use meromorphic Fredholm theory in order to show that S_{−λ}^{−1} extends meromorphically to a much bigger half-plane. The main point here is that J_{λ} can be used to construct a nice family of parametrices for S_{−λ}. Now (3) and (4) give the continuation of S_{λ} and ext, respectively, to this half-plane.

The remaining two steps are purely algebraic in nature. In the second step we embed G, hence Γ, into the isometry group of a symmetric space of sufficiently large dimension. In this higher dimensional situation we can apply step one. By the first two steps we obtain the meromorphic continuation of ext and S_{λ} to a half-plane which is independent of δ_{Γ}. This ”embeddding trick” has appeared already in [36] in the context of the meromorphic continuation of Eisenstein series. Unfortunately, it is not applicable to the exceptional case X = 𝕆H^{2}.

In the third step we use tensoring with finite-dimensional G-representations in order to embed the bundle V_{B}(σ_{λ}, φ) into a bundle of the form V_{B}(1_{μ}, φ′) for a suitable representation φ′ of Γ and μ ∈ 𝔞_{ℂ}^{∗} belonging to a region where ext is already known to be meromorphic. In this step it is crucial to allow non-unitary twists. Note that this method of meromorphic continuation is independent of any spectral theoretic argument.

In Section 6 we show how ext can be used to construct all invariant distributions ϕ ∈ ΓC^{−∞}(∂X, V (σ_{λ}, φ)), in particular those supported on the limit set. Indeed, it follows from res ∘ ext = id that at points λ ∈ 𝔞_{ℂ}^{∗}, where ext has a pole, the leading singular part of its Laurent expansion at λ maps C^{−∞}(B, V_{B}(σ_{λ}, φ)) to ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). This construction may be viewed as a generalization of the construction of the Patterson-Sullivan measure. The main result of this section is Theorem 6.1 stating the discreteness of the set of ”resonances” {λ ∈ 𝔞_{ℂ}^{∗} | ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0} and that for each λ ∈ 𝔞_{ℂ}^{∗} the space ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is finite-dimensional. In contrast to the meromorphic continuation of ext the proof of this theorem also requires analysis on X, in particular a detailed knowledge of asymptotics of Poisson transforms and a variant of Green’s formula. Note that the asymptotic formulas we use are simple consequences of asymptotic expansions of matrix coefficients known from representation theory. They are strong enough to imply also the asymptotics of Eisenstein series which will play a decisive role in the final proof of the Plancherel theorem. It also follows that the scattering matrix defined by (2) is really a scattering matrix in the sense that it determines the relation between the two leading asymptotic terms of the Eisenstein series. In Section 7 we return to the assumption that φ is unitary which allows us to gain more detailed information concerning the location of the singularities of ext. In particular, ext, and hence the Eisenstein series, are regular at non-zero imaginary λ.

Note that only a part of the results obtained up to Section 6 is really needed for the derivation of the Plancherel theorem. However, we believe that the present version of scattering theory also provides an adequate foundation for more ambitious tasks in analysis on Kleinian manifolds as there are trace formulas, the investigation of Selberg and Ruelle zeta functions or Paley-Wiener theorems. For instance, in [13] we have used this approach in the case of real hyperbolic manifolds in order to prove a conjecture of Patterson concerning the relation between invariant distributions supported on the limit set and the singularities of the (untwisted) Selberg zeta function. In a paper which is in preparation we extend this approach to scattering theory to geometrically finite groups Γ.

The spectral theoretic part of the paper starts with Section 8. The problem we have to solve is twofold: first to produce a certain amount of eigensections and wave packets of them, and second to show that they span the whole Hilbert space. While the first task is almost standard once the Eisenstein series are constructed the second requires additional arguments. In particular, one has to show the absence of the singular continuous spectrum. Usually, the limiting absorption principle (e.g. [48]) or commutator methods (see e.g. [24]) are employed at this point. Here we use a completely different method proposed by Bernstein [8]. This method, brought to our attention by Delorme’s proof of the Plancherel theorem for reductive symmetric spaces [19], rests on a theory of appropriate Schwartz spaces for Y (or Γ\G). It leads to the notion of tempered eigensections or, switching to representation theoretic language, tempered invariant distribution vectors of unitary representations of G. These notions will be discussed in Section 8. The crucial point is that a priori only tempered eigensections can occur in the spectral decomposition. In Section 9 we relate tempered invariant distribution vectors to invariant distributions on ∂X, in particular those supported on the limit set. Combining the results of Sections 6 and 7 with knowledge of the structure of the unitary dual Ĝ of G we obtain a classification of tempered invariant distribution vectors. This classification enables us to complete the exhaustion part of the proof of the Plancherel theorem in Section 11.

Some readers may have noticed an earlier version of this paper which appeared as an e-print more than two years ago. At that time we were not able to continue ext to all of 𝔞_{ℂ}^{∗}, but only to the complement of a set of integer points. In particular, in order to conclude finiteness of the discrete spectrum we were forced to combine several not very natural arguments. Now the discreteness result Theorem 6.1 which is a consequence of the meromorphy of ext gives (among other things) a very satisfactory understanding of the finiteness of the discrete spectrum. In addition, the point of view in the present version is more representation theoretic. In our opinion, this makes the fine structure of the spectrum much more transparent.

Acknowledgement: We thank R. Mazzeo and P. Perry for discussing of parts of this work.

2 Geometric preparations

Let G be a connected, linear, real simple Lie group of rank one, G = KAN be an Iwasawa decomposition of G, 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 be the corresponding Iwasawa decomposition of the Lie algebra 𝔤, M := Z_{K}(A) be the centralizer of A in K, and P := MAN be a minimal parabolic subgroup. The group G acts isometrically on the rank-one symmetric space X := G/K. Let ∂X := G/P = K/M be its geodesic boundary. We consider X̄ := X ∪∂X as a compact manifold with boundary.

By the classification of symmetric spaces with strictly negative sectional curvature X is one of the following spaces:

• a real hyperbolic space ℝH^{n}, n ≥ 1,

• a complex hyperbolic space ℂH^{n}, n ≥ 2,

• a quaternionic hyperbolic space ℍH^{n}, n ≥ 2,

• or the Cayley hyperbolic plane 𝕆H^{2},

and G is a linear group finitely covering the orientation-preserving isometry group of X.

We consider a torsion-free discrete subgroup Γ ⊂ G such that ∂X admits a Γ-invariant partition ∂X = Ω ∪ Λ, where Ω ≠ ∅ is open and Γ acts freely and cocompactly on X ∪ Ω. The closed subset Λ is called the limit set of Γ. The locally symmetric space Y := Γ\X is a complete Riemannian manifold of infinite volume without cusps. It can be compactified by adjoining the geodesic boundary B := Γ\Ω.

A subgroup Γ satisfying this assumption is often called convex-cocompact since it acts cocompactly on the convex hull of the limit set. The quotient Y can be called a Kleinian manifold generalizing the corresponding notion for three-dimensional hyperbolic manifolds.

By 𝔞_{ℂ}^{∗} we denote the complexified dual of 𝔞. If λ ∈ 𝔞_{ℂ}^{∗}, then we set a^{λ} := e^{⟨λ,log(a)⟩} ∈ ℂ. Let α be the short root of 𝔞 in 𝔫. We set A_{+} := {a ∈ A | a^{α} ≥ 1}. Define ρ ∈ 𝔞^{∗} as usual by ρ(H) := \frac{1}{2}tr(ad(H)_{|𝔫}), ∀H ∈ 𝔞. We have

\begin{matrix}X & ℝH^{n} & ℂH^{n} & ℍH^{n} & 𝕆H^{n} \\ ρ & \frac{n−1}{2}α & nα & (2n + 1)α & 11α \end{matrix} .

We adopt the following conventions about the notation for points of X and ∂X. A point x ∈ ∂X can equivalently be denoted by a subset kM ⊂ K or gP ⊂ G representing this point in ∂X = K/M or ∂X = G/P. If F ⊂ ∂X, then F M := ⋃_{kM∈F} kM ⊂ K. Analogously, we can denote b ∈ X by gK ⊂ G, where gK represents b in X = G/K.

Lemma 2.1 For any compact F ⊂ Ω we have ♯(Γ ∩ (F M)A_{+}K) < ∞.

Proof. Note that (F M)A_{+}K ∪ F ⊂ X ∪ Ω is compact. Thus its intersection with the orbit ΓK of the origin of X is finite. □

The geometry of the action of Γ on X ∪∂X can be studied in a uniform way using the various decompositions of G. Any element g ∈ G has a Cartan decomposition g = k_{g}a_{g}h, k_{g}, h ∈ K, a_{g} ∈ A_{+}, where a_{g} and k_{g}M ∈ K/M are uniquely determined by g. Let g = κ(g)a(g)n(g), κ(g) ∈ K, a(g) ∈ A, n(g) ∈ N be defined with respect to the given Iwasawa decomposition. The function G × K ∋ (g, k) ↦ a(g^{−1}k) descends to X × ∂X.

Given a normalization of the invariant distance d on X we identify A_{+} with [1, ∞) such that a = e^{d(eK,aK)}. Then for any g ∈ G and k ∈ K we have

a_{g} = e^{d(eK,gK)} and a(g^{−1}k) = e^{±d(eK,HS_{gK,kM})} ,  (5)

where HS_{gK,kM} = kNk^{−1}gK is the horosphere passing through gK ∈ X and kM ∈ ∂X. The sign ± is positive (negative) if eK lies inside (outside) the corresponding horoball.

Definition 2.2 The critical exponent δ_{Γ} ∈ 𝔞^{∗} of Γ is the smallest element such that ∑_{g∈Γ} a_{g}^{−(λ+ρ)} converges for all λ ∈ 𝔞^{∗} with λ > δ_{Γ}. If Γ is the trivial group, then we set δ_{Γ} := −∞.

Equation (5) shows that this definition of δ_{Γ} differs from the usual one by a ρ-shift, only.

The critical exponent δ_{Γ} has been extensively studied, in particular by Patterson [47], Sullivan [52], and Corlette [18]. From these papers we know that δ_{Γ} ∈ [−ρ, ρ), if Γ is non-trivial. Moreover, δ_{Γ} + ρ = dim_{H}(Λ)α, where dim_{H}(Λ) denotes the Hausdorff dimension of the limit set with respect to the natural class of sub-Riemannian metrics on ∂X (the Hausdorff dimension of the empty set is by definition −∞). If X is a quaternionic hyperbolic space or the Cayley hyperbolic plane, then δ_{Γ} can not be arbitrary close to ρ. In these cases we have δ_{Γ} ≤ (2n − 1)α and δ_{Γ} ≤ 5α, respectively [18].

We now collect some facts concerning the relation between the Cartan and the Iwasawa decomposition. First of all Equation (5) implies

a(g^{−1}k) ≤ a_{g} for all g ∈ G, k ∈ K .  (6)

Lemma 2.3 Let k_{0}M ∈ ∂X. For any compact W ⊂ (∂X \ k_{0}M)M and any neighbourhood U ⊂ K of k_{0}M satisfying W ⊂ (∂X \ clo(U)M)M there exists a constant C > 0, such that

Ca_{g} ≤ a(g^{−1}k) ≤ a_{g}

for all g = k_{g}a_{g}h ∈ KA_{+}K with k_{g} ∈ W, and all k ∈ U.

Proof. The upper bound is given by (6). We prove the lower bound. Let w ∈ N_{K}(M) represent the non-trivial element of the Weyl group of (𝔤, 𝔞). Set N̄ = θ(N), where θ is the Cartan involution of G fixing K. Since the set W^{−1}clo(U)M is compact and disjoint from M, there is a precompact open V ⊂ N̄ such that W^{−1}clo(U)M ⊂ wκ(V)M.

Let k ∈ U. Then we have k_{g}^{−1}k = wκ(n̄)m for n̄ ∈ V, m ∈ M. Using that a(n̄) ≥ 1 for all n̄ ∈ N̄ (see e.g. [28], Ch. IV, Cor. 6.6.) we obtain

a(g^{−1}k) = a(a_{g}^{−1}k_{g}^{−1}k)
= a(a_{g}^{−1}wκ(n̄)m)
= a(a_{g}κ(n̄))
= a(a_{g}n̄n(n̄)^{−1}a(n̄)^{−1})
= a(a_{g}n̄a_{g}^{−1})a(n̄)^{−1}a_{g}
≥ a(n̄)^{−1}a_{g} .

Since V is precompact we have C := inf_{n̄∈V} a(n̄)^{−1} > 0. It follows that a(g^{−1}k) ≥ Ca_{g} . □

As a corollary we obtain a certain converse of the triangle inequality.

Corollary 2.4 Let k_{0}, W, and U be as in Lemma 2.3. Then there exists a constant C > 0 such that for all a ∈ A, k ∈ U, and g ∈ G with k_{g} ∈ W

a_{g^{−1}ka} ≥ Ca_{g}a .

Proof. Combining Equation (6) with Lemma 2.3 we obtain

a_{g^{−1}ka} = a_{(g^{−1}ka)^{−1}} ≥ a(g^{−1}ka) = a(g^{−1}k)a ≥ Ca_{g}a .
□

A word concerning normalizations: The basic object will be a fixed invariant Riemannian metric on X. The exponential map then induces a metric on 𝔞, hence on 𝔞^{∗}. Throughout the paper we will often isometrically identify 𝔞 with ℝ, 𝔞_{ℂ}^{∗} with ℂ, A_{+} with [1, ∞) such that (5) holds. Haar measures will be normalized as follows: The measures on A, 𝔞^{∗} are fixed by the above metric. Compact groups will always have total mass 1. The Haar measure of G is given by dg = dk dx, where dk is the Haar measure of K and dx is the Riemannian measure on X = G/K. Finally, we will normalize the Haar measure on N̄ such that

∫N̄ a(n̄)^{−2ρ} dn̄ = 1 .

3 Analytic preparations

Let (σ, V_{σ}) be a finite-dimensional unitary representation of M. For λ ∈ 𝔞_{ℂ}^{∗} we form the representation σ_{λ} of P on V_{σ_{λ}} := V_{σ}, which is given by σ_{λ}(man) := σ(m)a^{ρ−λ}. Let V (σ_{λ}) := G×_{P} V_{σ_{λ}} be the associated homogeneous bundle over ∂X = G/P. It induces a bundle on B = Γ\Ω defined by V_{B}(σ_{λ}) := Γ\V (σ_{λ})_{|Ω}.

Let σ̃ be the dual representation to σ. Then there are natural pairings

V (σ̃_{−λ}) ⊗ V (σ_{λ}) → Λ^{max}T^{∗}∂X
V_{B}(σ̃_{−λ}) ⊗ V_{B}(σ_{λ}) → Λ^{max}T^{∗}B .

The orientation of ∂X induces one of B. Employing these pairings and integration with respect to the fixed orientation we obtain identifications

C^{−∞}(∂X, V (σ_{λ})) = C^{∞}(∂X, V (σ̃_{−λ}))′
C^{−∞}(B, V_{B}(σ_{λ})) = C^{∞}(B, V_{B}(σ̃_{−λ}))′ .

As a K-homogeneous bundle we have a canonical identification V (σ_{λ}) ≅ K ×_{M} V_{σ}. Thus ⋃_{λ∈𝔞_{ℂ}^{∗}} V (σ_{λ}) → 𝔞_{ℂ}^{∗} × ∂X has the structure of a trivial holomorphic family of bundles.

Let π^{σ,λ} denote the representation of G on the space of sections of V (σ_{λ}) given by the leftregular representation. Then π^{σ,λ} is called a principal series representation of G. Note that there are different globalizations of this representation which are distinguished by the regularity of the sections (smooth, distribution etc.).

For any small open subset U ⊂ B and diffeomorphic lift Ũ ⊂ Ω the restriction V_{B}(σ_{λ})_{|U} is canonically isomorphic to V (σ_{λ})_{|Ũ}. Let {U_{α}} be a cover of B by open sets as above. Then

⋃_{λ∈𝔞_{ℂ}^{∗}} V_{B}(σ_{λ}) → 𝔞_{ℂ}^{∗} × B

can be given the structure of a holomorphic family of bundles by gluing the trivial families

⋃_{λ∈𝔞_{ℂ}^{∗}} V_{B}(σ_{λ})_{|U} ≅ ⋃_{λ∈𝔞_{ℂ}^{∗}} V (σ_{λ})_{|Ũ}

together using the holomorphic families of gluing maps induced by π^{σ,λ}(g), g ∈ Γ.

The structure of a holomorphic family of bundles allows us to consider holomorphic or smooth or continuous families of sections 𝔞_{ℂ}^{∗} ∋ μ ↦ f_{μ} ∈ C^{±∞}(∂X, V (σ_{μ})), 𝔞_{ℂ}^{∗} ∋ μ ↦ f_{μ} ∈ C^{±∞}(B, V_{B}(σ_{μ})), respectively.

Let (φ, V_{φ}) be a finite-dimensional representation of Γ. We form the bundle V (σ_{λ}, φ) := V (σ_{λ}) ⊗ V_{φ} on ∂X carrying the tensor product action of Γ and define V_{B}(σ_{λ}, φ) := Γ\(V (σ_{λ}) ⊗ V_{φ})_{|Ω}. Then we have the spaces of sections C^{±∞}(∂X, V (σ_{λ}, φ)) and C^{±∞}(B, V_{B}(σ_{λ}, φ)) as well as the various notions of 𝔞_{ℂ}^{∗}-parametrized families of sections.

When dealing with holomorphic families of vectors in topological vector spaces we will employ the following functional analytic facts. Let F, G, H … be complete locally convex topological vector spaces. A locally convex vector space is called a Montel space if its closed bounded subsets are compact. A Montel space is reflexive, i.e., the canonical map into its bidual is an isomorphism. Moreover, the dual space of a Montel space is again a Montel space ([53], Prop. 36.10.).

Fact 3.1 ([53], Prop. 34.4.) The space of smooth sections of a vector bundle and its topological dual are Montel spaces.

We equip Hom(F, G) with the topology of uniform convergence on bounded sets. Let V ⊂ ℂ be open. A map f : V → Hom(F, G) is called holomorphic if for any z_{0} ∈ V there is a sequence f_{i} ∈ Hom(F, G) such that f(z) = ∑_{n=0}^{∞} f_{i}(z − z_{0})^{i} converges for all z close to z_{0}. Let f : V \ {z_{0}} → Hom(F, G) be holomorphic and f(z) = ∑_{n=−N}^{∞} f_{i}(z − z_{0})^{i} for all z ≠ z_{0} close to z_{0}. Then we say that f is meromorphic and has a pole of order N at z_{0}. If f_{i}, i = −N, …, −1, are finite-dimensional, then f has, by definition, a finite-dimensional singularity. We call a subset A ⊂ F × G′ sufficiently large if for B ∈ Hom(F, G) the condition ⟨ϕ, Bψ⟩ = 0, ∀(ψ, ϕ) ∈ A, implies B = 0. Proofs of the following facts can be found in [13], Section 2.2.

Fact 3.2 The following assertions are equivalent :

1. f : V → Hom(F, G) is holomorphic.

2. f is continuous and there is a sufficiently large subset A ⊂ F × G′ such that for all (ψ, ϕ) ∈ A the function V ∋ z ↦ ⟨ϕ, f(z)ψ⟩ is holomorphic.

Fact 3.3 Let f_{i} : V → Hom(F, G) be a sequence of holomorphic maps. Moreover let f : V → Hom(F, G) be continuous such that for a sufficiently large subset A ⊂ F × G′ the functions ⟨ϕ, f_{i}ψ⟩, (ψ, ϕ) ∈ A, converge locally uniformly in V to ⟨ϕ, fψ⟩. Then f is holomorphic, too.

Fact 3.4 Let f : V → Hom(F, G) be continuous. Then the adjoint f′ : V → Hom(G′, F′) is continuous. If f is holomorphic, then so is f′.

Fact 3.5 Assume that F is a Montel space. Let f : V → Hom(F, G) and f_{1} : V → Hom(G, H) be continuous. Then f_{1} ∘ f : V → Hom(F, H) is continuous. If f, f_{1} are holomorphic, so is f_{1} ∘ f.

Let H be a Hilbert space and F ⊂ H be a Fréchet space such that the embedding is continuous and compact. In the application we have in mind H will be some L^{2}- space of sections of a vector bundle over a compact closed manifold and F be the Fréchet space of smooth sections of this bundle. The continuous maps Hom(H, F) will be called smoothing operators.

Let V ⊂ ℂ be open and connected, and V ∋ z → R(z) ∈ Hom(H, F) be a meromorphic family of smoothing operators with at most finite-dimensional singularities. Note that R(z) is a meromorphic family of compact operators on H in a natural way.

Lemma 3.6 If 1 − R(z) is invertible at some point z ∈ V where R(z) is regular, then

(1 − R(z))^{−1} = 1 − S(z) ,

where V ∋ z → S(z) ∈ Hom(H, F) is a meromorphic family of smoothing operators with at most finite-dimensional singularities.

Proof. We apply Reed-Simon IV [50], Theorem XIII.13 in order to conclude that (1−R(z))^{−1} is a meromorphic family of operators on H having at most finite-dimensional singularities. Making the ansatz (1 − R(z))^{−1} = 1 − S(z), where apriori S(z) is a meromorphic familiy of bounded operators on H with finite-dimensional singularities, we obtain S = −R − R ∘ S. This shows that S is a meromorpic family in Hom(H, F). □

4 Push-down and extension

Distribution sections of V_{B}(σ_{λ}, φ) can be identified with Γ-invariant sections of V (σ_{λ}, φ)_{|Ω}. In order to extend these distributions across the limit set in an invariant way we will construct the extension map

ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) .

We first introduce its ”adjoint” which is the push-down

π_{∗} : C^{∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(B, V_{B}(σ_{λ}, φ)) .

Using the identification C^{∞}(B, V_{B}(σ_{λ}, φ)) = ^{Γ}C^{∞}(Ω, V (σ_{λ}, φ)) we define π_{∗} by

π_{∗}(f)(kM) = ∑_{g∈Γ}(π(g)f)(kM), kM ∈ Ω ,  (7)

if the sum converges. Here π(g) is the action induced by π^{σ,λ}(g) ⊗ φ(g).

Note that the universal enveloping algebra U(𝔤) is a filtered algebra. Let U(𝔤)_{m}, m ∈ ℕ_{0}, be the space of elements of degree less or equal than m. Choose some norm on V_{φ}. It induces a norm |.| on V_{σ} ⊗ V_{φ}. For any m and bounded subset A ⊂ U(𝔤)_{m} we define the seminorm ρ_{m,A} on C^{∞}(∂X, V (σ_{λ}, φ)) by

ρ_{m,A}(f) := sup_{X∈A,k∈K} |f(κ(kX))| ,

where we consider f as a function on K with values in V_{σ} ⊗ V_{φ}. These seminorms define the Fréchet topology of C^{∞}(∂X, V (σ_{λ}, φ)) (in fact a countable set of such seminorms is sufficient).

In order to describe the Fréchet topology on C^{∞}(B, V_{B}(σ_{λ}, φ)) we fix an open cover {U_{α}} of B such that each U_{α} has a diffeomorphic lift Ũ_{α} ⊂ Ω. Then we have canonical isomorphisms

C^{∞}(Ũ_{α}, V (σ_{λ}, φ)) ≅ C^{∞}( U_{α}, V_{B}(σ_{λ}, φ)) .

For any U ∈ {U_{α}} we define the topology of C^{∞}(Ũ, V (σ_{λ}, φ)) using the seminorms

ρ_{U,m,A}(f) := sup_{X∈A,k∈ŨM} |f(κ(kX))| ,

where m ∈ ℕ_{0} and A ⊂ U(𝔤)_{m} is bounded. Since C^{∞}(B, V_{B}(σ_{λ}, φ)) maps to C^{∞}(U_{α}, V_{B}(σ_{λ}, φ)) by restriction for each α we obtain a system of seminorms defining the Fréchet topology of C^{∞}(B, V_{B}(σ_{λ}, φ)).

Since Γ is finitely generated we can find an element μ ∈ 𝔞^{∗}, μ ≥ 0, and a constant C such that

||φ(g)|| ≤ Ca_{g}^{μ} for all g ∈ Γ .  (8)

Definition 4.1 Let δ_{φ} ∈ 𝔞^{∗} be the infimum of all μ ∈ 𝔞^{∗} satisfying Equation (8) for some C. It is independent of the chosen norm. We call δ_{φ} the exponent of (φ, V_{φ}).

Note that unitary representations have zero exponents. If φ̃ is the dual representation of φ, then δφ̃ = δ_{φ}. Furthermore, if φ is the restriction of a finite-dimensional representation of G with highest 𝔞-weight μ, then δ_{φ} = μ.

Lemma 4.2 If Re(λ) < −(δ_{Γ} + δ_{φ}), then the sum (7) converges for f ∈ C^{∞}(∂X, V (σ_{λ}, φ)) and defines a continuous map

π_{∗} : C^{∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(B, V_{B}(σ_{λ}, φ)) .

Moreover, π_{∗} depends holomorphically on λ.

Proof. Consider U ∈ {U_{α}}. We want to estimate

C^{∞}(∂X, V (σ_{λ}, φ)) ∋ f ↦ resŨ ∘ π(g)f ∈ C^{∞}(Ũ, V (σ_{λ}, φ)) .

Let Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) be the coproduct and write Δ(X) = ∑_{i} X_{i} ⊗ Y_{i}. Fix l ∈ ℕ_{0} and a bounded set A ∈ U(𝔤)_{l}. Then for any ϵ > 0 there is another bounded set A_{1} ⊂ U(𝔤)_{l} depending on A such that

ρ_{U,l,A}(res_{ŨM} ∘ π(g)f) = sup_{X∈A,k∈Ũ} |(π(g)f)(κ(kX))|
= sup_{X∈A,k∈ŨM} | ∑_{i} a(g^{−1}κ(kX_{i}))^{λ−ρ}(id ⊗ φ(g))f(κ(g^{−1}κ(kY_{i})))|
≤ a_{g}^{δ_{φ}+ϵ} sup_{X∈A_{1},k∈ŨM} |a(g^{−1}kX)^{λ−ρ}| sup_{X∈A_{1},k∈ŨM} |f(κ(g^{−1}kX))| . (9)

The Poincaré-Birkhoff-Witt theorem gives a decomposition U(𝔤) = U(𝔫̄)U(𝔪)U(𝔞) ⊕ U(𝔤)𝔫. Let q : U(𝔤) → U(𝔫̄)U(𝔪)U(𝔞) be the associated projection. Then for g ∈ G and X ∈ U(𝔤) we have κ(gX) = κ(gq(X)), a(gX) = a(gq(X)).

Let U_{1} ⊂ Ω be an open neighbourhood of Ũ. Then by Lemma 2.1 the intersection Γ ∩ U_{1}MA_{+}K is finite. Let W := (∂X \U_{1})M. As in the proof of Lemma 2.3 we can find a compact A_{+}-invariant set V ⊂ 𝔫̄ such that W^{−1}ŨM ⊂ wκ(V)M. For g = ha_{g}h′ ∈ W A_{+}K and k ∈ ŨM we obtain h^{−1}k = wκ(n̄)m for some n̄ ∈ V, m ∈ M.

Let X ∈ U(𝔤). Then

κ(g^{−1}kX) = κ(h^{′−1}a_{g}^{−1}h^{−1}kX)
= h^{′−1}κ(a_{g}^{−1}wκ(n̄)mX)
= h^{′−1}wκ(a_{g}n̄n(n̄)^{−1}a(n̄)^{−1}mX)
= h^{′−1}wκ(a_{g}n̄a_{g}^{−1}a_{g}[n(n̄)^{−1}a(n̄)^{−1}mXm^{−1}a(n̄)n(n̄)]a_{g}^{−1})m
= h^{′−1}wκ(a_{g}n̄a_{g}^{−1}a_{g}q(n(n̄)^{−1}a(n̄)^{−1}mXm^{−1}a(n̄)n(n̄))a_{g}^{−1})m .

Since V is compact the sets n(V)^{−1}a(V)^{−1}MA_{1}Ma(V)n(V) =: A_{2} ⊂ U(𝔤)_{l} and q(A_{2}) are bounded. Conjugating q(A_{2}) with A_{+} gives clearly another bounded set A_{3} ⊂ U(𝔤)_{l}. We can find a bounded set A_{4} ⊂ U(𝔤)_{l} such that κ(a_{g}n̄a_{g}^{−1}A_{3}) ⊂ κ(κ(a_{g}n̄a_{g}^{−1})A_{4}) for all a_{g} ∈ A_{+}. This implies for g ∈ W A_{+}K that

sup_{X∈A_{1},k∈ŨM} |f(κ(g^{−1}kX))| ≤ ρ_{l,A_{4}}(f) .  (10)

We also have

a(g^{−1}kX) = a(h^{′−1}a_{g}^{−1}h^{−1}kX)
= a(a_{g}^{−1}wκ(n̄)mX)
= a(a_{g}κ(n̄)mXm^{−1})
= a(a_{g}n̄n(n̄)^{−1}a(n̄)^{−1}mXm^{−1})
= a(a_{g}n̄a_{g}^{−1}a_{g}n(n̄)^{−1}a(n̄)^{−1}mXm^{−1}a(n̄)n(n̄)a_{g}^{−1})a(n̄)^{−1}a_{g} .
= a(a_{g}n̄a_{g}^{−1}a_{g}q(n(n̄)^{−1}a(n̄)^{−1}mXm^{−1}a(n̄)n(n̄))a_{g}^{−1})a(n̄)^{−1}a_{g} .

As in the proof of Lemma 2.3 there is a constant C < ∞ such that

|a(a_{g}n̄a_{g}^{−1}a_{g}q(n(n̄)^{−1}a(n̄)^{−1}mXm^{−1}a(n̄)n(n̄))a_{g}^{−1})^{λ−ρ}|a(n̄)^{ρ−λ} < C

for all a_{g} ∈ A_{+}, n̄ ∈ V, m ∈ M, and X ∈ A_{1}. It follows that

sup_{X∈A_{1},k∈ŨM} |a(g^{−1}kX)^{λ−ρ}| ≤ Ca_{g}^{λ−ρ}  (11)

for almost all g ∈ Γ. The estimates (9), (10) and (11) together imply that the sum

C^{l}(∂X, V (σ_{λ}, φ)) ∋ f ↦ ∑_{g∈Γ} resŨ ∘ π(g)f ∈ C^{l}(Ũ, V (σ_{λ}, φ))

converges for Re(λ) < −(δ_{Γ} + δ_{φ}) and defines a continuous map of Banach spaces. This map depends holomorphically on λ by Fact 3.3.

Combining these considerations for all U ∈ {U_{α}} and l ∈ ℕ_{0} we obtain that

π_{∗} : C^{∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(B, V_{B}(σ_{λ}, φ))

is defined and continuous for Re(λ) < −(δ_{Γ} + δ_{φ}). Moreover it is easy to see that π_{∗} depends holomorphically on λ. □

Definition 4.3 For Re(λ) > δ_{Γ} + δ_{φ} we define the extension map

ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

to be the adjoint of

π_{∗} : C^{∗}(∂X, V (σ̃_{−λ}, φ̃)) → C^{∗}(B, V_{B}(σ̃_{−λ}, φ̃)) .

This definition needs a justification. In fact, by Lemma 4.2 the extension exists, is continuous, and by Fact 3.4 it depends holomorphically on λ. Moreover, it is easy to see that the range of the adjoint of π_{∗} consists of Γ-invariant vectors.

We now consider the left-inverse of ext, the restriction

res : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{λ}, φ)) .

The space ^{Γ}C^{−∞}(Ω, V (σ_{λ}, φ)) of Γ-invariant vectors in C^{−∞}(Ω, V (σ_{λ}, φ)) can be canonically identified with the corresponding space C^{−∞}(B, V_{B}(σ_{λ}, φ)). Composing this identification with the restriction res_{Ω} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(Ω, V (σ_{λ}, φ)) we obtain the required restriction map res.

Lemma 4.4 There exists a continuous map

r̃es : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{λ}, φ)) ,

which depends holomorphically on λ and coincides with res on ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)).

Proof. Let φ̃ be the dual representation of φ. We exhibit r̃es as the adjoint of a continuous map

π^{∗} : C^{∞}(B, V_{B}(σ̃_{−λ}, φ̃)) → C^{∞}(∂X, V (σ̃_{−λ}, φ̃))

which depends holomorphically on λ. Then the lemma follows from Fact 3.4.

Let {U_{α}} be a finite open cover of B such that each U_{α} has a diffeomorphic lift Ũ_{α} ⊂ Ω. Choose a subordinated partition of unity {χ_{α}}. Pulling χ_{α} back to Ũ_{α} and extending the resulting function by 0 we obtain a function χ̃_{α} ∈ C^{∞}(∂X). We define

π^{∗}(f) := ∑_{α} χ̃_{α}f, f ∈ C^{∞}(B, V_{B}(σ̃_{−λ}, φ̃)) ,

where we consider f as an element of ^{Γ}C^{−∞}(Ω, V (σ̃_{−λ}, φ̃)). Then we set r̃es := (π^{∗})′. □

Lemma 4.5 We have res ∘ ext = id.

Proof. Recall the definition of π^{∗} from the proof of Lemma 4.4. Then π_{∗}π^{∗} is the identity on C^{∞}(B, V_{B}(σ̃_{−λ}, φ̃)). We obtain

res ∘ ext = r̃es ∘ ext = (π^{∗})′ ∘ (π_{∗})′ = (π_{∗}π^{∗})′ = id .
□

Let C^{−∞}(Λ, V (σ_{λ}, φ)) denote the space of distribution sections of V (σ_{λ}, φ) with support in the limit set Λ. Since Λ is Γ-invariant C^{−∞}(Λ, V (σ_{λ}, φ)) is a subrepresentation of the representation π^{σ,λ} ⊗ φ of Γ on C^{−∞}(∂X, V (σ_{λ}, φ)). Note that ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is exactly the kernel of res.

Lemma 4.6 If ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0 and if ext is defined, then we have ext ∘ res = id.

Proof.The assumption implies that res is injective. By Lemma 4.5 we have res(ext∘res−id) = 0. □

In order to apply this lemma we have to check its assumption. In the course of the paper we will prove several vanishing results for ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). Our first will be

Theorem 4.7 If Re(λ) > δ_{Γ} + δ_{φ}, then ^{Γ}C^{−ω}(Λ, V (σ_{λ}, φ)) = 0.

The proof of this theorem is the first of many instances in this paper where we employ the Poisson transform and its asymptotics.

Let γ be a finite-dimensional representation of K on V_{γ} and T ∈ Hom_{M}(V_{σ}, V_{γ}). We will view sections of V (γ) as functions from G to V_{γ} satisfying the usual K-invariance condition.

Definition 4.8 The Poisson transform

P := P_{λ}^{T} : C^{−∞}(∂X, V (σ_{λ})) → C^{∞}(X, V (γ))

is defined by

(P ϕ)(g) := ∫_{K} a(g^{−1}k)^{−(λ+ρ)}γ(κ(g^{−1}k))T ϕ(k)dk .

Here ϕ ∈ C^{−∞}(∂X, V (σ_{λ})) and the integral is a formal notation meaning that the distribution ϕ has to be applied to the smooth integral kernel.

The theory of the Poisson transform is well-known in the case σ = γ = 1 (see e.g. [51]). The general case has been worked out in [54], [58], [42]. The Poisson transform P intertwines the left-regular representations of G on C^{−∞}(∂X, V (σ_{λ})) and C^{∞}(X, V (γ)). If σ is irreducible, then the image of P consists of joint eigensections with respect to the action of the center Z of the universal enveloping algebra of 𝔤, where Z acts by the infinitesimal character of π^{σ,λ}. For any finite-dimensional representation (φ, V_{φ}) of Γ we denote the tensor product V (γ) ⊗ V_{φ} by V (γ, φ). Then the transformation P_{λ}^{T} ⊗ id, which we will also denote by P, intertwines the Γ-modules C^{−∞}(∂X, V (σ_{λ}, φ)) and C^{∞}(X, V (γ, φ)).

For the proof of Theorem 4.7 we can assume that σ is irreducible. The asymptotic properties of the Poisson transform will be discussed in detail at the beginning of section 6. At this point we need the following two facts. We fix a minimal K-type γ of the principal series representation C^{∞}(∂X, V (σ_{λ})) (see [31], Ch. XV for all that) and choose 0 ≠ T ∈ Hom_{M}(V_{σ}, V_{γ}). Assume that Re(λ) > 0. Then for ϕ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)) and f ∈ C^{∞}(∂X, V ((γ̃_{|M})_{−λ}, φ̃)) we have

lim_{a→∞} a^{ρ−λ} ∫_{K}⟨P ϕ(ka), f(k)⟩ dk = c_{σ}(λ)⟨ϕ,^{t}T f⟩ ,  (12)

where c_{σ}(λ) ≠ 0 (see Corollarry 6.3 below). In particular, P is injective. For any compact U ⊂ K \ supp(ϕ)M there are constants ϵ > 0, C < ∞ such that for all k ∈ U and a ∈ A_{+}

|P ϕ(ka)| ≤ Ca^{Re(λ)−ρ−ϵ} .  (13)

This follows from Lemma 6.2.

We now prove Theorem 4.7 under the additional hypotheses Re(λ) > 0. Choose a minimal K-type γ and an injective T ∈ Hom_{M}(V_{σ}, V_{γ}) as above. Let ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) and f ∈ C^{∞}(∂X, V ((γ̃_{|M})_{−λ}, φ̃)). We extend f to a section f̃ ∈ C^{∞}(X, V (γ̃, φ̃)) by f̃(k_{1}ak_{2}) = χ(a)γ̃(k_{2})^{−1}f(k_{1}), where χ ∈ C^{∞}(A_{+}) is some cut-off function satisfying χ ≡ 0 in a neighbourhood of 1 ∈ A_{+} and χ(a) = 1 for a ≫ 1. Let F be a fundamental domain of the action of Γ on X such that clo(F) ∩ ∂X ⊂ Ω is compact. There is a constant c ∈ ℝ such that

lim_{a→∞} a^{ρ−λ} ∫_{K}⟨P ϕ(ka), f(k)⟩ dk

= c lim_{a→∞} a^{−(λ+ρ)} ∫_{{x∈G | a≤a_{x}≤2a}}⟨P ϕ(x), f̃(x)⟩ dx
= c lim_{a→∞} a^{−(λ+ρ)} ∑_{g∈Γ} ∫_{{x∈G | a≤a_{gx}≤2a}∩F K}⟨(id ⊗ φ(g))P ϕ(x), f̃(gx)⟩ dx .

Choose ϵ > 0 such that the inequality (13) holds and ϵ′ := Re(λ) − (δ_{Γ} + δ_{φ} + 2ϵ) > 0. We now use Corollary 2.4 in order to estimate for g ∈ Γ

|a^{−(λ+ρ)} ∫_{{x∈G | a≤a_{gx}≤2a}∩F K}⟨(id ⊗ φ(g))P ϕ(x), f̃(gx)⟩ dx |
≤ C_{0} sup_{k∈K} |f(k)| a^{−(Re(λ)+ρ)} a_{g}^{δ_{φ}+ϵ} ∫_{{x∈G | a≤a_{g}a_{x}≤C_{1}a}∩F K} |P ϕ(x)| dx
≤ C_{2} a^{−(Re(λ)+ρ)} a_{g}^{δ_{φ}+ϵ} (a_{g}^{−1}a)^{Re(λ)+ρ−ϵ}
= C_{2} a_{g}^{−(δ_{Γ}+ρ+ϵ′)} a^{−ϵ} .

The constants C_{i}, i = 0, 1, 2 are independent of g ∈ Γ. Since ∑_{g∈Γ} a_{g}^{−(δ_{Γ}+ρ+ϵ′)} converges we obtain

|a^{ρ−λ} ∫_{K}⟨P ϕ(ka), f(k)⟩ dk| ≤ C_{3} a^{−ϵ} .

It follows that

lim_{a→∞} a^{ρ−λ} ∫_{K}⟨P ϕ(ka), f(k)⟩ dk = 0 .

Since f was arbitrary and ^{t}T is surjective we conclude from (12) that ϕ = 0.

It remains to consider the case Re(λ) ≤ 0. Let (π_{μ}, W_{μ}) be the irreducible finite-dimensional spherical representation (a representation which contains the trivial K-type) with highest weight μ ∈ 𝔞^{∗}. Then the highest 𝔞-weight space of W_{μ} is P -invariant, and it is isomorphic to V_{1_{−(μ+ρ)}} as a P -module. Hence we have an embedding of P -representations V_{σ_{λ}} ↪ V_{σ_{λ+μ}} ⊗ W_{μ |P}. This leads to an embedding of the G-homogeneous bundles over ∂X

V (σ_{λ}) ↪ V (σ_{λ+μ} ⊗ π_{μ}) ≅ V (σ_{λ+μ}) ⊗ W_{μ}

as well as of the corresponding spaces of sections. Note that G acts on V (σ_{λ+μ}) ⊗ W_{μ} with the tensor product action. Tensoring with V_{φ} we obtain an injective intertwining operator of Γ-modules

i_{μ} : C^{−∞}(∂X, V (σ_{λ}, φ)) ↪ C^{−∞}(∂X, V (σ_{λ+μ}, φ ⊗ π_{μ})) ,  (14)

which is local. Thus, if ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)), then i_{μ}(ϕ) ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ+μ}, φ ⊗ π_{μ})). Moreover, Re(λ + μ) > δ_{Γ} + δ_{φ} + μ = δ_{Γ} + δ_{φ⊗π_{μ}}. Now choose μ large enough such that Re(λ + μ) > 0. It follows from what we have shown above that i_{μ}(ϕ) = 0 and hence ϕ = 0. This finishes the proof of Theorem 4.7.

5 Meromorphic continuation of ext

The extension ext forms a holomorphic family of maps depending on λ ∈ 𝔞_{ℂ}^{∗} (we have omitted this dependence in order to simplify the notation) which is defined now for Re(λ) > δ_{Γ} + δ_{φ}. In the present section we construct the meromorphic continuation of ext to all of 𝔞_{ℂ}^{∗} for X ≠ 𝕆H^{2}. Our main tool is the scattering matrix which we introduce below. The scattering matrix for the trivial group Γ = {1} is the Knapp-Stein intertwining operator of the corresponding principal series representation. We first recall basic properties of the Knapp-Stein intertwining operators. Then we define the scattering matrix using the extension and the Knapp-Stein intertwining operators. We simultaneously obtain the meromorphic continuations of the scattering matrix and the extension map.

If σ is a representation of M, then its Weyl-conjugate σ^{w}, acting on the same vector space V_{σ}, is defined by σ^{w}(m) := σ(w^{−1}mw), where w ∈ N_{K}(𝔞) is a representative of the nontrivial element of the Weyl group W (𝔤, 𝔞) ≅ ℤ_{2}. The Knapp-Stein intertwining operators form meromorphic families of G-equivariant operators (see [32] and Lemmas 5.1 and 5.2 below)

Ĵ_{σ,λ}^{w} : C^{∗}(∂X, V (σ_{λ})) → C^{∗}(∂X, V (σ_{−λ}^{w})), ∗ = −∞, ∞ .

Here ˆ indicates that Ĵ_{σ,λ}^{w} is unnormalized.

In order to fix our conventions we give a definition of Ĵ_{σ,λ}^{w} as an integral operator acting on smooth functions for Re(λ) < 0. Its continuous extension to distributions is obtained by duality. For Re(λ) ≥ 0 it is defined by meromorphic continuation.

Consider f ∈ C^{∞}(∂X, V (σ_{λ})) as a function on G with values in V_{σ_{λ}} satisfying f(gp) = σ_{λ}(p)^{−1}f(g) for all p ∈ P. For Re(λ) < 0 the intertwining operator is defined by the convergent integral

(Ĵ_{σ,λ}^{w}f)(g) := ∫N̄ f(gwn̄)dn̄ .  (15)

Our first goal is to show that the intertwining operators form a meromorphic family of operators in the sense defined in Section 3. It is an easy consequence of the approach to the intertwining operators developed by Vogan-Wallach (see [56], Ch. 10). The only additional point we have to verify is that in the domain of convergence of (15) the operators Ĵ_{σ,λ}^{w} indeed form a continuous family.

Lemma 5.1 For Re(λ) < 0 the intertwining operators

Ĵ_{σ,λ}^{w} : C^{∞}(∂X, V (σ_{λ})) → C^{∞}(∂X, V (σ_{−λ}^{w}))

form a holomorphic family of continuous operators.

20 5 MEROMORPHIC CONTINUATION OF EXT

Proof. Let X_{i}, i = 1, …, dim(𝔨), be an orthonormal base of 𝔨. For any multiindex r = (i_{1}, …, i_{dim(𝔨)}) we set X_{r} = ∏_{l=1}^{dim(𝔨)} X_{l}^{i_{l}}, |r| = ∑_{l=1}^{dim(𝔨)} i_{l}, and for f ∈ C^{∞}(K, V_{σ_{λ}}) we define the seminorm

∥f∥_{r} = sup_{k∈K} |f(X_{r}k)| .

It is well known that the system {∥.∥_{r}}, r running over all multiindices, defines the Fréchet topology of C^{∞}(K, V_{σ_{λ}}) and by restriction the topology of C^{∞}(∂X, V (σ_{λ})).

We extend f ∈ C^{∞}(K, V_{σ_{λ}}) to a function f_{λ} on G by setting f_{λ}(kan) := f(k)a^{λ−ρ}. Then we have

Ĵ_{σ,λ}^{w}(f)(k) = ∫N̄ f_{λ}(kwn̄)dn̄ .

For any λ_{0} ∈ 𝔞_{ℂ}^{∗} with Re(λ) < 0 and δ > 0 we can find an ϵ > 0 such that for |λ − λ_{0}| < ϵ

∫N̄ |a(n̄)^{λ_{0}−ρ} − a(n̄)^{λ−ρ}|dn̄ < δ .

We obtain

∥Ĵ_{σ,λ_{0}}^{w}f − Ĵ_{σ,λ}^{w}f∥_{r} = sup_{k∈K} ∫N̄(f_{λ_{0}}(X_{r}kwn̄) − (f_{λ}(X_{r}kwn̄))dn̄
= sup_{k∈K} ∫N̄ f(X_{r}kwκ(n̄))(a(n̄)^{λ_{0}−ρ} − a(n̄)^{λ−ρ})dn̄
≤ ∥f∥_{r} ∫N̄ |a(n̄)^{λ_{0}−ρ} − a(n̄)^{λ−ρ}|dn̄
≤ δ∥f∥_{r}

This immediately implies that λ ↦ Ĵ_{σ,λ}^{w} is a continuous family of operators on the space of smooth functions. The fact that the family Ĵ_{σ,λ}^{w}, Re(λ) < 0, depends holomorphically on λ is now easy to check (apply [56], Lemma 10.1.3 and Fact 3.2). □

Lemma 5.2 The family of intertwining operators

Ĵ_{σ,λ}^{w} : C^{∞}(∂X, V (σ_{λ})) → C^{∞}(∂X, V (σ_{−λ}^{w}))

extends meromorphically to all of 𝔞_{ℂ}^{∗}.

Proof. We employ [56], Thm. 10.1.5, which states that there are polynomial maps b : 𝔞_{ℂ}^{∗} → ℂ and D : 𝔞_{ℂ}^{∗} → U(𝔤)^{K}, such that

b(λ)Ĵ_{σ,λ}^{w} = Ĵ_{σ,λ−4ρ}^{w} ∘ π^{σ,λ−4ρ}(D(λ)) .  (16)

This formula requires some explanation. We identify

C^{∞}(∂X, V (σ_{λ})) ≅ C^{∞}(K, V_{σ})^{M}

canonically. Then all operators act on the same space C^{∞}(K, V_{σ})^{M}.

If we know that Ĵ_{σ,λ}^{w} is meromorphic up to Re(λ) < μ, then we conclude that

Ĵ_{σ,λ}^{w} = \frac{1}{b(λ)}Ĵ_{σ,λ−4ρ}^{w} ∘ π^{σ,λ−4ρ}(D(λ))

is meromorphic up to Re(λ) < μ + 4ρ. Thus the lemma follows from Lemma 5.1. □

Lemma 5.3 Let χ, ϕ ∈ C^{∞}(∂X) such that supp(ϕ) ∩ supp(χ) = ∅. Then χĴ_{σ,λ}^{w}ϕ is a holomorphic family of smoothing operators. In particular, the residues of Ĵ_{σ,λ}^{w} are differential operators.

Proof. Since supp(ϕ) ∩ supp(χ) = ∅, there exists a compact set V ⊂ N̄ such that

κ(supp(χ)Mw(N̄ \ V))M ⊂ (∂X \ supp(ϕ))M .

For Re(λ) < 0 and f ∈ C^{∞}(∂X, V (σ_{λ})) we have (viewing f as a function on K with values in V_{σ_{λ}})

(χĴ_{σ,λ}^{w}ϕf)(k) = ∫N̄ χ(k)f(κ(kwn̄))ϕ(κ(kwn̄))a(n̄)^{λ−ρ}dn̄
= ∫_{V} χ(k)f(κ(kwn̄))ϕ(κ(kwn̄))a(n̄)^{λ−ρ}dn̄ .

The right-hand side of this equation extends to all of 𝔞_{ℂ}^{∗} and defines a holomorphic family of operators. This proves the first part of the lemma. It in particular implies that the residues of Ĵ_{σ,λ}^{w} are local operators. Hence the second assertion follows. □

We need the following consequence of Lemma 5.3. Let W ⊂ ∂X be a closed subset and let

G_{λ} := {f ∈ C^{−∞}(∂X, V (σ_{λ}))|f_{|∂X\W} ∈ C^{∞}(∂X \ W, V (σ_{λ}))} .

We equip G_{λ} with the weakest topology such that the maps G_{λ} ↪ C^{−∞}(∂X, V (σ_{λ})) and G_{λ} → C^{∞}(∂X \ W, V (σ_{λ})) are continuous. Let U ⊂ Ū ⊂ ∂X \ W be open.

Corollary 5.4 The composition

res_{U} ∘ Ĵ_{σ,λ}^{w} : G_{λ} → C^{∞}(U, V (σ_{−λ}^{w}))

is well-defined and depends meromorphically on λ. □

Below we shall work with a slight modification of Ĵ_{σ,λ}^{w} not depending on the particular representative w and having the intertwining properties we wish.

If σ is equivalent to σ^{w}, then we say that σ is Weyl-invariant. Unless indicated otherwise σ shall always denote a Weyl-invariant representation of M which is either irreducible or of the form σ′ ⊕ σ^{′w} with σ′ irreducible and not Weyl-invariant. In both cases the representation of M on V_{σ} can be extended to a representation of N_{K}(𝔞) which we also denote by σ. This extension is unique up to a character of the Weyl group, i.e., the two possible choices of σ(w) can differ by a sign, only. We fix such an extension and define the G-intertwining operator

Ĵ_{σ,λ} : C^{±∞}(∂X, V (σ_{λ})) → C^{±∞}(∂X, V (σ_{−λ}))

by Ĵ_{σ,λ} := σ(w)Ĵ_{σ,λ}^{w}. Then the operator Ĵ_{σ,λ} does not depend on the choice of w.

In order to define normalized intertwining operators we first have to recall properties of cfunctions and the functional equation of Ĵ_{σ,λ}. Let γ be a finite-dimensional representation of K and let T ∈ Hom_{M}(V_{σ}, V_{γ}). We define T^{w} ∈ Hom_{M}(V_{σ}, V_{γ}) by

T^{w} := γ(w)T σ(w^{−1}) .

T^{w} does not depend on the choice of the representative w. In a similar manner for T ∈ End_{M}(V_{γ}) we define T^{w} := γ(w)T γ(w^{−1}). Let c_{γ} : 𝔞_{ℂ}^{∗} → End_{M}(V_{γ}) be the meromorphic function given for Re(λ) > 0 by

c_{γ}(λ) := ∫N̄ a(n̄)^{−(λ+ρ)}γ(κ(n̄)) dn̄ .  (17)

Lemma 5.5 Let T ∈ Hom_{M}(V_{σ}, V_{γ}) and define T^{♯} ∈ Hom_{K}(C^{−∞}(∂X, V (σ_{λ})), V_{γ}) by T^{♯}(f) := (P_{λ}^{T} f)(1), f ∈ C^{−∞}(∂X, V (σ_{λ})).

1. We have T^{♯} ∘ Ĵ_{σ,λ} = [(c_{γ}(λ)T)^{w}]^{♯}.

2. The Poisson transform satisfies the following functional equation

P_{λ}^{T} ∘ Ĵ_{−λ} = P_{−λ}^{(c_{γ}(λ)T)^{w}} .  (18)

3. There is a meromorphic function p_{σ} : 𝔞_{ℂ}^{∗} → ℂ such that

Ĵ_{σ,−λ} ∘ Ĵ_{σ,λ} = \frac{1}{p_{σ}(λ)}id .  (19)

4. We have

c_{γ}(−λ)^{w}c_{γ}(λ)T = \frac{1}{p_{σ}(λ)}T .  (20)

5. The restriction of Ĵ_{σ̃,λ} to C^{∞}(∂X, V (σ̃_{λ})) coincides with the adjoint of Ĵ_{σ,−λ}.

6. We have

c_{γ}(λ)^{∗} = c_{γ}(λ̄)^{w} .  (21)

Proof. The identity 1. immediately follows from (18). The latter can be read off from the asymptotics of the Poisson transforms (38) and (39). The functional equation

Ĵ_{σ^{w},−λ}^{w^{−1}} ∘ Ĵ_{σ,λ}^{w} = \frac{1}{p_{σ}(λ)}id

can be found in [32]. We compute

Ĵ_{σ,−λ} ∘ Ĵ_{σ,λ} = σ(w) ∘ Ĵ_{σ,−λ}^{w} ∘ σ(w) ∘ Ĵ_{σ,λ}^{w}
= σ(w^{−1}) ∘ Ĵ_{σ,−λ}^{w^{−1}} ∘ σ(w) ∘ Ĵ_{σ,λ}^{w}
= Ĵ_{σ^{w},−λ}^{w^{−1}} ∘ σ(w^{−1}) ∘ σ(w) ∘ Ĵ_{σ,λ}^{w}
= Ĵ_{σ^{w},−λ}^{w^{−1}} ∘ Ĵ_{σ,λ}^{w}
= \frac{1}{p_{σ}(λ)}id .

Equation (20) can be obtained combining (19) with (18). The relation 5. follows from [32], Lemma 24. and (21) is a consequence of 5. and (20). □

Explicit formulas for the Plancherel density p_{σ} can be found e.g. in [32], Ch. 12.

For any (irreducible) σ ∈ M̂ we fix a minimal K-type γ_{σ} ∈ K̂ of π^{σ,λ}. Note that Hom_{M}(V_{σ}, V_{γ_{σ}}) is one-dimensional. Therefore we can define a meromorphic function c_{σ} : 𝔞_{ℂ}^{∗} ↦ ℂ such that

c_{γ_{σ}}(λ)T = c_{σ}(λ) \{ \begin{matrix}T^{w}, & σ = σ^{w} \\ T, & σ ≠ σ^{w} \end{matrix} .  (22)

If σ is of the form σ′ ⊕σ^{′w} for σ′ not Weyl-invariant, then we define c_{σ}(λ) ∈ End_{M}(V_{σ}) such that it acts on the M-isotypic components V_{σ}(σ′) (V_{σ}(σ^{′w})) as multiplication by c_{σ′}(λ) (c_{σ^{′w}}(λ)).

Let now σ be a Weyl-invariant representation of M satisfying our general convention introduced above. We define the normalized intertwining operator by

J_{σ,λ} := Ĵ_{σ,λ}c_{σ}(−λ)^{−1} .

Combining (19) and (20) we obtain the following functional equation:

J_{σ,−λ} ∘ J_{σ,λ} = id .  (23)

We shall often omit the subscript σ in the notation of the intertwining operators. Tensoring with a finite-dimensional representation (φ, V_{φ}) of Γ we obtain a meromorphic-family of Γintertwining operators which we denote by the same symbol

Ĵ_{λ} := Ĵ_{σ,λ} ⊗ id : C^{±∞}(∂X, V (σ_{λ}, φ)) → C^{±∞}(∂X, V (σ_{−λ}, φ))

and its normalized version J_{λ}. Sometimes we are forced to consider also intertwining operators for irreducible, non-Weyl-invariant representations σ. Then Ĵ_{σ,λ} denotes the restriction of Ĵ_{σ⊕σ^{w},λ} to C^{±∞}(∂X, V (σ_{λ}, φ)).

We now turn to the definition of the (normalized) scattering matrix as a family of operators

Ŝ_{λ} : C^{∗}(B, V_{B}(σ_{λ}, φ)) → C^{∗}(B, V_{B}(σ_{−λ}, φ)), ∗ = ∞, −∞
S_{λ} : C^{∗}(B, V_{B}(σ_{λ}, φ)) → C^{∗}(B, V_{B}(σ_{−λ}, φ)) .

Definition 5.6 For Re(λ) > δ_{Γ} + δ_{φ} we define

Ŝ_{λ} := res ∘ Ĵ_{λ} ∘ ext , S_{λ} := res ∘ J_{λ} ∘ ext .  (24)

Lemma 5.7 For Re(λ) > δ_{Γ}+δ_{φ} the scattering matrix forms a meromorphic family of operators

C^{±∞}(B, V_{B}(σ_{λ}, φ)) → C^{±∞}(B, V_{B}(σ_{−λ}, φ)) .

If Ŝ_{λ} is singular and Re(λ) > δ_{Γ} + δ_{φ}, then the residue of Ŝ_{λ} is a differential operator.

Proof. The assertion for the scattering matrix acting on distributions follows from the holomorphy of ext, Lemma 5.2, Lemma 5.3, Lemma 4.4, and Fact 3.5. By Corollary 5.4 it restricts nicely to the space of smooth sections. The last assertion is a consequence of Lemma 5.3. □

Lemma 5.8 If Re(λ) > δ_{Γ} + δ_{φ}, then the adjoint

tS_{λ} : C^{∞}(B, V_{B}(σ̃_{λ}, φ̃)) → C^{∞}(B, V_{B}(σ̃_{−λ}, φ̃))

of

S_{λ} : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{−λ}, φ))

coincides with the restriction of

S_{λ} : C^{−∞}(B, V_{B}(σ̃_{λ}, φ̃)) → C^{−∞}(B, V_{B}(σ̃_{−λ}, φ̃))

to C^{∞}(B, V_{B}(σ̃_{λ}, φ̃)).

Proof. We employ the fact (Lemma 5.5, 5.) that the corresponding relation holds for the intertwining operators (step (25) below). Recall the definition of π^{∗} from the proof of Lemma 4.4. In the domain of convergence of π_{∗} we have

⟨ϕ, π_{∗}(h)⟩ = ∑_{g∈Γ}⟨π^{∗}(ϕ), π(g)h⟩ ,

where ϕ ∈ C^{∞}(B, V_{B}(σ_{λ}, φ)), h ∈ C^{∞}(∂X, V (σ̃_{−λ}, φ̃)), and π(g) = π^{σ̃,−λ}(g) ⊗ φ̃(g). We will use this formula in step (26) below.

Let ϕ ∈ C^{∞}(B, V_{B}(σ_{λ}, φ)), f ∈ C^{∞}(B, V_{B}(σ̃_{λ}, φ̃)), and consider ϕ as a distribution section. Then

⟨ϕ,^{t}S_{λ}f⟩ = ⟨S_{λ}ϕ, f⟩
= ⟨(res ∘ J_{λ} ∘ ext)ϕ, f⟩
= ⟨(r̃es ∘ J_{λ} ∘ ext)ϕ, f⟩
= ⟨(J_{λ} ∘ ext)ϕ, π^{∗}f⟩
= ⟨extϕ, (^{t}J_{λ} ∘ π^{∗})f⟩
= ⟨extϕ, (J_{λ} ∘ π^{∗})f⟩  (25)
= ⟨ϕ, (π_{∗} ∘ J_{λ} ∘ π^{∗})f⟩
= ∑_{g∈Γ}⟨π^{∗}ϕ, (π(g) ∘ J_{λ} ∘ π^{∗})f⟩  (26)
= ∑_{g∈Γ}⟨π^{∗}ϕ, (J_{λ} ∘ π(g) ∘ π^{∗})f⟩
= ∑_{g∈Γ}⟨(π(g) ∘ J_{λ} ∘ π^{∗})ϕ, π^{∗}f⟩  (27)
= ⟨ϕ, S_{λ}f⟩ .  (28)

In order to obtain (28) from (27) we do the transformations backwards with the roles of ϕ and f interchanged. □

Lemma 5.9 If |Re(λ)| < −(δ_{Γ} +δ_{φ}), then the scattering matrix satisfies the functional equation (viewed as an identity of meromorphic families of operators)

S_{−λ} ∘ S_{λ} = id .

Proof. We employ Lemma 4.6, Theorem 4.7, and (23) in order to compute

S_{−λ} ∘ S_{λ} = res ∘ J_{−λ} ∘ ext ∘ res ∘ J_{λ} ∘ ext
= res ∘ J_{−λ} ∘ J_{λ} ∘ ext
= res ∘ ext
= id .
□

The main result of this section is

Theorem 5.10 Let X ≠ 𝕆H^{2}. Then the scattering matrix

S_{λ} : C^{±∞}(B, V_{B}(σ_{λ}, φ)) → C^{±∞}(B, V_{B}(σ_{−λ}, φ))

and the extension map

ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) ,

initially defined for Re(λ) > δ_{Γ} +δ_{φ}, have meromorphic continuations to all of 𝔞_{ℂ}^{∗}. In particular, we have

ext = J_{−λ} ∘ ext ∘ S_{λ} ,  (29)
S_{−λ} ∘ S_{λ} = id .  (30)

Moreover, ext has at most finite-dimensional singularities.

The proof of the theorem will occupy the remainder of this section. We first use the meromorphic Fredholm theory in order to understand a special case. We introduce the element β ∈ 𝔞^{∗} as follows

\begin{matrix}X & ℝH^{n} & ℂH^{n} & ℍH^{n} & 𝕆H^{n} \\ β & 0 & 0 & 2α & 6α \end{matrix} .

Lemma 5.11 If δ_{Γ} + δ_{φ} < 0 and M̂ ∋ σ = 1 is the trivial representation, then S_{λ} and ext have meromorphic continuations to W := {λ ∈ 𝔞_{ℂ}^{∗}|Re(λ) > −ρ+β}. On W the remaining assertions of Theorem 5.10 hold true.

Proof. We construct the meromorphic continuation of

S_{λ} : C^{∞}(B, V_{B}(1_{λ}, φ)) → C^{∞}(B, V_{B}(1_{−λ}, φ)) ,

and then we extend this continuation to distributions by duality using Lemma 5.8. The idea is to set S_{λ} := S_{−λ}^{−1} for Re(λ) < −(δ_{Γ} + δ_{φ}) and to show that S_{−λ}^{−1} forms a meromorphic family on

U := {λ ∈ 𝔞_{ℂ}^{∗} | max{δ_{Γ} + δ_{φ}, −ρ + β} < Re(λ) < ρ − β} .

Let {U_{α}} be a finite open covering of B and let Ũ_{α} be diffeomorphic lifts of U_{α}. Choose a subordinated partition of unity {ϕ_{α}}. Let ϕ̃_{α} be the corresponding compactly supported function on Ũ_{α}. Choose cut-off functions ψ̃_{α} such that supp(ψ̃_{α}) ⊂ Ũ_{α} and supp(1 − ψ̃_{α}) ∩ supp(ϕ̃_{α}) = ∅. Then we define χ ∈ ^{Γ}C^{∞}(Ω × Ω) by

χ(x, y) := ∑_{α} ∑_{g∈Γ} ψ̃_{α}(gx)ϕ̃_{α}(gy) .

Let

J_{λ}^{diag} : C^{∞}(B, V_{B}(1_{λ}, φ)) → C^{∞}(B, V_{B}(1_{−λ}, φ))

be the meromorphic family of operators obtained by multiplying the distribution kernel of Ĵ_{λ} by χ. If f ∈ C^{∞}(B, V_{B}(1_{λ}, φ)), then

(J_{λ}^{diag})f = ∑_{α} ψ̃_{α}J_{λ}(ϕ̃_{α}f)

using the canonical identifications (see the proof of Lemma 4.4).

For λ ∈ U define

R(λ) := J_{−λ}^{diag} ∘ S_{λ} − id .  (31)

The inverse of the normalized scattering matrix for λ ∈ U should be given by

S_{λ}^{−1} = (id + R(λ))^{−1} ∘ J_{−λ}^{diag} .  (32)

It exists as a meromorphic family if (id + R(λ))^{−1} does.

We apply the meromorphic Fredholm theory (Lemma 3.6) in order to invert id + R(λ) for λ ∈ U and to conclude that (id + R(λ))^{−1} is meromorphic.

We check the assumption of Lemma 3.6. We choose a Hermitian metric on V_{B}(1_{0}, φ) and a volume form on B. The Hilbert space H of Proposition 3.6 is L^{2}(B, V_{B}(1_{0}, φ)) defined using these choices. The Fréchet space F is just C^{∞}(B, V_{B}(1_{0}, φ)). Implicitly, we identify the spaces C^{∞}(B, V_{B}(1_{λ}, φ)) with C^{∞}(B, V_{B}(1_{0}, φ)) using a trivialization of the holomorphic family of bundles {V_{B}(1_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}}.

We need the following well-known fact.

Lemma 5.12 If |Re(λ)| < ρ − β, then the spherical intertwining operator

J_{1,λ} : C^{∞}(∂X, V (1_{λ})) → C^{∞}(∂X, V (1_{−λ}))

is regular and invertible. Moreover, J_{1,0} = id.

Proof. If |Re(λ)| < ρ − β, then the principal series representation C^{∞}(∂X, V (1_{λ})) is irreducible (see e.g. [29], Ch. VI, Thm. 3.6). By definition of the normalized intertwining operator its restriction to the minimal K-type is regular and bijective. Now the kernel of the leading term in the Laurent expansion of J_{1,μ} at μ = λ is a subrepresentation of C^{∞}(∂X, V (1_{λ})). It follows that all singular terms in the Laurent expansion vanish and that J_{1,λ} is bijective. By Schur’s Lemma there is μ ∈ ℂ such that J_{1,0} = μ id. Our normalization of J_{1,.} implies that μ = 1. □

It follows that J_{−λ}^{diag} as well as S_{λ} = res ∘ J_{λ} ∘ ext are regular on U. The difference J_{−λ}^{off} := res ∘ J_{−λ} − J_{−λ}^{diag} ∘ res is a continuous map from ^{Γ}C^{−∞}(∂X, V (1_{−λ}, φ)) to C^{∞}(B, V_{B}(1_{λ}, φ)) by Lemma 5.3. Since R(λ) = −J_{−λ}^{off} ∘J_{λ} ∘ext, the family R(λ) is a holomorphic family of smoothing operators on U. In addition, R(0) = 0. Thus id + R(λ) is invertible at λ = 0.

We now have verified the assumptions of Lemma 3.6. We conclude that the family S_{λ}^{−1} is meromorphic family on U with finite-dimensional singularities.

Since δ_{Γ} + δ_{φ} < 0, we have −U ∪ {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > δ_{Γ} + δ_{φ}} = W. Furthermore, by Lemma 5.9 we have S_{λ} = S_{−λ}^{−1} on −U ∩ {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > δ_{Γ} + δ_{φ}}. Thus, setting S_{λ} := S_{−λ}^{−1} for −λ ∈ U we obtain a well-defined continuation of S_{λ} to all of W. By duality this continuation extends to distributions still having finite-dimensional singularities. Moreover, the functional equation (30) holds by definition.

It remains to consider the extension map. We employ the scattering matrix in order to define for λ ∈ W, Re(λ) < −(δ_{Γ} + δ_{φ})

ext_{1} := J_{−λ} ∘ ext ∘ S_{λ} .

We claim that ext = ext_{1}. In fact, since res is injective on {|Re(λ)| < −(δ_{Γ} + δ_{φ})} by Theorem 4.7, the computation

res ∘ ext_{1} = res ∘ J_{−λ} ∘ ext ∘ S_{λ} = S_{−λ} ∘ S_{λ} = id

and Lemma 4.5 imply the claim.

We now have constructed a meromorphic continuation of ext to all of W. The relation (29) between the scattering matrix and ext follows by meromorphic continuation. This equation also implies that ext has at most finite-dimensional singularities. Thus the proof of Lemma 5.11 is complete. □

In the next step we drop the assumption δ_{Γ} + δ_{φ} < 0 employing the fact that for X ≠ 𝕆H^{2} the symmetric space X belongs to a series.

Lemma 5.13 If X ≠ 𝕆H^{2}, then Lemma 5.11 holds true without the assumption δ_{Γ} + δ_{φ} < 0.

Proof. X belongs to a series of rank-one symmetric spaces. First we assume that G^{n} belongs to the list {Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)}. Then we have a sequence of real, semisimple, linear Lie groups … ⊂ G^{n} ⊂ G^{n+1} ⊂ … inducing embeddings of the corresponding Iwasawa constituents K^{n} ⊂ K^{n+1}, N^{n} ⊂ N^{n+1}, M^{n} ⊂ M^{n+1} such that A = A^{n} = A^{n+1}. Furthermore, there are totally geodesic embeddings of the symmetric spaces X^{n} ⊂ X^{n+1} inducing embeddings of their boundaries ∂X^{n} ⊂ ∂X^{n+1}. If Γ ⊂ G^{n} is convex-cocompact then it is still convexcocompact viewed as a subgroup of G^{n+1}. We obtain embeddings Ω^{n} ⊂ Ω^{n+1} inducing B^{n} ⊂ B^{n+1} while the limit set Λ^{n} is identified with Λ^{n+1}. Let ρ^{n}(H) = \frac{1}{2}tr(ad(H)_{|𝔫^{n}}), H ∈ 𝔞.

The exponent of Γ now depends on n and is denoted by δ_{Γ}^{n}. We have the relation δ_{Γ}^{n+1} = δ_{Γ}^{n} − ζ, where ζ := ρ^{n+1} − ρ^{n} > 0. Thus δ_{Γ}^{n+m} → −∞ as m → ∞. Hence, taking m large enough we obtain δ_{Γ}^{n+m} + δ_{φ} < 0. The aim of the following discussion is to show how the meromorphic continuation of ext^{n+1} leads to the continuation of ext^{n}.

Let P^{n} := M^{n}A^{n}N^{n}, V (1_{λ}, φ)^{n} := G^{n} ×_{P^{n}} V_{1_{λ}} ⊗ V_{φ}, and V_{B^{n}}(1_{λ}, φ) = Γ\V (1_{λ}, φ)_{|Ω^{n}}^{n}. Here as always (φ, V_{φ}) is a finite-dimensional representation of Γ. The representation V_{1_{λ}} of P^{n+1} restricts to the representation V_{1_{λ−ζ}} of P^{n}. This induces isomorphisms of bundles

V (1_{λ}, φ)_{|∂X^{n}}^{n+1} ≅ V (1_{λ−ζ}, φ)^{n}, V_{B^{n+1}}(1_{λ}, φ)_{|B^{n}} ≅ V_{B^{n}}(1_{λ−ζ}, φ) .

Let

i^{∗} : C^{∞}(B^{n+1}, V_{B^{n+1}}(1_{λ}, φ̃)) → C^{∞}(B^{n}, V_{B^{n}}(1_{λ−ζ}, φ̃)) ,
j^{∗} : C^{∞}(∂X^{n+1}, V (1_{λ}, φ̃)^{n+1}) → C^{∞}(∂X^{n}, V (1_{λ−ζ}, φ̃)^{n})

denote the maps given by restriction of sections. Note that j^{∗} is G^{n}-equivariant. The adjoint maps define the push-forward of distribution sections

i_{∗} : C^{−∞}(B^{n}, V_{B^{n}}(1_{λ}, φ)) → C^{−∞}(B^{n+1}, V_{B^{n+1}}(1_{λ−ζ}, φ)) ,
j_{∗} : C^{−∞}(∂X^{n}, V (1_{λ}, φ)^{n}) → C^{−∞}(∂X^{n+1}, V (1_{λ−ζ}, φ)^{n+1}) .

If ϕ ∈ C^{−∞}(B^{n}, V_{B^{n}}(1_{λ}, φ)), then the push forward i_{∗}ϕ has support in B^{n} ⊂ B^{n+1}. Since res^{n+1} ∘ ext^{n+1} = id we have supp(ext^{n+1} ∘ i_{∗})(ϕ) ⊂ Λ^{n+1} ∪ Ω^{n} = ∂X^{n}.

Assume that ext^{n+1} is meromorphic on W^{n+1} := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > −ρ^{n+1} + β}. We are now going to continue ext^{n} using i_{∗}, ext^{n+1} and a left inverse of j_{∗}. As in previous occasions we trivialize the family {V (1_{λ})^{n+1}}_{λ}. We identify C^{∞}(∂X^{n+1}, V (1_{λ}, φ̃)^{n+1}) with C^{∞}(∂X^{n+1}) ⊗ Ṽ_{φ} for all λ ∈ ℂ. Let U ⊂ ∂X^{n+1} be a tubular neighbourhood of ∂X and fix a diffeomorphism T : (−1, 1) × ∂X^{n} →^{≅} U. Let χ ∈ C_{c}^{∞}((−1, 1)) be a cut-off function satisfying χ(0) = 1. Then we define a continuous extension t : C^{∞}(∂X^{n}) ⊗ Vφ̃ → C^{∞}(∂X^{n+1}) ⊗ Vφ̃ by

tf(x) := \{ \begin{matrix}χ(r)f(x′) & if x = T (r, x′) ∈ U, & r ∈ (−1, 1), x′ ∈ ∂X^{n} \\ 0 & if x ∉ U \end{matrix} .

Let t′ : C^{−∞}(∂X^{n+1}, V (1_{λ−ζ}, φ)^{n+1}) → C^{−∞}(∂X^{n}, V (1_{λ}, φ)^{n}) be the adjoint of t. Then t′ ∘j_{∗} = id. Now we can define

ẽxt^{n}ϕ := (t′ ∘ ext^{n+1} ∘ i_{∗})(ϕ) .

Then

ẽxt^{n} : C^{−∞}(B^{n}, V_{B^{n}}(1_{λ}, φ)) → C^{−∞}(∂X^{n}, V (1_{λ}, φ)^{n})

is a meromorphic family on W^{n+1} +ζ = W^{n} of continuous maps with at most finite-dimensional singularities.

In order to prove that ẽxt^{n} provides the desired meromorphic continuation it remains to show that it coincides with ext^{n} in the region Re(λ) > δ_{Γ}^{n} + δ_{φ}. If Re(λ) > δ_{Γ}^{n} + δ_{φ}, then Re(λ) − ζ > δ_{Γ}^{n+1} + δ_{φ}, and the push-down maps π_{∗,−λ}^{n}, π_{∗,−λ+ζ}^{n+1} are defined. It is easy to see from the definition of the push-down that in the domain of convergence

i^{∗} ∘ π_{∗}^{n+1} = π_{∗}^{n} ∘ j^{∗} .

Taking adjoints we obtain ext^{n+1} ∘ i_{∗} = j_{∗} ∘ ext^{n}. Therefore we have

ẽxt^{n} = t′ ∘ ext^{n+1} ∘ i_{∗} = t′ ∘ j_{∗} ∘ ext^{n} = ext^{n} .

It follows by meromorphy that im(ẽxt^{n}) consists of Γ-invariant sections for all λ ∈ W^{n}.

We define the meromorphic continuation of the scattering matrix by (24). Then it is easy to see that the scattering matrix has the properties as asserted.

If G^{n} does not belong to the list {Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)}, then there is a finite covering p : G̃^{n} → G^{n} with G̃^{n} ∈ {Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)}. In this case one can find a normal subgroup Γ^{0} ⊂ Γ of finite index and a discrete subgroup Γ̃^{0} ⊂ G̃^{n} such that p induces an isomorphism from Γ̃^{0} to Γ^{0}. Indeed, using Selberg’s Lemma we can take a torsion-free subgroup Γ̃^{0} of p^{−1}(Γ) of finite index and set Γ^{0} := p(Γ̃^{0}). We can apply the concept of embedding to the subgroup Γ̃^{0}. In order to transfer results for Γ̃^{0} to Γ we use averages over the finite group Γ/Γ^{0}. This finishes the proof of Lemma 5.13. □

We now use tensoring with finite-dimensional G-representations in order to complete the proof of Theorem 5.10. For a moment let σ ∈ M̂. Then the theory of highest weights for G implies that there are sequences μ_{i} ∈ 𝔞^{∗}, μ_{i} → ∞ and π_{σ,μ_{i}} of finite-dimensional irreducible representations of G such that π_{σ,μ_{i}} has highest 𝔞-weight μ_{i}, and the representation of M on the corresponding highest weight space is equivalent to σ. More details on this can be found e.g. in [11], pp. 39-41. If σ = σ′ ⊕ σ^{′w} for some non-Weyl-invariant σ′ ∈ M̂, then we set π_{σ,μ_{i}} := π_{σ′,μ_{i}} ⊕π_{σ^{′w},μ_{i}}. As in the proof of Theorem 4.7, Equation (14), we obtain an embedding of bundles

V (σ_{λ}, φ) ↪ V (1_{λ+μ_{i}}, φ ⊗ π_{σ,μ_{i}})  (33)

as well as an injective, local, Γ-intertwining operator

i_{σ,μ_{i}} : C^{−∞}(∂X, V (σ_{λ}, φ)) ↪ C^{−∞}(∂X, V (1_{λ+μ_{i}}, φ ⊗ π_{σ,μ_{i}})) .  (34)

It induces the embedding

i_{σ,μ_{i}}^{B} : C^{−∞}(B, V_{B}(σ_{λ}, φ)) ↪ C^{−∞}(B, V_{B}(1_{λ+μ_{i}}, φ ⊗ π_{σ,μ_{i}})) .

Let i_{σ,μ_{i}}′ and i_{σ,μ_{i}}^{B}′ be the adjoint operators, i.e., the projections onto the spaces smooth sections of the corresponding dual bundles. Then in the domain of convergence of the push-down map Re(λ) < −(δ_{Γ} + δ_{φ}) we have

i_{σ,μ_{i}}^{B}′ ∘ π_{∗} = π_{∗} ∘ i_{σ,μ_{i}}′ .

Note that the domains of convergence of both sides coincide. It follows that for Re(λ) > δ_{Γ} +δ_{φ}, ϕ ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)),

ext ∘ i_{σ,μ_{i}}^{B}(ϕ) = i_{σ,μ_{i}} ∘ ext(ϕ) .  (35)

Now let ν ∈ 𝔞∗ be arbitrary, and let W_{ν} := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > ν} be the corresponding half-plane. Choose μ_{i} large enough such that Re(λ) + μ_{i} > −ρ + β for all λ ∈ W_{ν}. By Lemma 5.13 the extension map on the left hand side of (35) has a meromorphic continuation to W_{ν} with finite-dimensional singularities. Moreover,

ext ∘ i_{σ,μ_{i}}^{B} (C^{−∞}(B, V_{B}(σ_{λ}, φ))) ⊂ i_{σ,μ_{i}} (C^{−∞}(∂X, V (σ_{λ}, φ))) .

In fact, this is true for λ in the domain of convergence by (35), hence on all of W_{ν} by meromorphy. Therefore we can define for ϕ ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)), λ ∈ W_{ν}

ext(ϕ) := (i_{σ,μ_{i}})^{−1} ∘ ext ∘ i_{σ,μ_{i}}^{B}(ϕ) .

This gives the desired meromorphic continuation of ext to W_{ν}. Now (35) holds on all of W_{ν}. It follows that the singularities of ext are at most finite-dimensional. Since ν was arbitrary ext is meromorphic on all of 𝔞_{ℂ}^{∗}.

We define the meromorphic continuation of the scattering matrix by (24). Using Lemma 4.6 and Theorem 4.7 it is easy to verify the functional equations (30) and (29) for Re(λ) < −(δ_{Γ}+δ_{φ}). By meromorphy they hold on all of 𝔞_{ℂ}^{∗}. This finishes the proof of Theorem 5.10. □

6 Invariant distributions on the limit set

In the present section we study the spaces ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) of invariant distributions which are supported on the limit set. The main result of this section is

Theorem 6.1 Let X ≠ 𝕆H^{2}. Fix finite-dimensional representations σ of M and φ of Γ. Then

1. The set {λ ∈ 𝔞_{ℂ}^{∗} |^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0} is discrete.

2. For each λ ∈ 𝔞_{ℂ}^{∗} the space ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is finite-dimensional.

The proof is based on the following observation: Assume that ext is singular at λ ∈ 𝔞_{ℂ}^{∗}. Since res∘ext(ϕ_{μ}) = ϕ_{μ} is regular for any holomorphic family μ ↦ ϕ_{μ} ∈ C^{−∞}(B, V_{B}(σ_{μ}, φ)) the leading singular part of the Laurent expansion of ext(ϕ_{μ}) at μ = λ belongs to^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). We show that for almost all λ these spaces are generated by the singular parts of ext.

First we need detailed knowledge of the asymptotics of the Poisson tansform.

Lemma 6.2 Let γ be a finite-dimensional representation of K, T ∈ Hom_{M}(V_{σ}, V_{γ}), and let w ∈ N_{K}(𝔞) represent the non-trivial element of the Weyl group W (𝔤, 𝔞).

1. Let f ∈ C^{∞}(∂X, V (σ_{λ})). If Re(λ) > 0, then there exists ϵ > 0 (depending on λ but not on f) such that for a → ∞ we have

(P_{λ}^{T} f)(ka) = a^{λ−ρ}c_{γ}(λ)T f(k) + O(a^{Re(λ)−ρ−ϵ})

uniformly in k ∈ K. If |Re(λ)| < \frac{1}{2}|α| and λ ≠ 0, then

(P_{λ}^{T} f)(ka) = a^{λ−ρ}c_{γ}(λ)T f(k) + a^{−λ−ρ}T^{w}(Ĵ_{λ}f)(k) + O(a^{− \frac{α}{2} −ρ−ϵ})  (36)

uniformly in k ∈ K. The remainder depends jointly continuously on λ and f.

If Ĵ_{0} is regular, then (36) remains valid for λ = 0. Otherwise we have

(P_{0}^{T} f)(ka) = log(a)a^{−ρ} (2res_{λ=0}(c_{γ}(λ))T f(k))
+a^{−ρ} (c_{γ}^{0}T f(k) + T^{w}(Ĵ_{0}^{0}f)(k)) + O(a^{−ρ−α}) ,

where c_{γ}^{0}, Ĵ_{0}^{0} denote the constant terms in the Laurent expansions of c_{γ} and Ĵ_{λ} at λ = 0.

2. Let ∂X = U ∪ Q, where U is open and Q := ∂X \ U. Let f ∈ C^{−∞}(∂X, V (σ_{λ})) with suppf ⊂ Q. Then there exist smooth functions ψ_{n}, n ∈ ℕ, on U such that

(P_{λ}^{T} f)(ka) = a^{−(λ+ρ)}T^{w}(Ĵ_{λ}f)(k) + ∑_{n≥1} a^{−(λ+ρ)−nα}ψ_{n}(k) , k ∈ U .  (37)

The series converges uniformly for a ≫ 0 and k in compact subsets of U. In particular, for a → ∞ we have

(P_{λ}^{T} f)(ka) = a^{−λ−ρ}T^{w}(Ĵ_{λ}f)(k) + O(a^{−λ−ρ−α})

uniformly as kM varies in compact subsets of U.

3. Let U, Q be as in 2. and f ∈ C^{−∞}(∂X, V (σ_{λ})) such that res_{U}f ∈ C^{∞}(U, V (σ_{λ})). If Re(λ) > 0, then there exists ϵ > 0 such that for a → ∞ we have

(P_{λ}^{T} f)(ka) = a^{λ−ρ}c_{γ}(λ)T f(k) + O(a^{λ−ρ−ϵ})

uniformly as kM varies in compact subsets of U. If |Re(λ)| < \frac{1}{2}|α|, λ ≠ 0, then we have for a → ∞

(P_{λ}^{T} f)(ka) = a^{λ−ρ}c_{γ}(λ)T f(k) + a^{−λ−ρ}T^{w}(Ĵ_{λ}f)(k) + O(a^{− \frac{α}{2} −ρ−ϵ})

uniformly as kM varies in compact subsets of U. The remainder depends jointly continuously on λ and f.

The asymptotic expansions can be differentiated with respect to a.

Proof. Assertion 1 is a consequence of the general results concerning the asymptotics of matrix coefficients of admissible representations including their dependence on parameters ([55], Thm. 4.4.3, [56], 12.4., 12.5., 12.6., [42]) combined with the limit formulas for the Poisson transform (see [54] or [42], also [55], Thm. 5.3.4)

lim_{a→∞} a^{ρ−λ}(P_{λ}^{T} f)(ka) = c_{γ}(λ)T f(k),  Re(λ) > 0 ,  (38)
lim_{a→∞} a^{ρ+λ}(P_{λ}^{T} f)(ka) = T^{w}(Ĵ_{λ}f)(k),  Re(λ) < 0 .  (39)

3. is a consequence of 1. and 2.. Indeed let W, W_{1} ⊂ U be a compact subsets such that W ⊂ int(W_{1}). Let χ ∈ C_{c}^{∞}(U) be such that χ_{|W_{1}} ≡ 1. Then we can write f = χf + (1 − χ)f, where χf is smooth and supp(1 − χ)f ⊂ ∂X \ int(W_{1}). We now apply 1. to χf and 2. to (1 − χ)f for kM ∈ W.

It remains to prove assertion 2. We imitate the argument of [6], Thm. 4.8. which proves the assertion for the case σ = 1.

Let w ∈ N_{K}(𝔞) be a representative of the non-trivial element of W (𝔤, 𝔞). In the following computation we write the pairing of a distribution with a smooth function as an integral.

(P_{λ}^{T} f)(ka) = ∫_{K} a(a^{−1}k^{−1}h)^{−(λ+ρ)}γ(κ(a^{−1}k^{−1}h))T f(h)dh
= ∫N̄ a(a^{−1}wκ(n̄))^{−(λ+ρ)}γ(κ(a^{−1}wκ(n̄)))T f(kwκ(n̄))a(n̄)^{−2ρ}dn̄
= ∫N̄ a(an̄a^{−1})^{−(λ+ρ)}a^{−(λ+ρ)}a(n̄)^{λ+ρ}γ(w)γ(κ(an̄a^{−1}))T f(kwκ(n̄))a(n̄)^{−2ρ}dn̄
= a^{−(λ+ρ)}γ(w) ∫N̄ a(n̄)^{λ−ρ}a(an̄a^{−1})^{−(λ+ρ)}γ(κ(an̄a^{−1}))T f(kwκ(n̄))dn̄ . (40)

For z ∈ ℝ^{+} define a_{z} ∈ A by z = a_{z}^{−α}. We consider the map Φ : (0, ∞) × N̄ ∋ (z, n̄) ↦ a_{z}n̄a_{z}^{−1} ∈ N̄ which according to the decomposition of 𝔫 into root spaces 𝔫 = 𝔫_{α} ⊕ 𝔫_{2α} can also be written as

Φ(z, exp(X + Y)) := exp(zX + z^{2}Y), X ∈ 𝔫_{α}, Y ∈ 𝔫_{2α} .

Thus Φ and hence (z, n̄) ↦ a(a_{z}n̄a_{z}^{−1})^{−(λ+ρ)}γ(κ(a_{z}n̄a_{z}^{−1})) extend analytically to ℝ × N̄. Taking the Taylor expansion with respect to z at z = 0 we obtain

a(a_{z}n̄a_{z}^{−1})^{−(λ+ρ)}γ(κ(a_{z}n̄a_{z}^{−1})) = id + ∑_{n≥1} A_{n}(n̄)z^{n} .

Here A_{n} : N̄ → End(V_{γ}) are analytic and the series converges in the spaces of smooth functions on N̄ with values in End(V_{γ}).

Inserting this expansion into (40) we obtain

(P_{λ}^{T} f)(ka) = a^{−(λ+ρ)}T^{w}(Ĵ_{λ}f)(k) + ∑_{n≥1} a^{−(λ+ρ)−nα}ψ_{n}(k) ,

where

ψ_{n}(k) := γ(w) ∫N̄ A_{n}(n̄)T a(n̄)^{λ−ρ}f(kwκ(n̄))dn̄ .

Note that k ↦ f(kwκ(.)) is a smooth family of distributions with compact support in N̄. Thus ψ_{n} is smooth. This finishes the proof of the lemma. □

Corollary 6.3 Let ϕ ∈ C^{−∞}(∂X, V (σ_{λ})) and f ∈ C^{∞}(∂X, V ((γ̃_{|M})_{−λ})). If Re(λ) > 0, then there exists ϵ > 0 such that for a → ∞ we have

∫_{K}⟨P_{λ}^{T} ϕ(ka), f(k)⟩ dk = a^{λ−ρ}⟨ϕ,^{t}(c_{γ}(λ)T)f⟩ + O(a^{λ−ρ−ϵ}) .

There are corresponding formulas for Re(λ) = 0.

Proof.The argument is adapted from [51], 5.1. Let v_{i} be a basis of Vγ̃, and write f(k) = ∑ f_{i}(k)v_{i}. Defineˇ : C^{∞}(K) → C^{∞}(K) by ȟ(k) := h(k^{−1}). Let ∗ be the convolution on the group K. Then we have

∫_{K}⟨P_{λ}^{T} ϕ(ka), f(k)⟩ dk = ∑_{i}⟨v_{i}, ∫_{K} P_{λ}^{T} ϕ(ka)f_{i}(k) dk⟩
= ∑_{i}⟨v_{i}, (f̌_{i} ∗ P_{λ}^{T} ϕ(.a))(1)⟩
= ∑_{i}⟨v_{i}, P_{λ}^{T} (f̌_{i} ∗ ϕ)(a)⟩ .

Now observe that f̌_{i} ∗ ϕ ∈ C^{∞}(∂X, V (σ_{λ})) and apply Lemma 6.2. □

Now we show a variant of Green’s formula. We need nice cut-off functions which exist by the following lemma. Let Δ_{X} be the Laplace-Beltrami operator of X.

Lemma 6.4 There exists a cut-off function χ ∈ C_{c}^{∞}(X ∪ Ω) such that

1. ∑_{g∈Γ} g^{∗}χ = 1,

2. sup_{gK∈X} a_{g}^{α} |dχ(gK)| < ∞,

3. sup_{gK∈X} a_{g}^{α} |Δ_{X}χ(gK)| < ∞,

4. sup_{gK∈X} |χ(DgK)| < ∞, ∀D ∈ U(𝔤).

We denote the restriction of χ to Ω by χ_{∞}.

Proof. Let W ⊂ X ∪ Ω be a compact subset such that ⋃_{g∈Γ} gW = X ∪ Ω. Then we choose a cut-off function ψ ∈ C_{c}^{∞}(X ∪ Ω) such that ψ_{|W} = 1. We define

χ := \frac{ψ}{∑_{g∈Γ} g^{∗}ψ} .

Note that χ is well-defined since ∑_{g∈Γ} g^{∗}ψ never vanishes on X ∪ Ω. Property 1 is obvious by the definition of χ.

In the following we consider χ as an element of C^{∞}(X̄). Since G acts smoothly on the compact manifold X̄ we have L^{♯} ∈ C^{∞}(X̄, T X̄), where L^{♯} denotes the fundamental vector field corresponding to L ∈ 𝔤. If Y ∈ C^{∞}(X̄, T X̄), then Y (C^{∞}(X̄)) ⊂ C^{∞}(X̄). Thus the left action of 𝔤 and hence of U(𝔤) preserves C^{∞}(X̄). This shows assertion 4.

We have dχ ∈ C^{∞}(X̄, T^{∗}X̄). The tensor field in C^{∞}(X, S^{2}T X) which is dual to the Riemannian metric of X has a continuous extension to X̄ vanishing of second order at ∂X. We conclude that |dχ| vanishes of first order at the boundary of X̄. This shows assertion 2.

Since the coefficients of Δ_{X} vanish at ∂X of at least first order assertion 3 follows. □

Let ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). Then res ∘ Ĵ_{λ}(ϕ) ∈ C^{∞}(B, V_{B}(σ_{−λ}, φ)) is well-defined even if Ĵ_{μ} has a pole at μ = λ. In the latter case the residue of Ĵ_{μ} at μ = λ is a differential operator D_{λ} (see Lemma 5.3) and res ∘ D_{λ}(ϕ) = 0.

Proposition 6.5 If ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) and f ∈ ^{Γ}C^{−∞}(∂X, V (σ̃_{λ}, φ̃)), then

⟨res ∘ Ĵ_{λ}(ϕ), res(f)⟩ = 0 .

Proof. At first we need

Lemma 6.6 The space

ΓC_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)) := {f ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) | f_{|Ω} ∈ C^{∞}(Ω, V (σ_{λ}, φ))}

is dense in ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)).

Proof. By Theorem 5.10 ext has an at most finite-dimensional singularity at λ. Thus there is a finite-dimensional subspace W ⊂ C^{∞}(B, V_{B}(σ̃_{−λ}, φ̃)) such that

ext_{|W^{⊥}} : W^{⊥} → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

is a well-defined continuous map, where W^{⊥} := {ϕ ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)) | ⟨ϕ, W ⟩ = {0}}. Since C^{∞}(B, V_{B}(σ_{λ}, φ)) ⊂ C^{−∞}(B, V_{B}(σ_{λ}, φ)) is dense we can choose a complement W̃ ⊂ C^{∞}(B, V_{B}(σ_{λ}, φ)) such that C^{−∞}(B, V_{B}(σ_{λ}, φ)) = W^{⊥} ⊕ W̃.

Let f ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)). Then we can write res(f) = g = g^{⊥} ⊕ g̃, g^{⊥} ∈ W^{⊥}, g̃ ∈ W̃. Now res(f − ext(g^{⊥})) = g̃. It follows that f − ext(g^{⊥}) ∈ ^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)). Let now g_{i} ∈ C^{∞}(B, V_{B}(σ_{λ}, φ)) be a sequence such that lim_{i→∞} g_{i} = g. Then we can decompose g_{i} = g_{i}^{⊥} + g̃_{i}. The sections g_{i}^{⊥} are smooth since g_{i} and g̃_{i} ∈ W̃ are so. It follows that ext(g_{i}^{⊥}) ∈ ^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)). By continuity of ext_{|W^{⊥}} we have ext(g^{⊥}) = lim_{i→∞} ext(g_{i}^{⊥}). The assertion of the lemma now follows from f = f − ext(g^{⊥}) + lim_{i→∞} ext(g_{i}^{⊥}). □

We now prove Proposition 6.5 for the case σ = 1, Re(λ) > 0. We consider the Poisson transforms, both denoted by P,

P = P_{λ} ⊗ id : C^{−∞}(Λ, V (1_{λ}, φ)) → C^{∞}(X, V (1, φ))

and

P = P_{λ} ⊗ id : C^{−∞}(Λ, V (1_{λ}, φ̃)) → C^{∞}(X, V (1, φ̃)) .

Let D := Δ_{X} ⊗ id − ρ^{2} + λ^{2} be the shifted Laplace operator acting on C^{∞}(X, V (1, φ)) and C^{∞}(X, V (1, φ̃)), respectively. Then we have D ∘ P = 0. Let χ be a cut-off function as in Lemma 6.4. By B_{R} we denote the metric R-ball centered at the origin of X. The following is an application of Green’s formula:

0 = ∫_{B_{R}} χ(x) (⟨DP ϕ(x), P f(x)⟩ − ⟨P ϕ(x), DP f(x)⟩) dx
= ∫_{B_{R}} (⟨Dχ(x)P ϕ(x), P f(x)⟩ − ⟨χ(x)P ϕ(x), DP f(x)⟩ − ⟨[D, χ]P ϕ(x), P f(x)⟩) dx
= − ∫_{∂B_{R}} (⟨∇_{n}χ(y)P ϕ(y), P f(y)⟩ − ⟨χ(y)P ϕ(y), ∇_{n}P f(y)⟩) dy
− ∫_{B_{R}}⟨[D, χ]P ϕ(x), P f(x)⟩ dx ,  (41)

where n is the exterior unit normal vector field at ∂B_{R}.

By Lemma 6.6 we can assume that f_{|Ω} is smooth. Then Lemma 6.2, 2. and 3. combined with properties 2 and 3 of χ implies that |⟨[D, χ]P ϕ, P f⟩| is integrable over all of X. From Lemma 6.4, property 1, and the Γ-invariance of ⟨P ϕ, P f⟩ it follows that ∫_{X}⟨[D, χ]P ϕ, P f⟩ dx = 0. Taking the limit R → ∞ in (41) we obtain by Lemma 6.2, 2. and 3.

0 = (λ + ρ) ∫_{∂X} χ_{∞}(k)⟨(Ĵ_{λ}ϕ)(k), c_{1}(λ)f(k)⟩ dk
+(λ − ρ) ∫_{∂X} χ_{∞}(k)⟨(Ĵ_{λ}ϕ)(k), c_{1}(λ)f(k)⟩ dk
= 2λc_{1}(λ) ∫_{∂X} χ_{∞}(k)⟨(Ĵ_{λ}ϕ)(k), f(k)⟩ dk
= 2λc_{1}(λ)⟨res ∘ Ĵ_{λ}(ϕ), res(f)⟩ .

This is the assertion of the proposition in our special case since c_{1}(λ) ≠ 0 for Re(λ) > 0.

Note that almost the same proof would also work for general σ and Re(λ) ≥ 0. But for Re(λ) < 0 we have to employ tensoring with finite-dimensional representations, anyway. This method will reduce matters to the case σ = 1, Re(λ) > 0 treated above. As in the proof of Theorem 5.10 we consider the finite-dimensional representation π_{σ,μ} of G and the embedding

i_{σ,μ} : C^{−∞}(∂X, V (σ_{λ}, φ)) ↪ C^{−∞}(∂X, V (1_{λ+μ}, φ ⊗ π_{σ,μ})) .

In addition, the projection onto the lowest weight space of π_{σ,μ} induces a surjection

p_{σ,μ} : C^{−∞}(∂X, V (1_{−λ−μ}, φ ⊗ π_{σ,μ})) → C^{−∞}(∂X, V (σ_{−λ}, φ)) .

Then we have the following identity of meromorphic families of operators (see [56], 10.2.6)

Ĵ_{σ,λ} ⊗ id_{V_{φ}} = p_{σ,μ} ∘ (Ĵ_{1,λ+μ} ⊗ id_{V_{φ}⊗V_{πσ,μ}} ) ∘ i_{σ,μ} .  (42)

We also consider the induced operators on B denoted by i_{σ,μ}^{B} and p_{σ,μ}^{B}. Now let ϕ and f be as in the statement of the proposition. Choose μ ∈ 𝔞^{∗} large enough such that π_{σ,μ} exists and Re(λ + μ) > 0. Then we have by the first part of the proof

⟨res ∘ Ĵ_{λ}(ϕ), res(f)⟩ = ⟨res ∘ p_{σ,μ} ∘ Ĵ_{λ+μ} ∘ i_{σ,μ}(ϕ), res(f)⟩
= ⟨p_{σ,μ}^{B} ∘ res ∘ Ĵ_{λ+μ} ∘ i_{σ,μ}(ϕ), res(f)⟩
= ⟨res ∘ Ĵ_{λ+μ} ∘ i_{σ,μ}(ϕ), i_{σ̃,μ}^{B} ∘ res(f)⟩
= ⟨res ∘ Ĵ_{λ+μ} ∘ i_{σ,μ}(ϕ), res ∘ i_{σ̃,μ}(f)⟩
= 0 .

This proves the proposition. □

Let 𝔪 denote the Lie algebra of M. We choose a Cartan subalgebra 𝔱 of 𝔪. Then 𝔱 ⊕ 𝔞 =: 𝔥 is a Cartan algebra of 𝔤. Via the Harish-Chandra isomorphism characters of Z are parametrized by elements of 𝔥_{ℂ}^{∗}/W (𝔤_{ℂ}, 𝔥_{ℂ}), where W (𝔤_{ℂ}, 𝔥_{ℂ}) is the Weyl group of (𝔤_{ℂ}, 𝔥_{ℂ}). A character χ_{ν}, ν ∈ 𝔥_{ℂ}^{∗}, is called integral, if

2\frac{⟨ν, ε⟩}{⟨ε, ε⟩} ∈ ℤ  (43)

for all roots ε of (𝔤_{ℂ}, 𝔥_{ℂ}).

We further choose a positive root system of (𝔪_{ℂ}, 𝔱_{C}). Let ρ_{m} denote half of the sum of these positive roots. For σ ∈ M̂ let μ_{σ} ∈ i𝔱^{∗} be its highest weight. The infinitesimal character of the principal series representation π^{σ,λ} of G, σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗}, is now given by χ_{μ_{σ}+ρ_{m}−λ}. If X ≠ ℝH^{2}, then for σ ∈ M̂ we define the lattice

I_{σ} := {λ ∈ 𝔞^{∗} | χ_{μ_{σ}+ρ_{m}−λ} is integral } .

If X = ℝH^{2} and G = SL(2, ℝ), then M ≅ ℤ_{2}. Let ±1 denote the trivial (+), resp. non-trivial (-) irreducible representation of M. We define

I_{1} := (\frac{1}{2} + ℤ)α, I_{−1} := ℤα .

If G = P SL(2, ℝ), then M = \{1\}, and we define I_{1} := (\frac{1}{2} + ℤ)α.

If λ ∉ I_{σ}, then the principal series representation π^{σ,λ} is irreducible (see [17], 4.3.3). Note that I_{σ^{w}} = I_{σ}, so the definition is compatible with our previous convention concerning the Weylinvariance of σ. Let I_{𝔞} ⊂ 𝔞^{∗} be the ℤ-module generated by the short root α, if 2α is a root, or by α/2, if not. Then all poles of Ĵ_{λ} are located in I_{𝔞} ([32], Thm. 3 and Prop. 43). Note that for any σ we have that I_{σ} ⊂ I_{𝔞} is a sublattice of index 2.

Lemma 6.7 Assume that σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗} satisfy one of the following conditions

1. Re(λ) ≥ 0 and σ = 1.

2. Re(λ) ≥ 0, G = SL(2, ℝ) and σ = −1.

3. Re(λ) ≥ 0 and λ ∉ I_{σ}.

4. Re(λ) < 0 and λ ∉ I_{𝔞}.

Let U ⊂ ∂X be a non-empty open subset, and let ϕ ∈ C^{−∞}(∂X, V (σ_{λ})) be such that ϕ_{|U} = 0 and (Ĵ_{σ,λ}ϕ)_{|U} = 0. Then ϕ = 0.

Before turning to the proof we recall that Lemma 5.3 implies that (Ĵ_{σ,λ}ϕ)_{|U} is well-defined even if Ĵ_{σ,λ} has a pole.

Proof. We modify an argument given by van den Ban-Schlichtkrull [6] for the case σ = 1.

Assume that (σ, λ) satisfies condition 4. Since Re(λ) < 0, the operator Ĵ_{σ,λ} is regular and non-vanishing. Since λ ∉ I_{𝔞} ⊃ I_{σ} the principal series representation π^{σ,λ} is irreducible. This implies that Ĵ_{σ,λ} is bijective. Moreover, Ĵ_{σ,−λ} is regular, since −λ ∉ I_{𝔞}. Thus, by the functional equation (23) we can reduce the proof to case 1, 2 or 3 replacing ϕ by Ĵ_{σ,λ}(ϕ).

We now assume that Re(λ) ≥ 0, σ = 1 and ϕ ≢ 0. Then the Poisson transform P_{λ}ϕ ∈ C^{∞}(X) does not vanish and is annihilated by D := Δ_{X} − ρ^{2} + λ^{2}. Without loss of generality we can assume that M ∈ U. Since P_{λ}ϕ is real analytic and not identically zero the expansion (37) has non-trivial terms. Let m be the smallest integer such that ψ_{m} ≢ 0 near M, where we set ψ_{0} := Ĵ_{1,λ}ϕ. We have to show that m = 0.

With respect to the coordinates k, a the operator D has the form D = D_{0} + a^{−2α}R(a, k), where D_{0} is a constant coefficient operator on A and R is a differential operator with coefficients which remain bounded if a → ∞ (see [28], Ch. IV, §5, (8)). Moreover, it is known that D_{0} coincides with the N̄-radial part of D.

We consider the N̄-invariant function f ∈ C^{∞}(X) defined by f(n̄a) := a^{−(λ+ρ+mα)}. Since D annihilates the asymptotic expansion (37) we have Df = D_{0}f = 0. On the other hand, f satisfies (Δ_{X} − ρ^{2} + (λ + mα)^{2})f = 0. Hence (λ + mα)^{2} = λ^{2}. Since Re(λ) ≥ 0 we conclude that m = 0.

The proof for general σ proceeds similarly. Let Re(λ) ≥ 0 and λ ∉ I_{σ}. In particular, the principal series representation π^{σ,λ} is irreducible. We choose 0 ≠ T ∈ Hom_{M}(V_{σ}, V_{γ}) for a suitable γ ∈ K̂. Then P := P_{λ}^{T} is injective, and the range of P can be identified with the kernel of a certain invariant differential operator D : C^{∞}(X, V (γ)) → C^{∞}(X, V (γ′)) for some representation γ′ of K (see [10], Sec.3).

We now assume ϕ ≢ 0. Moreover, without loss of generality we can assume that M ∈ U. Since P ϕ is real analytic, the expansion (37) does not vanish. Let m be the smallest integer such that ψ_{m} ≢ 0 near M (where ψ_{0} := T^{w}Ĵ_{σ,λ}ϕ). Again, D = D_{0} + a^{−α}R(a, k), where D_{0} is the constant coefficient operator on A given by the N̄-radial part of D, and R remains bounded if a → ∞ (see [57], Thm. 9.1.2.4).

Choose k ∈ K near 1 and σ′ ⊂ γ_{|M} such that that there exists an orthogonal projection S ∈ Hom_{M}(V_{γ}, V_{σ′}) with Sγ(k)ψ_{m}(k) =: v ≠ 0. Consider the N̄-invariant section f ∈ C^{∞}(X, V (γ)) defined by

f(n̄a) := a^{−(λ+ρ+mα_{1})}S^{∗}v .

Since D annihilates the asymptotic expansion (37), one can check that Df = D_{0}f = 0 and thus f = P ϕ_{1} for some N̄-invariant ϕ_{1} ∈ C^{−∞}(∂X, V (σ_{λ})).

Now f = P_{λ+mα_{1}}^{S}δv, where δv ∈ C^{−∞}(∂X, V (σ′_{λ+mα_{1}})) is the delta distribution at 1 with vector part v. Since D and P_{λ+mα_{1}}^{S} are G-equivariant and δv generates the G-module C^{−∞}(∂X, V (σ′_{λ+mα})), we obtain a non-trivial intertwining operator I from C^{−∞}(∂X, V (σ′_{λ+mα})) to the kernel of D, hence to C^{−∞}(∂X, V (σ_{λ})). This implies

χ_{μ_{σ}+ρ_{m}−λ} = χ_{μ_{σ′}+ρ_{m}−λ−mα} .  (44)

We conclude that χ_{μ_{σ′}+ρ_{m}−λ−mα} is not integral and thus π^{σ′,λ+mα} is irreducible. Hence I is an isomorphism. Counting K-types one finds that σ = σ′ or σ = σ^{′w}. It follows that |μ_{σ} + ρ_{m}|^{2} = |μ_{σ′} + ρ_{m}|^{2}, and hence λ^{2} = (λ + mα)^{2}. The condition Re(λ) ≥ 0 implies that m = 0. Hence (Ĵ_{σ,λ}ϕ)_{|U} ≠ 0.

In case G = SL(2, ℝ) and σ = −1 we argue as above in order to obtain (44) which is in this case equivalent to λ^{2} = (λ + mα)^{2}. This completes the proof of the lemma. □

The above argument can be extended to cover also some cases of σ ≠ 1, Re(λ) ≥ 0 with λ ∈ I_{σ}. This would lead to stronger vanishing results for ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) below. But there exist examples of σ ∈ M̂ and λ ∈ 𝔞^{∗} with Re(λ) ≥ 0 and λ ∈ I_{σ}, where the assertion of Lemma 6.7 is false. This is connected with the non-triviality of the spaces U_{Λ}(σ_{λ}, φ) introduced in Definition 7.5 below.

Corollary 6.8 Under the assumptions of Lemma 6.7

res_{Ω} ∘ Ĵ_{σ,λ} : C^{−∞}(Λ, V (σ_{λ})) → C^{∞}(Ω, V (σ_{−λ}))

is injective. □

For the remainder of this section we assume X ≠ 𝕆H^{2}. The following corollary now gives the first part of Theorem 6.1.

Corollary 6.9 For fixed σ and φ the set {λ ∈ 𝔞_{ℂ}^{∗} |^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0} is discrete.

Proof. Assume that ext : C^{−∞}(B, V_{B}(σ̃_{λ}, φ̃)) → ^{Γ}C^{−∞}(∂X, V (σ̃_{λ}, φ̃)) is regular at λ. Then res : ΓC^{−∞}(∂X, V (σ̃_{λ}, φ̃)) → C^{−∞}(B, V_{B}(σ̃_{λ}, φ̃)) is surjective. It follows from Proposition 6.5 that res ∘ Ĵ_{λ} (^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ))) = 0. Assume in addition that λ ∉ I_{𝔞}. Then ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0 by Corollary 6.8. We conclude that ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0 implies that ext is singular at λ or λ ∈ I_{𝔞}. □

We denote by O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)) and O_{λ}C^{−∞}(B, V_{B}(σ_{.}, φ)) the spaces of germs at λ of holomorphic families μ ↦ f_{μ} ∈ C^{−∞}(∂X, V (σ_{μ}, φ)) and μ ↦ f_{μ} ∈ C^{−∞}(B, V_{B}(σ_{μ}, φ)), respectively.

Definition 6.10 Let

O_{λ}^{0}C^{−∞}(B, V_{B}(σ_{.}, φ)) := {f_{μ} ∈ O_{λ}C^{−∞}(B, V_{B}(σ_{.}, φ)) | (μ − λ)extf_{μ} ∈ O_{λ}C^{−∞}(∂X, V (σ_{.}, φ))} .

We define the space of invariant distributions on the limit set which are generated by the singular parts of ext by

E_{Λ}(σ_{λ}, φ) := {res_{μ=λ}ext(f_{μ}) |f_{μ} ∈ O_{λ}^{0}C^{−∞}(B, V_{B}(σ_{.}, φ))} .

Proposition 6.11 The space E_{Λ}(σ_{λ}, φ) is finite-dimensional and

E_{Λ}(σ_{λ}, φ) ⊂ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) .  (45)

Assume that (σ, λ) satisfies the assumptions of Lemma 6.7. Then equality holds in (45) and ext is regular at λ if and only if ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0.

Proof. Since ext has at most finite-dimensional singularities E_{Λ}(σ_{λ}, φ) is finite-dimensional.

(45) follows from the meromorphy of ext and the equation res ∘ ext = id. It remains to show that under the assumptions of Lemma 6.7

dim E_{Λ}(σ_{λ}, φ) = dim^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) .

The main step in the proof is to show that dim E_{Λ}(σ_{λ}, φ) ≥ dim coker(res) (see Formula (47) below). This is more or less obvious if ext has a pole of first order at λ. The general case is more involved.

Let L_{λ} be the multiplication operator on O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)) and O_{λ}C^{−∞}(B, V_{B}(σ_{.}, φ)) given by (L_{λ}f)_{μ} := (μ − λ)f_{μ}. O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)) becomes a Γ-module by (π(g)f)_{μ} := (π^{σ,μ} ⊗ φ)(g)f_{μ}, g ∈ Γ. The Γ-action commutes with L_{λ}. The restriction map induces a morphism of ℂ[L_{λ}]-modules

O_{λ}res : ^{Γ}O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)) → O_{λ}C^{−∞}(B, V_{B}(σ_{.}, φ)) .

By Corollary 6.9 and Lemma 4.6 we have for f ∈ ^{Γ}O_{λ}C^{−∞}(∂X, V (σ_{.}, φ))

ext ∘_{O_{λ}}res(f) = f .  (46)

Let k be the order of the pole of ext at λ. Set

O_{λ,k}C^{−∞}(∂X, V (σ_{.}, φ)) := coker (L_{λ}^{k} : O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)) → O_{λ}C^{−∞}(∂X, V (σ_{.}, φ))) .

Then we consider the singular part of ext as an operator between ℂ[L_{λ}]-modules

ext^{<0} : coker_{O_{λ}}res → ^{Γ}O_{λ,k}C^{−∞}(∂X, V (σ_{.}, φ))

given by

ext^{<0}([f]) := ext(L_{λ}^{k}f) mod imL_{λ}^{k} , f ∈ O_{λ}C^{−∞}(B, V_{B}(σ_{.}, φ)) ,

which is well-defined by (46). Assume that ext^{<0}([f]) = 0. Then ext(L_{λ}^{k}f) = L_{λ}^{k}g for some g ∈ O_{λ}C^{−∞}(∂X, V (σ_{.}, φ)). It follows that f = _{O_{λ}}res(g). We conclude that ext^{<0} is injective. In particular, since ext^{<0} is a finite-dimensional operator, the space coker_{O_{λ}}res is finitedimensional. The map

E_{Λ}(σ_{λ}, φ) ∋ res_{μ=λ}ext(f_{μ}) ↦ ext^{<0}([f]) ∈ ^{Γ}O_{λ,k}C^{−∞}(∂X, V (σ_{.}, φ)), f ∈ O_{λ}^{0}C^{−∞}(B, V_{B}(σ_{.}, φ)),

is well-defined and identifies E_{Λ}(σ_{λ}, φ) with ker (L_{λ} : im ext^{<0} → im ext^{<0}). We also consider the usual point-wise restriction map

res : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(B, V_{B}(σ_{λ}, φ))

and the surjection

coker (L_{λ} : coker_{O_{λ}}res → coker_{O_{λ}}res) → coker(res)

induced by the point evaluation at λ. Summarizing the above discussion we obtain

d := dim E_{Λ}(σ_{λ}, φ) = dim ker (L_{λ} : im ext^{<0} → im ext^{<0})
= dim coker (L_{λ} : im ext^{<0} → im ext^{<0})
= dim coker (L_{λ} : coker_{O_{λ}}res → coker_{O_{λ}}res)
≥ dim coker(res) .  (47)

Set d̃:= dim E_{Λ}(σ̃_{λ}, φ̃). If (σ, λ) satisfies the assumptions of Lemma 6.7, then so does (σ̃, λ). Thus Proposition 6.5 combined with Corollary 6.8 implies

dim coker(res) ≥ dim^{Γ}C^{−∞}(Λ, V (σ̃_{λ}, φ̃)) .

It eventually follows from (45) that

d ≥ dim coker(res) ≥ dim^{Γ}C^{−∞}(Λ, V (σ̃_{λ}, φ̃)) ≥ d̃ .

Changing the roles of (σ, φ) and (σ̃, φ̃) we obtain d = d̃= dim^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). This finishes the proof of the proposition. □

Corollary 6.12 For each pair (σ_{λ}, φ) the space ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is finite-dimensional.

Proof. Choose μ ∈ 𝔞^{∗} large enough such that the finite-dimensional representation π_{σ,μ} exists and Re(λ + μ) ≥ 0. Recall the definition (34) of the embedding i_{σ,μ}. Since (1, λ + μ) satisfies the assumptions of Lemma 6.7 we obtain by Proposition 6.11 that

i_{σ,μ} (^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ))) ⊂ ^{Γ}C^{−∞}(Λ, V (1_{λ+μ}, φ ⊗ π_{σ,μ})) = E_{Λ}(1_{λ+μ}, φ ⊗ π_{σ,μ}) .

The corollary now follows since the space on the right hand side is finite-dimensional. □

Thus we have completed the proof of Theorem 6.1. It is perhaps worth noting that the finitedimensionality of ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) can also be proved without referring to the meromorphy of ext. In fact, using the asymptotic expansion (37) it is not difficult to show that there exists a number k ∈ ℕ_{0} (depending on λ) and a continuous embedding

i : ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ↪ (C^{k}(∂X, V (σ̃_{−λ}, φ̃)))′ .

Furthermore, the Banach space topology on ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) induced by i coincides with the topology as a closed subspace of the Montel space C^{−∞}(∂X, V (σ_{λ}, φ)). It follows that the unit ball in the Banach space ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is compact. This implies Corollary 6.12. However, arguments of this type seem not to be sufficient in order to establish results like Corollary 6.9.

7 Consequences of unitarity

From now on we assume that the finite-dimensional representation (φ, V_{φ}) of Γ is unitary. Then δ_{φ} = 0. If X = 𝕆H^{2}, then in addition we assume that δ_{Γ} < 0. Hence in any case ext is meromorphic on a half-plane {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > −ϵ} for some ϵ > 0. The aim of this section is to work out the consequences of the unitarity of φ for the singularities of ext in that region. In particular, for fixed σ and φ we show that the set of parameters λ with Re(λ) ≥ 0 and allowing non-trivial invariant distributions with support on the limit set is finite and real. This result will be the main ingredient in the proof of the finiteness of the discrete spectrum of Z on L^{2}(Y, V_{Y} (γ, φ)), where V_{Y} (γ, φ) is the hermitian vector bundle over Y defined by V_{Y} (γ, φ) := Γ\V (γ, φ).

In contrast to bilinear pairings ⟨., .⟩ sesquilinear pairings will always be written as (., .). By convention these pairings are ℂ-linear with respect to the first variable.

Since σ and φ are unitary representations, for λ ∈ i𝔞^{∗} we have a positive definite conjugate linear pairing (V_{σ_{λ}} ⊗ V_{φ}) ⊗ (V_{σ_{λ}} ⊗ V_{φ}) → V_{1_{−ρ}}. Since V_{B}(1_{−ρ}) ≅ Λ^{max}T^{∗}B integration gives a natural L^{2}-scalar product on C^{∞}(B, V_{B}(σ_{λ}, φ)). Let L^{2}(B, V_{B}(σ_{λ}, φ)) be associated Hilbert space. Using Lemma 5.8 we see that the adjoint S_{λ}^{∗} with respect to this Hilbert space structure is just Sλ̄ = S_{−λ}.

Lemma 7.1 If Re(λ) = 0, then S_{λ} is regular and unitary.

Proof. Let λ be imaginary such that S_{±λ} are regular. Let f ∈ C^{∞}(B, V_{B}(σ_{λ}, φ)). Then by the functional equation (30)

∥S_{λ}f∥_{L^{2}(B,V_{B}(σ_{λ},φ))}^{2} = (S_{−λ} ∘ S_{λ}f, f)_{L^{2}(B,V_{B}(σ_{λ},φ))} = ∥f∥_{L^{2}(B,V_{B}(σ_{λ},φ))}^{2} .

By meromorphy of S_{λ} this equation implies that S_{λ} is regular and unitary on i𝔞^{∗}. □

Lemma 7.2 If Re(λ) = 0 and λ ≠ 0, then

ext : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))

is regular and ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0.

Proof. Note that (σ, λ) satisfies the assumptions of Lemma 6.7. Assume that ext is singular at a non-zero imaginary λ. The leading singular part of ext maps to distributions which are supported on the limit set Λ. Thus by Corollary 6.8 the scattering matrix Ŝ_{λ} = res ∘ Ĵ_{λ} ∘ ext is singular at λ, too. Since Ŝ_{λ} = S_{λ}c_{σ}(−λ), and c_{σ} is regular on i𝔞^{∗} \ {0}, this contradicts Lemma 7.1. Thus ext is regular at λ. It follows by Proposition 6.11 that ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0. □

Proposition 7.3

1. If Re(λ) > 0, then for any finite-dimensional representation γ of K and T ∈ Hom_{M}(V_{σ}, V_{γ}) the Poisson transform P_{λ}^{T} maps ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) to L^{2}(Y, V_{Y} (γ, φ)). If ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0, then λ is real.

2. If ext : C^{−∞}(B, V_{B}(σ_{μ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{μ}, φ)) is singular at μ = λ, Re(λ) > 0, then λ is real and the order of the pole is 1.

3. If p_{σ}(0) ≠ 0, i.e., the intertwining operator Ĵ_{0} is regular (see (23)), then ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = 0 for Re(λ) > 0, and ext is regular in that region.

Proof. Let γ be a finite-dimensional representation of K and T ∈ Hom_{M}(V_{σ}, V_{γ}). Let ϕ ∈ ΓC^{−∞}(Λ, V (σ_{λ}, φ)). By Lemma 6.2 for any compact subset F ⊂ Ω there exists a constant C such that for a ∈ A_{+}, k ∈ F M

|P_{λ}^{T} ϕ(ka)| ≤ Ca^{−λ−ρ} .  (48)

In particular, P_{λ}^{T} ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)). This shows the first part of assertion 1.

Now let γ ∈ K̂ be a minimal K-type of π^{σ,λ} and T be injective. Then P_{λ}^{T} is injective. There is a real constant c(σ) (see [11]) such that (−Ω_{G} + c(σ) + λ^{2}) ∘ P_{λ}^{T} = 0.

If ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)), then P_{λ}^{T} ϕ is a square-integrable eigenfunction of −Ω_{G} + c(σ) + λ^{2} on Y. Since Y is complete Ω_{G} is essentially selfadjoint on the domain C_{c}^{∞}(Y, V_{Y} (γ, φ)). Its selfadjoint closure has the domain of definition {f ∈ L^{2}(Y, V_{Y} (γ, φ)) | Ω_{G}f ∈ L^{2}(Y, V_{Y} (γ, φ))}. In particular, Ω_{G} can not have non-trivial eigenvectors in L^{2}(Y, V_{Y} (γ, φ)) to eigenvalues with non-trivial imaginary part. Since Re(λ) > 0, λ has to be real. This proves the second part of assertion 1.

Assume in addition that p_{σ}(0) ≠ 0. Then there is a representation γ^{s} of K, a K-equivariant embedding i : γ ↪ γ^{s} and a locally invariant Dirac operator D(σ) acting on C_{c}^{∞}(Y, V_{Y} (γ^{s}, φ)) such that

D(σ)^{2} = −Ω_{G} + c(σ) .  (49)

(If X is e.g. an odd-dimensional hyperbolic space , then σ = σ′⊕σ^{′w} for some non Weyl-invariant σ′ ∈ M̂ and γ^{s} is constructed in [11], see pp. 28-29 for (49).) Since D(σ) is selfadjoint −Ω_{G}+c(σ) is non-negative. It follows that Re(λ) = 0 which contradicts our assumption. Alternatively, one could use the fact ([3], Thm. 6.1) that in case p_{σ}(0) ≠ 0 for Re(λ) > 0 the principal series π^{σ,λ} does not contain unitary subrepresentations. Compare Proposition 9.2 below. This proves assertion 3.

Now let f_{μ} ∈ C^{−∞}(B, V_{B}(σ_{μ}, φ)), μ ∈ 𝔞_{ℂ}^{∗}, be a holomorphic family such that ext(f_{μ}) has a pole of order k ≥ 1 at μ = λ, Re(λ) > 0. Let 0 ≠ ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) be the leading singular part of ext(f_{μ}) at μ = λ. In particular, λ is real and P_{λ}^{T} ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)) by assertion 1.

Since the leading singular part of ext is a finite-dimensional operator we may assume that f_{μ} ∈ C^{∞}(B, V_{B}(σ_{μ}, φ)) without changing ϕ. Then by Lemma 6.2 for any compact subset F ⊂ Ω and μ_{0} > 0 there exist constants C_{1}, C_{2}, C_{3} such that for a ∈ A_{+}, k ∈ F M, μ_{0} ≤ μ < λ,

|(μ − λ)^{k}P_{μ}^{T} ext(f_{μ})(ka)| ≤ C_{1}|μ − λ|^{k}a^{μ−ρ} + C_{2}a^{−μ−ρ}  (50)
≤ C_{3}(1 + log a)^{−k}a^{λ−ρ} .

There is a constant C_{4} such that for μ_{0} ≤ μ < λ

|μ − λ|^{k}|P_{μ}^{T} ext(f_{μ})(x)| |P_{λ}^{T} ϕ(x)| ≤ C_{4}(1 + log(a_{x}))^{−k}a_{x}^{−2ρ}, x ∈ F .

We now assume that k ≥ 2. Then

∫_{F K}(1 + log(a_{x}))^{−k}a_{x}^{−2ρ}dx < ∞ ,

and we obtain by Lebesgue’s Theorem of dominated convergence

∥P_{λ}^{T} ϕ∥_{L^{2}(Y,V_{Y} (γ,φ))}^{2} = ( lim_{{μ→λ \atop μ<λ}}(μ − λ)^{k}P_{μ}^{T} ext(f_{μ}), P_{λ}^{T} ϕ)_{L^{2}(Y,V_{Y} (γ,φ))}
= lim_{{μ→λ \atop μ<λ}}(μ − λ)^{k}(P_{μ}^{T} ext(f_{μ}), P_{λ}^{T} ϕ) .

On the other hand the estimates (48) and (50) allow for partial integration, and we obtain for μ < λ

(P_{μ}^{T} ext(f_{μ}), P_{λ}^{T} ϕ) = \frac{1}{λ^{2} − μ^{2}} ((−Ω_{G} + c(σ) + λ^{2})P_{μ}^{T} ext(f_{μ}), P_{λ}^{T} ϕ)

= \frac{1}{λ^{2} − μ^{2}} (P_{μ}^{T} ext(f_{μ}), (−Ω_{G} + c(σ) + λ^{2})P_{λ}^{T} ϕ)
= 0 .

Hence ∥P_{λ}^{T} ϕ∥_{L^{2}(Y,V_{Y} (γ,φ))} = 0. Since P_{λ}^{T} is injective this contradicts ϕ ≠ 0. We conclude that k = 1. This proves assertion 2. □

Proposition 7.4

1. The order of the pole of ext at 0 is at most 1.

2. If p_{σ}(0) ≠ 0, then ext is regular at 0.

3. If p_{σ}(0) = 0, then the residue res_{μ=0}ext vanishes on the 1-eigenspace of the involution S_{0} and identifies the −1-eigenspace with E_{Λ}(σ_{0}, φ).

Proof. The proof is analogous to the proof of Proposition 7.3. Let γ be the representation of K on the sum of the K-isotypic components of C^{∞}(∂X, V (σ_{0})) that correspond to all minimal Ktypes of C^{∞}(∂X, V (σ_{0})). (γ is irreducible iff p_{σ}(0) = 0.) Then one can find T ∈ Hom_{M}(V_{σ}, V_{γ}) such that the Poisson transform

P_{μ} := P_{μ}^{T} ⊗ id : C^{−∞}(∂X, V (σ_{μ}, φ)) → C^{∞}(X, V (γ, φ))

is injective for μ in a neighbourhood of 0.

Let μ ↦ f_{μ} ∈ C^{∞}(B, V_{B}(σ_{μ}, φ)) be a holomorphic family defined on such a neighbourhood. Assume that extf_{μ} has a pole of order k at 0. We want to study the leading singular part ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{0}, φ)) of extf_{μ} at μ = 0 via the leading singular part P_{0}ϕ of P_{μ}(extf_{μ}). For μ on a sufficiently small pointed disc D around 0 we obtain by Lemma 6.2, 3.,

(P_{μ}(extf_{μ}))(ka) = a^{μ−ρ}c_{γ}(μ)T f_{μ}(k) + a^{−μ−ρ}T^{w}(Ŝ_{μ}f_{μ})(k) + O(a^{−ρ−ϵ})  (51)

for a → ∞ uniformly in μ and kM varying in compact subsets D_{0} ⊂ D, F ⊂ Ω.

If p_{σ}(0) ≠ 0, then c_{γ}(μ) and Ŝ_{μ} are regular at μ = 0. We multiply (51) by μ^{k} and apply Cauchy’s integral formula in order to conclude that for any compact subset F ⊂ Ω there exists a constant C such that for a ∈ A_{+}, k ∈ F M

|P_{0}ϕ(ka)| ≤ Ca^{−ρ−ϵ} .  (52)

If p_{σ}(0) = 0, then c_{γ}(μ) and Ŝ_{μ} have only first order poles at μ = 0. If k ≥ 2, then we can argue as above in order to obtain (52).

In particular, P_{0}ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)). By (51) and (52) the pairing (P_{μ}ext(f_{μ}), P_{0}ϕ) is defined for 0 < |μ| < ϵ. As in the proof of Proposition 7.3 we show by partial integration that this pairing vanishes and that ∥P_{0}ϕ∥_{L^{2}(Y,V_{Y} (γ,φ))}^{2} = 0. Hence ϕ = 0 unless k = 1 and p_{σ}(0) = 0.

It remains to show the last assertion of the proposition. Thus assume k = 1 and p_{σ}(0) = 0. Applying the residue theorem and (22) to (51) we obtain

(P_{0}ϕ)(ka) = a^{−ρ}(res_{μ=0}c_{σ}(μ))T^{w}(f_{0}(k) − S_{0}f_{0}(k)) + O(a^{−ρ−ϵ}) .

Thus the leading asymptotic coefficient of P_{0}ϕ vanishes iff S_{0}f_{0} = f_{0}. In this case P_{0}ϕ satisfies (52) which implies that ϕ = 0 as above. We conclude that ker res_{μ=0}ext = ker(S_{0} − id). It follows from the definition of E_{Λ}(σ_{0}, φ) that

res_{μ=0}ext : ker(S_{0} + id) → E_{Λ}(σ_{0}, φ)

is an isomorphism. The proof of the proposition is now complete. □

For 0 ∉ I_{σ} Proposition 7.4 could also have been proved by a refinement of the proof of Lemma 7.2.

Recall Definition 6.10 of E_{Λ}(σ_{λ}, φ). For Re(λ) ≥ 0 we have just proved that this space coincides with the image of the residue of ext at λ.

Definition 7.5 For all λ ∈ 𝔞_{ℂ}^{∗} we define the space of ”stable” invariant distributions supported on the limit set by

U_{Λ}(σ_{λ}, φ) := {ϕ ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) | res ∘ Ĵ_{σ,λ}(ϕ) = 0} .

Corollary 6.8 implies that if Re(λ) ≥ 0 and U_{Λ}(σ_{λ}, φ) is non-trivial, then λ ∈ I_{σ}. We can now refine Proposition 6.11.

Proposition 7.6 For Re(λ) ≥ 0 we have

ΓC^{−∞}(Λ, V (σ_{λ}, φ)) = E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ) .

Proof.We assume that λ ∈ 𝔞^{∗}. Otherwise there is nothing to show. Let ϕ ∈ E_{Λ}(σ_{λ}, φ)∩U_{Λ}(σ_{λ}, φ). Let P be an injective Poisson transform as in the proofs of Propositions 7.3 and 7.4. Using Lemma 6.2, 1. in case λ = 0 and 2. in case λ > 0, we see that there exists a constant ϵ > 0 such that for any compact F ⊂ Ω there exists a constant C such that for a ∈ A_{+}, k ∈ F M

|P ϕ(ka)| ≤ Ca^{−λ−ρ−ϵ} .

As in the proof of Proposition 7.4 this implies ϕ = 0. Thus res ∘ Ĵ_{λ} is injective on E_{Λ}(σ_{λ}, φ). It remains to show that

dim E_{Λ}(σ_{λ}, φ) = dim res ∘ Ĵ_{λ} (^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)))  (53)

By (47) we have dim E_{Λ}(σ_{λ}, φ) ≥ dim coker(res). On the other hand Proposition 6.5 implies that

dim coker(res) ≥ dim res ∘ Ĵ_{λ} (^{Γ}C^{−∞}(Λ, V (σ̃_{λ}, φ̃))) .

Since λ is real and φ is unitary the space on the right hand side is conjugate linear isomorphic to res ∘ Ĵ_{λ} (^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ))). This implies (53). □

Definition 7.7 Define

P S(σ, φ) := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) ≥ 0, ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ≠ 0} ,
P S_{res}(σ, φ) := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) ≥ 0, E_{Λ}(σ_{λ}, φ) ≠ 0} ,
P S_{U}(σ, φ) := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) ≥ 0, U_{Λ}(σ_{λ}, φ) ≠ 0}.

Proposition 7.8 P S(σ, φ) = P S_{res}(σ, φ) ∪ P S_{U}(σ, φ) is a finite subset of the interval [0, δ_{Γ}] ⊂ 𝔞^{∗}, and we have P S_{U}(σ, φ) ⊂ I_{σ}. The space

⨁_{λ∈P S(σ,φ)}^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ))

is finite-dimensional. If λ ∈ P S(σ, φ) \ I_{σ}, then ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) is spanned by the residue of ext at λ.

If p_{σ}(0) ≠ 0, then P S(σ, φ) = P S_{U}(σ, φ) ⊂ {0}.

If δ_{Γ} > 0, then δ_{Γ} ∈ P S_{res}(1, 1) and dim^{Γ}C^{−∞}(Λ, V (1_{δ_{Γ}})) = 1.

Proof. Combine the results of the present section with Theorem 6.1 and Theorem 4.7. The last assertion follows from the construction of the Patterson-Sullivan measure ([47], [52], [18]) and the uniqueness of the eigenfunction f ∈ L^{2}(Y) corresponding to the smallest eigenvalue of Δ. □

Corollary 7.9 If X is an odd-dimensional hyperbolic space and σ is a faithful representation of M = Spin(n − 1), then P S(σ, φ) is empty.

Proof. The condition on σ ensures that 0 ∉ I_{σ} and that p_{σ}(0) ≠ 0 ([32], Ch.12). □

8 Abstract harmonic analysis on Γ\G

Let (φ, V_{φ}) be a unitary representation of Γ. We consider the Hilbert space

L^{2}(Γ\G, φ) := {f : G → V_{φ} | f(gx) = φ(g)f(x) ∀g ∈ Γ, x ∈ G, ∫_{Γ\G} |f(x)|^{2} dx < ∞} .

48 8 ABSTRACT HARMONIC ANALYSIS ON Γ\G

The action of G on Γ\G by right translations induces a unitary representation of G on L^{2}(Γ\G, φ). The abstract Plancherel theorem (see e.g. [56], Thm. 14.10.5) provides a direct integral decomposition of this representation into irreducibles

L^{2}(Γ\G, φ) ≅^{α} ∫ ⊕ ; Ĝ M_{π} ˆ⊗V_{π} dκ(π) .  (54)

Here Ĝ denotes the unitary dual of G, V_{π} carries an irreducible unitary representation belonging to the class π ∈ Ĝ, the Hilbert space M_{π} is called the multiplicity space of π in L^{2}(Γ\G, φ), and κ is a Borel measure on Ĝ called the Plancherel measure. The isomorphism (54) is a unitary equivalence of representations, where G acts on the right hand side by id_{M_{π}} ⊗ π. Throughout the paper we shall not distinguish between an element π ∈ Ĝ and a particular representative (π, V_{π}). Note that only the measure class of κ is uniquely defined. Strictly speaking, the spaces M_{π} are defined on a complement of a set of measure zero, only.

If Γ is the trivial group, then M_{π} ≅ V′_{π}, and M_{π} ˆ⊗V_{π} can be identified with the space L^{2}(V_{π}) of Hilbert-Schmidt operators on V_{π} which has a canonical scalar product. Then a canonical choice for α is α(f)(π) := π(f) ∈ L^{2}(V_{π}), f ∈ L^{2}(G), π ∈ Ĝ. This determines the Plancherel measure. In contrast, for non-trivial Γ there is no obvious normalization of the unitary equivalence α, of the Hilbert space structure on M_{π} ˆ⊗V_{π} and of the Plancherel measure.

The goal of this paper can now be phrased as follows: Make the equivalence (54) and all its ingredients as explicit as possible. In the present section we specify this task. In particular, following the approach of Bernstein [8] we identify M_{π} with a subspace of the ”tempered” Γinvariant functionals on the space of smooth vectors V_{π,∞} of V_{π}. This provides the bridge to the results of Section 7.

We introduce the Harish-Chandra Schwartz space C(G, V_{φ}) of G in a way suitable for our purposes. Fix a base {X_{i}} of 𝔤 and let I_{N}, N ∈ ℕ_{0}, denote the set of all multiindices I = (i_{1}, …, i_{dim(𝔤)}), |I| ≤ N. A multiindex I ∈ I_{N} defines an element X_{I} = X_{1}^{i_{1}} … X_{dim(𝔤)}^{i_{dim(𝔤)}} ∈ U(𝔤). For N ∈ ℕ_{0} and a V_{φ}-valued smooth function f on G we set

q_{N}(f)^{2} := ∑_{(I,J)∈I_{N} ×I_{N}} ∫_{G} | log(a_{g})^{N}f(X_{I}gX_{J})|^{2} dg .

Then C(G, V_{φ}) = {f | q_{N}(f) < ∞ ∀N ∈ ℕ_{0}}. The seminorms q_{N}, N ∈ ℕ_{0} induce the structure of a Fréchet space on C(G, V_{φ}). For fixed N ∈ ℕ_{0} we define the Hilbert space C^{N}(G, V_{φ}) as the completion of C(G, V_{φ}) with respect to q_{N}. Then we have continuous inclusions C(G, V_{φ}) ⊂ C^{N}(G, V_{φ}) ⊂ C^{0}(G, V_{φ}) = L^{2}(G, V_{φ}).

Let χ be the cut-off function χ constructed in Lemma 6.4. We consider χ as a right Kinvariant function on G.

Definition 8.1 We define the Schwartz space on Γ\G by

C(Γ\G, φ) := {f ∈ L^{2}(Γ\G, φ) | χf ∈ C(G, V_{φ})} .

It inherits the structure of a Fréchet space from C(G, V_{φ}). Similarly, we define intermediate Hilbert spaces C^{N}(Γ\G, φ), N ∈ ℕ_{0}, by

C^{N}(Γ\G, φ) := {f ∈ L^{2}(Γ\G, φ) | χf ∈ C^{N}(G, V_{φ})} .

Define C^{−N}(Γ\G, φ) to be the conjugate linear dual of C^{N}(Γ\G, φ). The space of tempered distributions C′(Γ\G, φ) is then by definition the conjugate linear dual of C(Γ\G, φ).

Let (γ, V_{γ}) be a finite-dimensional unitary representation of K. Then L^{2}(Y, V_{Y} (γ, φ)) ≅ [L^{2}(Γ\G, φ) ⊗ V_{γ}]^{K}. Using this isomorphism we define for ∗ ∈ {∅, N, −N, ′}

C^{∗}(Y, V_{Y} (γ, φ)) := [C^{∗}(Γ\G, φ) ⊗ V_{γ}]^{K} .

Lemma 8.2 If N is sufficiently large, then the inclusion

i : C^{N}(Γ\G, φ) ↪ L^{2}(Γ\G, φ)

is Hilbert-Schmidt.

Proof. Consider the inclusion j : C^{N}(G, V_{φ}) ↪ L^{2}(G, V_{φ}). It is Hilbert-Schmidt for N sufficiently large (see e.g. [8]). Let χ′ ∈ C_{c}^{∞}(X ∪ Ω) be a second cut-off function such that χ′χ = χ. Again, we consider χ′ as a function on G. Then p_{χ′} : L^{2}(G, V_{φ}) → L^{2}(Γ\G, φ) defined by p_{χ′}(f)(x) := ∑_{g∈Γ} φ(g)χ′(g^{−1}x)f(g^{−1}x) is continuous. Let m_{χ} : C^{N}(Γ\G, φ) → C^{N}(G, V_{φ}) be the operator induced by multiplication with χ. Now we can factorize i over the Hilbert-Schmidt operator j: i = p_{χ′} ∘ j ∘ m_{χ}. The lemma follows. □

In the following we choose N sufficiently large. Fix an operator α providing the unitary equivalence (54). By a theorem of Gelfand/Kostyuchenko (see [8]) Lemma 8.2 implies that the composition

C^{N}(Γ\G, φ) ↪ L^{2}(Γ\G, φ) ⟶^{α} ∫ ⊕ ; Ĝ M_{π} ˆ⊗V_{π} dκ(π)

is pointwise defined, i.e., there exists a collection of continuous maps

α_{π} : C^{N}(Γ\G, φ) → M_{π} ˆ⊗V_{π}, π ∈ Ĝ

such that for f ∈ C^{N}(Γ\G, φ) we have α(f)(π) = α_{π}(f). By changing α_{π} on a set of π’s of measure zero (w.r.t. κ) we can assume that for all π ∈ Ĝ the map α_{π} is an intertwining operator of G-representations. Let

β_{π} : M_{π} ˆ⊗V_{π} → C^{−N}(Γ\G, φ)

denote the adjoint of α_{π}. Note that C^{−N}(Γ\G, φ) ⊂ C′(Γ\G, φ). The composition of β_{π} with this inclusion will also be denoted by β_{π}.

Let (π, V_{π}) be a representation of G on a reflexive Banach space, and let (π′, V_{π′}) be its dual representation. The space of distribution vectors V_{π,−∞} of V_{π} is by definition (V_{π′,∞})′, where the subscript ∞ indicates the transition to the subspace of smooth vectors, and the second dualization is with respect to the canonical Fréchet topology on V_{π′,∞}. Then we have the following inclusions of G-representations: V_{π,∞} ⊂ V_{π} ⊂ V_{π,−∞}.

Definition 8.3 Let (π, V_{π}) be a representation of G on a reflexive Banach space, and let (φ, V_{φ}) be a finite-dimensional unitary representation of Γ. An invariant distribution vector ϕ ∈ ^{Γ}(V_{π,−∞} ⊗ V_{φ}) is called tempered (square integrable, resp.) if for all v ∈ V_{π′,∞} the function

G ∋ g ↦ c_{ϕ,v}(g) := ⟨ϕ, π′(g)v⟩ ∈ V_{φ}

belongs to C′(Γ\G, φ) (L^{2}(Γ\G, φ)). By ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{d} ⊂ ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{temp} ⊂ ^{Γ}(V_{π,−∞} ⊗ V_{φ}) we denote the linear subspaces of square integrable and tempered invariant distribution vectors.

If (π, V_{π}) is an admissible representation of finite length, we have the following characterizations and consequences of temperedness.

Lemma 8.4 Let (π, V_{π}) be an admissible G-representation of finite length on a reflexive Banach space, and let V_{π′,K} ⊂ V_{π′,∞} be the underlying (𝔤, K)-module of K-finite elements of the dual representation V_{π′}. If S ⊂ V_{π′,K} is a generating set and ϕ ∈ ^{Γ}(V_{π,−∞} ⊗ V_{φ}), then the following conditions are equivalent:

1. c_{ϕ,v} ∈ C′(Γ\G, φ) for all v ∈ S.

2. ϕ ∈ ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{temp} .

3. c_{ϕ,v} ∈ C′(Γ\G, φ) for all v ∈ V_{π′,∞}, and the map c_{ϕ,.} : V_{π′,∞} → C′(Γ\G, φ) is continuous.

The analogous assertions for ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{d} and L^{2}(Γ\G, φ) hold true, too.

Proof. The lemma is a consequence of the globalization theory of Casselman and Wallach ([15], [56], Ch. 11). The conclusions 3. ⇒ 2. ⇒ 1. are obvious. We outline the argument for 1.⇒ 3.

Assume 1. Since the (𝔤, K)-module V_{π′,K} is finitely generated and admissible, and K-finite matrix coefficients satisfy elliptic differential equations one can show that there exists N ∈ ℕ_{0} such that c_{ϕ,v} ∈ C^{−N}(Γ\G, φ)_{K} for all v ∈ V_{π′,K}. Here C^{−N}(Γ\G, φ)_{K} denotes the subspace of K-finite smooth vectors of C^{−N}(Γ\G, φ). But C^{−N}(Γ\G, φ) is a Hilbert space on which G acts continuously. The theorem of Casselman and Wallach now states that the (𝔤, K)-module homomorphism V_{π′,K} ∋ v ↦ c_{ϕ,v} ∈ C^{−N}(Γ\G, φ)_{K} extends to a continuous homomorphism V_{π′,∞} → C^{−N}(Γ\G, φ)_{∞}. Condition 3 follows. The argument for ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{d} is essentially the same. □

The notion of temperedness is compatible with the notion of a tempered (irreducible) representation (π, V) of G as follows. For a moment let Γ and V_{φ} be trivial, and define V_{−∞,temp} as above. Then π is tempered iff V_{−∞,temp} ⊂ V_{−∞} is dense. Similarly, π is square integrable, i.e., belongs to the discrete series, iff V_{−∞,d} ⊂ V_{−∞} is dense. The above lemma in mind it is not difficult to see that this characterization of temperedness and square-integrability is equivalent to the various definitions appearing in the literature. In fact, more is true:

Lemma 8.5 If (π, V) is an admissible G-representation of finite length on a reflexive Banach space, then π is tempered iff V_{−∞,temp} = V_{−∞}. Moreover, if v ∈ V′_{∞} is fixed, then the matrix coefficient map

V_{−∞} ∋ ϕ ↦ c_{ϕ,v} ∈ C′(G)

is continuous.

Proof. The lemma follows from Lemma 10 in [2] which is based on subtle estimates of K-finite matrix coefficients. However, we would like to indicate a different argument which is more in the spirit of the proof of Lemma 8.4.

Fix v ∈ V′_{∞}. Then for all ϕ ∈ V_{K} the asymptotic expansions of matrix coefficients ([55], 4.4.3.) give c_{ϕ,v} ∈ C′(G). Thus by Lemma 8.4 we have v ∈ V′_{−∞,temp}. As in the proof of Lemma 8.4 the map ϕ ↦ c_{ϕ,v} extends to a continuous map c_{.,v} : V_{∞} → C^{−N}(G)_{∞} for some N ∈ ℕ_{0}. Since C^{−N}(G) is a Hilbert space this map has a continuous right inverse, the adjoint of which is the continuous extension of c_{.,v} to V_{−∞}

c_{.,v} : V_{−∞} → C^{N}(G)_{−∞} ⊂ C′(G) .

This proves the lemma. □

We now return to the discussion of the map β_{π}.

Lemma 8.6 For any π ∈ Ĝ there is an embedding

i_{π} : M_{π} ↪ ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{temp}

such that for m ∈ M_{π} and v ∈ V_{π,∞} we have β_{π}(m ⊗ v) = c_{i_{π}(m),v}.

Proof. Let N be sufficiently large. Fix m ∈ M_{π} and consider the G-intertwining operator F_{m} : V_{π} → C^{−N}(Γ\G, φ) given by F_{m}(v) := β_{π}(m ⊗ v). Then

F_{m}(V_{π,∞}) ⊂ C^{−N}(Γ\G, φ)_{∞} ⊂ C^{−N}(Γ\G, φ) ∩ C^{∞}(Γ\G, φ) .

In particular, elements of F_{m}(V_{π,∞}) can be evaluated at the identity e ∈ G. We define i_{π} by ⟨i_{π}(m), v⟩ := F_{m}(v)(e), v ∈ V_{π,∞}. The assertion of the lemma is now obvious. □

Our first concretization of the abstract Plancherel decomposition (54) is given by the following corollary.

Corollary 8.7 There exists a collection of Hilbert spaces N_{π} ⊂ ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{temp}, π ∈ Ĝ, and a direct integral

∫ ⊕ ; Ĝ N_{π} ˆ⊗V_{π} dκ(π)

such that the following holds:

1. The matrix coefficient map c : ^{Γ}(V_{π′,−∞} ⊗V_{φ})_{temp} ⊗V_{π,∞} → C′(Γ\G, φ) gives rise to a map

c_{π} : N_{π} ˆ⊗V_{π} → C′(Γ\G, φ) .

2. Let F_{π} : C(Γ\G, φ) → N_{π} ˆ⊗V_{π} be the adjoint of c_{π}. Then the collection of maps F_{π} extends to a unitary equivalence

F : L^{2}(Γ\G, φ) ⟶^{≅} ∫ ⊕ ; Ĝ N_{π} ˆ⊗V_{π} dκ(π) .

3. If ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} ≠ 0, then N_{π} = ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d}.

In particular, the Plancherel measure κ is supported on the set

{π ∈ Ĝ |^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{temp} ≠ 0}

and κ({π}) ≠ 0 iff ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} ≠ 0.

The Plancherel measure and the scalar product on the subspace ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} ⊂ N_{π} can be chosen such that κ({π}) = 1, if (V_{π′,−∞} ⊗ V_{φ})_{d} ≠ 0, and such that c_{π} induces an isometric embedding of ^{Γ}(V_{π′,−∞} ˆ⊗V_{φ})_{d} ⊗ V_{π} into L^{2}(Γ\G, φ).

Proof. Set N_{π} := i_{π}(M_{π}). □

The plan of the rest of the paper is now as follows. In the next section we determine the spaces (V_{π,−∞} ⊗ V_{φ})_{temp} and (V_{π,−∞} ⊗ V_{φ})_{d} for all π ∈ Ĝ. This is based on the results of Section 7. In Section 10 we study wave packets of Eisenstein series. It turns out that they span the orthogonal complement to the discrete subspace

L^{2}(Γ\G, φ)_{d} := ⨁_{{π∈Ĝ | (V_{π,−∞}⊗V_{φ})_{d}≠0}}^{Hilbert} im c_{π} ⊂ L^{2}(Γ\G, φ) .

The proof of this fact heavily depends on our a priori knowledge of the support of the Plancherel measure. The last section contains the summary of our results, including the determination of the scalar products on N_{π} and of the Placherel measure, as well as the consequences for the spectral theory of the Casimir operator acting on sections of the locally homogeneous vector bundle V_{Y} (γ, φ) over the Kleinian manifold Y.

9 Tempered invariant distribution vectors

In this section we determine the tempered and square integrable invariant distribution vectors for all π ∈ Ĝ.

First we need a rough classification of the unitary dual Ĝ. Recall the notions of temperedness and square integrability of an irreducible representation (see the discussion following Lemma 8.4). The unitary dual is a disjoint union of the dicrete series, the unitary principal series, and the complementary series

Ĝ = Ĝ_{d} ∪ Ĝ_{u} ∪ Ĝ_{c} ,

where

Ĝ_{d} := {π ∈ Ĝ | V_{π} is square integrable} ,
Ĝ_{u} := {π ∈ Ĝ | V_{π} is tempered} \ Ĝ_{d} ,
Ĝ_{c} := {π ∈ Ĝ | V_{π} is not tempered} .

The discrete series Ĝ_{d} has been determined by Harish-Chandra. It is empty iff X = H^{n}, n odd. In the other cases one can choose a Cartan subalgebra 𝔥 of 𝔤 which is contained in 𝔨. An infinitesimal character χ_{λ}, λ ∈ 𝔥_{ℂ}^{∗}, is called regular if no expression of the form (43) vanishes. Let W (𝔨_{ℂ}, 𝔥_{ℂ}) be the Weyl group of K. Then Ĝ_{d} can be parametrized by the Harish-Chandra parameters

{λ ∈ 𝔥_{ℂ}^{∗} | χ_{λ} is regular and integral}/W (𝔨_{ℂ}, 𝔥_{ℂ})  (55)

such that Z acts on π ∈ Ĝ_{d} with infinitesimal character χ_{λ}, and π has the minimal K-type with highest weight λ + ρ_{𝔤} − 2ρ_{𝔨} (see e.g. [55], Ch. 6 and Ch. 8). Here ρ_{𝔤} and ρ_{𝔨} are the half sums of the positive (w.r.t. λ) roots of 𝔥 in 𝔤 and 𝔨, respectively. In particular, for any γ ∈ K̂ there are only finitely many discrete series representations containing the K-type γ. Strictly speaking, (55) parametrizes the discrete series representations for the linear group G with Lie algebra 𝔤 which has a simply connected complexification G_{ℂ}. In general, the parametrization remains valid, if one sharpens the notion of integrality.

The set Ĝ_{u} consists of the unitary principal series representations π^{σ,λ}, σ ∈ M̂, Re(λ) = 0. They are irreducible unless σ = σ^{w}, p_{σ}(0) = 0 and λ = 0. In the latter case we have π^{σ,0} = π^{σ,+} ⊕ π^{σ,−}, where the irreducible representations π^{σ,±}, called the non-degenerate limits of discrete series, are the ±1-eigenspaces of J_{σ,0}. All equivalences between these representations are induced by the intertwining operators J_{σ,λ}. For all that see [31], Ch. XIV.

Though also Ĝ_{c} is completely known (see [4] and the references therein) for our purposes less information is sufficient. The Langlands classification (see [55], Ch. 5) associates to any irreducible non-tempered representation (π, V_{π}) a unique Langlands parameter (σ, λ), σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗}, Re(λ) > 0, such that V_{π,±∞} is equivalent to the unique irreducible subrepresentation of the principal series representation π^{σ,λ} acting on C^{±∞}(∂X, V (σ_{λ})). We denote this subrepresentation by (π̄^{σ,λ}, I_{±∞}^{σ,λ}). It is the image of J_{σ^{w},−λ}. It is also the unique irreducible quotient of C^{∞}(∂X, V (σ_{−λ}^{w})). If π ∈ Ĝ_{c}, then σ = σ^{w}, λ ∈ 𝔞^{∗} (e.g. [31], Thm. 16.6.) and p_{σ}(0) ≠ 0 ([3], Thm. 6.1). An invariant pre-Hilbert structure (., .) on I_{∞}^{σ,λ} realized as a quotient of C^{∞}(∂X, V (σ_{−λ})) can now be described as follows: Let (., .)_{0} be the invariant sesquilinear pairing between C^{∞}(∂X, V (σ_{−λ})) and C^{∞}(∂X, V (σ_{λ})). Then

([f], [g]) := (f, J_{σ,−λ}(g))_{0} ,  (56)

where we have represented [f], [g] ∈ I_{∞}^{σ,λ} by f, g ∈ C^{∞}(∂X, V (σ_{−λ})). Indeed, since^{t}J_{σ,−λ} = J_{σ,−λ} the pairing (56) is hermitian. By I^{σ,λ} we denote the Hilbert space completion of I_{∞}^{σ,λ} with respect to (., .).

We recall the relation between the Poisson transform P_{λ}^{T} and the matrix coefficients of principal series representations. For later reference we will state it as a lemma. Its verification is a standard computation with integral formulas.

Lemma 9.1 Let φ, σ, γ be finite-dimensional representations of Γ, M, and K, respectively, λ ∈ 𝔞_{ℂ}^{∗} and T ∈ Hom_{M}(V_{σ}, V_{γ}). We consider the Poisson transform

P_{λ}^{T} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(X, V (γ, φ)) ≅ [C^{∞}(G, V_{φ}) ⊗ V_{γ}]^{K} .

Then for any v ∈ Vγ̃, ϕ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)), g ∈ G we have

⟨P_{λ}^{T} ϕ(g), v⟩ = c_{ϕ,v_{T}} (g) ∈ V_{φ} ,

where v_{T} ∈ C^{∞}(∂X, V (σ̃_{−λ})) is the element defined by Frobenius reciprocity v_{T} (k) := ^{t}T γ̃(k^{−1})v, k ∈ K. □

Fix a finite-dimensional unitary representation (φ, V_{φ}) of Γ. Recall the decomposition ΓC^{−∞}(Λ, V (σ_{λ}, φ)) = E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ) from Proposition 7.6.

Proposition 9.2 Let σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗} with Re(λ) > 0. Then

Γ(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d} = ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{temp} = ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) = E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ) .

If one of these spaces is non-zero, then π̄^{σ,λ} ∈ Ĝ_{c}.

Proof. Observe that ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{temp} ⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{temp}. Fix a minimal K-type γ of C^{∞}(∂X, V (σ_{λ})) together with a non-trivial T ∈ Hom_{M}(V_{σ}, V_{γ}). We consider the injective Poisson transform

P = P_{λ}^{T} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(X, V (γ, φ)) .

Recall the definition of v_{T} from Lemma 9.1. If ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{temp}, then by definition c_{ϕ,v_{T}} ∈ C′(Γ\G, φ) for all v ∈ Vγ̃, hence P ϕ ∈ C′(Y, V_{Y} (γ, φ)).

Let f ∈ C^{∞}(∂X, V ((γ̃_{|M})_{−λ}, φ̃)) with supp(f) ⊂ Ω. We want to show that ⟨ϕ,^{t}T f⟩ = 0. As in the proof of Theorem 4.7 we extend f to a section f̃ ∈ C^{∞}(X, V (γ̃, φ̃)). By Corollary 6.3 there is a constant C such that

⟨ϕ,^{t}T f⟩ = C lim_{a→∞} a^{ρ−λ} ∫_{K}⟨P ϕ(ka), f(k)⟩ dk = C lim_{n→∞} ∫_{G}⟨P ϕ(x), f̃_{n}(x)⟩ dx,

where f̃_{n}(x) := a_{x}^{−(λ+ρ)}ψ(log(a_{x} − n))f̃(x) for some ψ ∈ C_{c}^{∞}(0, 1) satisfying ∫_{0}^{1} ψ(t)dt = 1. Define F_{n} ∈ C_{c}^{∞}(Y, V_{Y} (γ̃, φ̃)) ⊂ C(Y, V_{Y} (γ̃, φ̃)) by

F_{n}(x) := ∑_{g∈Γ} φ̃(g)f̃_{n}(g^{−1}x) .

We claim that lim_{n→∞} F_{n} = 0 in C(Y, V_{Y} (γ̃, φ̃)). Using that supp(f) ⊂ Ω we find a finite subset L ⊂ Γ such that

χ(x)F_{n}(x) = χ(x) ∑_{g∈L} φ̃(g)f̃_{n}(g^{−1}x) = ∑_{g∈L} φ̃(g)((g^{−1})^{∗}χf̃_{n})(g^{−1}x) ,

where χ is the cut-off function as in Lemma 6.4. We have lim_{n→∞} f̃_{n} = 0 in C(X, V (γ̃, φ̃)) and hence (g^{−1})^{∗}χf_{n} → 0 in C(X, V (γ̃, φ̃)). This shows the claim.

Since P ϕ ∈ C′(Y, V_{Y} (γ, φ)) we obtain

⟨ϕ,^{t}T f⟩ = C lim_{n→∞} ∫_{Γ\G}⟨P ϕ(x), F_{n}(x)⟩ dx = C lim_{n→∞}⟨P ϕ, F_{n}⟩ = 0 .

This proves that ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{temp} ⊂ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)).

On the other hand, we have by Proposition 7.3 that P (^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ))) ⊂ L^{2}(Y, V_{Y} (γ, φ)). Since the elements v_{T}, v ∈ Vγ̃, generate the (𝔤, K)-module C^{∞}(∂X, V (σ̃_{−λ}))_{K} it follows from Lemma 8.4 and Lemma 9.1 that c_{ϕ,f} ∈ L^{2}(Γ\G, φ) for all f ∈ C^{∞}(∂X, V (σ̃_{−λ})). Thus ΓC^{−∞}(Λ, V (σ_{λ}, φ)) ⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{d}. It remains to show that ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{d} ⊂ Γ(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d}. Indeed, let ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{d} and consider the G-map

c_{ϕ,.} : C^{∞}(∂X, V (σ̃_{−λ})) → L^{2}(Γ\G, φ) .

Since the target space is a unitary representation of G the image of c_{ϕ,.} decomposes into a direct sum of irreducible representations. But C^{∞}(∂X, V (σ̃_{−λ})) has the unique irreducible quotient

C^{∞}(∂X, V (σ̃_{−λ}))/(I_{−∞}^{σ,λ})^{⊥} .

Thus c_{ϕ,.} factorizes over this quotient, and hence ϕ ∈ ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d}. Now, if ϕ is non-trival, then we can pull back the invariant pre-Hilbert structure from L^{2}(Γ\G, φ) to this quotient. By duality this induces an invariant scalar product on I_{∞}^{σ,λ}. Hence π̄^{σ,λ} ∈ Ĝ_{c}. In view of the chain of inclusions

Γ(I_{−∞}^{σ,λ} ⊗ V_{φ})_{temp} ⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{temp} ⊂ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) ⊂
⊂ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{d} ⊂ ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d}

the proof of the proposition is now complete. □

Corollary 9.3 The space ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d} is finite-dimensional. It is non-trivial iff λ belongs to the finite set P S(σ, φ) \ {0} ⊂ (0, δ_{Γ}].

Proof. Combine Proposition 9.2 with Proposition 7.8. □

The case σ = 1 is particularly interesting. In the following table we give the set of λ > 0 with I^{1,λ} ∈ Ĝ.

\begin{matrix}X & ℝH^{n} & ℂH^{n} & ℍH^{n} & 𝕆H^{n} \\ λ & (0, ρ] & (0, ρ] & (0, ρ − 2α] ∪ \{ρ\} & (0, ρ − 6α] ∪ \{ρ\} \end{matrix} .

If δ_{Γ} > 0, then by Proposition 9.2 the representation I^{1,δ_{Γ}} is unitary. This leads to the restriction of the set of possible values of δ_{Γ} found by Corlette [18] (see Section 2).

Lemma 9.4 If (π, V_{π}) ∈ Ĝ is tempered, then ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{temp} = ^{Γ}(V_{π−∞} ⊗ V_{φ}). Moreover, for any v ∈ V_{π̃,∞} the map

Γ(V_{π,−∞} ⊗ V_{φ}) ∋ ϕ ↦ c_{ϕ,v} ∈ C′(Γ\G, φ)

is continuous. If, in addition, π = π^{σ,λ}, λ ≠ 0 imaginary, then ext identifies C^{−∞}(B, V_{B}(σ_{λ}, φ)) with ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{temp}.

Proof. Observe that there is a natural inclusion ^{Γ}C′(G, V_{φ}) ↪ C′(Γ\G, φ) induced by the adjoint of the multiplication by χ. The first assertions of the lemma now follow from Lemma 8.5. The last one follows from Lemmas 4.6 and 7.2. □

Proposition 9.5 Let σ ∈ M̂. If λ ≠ 0 is imaginary, then ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ))_{d} = 0. For λ = 0 we have

ΓC^{−∞}(∂X, V (σ_{0}, φ))_{d} = U_{Λ}(σ_{0}, φ) .

If this finite-dimensional space is non-trivial, then σ ≠ 1 and 0 ∈ I_{σ}.

Proof. We claim that for Re(λ) = 0

ΓC^{−∞}(∂X, V (σ_{λ}, φ))_{d} ⊂ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)) .  (57)

We now prove the proposition assuming the claim. If λ ≠ 0, then by Lemma 7.2 we have ΓC^{−∞}(Λ, V (σ_{λ}, φ)) = 0 and hence C^{−∞}(∂X, V (σ_{λ}, φ))_{d} = 0. Now consider the case λ = 0. If ϕ ∈ C^{−∞}(∂X, V (σ_{λ}, φ))_{d}, then the claim implies res_{Ω}ϕ = 0. By the asymptotic expansion Lemma 6.2, 2., we see that ϕ is square-integrable iff res_{Ω} ∘ Ĵ_{0}ϕ = 0. This proves that ΓC^{−∞}(∂X, V (σ_{0}, φ))_{d} = U_{Λ}(σ_{0}, φ). If the latter space is non-trivial, then σ ≠ 1 and 0 ∈ I_{σ} by Corollary 6.8.

The proof of (57) is analogous to the proof of Proposition 9.2. Let ϕ ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)). One has to show that for some Poisson transform P the condition P ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)) implies res(ϕ) = 0. We give the argument for the most involved case that σ is Weyl-invariant, λ = 0, and Ĵ_{0} regular. The similar treatment of the remaining cases is left to the reader.

In fact, in this case we show at once that res(ϕ) = 0 and res(Ĵ_{0}ϕ) = 0. Recall that π^{σ,0} splits into the ±1-eigenspaces π^{σ,±} of J_{0}. Let γ_{±} be the minimal K-type of π^{σ,±}. Choose embeddings T^{±} ∈ Hom_{M}(V_{σ}, V_{γ_{±}}) and set P^{±} := P_{0}^{T^{±}} ⊗ id. Define t^{±} ∈ Hom_{M}(Vσ̃, Vγ̃_{±}) by t(T^{±})^{w} ∘ t^{±} = id_{Vσ̃}. Using (36) and Corollary 6.3 we have for f ∈ C^{∞}(∂X, V (σ̃_{0}, φ̃))

⟨c_{γ_{±}}(0)T^{±}ϕ + (T^{±})^{w}Ĵ_{0}ϕ, t^{±}f⟩ = lim_{a→∞} a^{ρ} ∫_{K}⟨P^{±}ϕ(ka), t^{±}f(k)⟩dk .  (58)

We can rewrite the left hand side as follows:

⟨c_{γ_{±}}(0)T^{±}ϕ + (T^{±})^{w}Ĵ_{0}ϕ, t^{±}f⟩ = ⟨±c_{σ}(0)(T^{±})^{w}ϕ + (T^{±})^{w}Ĵ_{0}ϕ, t^{±}f⟩
= ⟨±c_{σ}(0)ϕ + Ĵ_{0}ϕ,^{t}(T^{±})^{w}t^{±}f⟩
= ⟨±c_{σ}(0)ϕ + Ĵ_{0}ϕ, f⟩ .

We continue tranforming the right-hand side of (58).

⟨±c_{σ}(0)ϕ + Ĵ_{0}ϕ, f⟩ = lim_{n→∞} \frac{1}{n} ∫_{0}^{n} e^{tρ} ∫_{K}⟨P^{±}ϕ(k exp(tH)), t^{±}f(k)⟩ dk dt
= lim_{n→∞} ∫_{G}⟨P ϕ(x), f_{n}^{±}(x)⟩ dx,

where H ∈ 𝔞_{+} is the unit vector, and the compactly supported sections f_{n}^{±} ∈ L^{2}(X, V (γ̃_{±}, φ̃)) are defined by f_{n}^{±}(k_{1}ak_{2}) := \frac{1}{n}a^{−ρ}χ_{(0,n]}(log(a))γ̃_{±}(k_{2}^{−1})t^{±}f(k_{1}). Here χ_{(0,n]} denotes the characteristic function of the interval (0, n].

Assume now that supp(f) ⊂ Ω and consider F_{n}^{±} ∈ L^{2}(Y, V_{Y} (γ̃_{±}, φ̃)) given by

F_{n}^{±}(x) := ∑_{g∈Γ} φ̃(g)f_{n}^{±}(g^{−1}x) .

Then as in the proof of Proposition 9.2 we see that lim_{n→∞} F_{n}^{±} = 0 in L^{2}(Y, V_{Y} (γ̃_{±}, φ̃)). Since P^{±}ϕ ∈ L^{2}(Y, V_{Y} (γ_{±}, φ)) we obtain

⟨±c_{σ}(0)ϕ + Ĵ_{0}ϕ, f⟩ = lim_{n→∞} ∫_{Γ\G}⟨P^{±}ϕ(x), F_{n}^{±}(x)⟩ dx = lim_{n→∞}⟨P^{±}ϕ, F_{n}^{±}⟩ = 0 .

Since c_{σ}(0) ≠ 0 this proves res(ϕ) = 0 and res(Ĵ_{0}ϕ) = 0. □

The next lemma is independent of the theory of tempered invariant distribution vectors.

Lemma 9.6 Let (σ, λ), (τ, μ) ∈ M̂ × 𝔞_{ℂ}^{∗}, and let

A : C^{−∞}(∂X, V (σ_{λ})) → C^{−∞}(∂X, V (τ_{μ}))

be a G-intertwining operator. Let (φ, V_{φ}) be a finite-dimensional (not necessarily unitary) representation of Γ. We consider the operator

A_{Γ,φ} : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (τ_{μ}, φ))

induced by A ⊗ id. Then im(A_{Γ,φ}) is infinite-dimensional unless im(A) is finite-dimensional.

Proof. The operator A restricts to a continuous operator

A : C^{∞}(∂X, V (σ_{λ})) → C^{∞}(∂X, V (τ_{μ})) .

In fact, A induces an intertwining operator of the underlying (𝔤, K)-modules which canonically extends to a continuous G-map between the spaces of smooth sections by the globalization theory of Casselman and Wallach [56], Ch. 11.

If f is a distribution section of a vector bundle over some manifold U, then let singsupp(f) ⊂ U denote the singular support of f. We claim that singsupp A(f) ⊂ singsupp(f) for all f ∈ C^{−∞}(∂X, V (σ_{λ}, φ)).

Consider the delta distribution δv ∈ C^{−∞}(∂X, V (σ_{λ})) at x_{0} := MAN ∈ ∂X with vector part v ∈ V_{σ}. Since δv is N-invariant we have A(δv) ∈ ^{N}C^{−∞}(∂X, V (τ_{μ})). The Bruhat decomposition ∂X = Nwx_{0} ∪ {x_{0}} implies that A(δv) is smooth outside x_{0}. Now let f ∈ C^{−∞}(∂X, V (σ_{λ})). Then there exists f̃ ∈ C_{c}^{−∞}(G) such that singsupp(f̃)P = singsupp(f) and π^{σ,λ}(f̃)δv = f. It follows that A(f) = π^{τ,μ}(f̃)A(δv), and hence singsuppA(f) ⊂ singsupp(f̃)P {x_{0}} = singsupp(f). This shows the claim. In particular, A_{Γ,φ} maps ^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)) into ^{Γ}C_{Ω}^{−∞}(∂X, V (τ_{μ}, φ)) (see Lemma 6.6 for notation).

Assume now that dim im(A_{Γ,φ}) < ∞. By Lemma 6.6 the space A_{Γ,φ} (^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ))) is dense in im(A_{Γ,φ}), hence A_{Γ,φ} (^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ))) = im(A_{Γ,φ}). We conclude that im(A_{Γ,φ}) ⊂ ^{Γ}C_{Ω}^{−∞}(∂X, V (τ_{μ}, φ)).

Without loss of generality we may assume that x_{0} ∈ Ω. Choose 0 ≠ w ∈ V_{φ} and consider the delta distribution

T := ∑_{g∈Γ}(π^{σ,λ}(g) ⊗ φ(g))(δv ⊗ w) ∈ ^{Γ}C^{−∞}(Ω, V (σ_{λ}, φ)) ≅ C^{−∞}(B, V_{B}(σ_{λ}, φ)) .

Since the singular parts of ext are finite-dimensional we find a smooth section ϕ ∈ C^{∞}(B, V_{B}(σ_{λ}, φ)) such that f := ext(T − ϕ) ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) is defined. We decompose δv ⊗ w = f − (f − δv ⊗ w). Since x_{0} ∉ singsupp(f − δv ⊗ w) and x_{0} ∉ singsuppA_{Γ,φ}(f) we conlude that A(δv) is smooth at x_{0}, and hence A(δv) ∈ ^{N}C^{∞}(∂X, V (τ_{μ})).

Now

NC^{∞}(∂X, V (τ_{μ})) = H^{0}(𝔫, C^{∞}(∂X, V (τ_{μ}))) ≅ H^{0}(𝔫, C^{∞}(∂X, V (τ_{μ}))_{K}),

where the second equality is a special case of Casselman’s comparison theorem for 𝔫-cohomology (see e.g. [12] or [27]). Thus A(δv) is K-finite. There exists a finite-dimensional subspace E ⊂ U(𝔤) such that U(𝔤) = U(𝔨)ZE U(𝔫) (see [55], 3.7.1). Thus the space Z_{K} := U(𝔤)A(δv) = U(𝔨)EA(δv) is finite-dimensional.

Since δv generates C^{−∞}(∂X, V (σ_{λ})), the element A(δv) generates im(A). We conclude that im(A)_{K} = Z_{K} and thus dim im(A) < ∞. This finishes the proof of the lemma. □

Corollary 9.7 Let (φ, V_{φ}) be a finite-dimensional (not necessarily unitary) representation of Γ, and let V_{π} be an irreducible admissible representation of G. Then ^{Γ}(V_{π,−∞} ⊗ V_{φ}) is infinitedimensional unless V_{π} is finite-dimensional.

Proof. By Casselman’s subrepresentation theorem ([55], 3.8.3.) in conjunction with the functorial properties of the smooth globalization ([56], 11.6.7.) we find elements (σ, λ), (τ, μ) ∈ M̂ × 𝔞_{ℂ}^{∗} such that V_{π,−∞} is a quotient of C^{−∞}(∂X, V (σ_{λ})) and a submodule of C^{−∞}(∂X, V (τ_{μ})). Thus there is a non-trivial G-intertwining operator

A : C^{−∞}(∂X, V (σ_{λ})) → C^{−∞}(∂X, V (τ_{μ}))

satisfying im(A) ≅ V_{π,−∞}. If dim(V_{π}) = ∞, then by Lemma 9.6 the subspace im(A_{Γ,φ}) ⊂ Γ(V_{π,−∞} ⊗ V_{φ}) is infinite-dimensional. □

The following proposition completes our description of the tempered and square-integrable invariant distribution vectors.

Proposition 9.8 Let (π, V_{π}) ∈ Ĝ_{d}. Then for any unitary representation (φ, V_{φ}) of Γ the space Γ(V_{π,−∞} ⊗ V_{φ})_{d} is infinite-dimensional.

Proof. Let γ ∈ K̂ be the minimal K-type of V_{π}. Casselman’s subrepresentation theorem provides an embedding

β : V_{π,∞} → C^{∞}(∂X, V (σ_{−λ})) ,

where Hom_{M}(V_{σ}, V_{γ}) ≠ 0, and −(λ + ρ) is the leading exponent in the asymptotic expansion for a → ∞ of the matrix coefficients c_{v,ṽ}, v ∈ V_{π,∞}, ṽ ∈ V_{π′,K}. Since V_{π} is a discrete series representation we have 0 < λ ∈ 𝔞^{∗}. Forming the adjoint with respect to hermitian scalar products we obtain a projection

q : C^{−∞}(∂X, V (σ_{λ})) → V_{π,−∞} .

By functoriality we can extend β to a map between the corresponding spaces of distribution vectors and obtain a G-intertwining operator

A := β ∘ q : C^{−∞}(∂X, V (σ_{λ})) → C^{−∞}(∂X, V (σ_{−λ}))

satisfying im(A) ≅ V_{π,−∞}. As in Lemma 9.6 we consider the operator

A_{Γ,φ} : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → ^{Γ}C^{−∞}(∂X, V (σ_{−λ}, φ)) .

This lemma combined with Lemma 6.6 tells us that A_{Γ,φ} (^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ))) is infinitedimensional. Hence Z := (q⊗id) (^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ))) ⊂ ^{Γ}(V_{π,−∞}⊗V_{φ}) is infinite-dimensional, too. It remains to show that Z ⊂ ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{d}.

Choose an embedding t ∈ Hom_{K}(Vγ̃, V_{π′}), and define T ∈ Hom_{M}(V_{σ}, V_{γ}) by ⟨T (w), v⟩ := ⟨w, [^{t}q ∘ t(v)](e)⟩ for all w ∈ V_{σ}, v ∈ Vγ̃. Recall the definition of v_{T} from Lemma 9.1 and observe that v_{T} = ^{t}q(t(v)). We consider the Poisson transform

P := P_{λ}^{T} ⊗ id : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{∞}(X, V (γ, φ)) .

By Lemma 9.1 we find for all ϕ ∈ C^{−∞}(∂X, V (σ_{λ}, φ)), v ∈ Vγ̃, g ∈ G

⟨P ϕ(g), v⟩ = c_{ϕ,v_{T}} (g) = c_{(q⊗id)(ϕ),t(v)}(g) ∈ V_{φ} .  (59)

Since t(v) generates V_{π′,K} by Lemma 8.4 it suffices to show that P ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)) for all ϕ ∈ ^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)).

Let D be a closed neighbourhood of clo(F) ∩ Ω for some fundamental domain F ⊂ X of Γ, and let Q ⊂ ∂X \ D be a closed neighbourhood of Λ. Let χ ∈ C^{∞}(∂X) be a cut-off function with supp(χ) ⊂ Q, supp(1 − χ) ∩ Λ = ∅. Let ϕ ∈ ^{Γ}C_{Ω}^{−∞}(∂X, V (σ_{λ}, φ)). By Lemma 6.2, 3., there exists a constant C such that for a ≫ 0, k ∈ DM

|P (χϕ)(ka)| ≤ Ca^{−(λ+ρ)} .  (60)

On the other hand (1 − χ)ϕ ∈ C^{∞}(∂X, V (σ_{λ}, φ)). By (59) the function P ((1 − χ)ϕ) can be expressed in terms of matrix coefficients c_{s,t} of the discrete series representation V_{π}, where s ∈ V_{π,∞} and t ∈ V_{π′,K}. Thus P ((1−χ)ϕ) has an asymptotic expansion with leading exponent −(λ+ρ). In particular, it satisfies an estimate of the form (60), too. We conclude that P ϕ ∈ L^{2}(Y, V_{Y} (γ, φ)). This finishes the proof of the proposition. □

10 Eisenstein series, wave packets, and scalar products

In this section we consider the Eisenstein series and the wave-packet transform. For a moment we can drop the unitarity condition on φ. Let γ be a finite-dimensional unitary representation of K, σ ∈ M̂, T ∈ Hom_{M}(V_{σ}, V_{γ}), and let P_{λ}^{T}, λ ∈ 𝔞_{ℂ}^{∗}, be the associated Poisson transform (see Definition 4.8).

Definition 10.1 For ϕ ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)) we define the Eisenstein series E(λ, ϕ, T) ∈ C^{∞}(Y, V_{Y} (γ, φ)) by

E(λ, ϕ, T) := P_{λ}^{T} ∘ ext(ϕ) .

The Eisenstein series E(λ, ϕ, T) is an eigenvector of Z for the infinitesimal character of the principal series representation π^{σ,λ}. Theorem 5.10 and the functional equation of the Poisson transform (18) have the following immediate corollary.

Corollary 10.2 The Eisenstein series gives rise to a meromorphic family defined on 𝔞_{ℂ}^{∗} (or {Re(λ) > δ_{Γ} + δ_{φ}} in case that X = 𝕆H^{2}) of continuous maps

E(λ, ., T) : C^{−∞}(B, V_{B}(σ_{λ}, φ)) → C^{∞}(Y, V_{Y} (γ, φ))

with finite-dimensional singularities. It satisfies the functional equation

E(λ, Ŝ_{−λ}ϕ, T) = E(−λ, ϕ, (c_{γ}(λ)T)^{w}) .  (61)

From Lemma 6.2, 3., we gain detailed knowledge of the asymptotics of E(λ, ϕ, T)(y) for y → b ∈ B (see the discussion of (66) in the proof of Proposition 10.8 below).

We now return to our unitarity assumption on φ.

Corollary 10.3

1. The Eisenstein series is regular on {Re(λ) = 0, λ ≠ 0}.

2. If Re(λ) = 0, λ ≠ 0, then E(λ, ., T) maps C^{−∞}(B, V_{B}(σ_{λ}, φ)) continuously to C′(Y, V_{Y} (γ, φ)).

3. In the half plane {Re(μ) ≥ 0} the Eisenstein series has at most first-order poles which are located in the finite set P S_{res}(σ, φ) ⊂ [0, δ_{Γ}]. The residue at λ ∈ P S_{res}(σ, φ) \ {0}

res_{μ=λ}E(μ, ., T)

maps C^{−∞}(B, V_{B}(σ_{λ}, φ)) to L^{2}(Y, V_{Y} (γ, φ)).

Proof. 1. follows from Lemma 7.2. To see 2. note that π^{σ,λ} is tempered if Re(λ) = 0, that E(λ, ϕ, T) can be expressed in terms of matrix coefficients of π^{σ,λ} (Lemma 9.1), and apply 9.4. We have

res_{μ=λ}E(μ, ., T)(ϕ) = P_{λ}^{T} ∘ (res_{λ}ext)(ϕ) .  (62)

3. now follows from Propositions 7.3, 7.4, and 7.8. □

By (62) the following proposition can also be considered as the determination of the L^{2}-scalar product between residues of Eisenstein series.

We introduce a scalar product on Hom_{M}(V_{σ}, V_{γ}) by (T_{1}, T_{2})id_{V_{σ}} := T_{2}^{∗}T_{1}. This makes sense because of our standing assumption σ ∈ M̂. By (., .)_{B} we denote the natural sesquilinear pairing between C^{−∞}(B, V_{B}(σ_{λ}, φ)) and C^{∞}(B, V_{B}(σ_{−λ̄}, φ)).

Proposition 10.4 Let λ ∈ P S_{res}(σ, φ) \ {0}, let γ be a finite-dimensional representation of K, T_{1}, T_{2} ∈ Hom_{M}(V_{σ}, V_{γ}), ϕ_{1} = (res_{λ}ext)(f) ∈ E_{Λ}(σ_{λ}, φ) for some f ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)), and ϕ_{2} ∈ ^{Γ}C^{−∞}(Λ, V (σ_{λ}, φ)). Then

(P_{λ}^{T_{1}}ϕ_{1}, P_{λ}^{T_{2}}ϕ_{2})_{L^{2}(Y,V_{Y} (γ,φ))} = ω_{X} (c_{γ}(λ)T_{1}, T_{2}^{w}) (f, res ∘ Ĵ_{λ}ϕ_{2})_{B} .

Here ω_{X} = \frac{ω_{n}}{2^{r}α^{n−1}} , where n = dim X, ω_{n} = vol(S^{n−1}) = \frac{2π^{\frac{n}{2}}}{Γ( \frac{n}{2} )} and r ∈ ℕ is such that rα = 2ρ.

If γ is the minimal K-type of π^{σ,λ}, then

(c_{γ}(λ)T_{1}, T_{2}^{w}) = c_{σ}(λ)(T_{1}, T_{2}) .

Proof. First we can assume that f is smooth. We extend f to a holomorphic family μ ↦ f_{μ} ∈ C^{∞}(B, V_{B}(σ_{μ}, φ)) defined in a neighbourhood of λ. Let B_{R} ⊂ X be the ball of radius R, and let (., .)_{B_{R}} denote the scalar product in L^{2}(B_{R}, V (γ, φ)). Let χ ∈ C^{∞}(X) be the cut-off function constructed in Lemma 6.4. Set

S_{R}(μ) := (μ − λ) (P_{μ}^{T_{1}}ext(f_{μ}), χP_{λ}^{T_{2}}ϕ_{2})_{B_{R}} .

Then

(P_{λ}^{T_{1}}ϕ_{1}, P_{λ}^{T_{2}}ϕ_{2})_{L^{2}(Y,V_{Y} (γ,φ))} = lim_{R→∞} lim_{μ→λ} S_{R}(μ) .

Let D := −Ω_{G} + c(σ) + λ^{2} be the shifted Casimir operator. As in the proof of Proposition 7.3 we obtain

S_{R}(μ) = −\frac{1}{λ + μ} (DP_{μ}^{T_{1}}ext(f_{μ}), χP_{λ}^{T_{2}}ϕ_{2})_{B_{R}} .

While P_{μ}^{T_{1}}ext(f_{μ}) has a first order pole at μ = λ, DP_{μ}^{T_{1}}ext(f_{μ}) is regular, and its value at μ = λ equals DF, where F ∈ ^{Γ}C^{∞}(X, V (γ, φ)) is the constant term of the Laurent expansion of P_{μ}^{T_{1}}ext(f_{μ}) at λ.

Note that D = ∇^{∗}∇ + R for some selfadjoint endomorphism R of V (γ, φ), where ∇^{∗}∇ is the Bochner Laplacian associated to the invariant connection ∇ of V (γ, φ). Thus we can apply Green’s formula in the same spirit as in the proof of Proposition 6.5:

lim_{μ→λ} S_{R}(μ) = −\frac{1}{2λ} ((DF, χP_{λ}^{T_{2}}ϕ_{2})_{B_{R}} − (F, DχP_{λ}^{T_{2}}ϕ_{2})_{B_{R}} + (F, [D, χ]P_{λ}^{T_{2}}ϕ_{2})_{B_{R}})
= \frac{1}{2λ} ((∇_{n}F, χP_{λ}^{T_{2}}ϕ_{2})_{∂B_{R}} − (F, ∇_{n}χP_{λ}^{T_{2}}ϕ_{2})_{∂B_{R}} − (F, [D, χ]P_{λ}^{T_{2}}ϕ_{2})_{B_{R}}) . (63)

Applying Cauchy’s integral formula to the asymptotic expansion of P_{μ}^{T_{1}}ext(f_{μ}) given in Lemma 6.2, 3. we find some ϵ > 0 such that for a → ∞

F (ka) = a^{λ−ρ}c_{γ}(λ)T_{1}f(k) + O(a^{λ−ρ−ϵ})

uniformly as kM varies in compact subsets of Ω.

As in the proof of Proposition 6.5 we obtain (F, [D, χ]P_{λ}^{T_{2}}ϕ_{2})_{X} = 0. In order to perform the limit R → ∞ in (63) we use lim_{R→∞} e^{−2ρR}vol(∂B_{R}) = ω_{X} and the asymptotic expansions of F, P_{λ}^{T_{2}}ϕ_{2}, and obtain

(P_{λ}^{T_{1}}ϕ_{1}, P_{λ}^{T_{2}}ϕ_{2})_{L^{2}(Y,V_{Y} (γ,φ))} = \frac{ω_{X}}{2λ} ((λ − ρ) ∫_{∂X}(c_{γ}(λ)T_{1}f(k), χ_{∞}(k)T_{2}^{w}(Ĵ_{λ}ϕ_{2})(k)) dk
+(λ + ρ) ∫_{∂X}(c_{γ}(λ)T_{1}f(k), χ_{∞}(k)T_{2}^{w}(Ĵ_{λ}ϕ_{2})(k)) dk)
= ω_{X}(T_{2}^{w∗}c_{γ}(λ)T_{1}f, res ∘ Ĵ_{λ}ϕ_{2})_{B} .

This finishes the proof of the proposition. □

On I^{σ,λ} we consider consider the scalar product (56). Let (., .) be the scalar product on E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ) = ^{Γ}(I_{−∞}^{σ,λ} ⊗ V_{φ})_{d} induced by the matrix coefficient map (see Proposition 9.2 and Corollary 8.7).

Corollary 10.5 The decomposition E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ) is orthogonal with respect to (., .). If ϕ_{1}, ϕ_{2} ∈ E_{Λ}(σ_{λ}, φ), then

(ϕ_{1}, ϕ_{2}) = \frac{ω_{X}c_{σ}(λ)}{dim(V_{σ})}(f, res ∘ Ĵ_{λ}ϕ_{2}) ,

where f ∈ C^{−∞}(B, V_{B}(σ_{λ}, φ)) such that ϕ_{1} = (res_{λ}ext)(f).

Proof. Let γ be the minimal K-type of π^{σ,λ}. For T ∈ Hom_{M}(V_{σ}, V_{γ}) and v ∈ Vγ̃ let v_{T} ∈ C^{∞}(∂X, V (σ̃_{±λ})) be given by v_{T} (k) = ^{t}T γ̃(k^{−1})v. We have J_{σ̃,−λ}v_{T} = v_{T}. In fact, σ is Weylinvariant and thus for all ϕ ∈ C^{−∞}(∂X, V (σ_{−λ}))

⟨ϕ, J_{σ̃,−λ}v_{T} ⟩ = ⟨J_{σ,−λ}ϕ, v_{T} ⟩
= ⟨Ĵ_{σ,−λ}c_{σ}(λ)^{−1}ϕ, v_{T} ⟩
=^{9.1} c_{σ}(λ)^{−1}⟨P_{λ}^{T} ∘ Ĵ_{σ,−λ}(ϕ)(1), v⟩
=^{(18)} c_{σ}(λ)^{−1}⟨P_{−λ}^{(c_{γ}(λ)T)^{w}}(ϕ)(1), v⟩
=^{(22)} ⟨P_{−λ}^{T}(ϕ)(1), v⟩
= ⟨ϕ, v_{T} ⟩ .

By (56) we have

([v_{T}], [v_{T}]) = (v_{T} , J_{σ̃,−λ}v_{T})_{0}
= ∫_{K} ∥v_{T} (k)∥^{2}dk
= ∫_{K}(^{t}T γ̃(k^{−1})v,^{t}T γ̃(k^{−1})v)dk
= ∫_{K}(γ̃(k)(^{t}T)^{∗}^{t}T γ̃(k^{−1})v, v)dk
= \frac{dim(Vσ̃)}{dim(Vγ̃)}∥v∥^{2}∥T ∥^{2} .  (64)

Let {v^{i}}_{i=1}^{dim(Vγ̃)} be an othonormal base of Vγ̃. We compute using Proposition 10.4, Lemma 9.1

\frac{ω_{X}c_{σ}(λ)}{dim(V_{σ})}(f, res ∘ Ĵ_{λ}ϕ_{2}) = \frac{1}{∥T ∥^{2} dim(V_{σ})}(P_{λ}^{T} ϕ_{1}, P_{λ}^{T} ϕ_{2})_{L^{2}(Y,V_{Y} (γ,φ))}

= \frac{1}{∥T ∥^{2} dim(V_{σ})} ∑_{i=1}^{dim(V_{γ})}(c_{ϕ_{1},v_{T}^{i}} , c_{ϕ_{2},v_{T}^{i}})_{L^{2}(Γ\G,φ)}

= \frac{1}{∥T ∥^{2} dim(V_{σ})} ∑_{i=1}^{dim(V_{γ})}(ϕ_{1}, ϕ_{2})∥v_{T}^{i} ∥^{2}
= (ϕ_{1}, ϕ_{2}) .

This proves the corollary. □

Now we turn to the definition of the wave packet transform. Roughly speaking, a wave packet of Eisenstein series is an average of the Eisenstein series over imaginary parameters with, say, a smooth, compactly supported weight function with respect to the expected Plancherel measure p_{σ}(iμ)dμ. More precisely, let 𝔞_{+}^{∗} := {λ ∈ 𝔞^{∗} | ⟨λ, α⟩ > 0} be the open positive chamber in 𝔞^{∗}. Then the space of such weight functions H_{0}^{σ}(φ) is the linear space of smooth families 𝔞_{+}^{∗} ∋ μ ↦ ϕ_{iμ} ∈ C^{∞}(B, V_{B}(σ_{iμ}, φ)) with compact support in 𝔞_{+}^{∗} with respect to μ. Because of the functional equation (61) it will be sufficient to consider wave packets on the positive imaginary axis, only.

Let γ be a finite-dimensional unitary representation of K. We first define the wave packet transform on H_{0}^{σ}(φ) ⊗ Hom_{M}(V_{σ}, V_{γ}). Later we will extend it by continuity to a Hilbert space closure.

Definition 10.6 The wave packet transform is the map

E : H_{0}^{σ}(φ) ⊗ Hom_{M}(V_{σ}, V_{γ}) → C^{∞}(Y, V_{Y} (γ, φ))

given by

E(ϕ ⊗ T) := E(ϕ, T) := ∫_{𝔞_{+}^{∗}} E(iμ, ϕ_{iμ}, T)p_{σ}(iμ) dμ ,

where dμ is the Lebesgue measure on 𝔞_{+}^{∗} ≅ (0, ∞). The section E(ϕ, T), ϕ ∈ H_{0}^{σ}(φ), T ∈ Hom_{M}(V_{σ}, V_{γ}) is called a wave packet (of Eisenstein series).

Lemma 10.7 If T ∈ Hom_{M}(V_{σ}, V_{γ}), ϕ ∈ H_{0}^{σ}(φ), then E(ϕ, T) ∈ C(Y, V_{Y} (γ, φ)).

Proof. Set ψ_{iμ} := ext ϕ_{iμ} and define

P (ψ) := ∫_{𝔞_{+}^{∗}} P_{iμ}^{T}(ψ_{iμ})p_{σ}(iμ) dμ .

Let χ be the cut-off function constructed in Lemma 6.4. In view of Definition 8.1 of the Schwartz space we have to show that χP (ψ) ∈ C(X, V (γ, φ)). Let χ_{0} ∈ C_{c}^{∞}(X) be some cut-off function which is equal to 1 on some neighbourhood of eK ∈ X. Obviously, we have χχ_{0}P (ψ) ∈ C_{c}^{∞}(X, V (γ, φ)) ⊂ C(X, V (γ, φ)). It remains to show that

χ_{1}P (ψ) ∈ C(X, V (γ, φ)) ,

where χ_{1} = χ(1 − χ_{0}). Observe that the seminorms q_{D,N}, D ∈ U(𝔤), N ∈ N_{0} defined by

q_{D,N}(f)^{2} := ∫_{G} | log(a_{g})^{N}f(Dg)|^{2} dg

are sufficient in order to define the topology on C(X, V (γ, φ)). We thus have to show that g ↦ log(a_{g})^{N}(χ_{1}P (ψ))(Dg) is square-integrable. Let Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) denote the coproduct and write Δ(D) = ∑_{α} D_{α} ⊗ D′_{α} for some D_{α}, D′_{α} ∈ U(𝔤). Then (χ_{1}P (ψ))(Dg) = ∑_{α} χ_{1}(D_{α}g)P (ψ)(D′_{α}g). By Lemma 6.4, 4., g ↦ χ_{1}(Dg) is bounded for any D ∈ U(𝔤). It remains to show for any D ∈ U(𝔤) and cut-off function χ_{2} ∈ C^{∞}(X) having compact support in (X ∪ Ω) \ {eK} the function

log(a_{g})^{N}χ_{2}(g)P (ψ)(Dg) = log(a_{g})^{N}χ_{2}(ka) ∫_{𝔞_{+}^{∗}} P_{iμ}^{T}(π^{σ,iμ}(D)ψ_{iμ})(g)p_{σ}(iμ) dμ  (65)

is square integrable. By Lemma 6.2, 3., there exists ϵ > 0 such that for D ∈ U(𝔤), ka ∈ supp(χ_{2})

P_{iμ}^{T}(π^{σ,iμ}(D)ψ_{iμ})(ka) = a^{iμ−ρ}c_{γ}(iμ)T ψ_{iμ}(k)
+a^{−iμ−ρ}T^{w}(Ĵ_{iμ}(π^{σ,iμ}(D)ψ_{iμ}))(k)  (66)
+a^{−(ρ+ϵ)}R(iμ, π^{σ,iμ}(D)ψ_{iμ}, ka) ,

where the remainder function (μ, ka) ↦ R(iμ, π^{σ,iμ}(D)ψ_{iμ}, ka) is uniformly bounded.

Since the families ψ_{iμ} and Ĵ_{iμ}(π^{σ,iμ}(D)ψ_{iμ}) have compact support with respect to μ and are smooth in (μ, k) (as long kM ∈ Ω) by Lemma 5.3 each summand of (66) contributes to the function (65) a summand which is bounded by C_{N′}(1 + log(a_{g}))^{−N′}a_{g}^{−ρ} for any N′ ∈ ℕ. It follows that the function (65) is square integrable. This implies the lemma. □

Let σ^{i} ∈ M̂, T_{i} ∈ Hom_{M}(V_{σ^{i}}, V_{γ}), i = 1, 2, λ ∈ 𝔞_{+}^{∗}, ϕ ∈ H_{0}^{σ^{2}}(φ), and ψ ∈ C^{−∞}(B, V_{B}(σ^{1}_{iλ}, φ)). By Lemma 10.7 and Corollary 10.3 the pairing (E(iλ, ψ, T_{1}), E(ϕ, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))} between the Eisenstein series E(iλ, ψ, T_{1}) and the wave packet E(ϕ, T_{2}) is well-defined. The following proposition gives an explicit formula for this pairing and is the crucial step in the determination of the absolute continuous part of the Plancherel measure.

Proposition 10.8 We have

(E(iλ, ψ, T_{1}), E(ϕ, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))} = \{ \begin{matrix}2πω_{X} (T_{1}, T_{2}) (ψ, ϕ_{iλ})_{B} & σ^{1} = σ^{2} \\ 0 & σ^{1} ≠ σ^{2} \end{matrix} .

Proof. For fixed ψ and T_{i}, i = 1, 2, we consider the continuous linear functional (which is in fact a distribution section of a bundle over i𝔞_{+}^{∗} × B)

R : H_{0}^{σ^{2}}(φ) → ℂ

given by R(ϕ) := (E(iλ, ψ, T_{1}), E(ϕ, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))}. If D ∈ Z and σ ∈ M̂ we consider the polynomial χ_{μ_{σ}+ρ_{m}−μ}(D) on 𝔞_{ℂ}^{∗}. Since χ_{μ_{σ}+ρ_{m}−μ} is the infinitesimal character of π^{σ,μ} we have

0 = ((D^{∗} − χ_{μ_{σ^{1}}+ρ_{m}−iλ}(D^{∗}))E(iλ, ψ, T_{1}), E(ϕ, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))}
= (E(iλ, ψ, T_{1}), (D − χ̄_{μ_{σ^{1}}+ρ_{m}−iλ}(D^{∗}))E(ϕ, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))}
= (E(iλ, ψ, T_{1}), E(ϕ̃, T_{2}))_{L^{2}(Y,V_{Y} (γ,φ))} ,

where ϕ̃_{iμ} = (χ_{μ_{σ^{2}}+ρ_{m}−iμ}(D) − χ_{μ_{σ^{1}}+ρ_{m}−iλ}(D))ϕ_{iμ}. We conclude that multiplication by the polynomial (χ_{μ_{σ^{2}}+ρ_{m}−iμ}(D)−χ_{μ_{σ^{1}}+ρ_{m}−iλ}(D)) annihilates the functional R. Thus R is supported on the zero set of this polynomial. If σ^{1} ≠ σ^{2}, then

⋂_{D∈Z}{μ ∈ 𝔞_{+}^{∗} | (χ_{μ_{σ^{2}}+ρ_{m}−iμ}(D) − χ_{μ_{σ^{1}}+ρ_{m}−iλ}(D)) = 0} = ∅

and therefore R = 0. This proves the proposition in case σ^{1} ≠ σ^{2}.

Assume now that σ^{1} = σ^{2} =: σ. Observe that

⋂_{D∈Z}{μ ∈ 𝔞_{+}^{∗} | (χ_{μ_{σ}+ρ_{m}−iμ}(D) − χ_{μ_{σ}+ρ_{m}−iλ}(D)) = 0} = {λ}

and that the functional R is of the form R(ϕ) = r(ϕ_{iλ}) for some r ∈ C^{−∞}(B, V_{B}(σ_{iλ}, φ)), where r remains to be determined. We prefer to give a direct proof of the proposition in case σ^{1} = σ^{2}, which does not refer to this observation.

Because of the continuity of E(iλ, ., T_{1}) (see Corollary 10.3, 2.) we can assume that ψ ∈ C^{∞}(B, V_{B}(σ_{iλ}, φ)). We apply Green’s formula in a similar way as in the proofs of Propositions 6.5 and 10.4.

Let χ be the cut-off function as constructed in Lemma 6.4 and B_{R} the ball of radius R around the origin of X. For μ ∈ 𝔞_{+}^{∗} we consider

S_{R}(μ) := (E(iλ, ψ, T_{1}), χE(iμ, ϕ_{iμ}, T_{2}))_{B_{R}} .

We have thus to compute

lim_{R→∞} ∫_{0}^{∞} S_{R}(μ)p_{σ}(iμ) dμ .

If we set A := −Ω_{G} + c(σ), then we obtain

(λ^{2} − μ^{2})S_{R}(μ) = (A E(iλ, ψ, T_{1}), χE(iμ, ϕ_{iμ}, T_{2}))_{B_{R}} − (E(iλ, ψ, T_{1}), AχE(iμ, ϕ_{iμ}, T_{2}))_{B_{R}}
+(E(iλ, ψ, T_{1}), [A, χ]E(iμ, ϕ_{iμ}, T_{2}))_{B_{R}}
= −(∇_{n}E(iλ, ψ, T_{1}), χE(iμ, ϕ_{iμ}, T_{2}))_{∂B_{R}}  (67)
+(E(iλ, ψ, T_{1}), ∇_{n}χE(iμ, ϕ_{iμ}, T_{2}))_{∂B_{R}}  (68)
+([A, χ]E(iμ, ϕ_{iμ}, T_{1}), E(iλ, ψ, T_{2}))_{B_{R}} .  (69)

We now apply the asymptotic expansion (66) which holds on the support of χ in case that D = 1, and for ext(ϕ_{iμ}) and ext(ψ) in place of ψ_{iμ}. Note that for k ∈ ΩM we have ext(ϕ_{iμ})(k) = ϕ_{iμ}(k), Ĵ_{iμ}(ext(ϕ_{iμ}))(k) = Ŝ_{iμ}(ϕ_{iμ})(k), etc. We obtain for large R with a_{R} := e^{R}, ω_{R} := a_{R}^{−2ρ}vol(∂B_{R})

\frac{1}{ω_{R}} ((67) + (68)) = i(−λ − μ)a_{R}^{i(λ−μ)} ∫_{K} χ_{∞}(k)(c_{γ}(iλ)T_{1}ψ)(k), c_{γ}(iμ)T_{2}ϕ_{iμ}(k)) dk
+i(−λ + μ)a_{R}^{i(λ+μ)} ∫_{K} χ_{∞}(k)(c_{γ}(iλ)T_{1}ψ(k), T_{2}^{w}(Ŝ_{iμ}ϕ_{iμ})(k)) dk
+i(λ − μ)a_{R}^{i(−λ−μ)} ∫_{K} χ_{∞}(k)(T_{1}^{w}Ŝ_{iλ}ψ)(k), c_{γ}(iμ)T_{2}ϕ_{iμ}(k)) dk
+i(λ + μ)a_{R}^{i(−λ+μ)} ∫_{K} χ_{∞}(k)(T_{1}^{w}(Ŝ_{iλ}ψ)(k), T_{2}^{w}(Ŝ_{iμ}ϕ_{iμ})(k)) dk
+o(1) .

The remainder term o(1) contains integrals over K of terms involving the normal derivative of χ, the difference χ(ka_{R}) − χ_{∞}(k) and the function a_{R}^{−ρ−ϵ}R(iμ, ., ka_{R}) appearing in (66). We combine this remainder with the term (69) divided by ω_{R} to F (λ, μ, R). Since the asymptotic expansion (66) can be differentiated with respect to μ, there exists a constant C ∈ ℝ such that

|F (λ, μ, R)| + |\frac{d}{dμ}F (λ, μ, R)| < C, ∀R > 0, μ ∈ 𝔞_{+}^{∗} .  (70)

We can write for μ ≠ λ

\frac{1}{ω_{R}} S_{R}(μ) = i\frac{a_{R}^{i(λ−μ)}}{−λ + μ} (T_{2}^{∗}c_{γ}(iμ)^{∗}c_{γ}(iλ)T_{1}ψ, ϕ_{iμ})_{B} + i\frac{a_{R}^{i(λ+μ)}}{−λ − μ} (T_{2}^{w∗}c_{γ}(iλ)T_{1}ψ, Ŝ_{iμ}ϕ_{iμ})_{B}
+i\frac{a_{R}^{i(−λ−μ)}}{λ + μ} ((Ŝ_{iλ}ψ, T_{1}^{w∗}c_{γ}(iμ)T_{2}ϕ_{iμ}))_{B} + i\frac{a_{R}^{i(−λ+μ)}}{λ − μ} (T_{2}^{w∗}T_{1}^{w}Ŝ_{iλ}ψ, Ŝ_{iμ}ϕ_{iμ})_{B} (71)

+\frac{F (λ, μ, R)}{λ^{2} − μ^{2}} .

By the Lemma of Riemann-Lebesgue

lim_{R→∞} ∫_{0}^{∞} \frac{a_{R}^{i(λ+μ)}}{−λ − μ} (T_{2}^{w∗}c_{γ}(iλ)T_{1}ψ, Ŝ_{iμ}ϕ_{iμ})_{B} p_{σ}(iμ) dμ = 0 ,
lim_{R→∞} ∫_{0}^{∞} \frac{a_{R}^{i(−λ−μ)}}{λ + μ} (Ŝ_{iλ}ψ, T_{1}^{w∗}c_{γ}(iμ)T_{2}ϕ_{iμ})_{B} p_{σ}(iμ) dμ = 0 .

We set s := μ − λ and regroup the remaining terms of (71) to

\frac{a_{R}^{is} − a_{R}^{−is}}{is} (T_{2}^{∗}c_{γ}(iμ)^{∗}c_{γ}(iλ)T_{1}ψ, ϕ_{iμ})_{B}  (72)
− a_{R}^{is} \frac{1}{is} ((T_{2}^{∗}c_{γ}(iμ)^{∗}c_{γ}(iλ)T_{1}ψ, ϕ_{iμ})_{B} − (T_{1}, T_{2})(Ŝ_{iλ}ψ, Ŝ_{iμ}ϕ_{iμ})_{B})  (73)
+ \frac{F (λ, μ, R)}{λ^{2} − μ^{2}} .

Note that (72) is smooth at μ = λ. We claim that (73) is smooth at μ = λ, too.

By Ŝ_{iλ}^{∗} = Ŝ_{−iλ} and the functional equation of the scattering matrix (30) we obtain

Ŝ_{iλ}^{∗}Ŝ_{iλ} = \frac{1}{p_{σ}(iλ)}id .

The claim now follows from

T_{2}^{∗}c_{γ}(iλ)^{∗}c_{γ}(iλ)T_{1} = \frac{(T_{1}, T_{2})}{p_{σ}(iλ)}  (74)

which is a consequence of (20) and (21).

Now (71) forces also \frac{F (λ,μ,R)}{λ^{2}−μ^{2}} to be smooth at λ = μ. By (70) and Lebesgue’s Dominated Convergence Theorem we obtain

lim_{R→∞} ∫_{0}^{∞} \frac{F (λ, μ, R)}{λ^{2} − μ^{2}} p_{σ}(iμ) dμ = 0 .

If we integrate (73) with respect to s and perform the limit R → ∞, then the result vanishes by the Riemann-Lebesgue lemma.

We now use (74) and the identity of distributions lim_{r→∞} \frac{sin(rs)}{s} = πδ_{0}(s) in order to compute

lim_{R→∞} ∫_{0}^{∞} S_{R}(μ)p_{σ}(iμ) dμ = lim_{R→∞} ω_{R} ∫_{−∞}^{∞} \frac{a_{R}^{is} − a_{R}^{−is}}{is} (T_{2}^{∗}c_{γ}(iμ)^{∗}c_{γ}(iλ)T_{1}ψ, ϕ_{iμ})_{B} p_{σ}(iμ) ds
= 2π ω_{X} (T_{1}, T_{2}) (ψ, ϕ_{iλ})_{B} .

This proves the proposition. □

11 The Plancherel theorem and spectral decompositions

In this final section we obtain our explicit Plancherel theorem, i.e., the decomposition of L^{2}(Γ\G, φ). We use the scalar product formula of Proposition 10.8 in order to show that the subspace of L^{2}(Γ\G, φ) spanned by the wave packets of Eisenstein series is the absolute continuous subspace, that its complement is the discrete subspace, and that there is no singular continuous subspace. It is not surprising that the absolute continuous part of the Plancherel measure κ coincides with the absolute continuous part for L^{2}(G).

As a consequence of the decomposition of the Plancherel theorem we derive the spectral decomposition of L^{2}(Y, V_{Y} (γ, φ)) with respect to the invariant differential operators.

We first introduce and describe certain subspaces of ∫ ⊕ ; Ĝ M_{π} ˆ⊗V_{π} dκ(π) corresponding to the partition Ĝ = Ĝ_{d} ∪ Ĝ_{u} ∪ Ĝ_{c} (see the beginning of Section 9).

For each π ∈ Ĝ we fix the scalar product on ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} such that the matrix coefficient map c_{π} (see Corollary 8.7) is unitary.

We define the Hilbert space associated to discrete series Ĝ_{d} by

H_{cusp}(φ) := ⨁_{π∈Ĝ_{d}}^{Hilbert} ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} ˆ⊗V_{π} .

According to Proposition 9.2 and Corollary 10.5 for π = π̄^{σ,λ} ∈ Ĝ_{c} we have an orthogonal decomposition

Γ(V_{π′,−∞} ⊗ V_{φ})_{d} = C^{−∞}(Λ, V (σ̃_{λ})) = E_{Λ}(σ̃_{λ}, φ) ⊕ U_{Λ}(σ̃_{λ}, φ) .

The same corollary gives an alternative expression for the restriction of the scalar product to E_{Λ}(σ_{λ}, φ) in terms of the boundary geometry. This space is non-tivial iff λ > 0 belongs to the finite index set P S(σ̃, φ) = P S_{res}(σ̃, φ) ∪ P S_{U}(σ̃, φ) introduced in Definition 7.7. We define the Hilbert spaces

H_{res}(φ) := ⨁_{{σ∈M̂ | p_{σ}(0)=0}}^{Hilbert} ⨁_{λ∈P S_{res}(σ̃,φ)\{0}} E_{Λ}(σ̃_{λ}, φ) ⊗ I^{σ,λ}

0H_{U}(φ) := ⨁_{{σ∈M̂ | p_{σ}(0)=0}}^{Hilbert} ⨁_{λ∈P S_{U} (σ̃,φ)\{0}} U_{Λ}(σ̃_{λ}, φ) ⊗ I^{σ,λ} .

The sum H_{res}(φ) ⊕_{0}H_{U}(φ) is the Hilbert space associated to the complementary series Ĝ_{c}.

Now we consider the Hilbert space asssociated to the unitary principal series. First we discuss the contribution of π^{σ,0}. We decompose M̂ = ∪_{i=1}^{3}M̂_{i} such that

1. σ ∈ M̂_{1} iff it is Weyl-invariant and p_{σ}(0) = 0 (i.e. π^{σ,0} irreducible),

2. σ ∈ M̂_{2} iff it is Weyl-invariant and p_{σ}(0) ≠ 0 (i.e. π^{σ,0} = π^{σ,+} ⊕ π^{σ,−}), and

3. σ ∈ M̂_{3} iff is not Weyl-invariant.

We define

1H_{U}(φ) := ⨁_{{σ∈M̂_{1} | 0∈P S_{U} (σ̃,φ)}}^{Hilbert} U_{Λ}(σ̃_{0}, φ) ⊗ L^{2}(∂X, V (σ_{0})) .

If σ ∈ M̂_{2}, then L^{2}(∂X, V (σ_{0})) = V_{π^{σ,+}} ⊕ V_{π^{σ,+}}, and we define

U_{Λ}^{±}(σ̃_{0}, φ) := {f ∈ U_{Λ}(σ̃_{0}, φ) | ⟨f, g⟩ = 0 ∀g ∈ V_{π^{σ,∓},∞}}
= {f ∈ C^{−∞}(Λ, V (σ̃_{0}, φ)) | J_{0}(f) = ±f} .

We define

2H_{U}(φ) := ⨁_{{σ∈M̂_{2} | 0∈P S_{U} (σ̃,φ)}}^{Hilbert}(U_{Λ}^{+}(σ̃_{0}, φ) ⊗ V_{π^{σ,+}}) ⊕ (U_{Λ}^{−}(σ̃_{0}, φ) ⊗ V_{π^{σ,−}}) .

Let M̂_{4} ⊂ M̂_{3} be a set of representatives of M̂_{3}/W (𝔤, 𝔞). We define

3H_{U}(φ) := ⨁_{{σ∈M̂_{4} | 0∈P S_{U} (σ̃,φ)}}^{Hilbert} U_{Λ}(σ̃_{0}, φ) ⊗ L^{2}(∂X, V (σ_{0})) .

It seems to be natural to collect together all spaces connected with U_{Λ}(σ_{λ}, φ), λ ∈ P S_{U}(σ, φ), and to define

H_{U}(φ) := ⨁_{i=0}^{3} _{i}H_{U}(φ) .

The main contribution of unitary principal series is the Hilbert space

H_{ac}(φ) := ⨁_{σ∈M̂}^{Hilbert} H_{ac}^{σ}(φ) ,

where

H_{ac}^{σ}(φ) := ∫_{𝔞_{+}^{∗}}^{⊕} L^{2}(B, V_{B}(σ̃_{iλ}, φ)) ˆ⊗L^{2}(∂X, V (σ_{−iλ})) \frac{2πω_{X}}{dim(V_{σ})}p_{σ}(iλ) dλ .

For σ ∈ M̂ we define the Wave packet transform

W P_{σ} : Hσ̃ ; 0(φ) ⊗ L^{2}(∂X, V (σ_{−iλ}))_{K} → L^{2}(Γ\G, φ)

in the following way (the space Hσ̃ ; 0(φ) is defined just before Definition 10.6). Consider T ∈ Hom_{M}(Vσ̃, Vγ̃) for some γ ∈ K̂, v ∈ V_{γ}, and ϕ ∈ Hσ̃ ; 0(φ). We take the element v_{T} ∈ L^{2}(∂X, V (σ_{−iλ}))_{K} (see Lemma 9.1) and form ϕ ⊗ v_{T}. Then we define

W P_{σ}(ϕ ⊗ v_{T}) = ⟨E(ϕ, T), v⟩ .

We employ the extension ext in order to identify the space L^{2}(B, V_{B}(σ̃_{iλ}, φ)) with a subspace of Γ(C^{−∞}(∂X, V (σ̃_{iλ})) ⊗ V_{φ})_{temp} which is our candidate of N_{π^{σ,−iλ}}. Then by Lemma 9.1 the wave packet transform W P_{σ} is related to the family of matrix coefficient maps {c_{π^{σ,−iλ}}}_{λ∈𝔞_{+}^{∗}} by

W P_{σ}(ϕ ⊗ v_{T}) = \frac{2πω_{X}}{dim(V_{σ})} ∫_{𝔞_{+}^{∗}} c_{π^{σ,−iλ}}(ext(ϕ) ⊗ v_{T})p_{σ}(iλ) dλ .  (75)

Note that the elements of the form ϕ ⊗ v_{T} span the dense subspace Hσ̃ ; 0(φ) ⊗ L^{2}(∂X, V (σ_{−iλ}))_{K} of H_{ac}^{σ}(φ).

We are now able to state the Plancherel theorem for L^{2}(Γ\G, φ). Recall that in case X = 𝕆H^{2} we assume δ_{Γ} < 0.

Theorem 11.1 The direct sum of the matrix coefficient maps c_{π} : ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} ⊗ V_{π,∞} → L^{2}(Γ\G, φ), π ∈ Ĝ, and the wave packet transforms W P_{σ}, σ ∈ M̂, extends to a unitary equivalence of G-representations

H_{ac}(φ) ⊕ H_{cusp}(φ) ⊕ H_{res}(φ) ⊕ H_{U}(φ) ≅ L^{2}(Γ\G, φ) .

It gives rise to a corresponding decomposition

L^{2}(Γ\G, φ) = L^{2}(Γ\G, φ)_{ac} ⊕ L^{2}(Γ\G, φ)_{d} ,

where the discrete subspace

L^{2}(Γ\G, φ)_{d} := L^{2}(Γ\G, φ)_{cusp} ⊕ L^{2}(Γ\G, φ)_{res} ⊕ L^{2}(Γ\G, φ)_{U}

is the sum of the cuspidal, the residual, and the ”stable” part.

L^{2}(Γ\G, φ)_{d} is the sum of all irreducible subrepresentations of L^{2}(Γ\G, φ). L^{2}(Γ\G, φ)_{cusp} decomposes into discrete series representations of G, each discrete series representation of G occurs with infinite multiplicity. It is empty iff X = ℝH^{n}, n odd. The remaining part of L^{2}(Γ\G, φ)_{d} consists of non-discrete series representations of G with real infinitesimal character occuring with finite multiplicity. If δ_{Γ} < 0, then it is empty. If δ_{Γ} > 0 and φ = 1, then it contains the representation I^{1,δ_{Γ}} with multiplicity one.

L^{2}(Γ\G, φ)_{ac} decomposes into a sum of direct integrals corresponding to the unitary principal series representations of G, each occuring with infinite multiplicity.

The notions L^{2}(Γ\G, φ)_{cusp} and L^{2}(Γ\G, φ)_{res} are chosen in analogy with the case of groups Γ with finite covolume. Indeed, L^{2}(Γ\G, φ)_{res} is spanned by the residues of Eisenstein series. However, the ”cusp forms” forming the space L^{2}(Γ\G, φ)_{cusp} share the properties of the cusp forms in the sense of Harish-Chandra associated to the trivial group and not of those for groups with finite covolume. The appearance of L^{2}(Γ\G, φ)_{U} does not seem to have an analogue.

Proof of the theorem. It follows from Corollary 8.7, the determination of ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} in Section 9, and our definition of the scalar products on ^{Γ}(V_{π′,−∞} ⊗ V_{φ})_{d} that the matrix coefficient maps induce a G-equivariant unitary map of H_{cusp}(φ) ⊕ H_{res}(φ) ⊕ H_{U}(φ) onto the discrete subspace L^{2}(Γ\G, φ)_{d}. Next we show that the wave packet transform W P_{σ} extends to a unitary embedding of H_{ac}^{σ}(φ) into L^{2}(Γ\G, φ). We compute using Proposition 10.8 and (64)

∥W P_{σ}(ϕ ⊗ v_{T})∥_{L^{2}(Γ\G,φ)}^{2}
= ( ∫_{𝔞_{+}^{∗}}⟨E(iλ, ϕ_{iλ}, T), v⟩ p_{σ}(iλ)dλ , ⟨E(ϕ, T), v⟩)_{L^{2}(Γ\G,φ)}
= ∫_{𝔞_{+}^{∗}}⟨(E(iλ, ϕ_{iλ}, T), v⟩ , ⟨E(ϕ, T), v⟩)_{L^{2}(Γ\G,φ)} p_{σ}(iλ)dλ
= ∫_{𝔞_{+}^{∗}} ∫_{Γ\G} ∫_{K}⟨E(iλ, ϕ_{iλ}, T)(gk), v⟩\overline{⟨E(ϕ, T)(gk), v⟩} dk dg p_{σ}(iλ)dλ
= ∫_{𝔞_{+}^{∗}} ∫_{Γ\G} ∫_{K}⟨γ̃(k^{−1})E(iλ, ϕ_{iλ}, T)(g), v⟩\overline{⟨γ̃(k^{−1})E(ϕ, T)(g), v⟩} dk dg p_{σ}(iλ)dλ

= \frac{∥v∥^{2}}{dim(V_{γ})} ∫_{𝔞_{+}^{∗}} ∫_{Γ\G}(E(iλ, ϕ_{iλ}, T)(g), E(ϕ, T)(g)) dg p_{σ}(iλ)dλ

= \frac{2πω_{X}∥T ∥^{2}∥v∥^{2}}{dim(V_{γ})} ∫_{𝔞_{+}^{∗}}(ϕ_{iλ}, ϕ_{iλ})_{B} p_{σ}(iλ)dλ

= \frac{2πω_{X}∥v_{T} ∥^{2}}{dim(V_{σ})} ∫_{𝔞_{+}^{∗}}(ϕ_{iλ}, ϕ_{iλ})_{B} p_{σ}(iλ)dλ
= ∥ϕ ⊗ v_{T} ∥_{H_{ac}^{σ}(φ)}^{2} .

By a similar computation we see that the images of the W P_{σ}, σ ∈ M̂, are pairwise orthogonal. It follows from (75) that the extension of W P_{σ} to H_{ac}^{σ}(φ) is G-equivariant. As a subspace of the continuous subspace of L^{2}(Γ\G, φ) the image of W P_{σ} is also orthogonal to L^{2}(Γ\G, φ)_{d}.

It remains to show that the image of H_{ac}(φ) ⊕ H_{cusp}(φ) ⊕ H_{res}(φ) ⊕ H_{U}(φ) exhausts L^{2}(Γ\G, φ). Let f ∈ L^{2}(Γ\G, φ) be orthogonal to that image. We may and shall assume that it belongs to a K-isotypic component L^{2}(Γ\G, φ)(γ) for some γ ∈ K̂. By Corollary 8.7 we can compute the scalar product of f with a Schwartz function g ∈ C(Γ\G, φ) in the following way

(f, g) = ∫Ĝ(F(f)(π), F(g)(π))_{N_{π}⊗V_{π}} dκ(π)
= ∫Ĝ(F(f)(π), F_{π}(g))_{N_{π}⊗V_{π}} dκ(π)
= ∫Ĝ(c_{π}(F(f)(π)), g)_{L^{2}(Γ\G,φ)} dκ(π) .  (76)

Since f is orthogonal to L^{2}(Γ\G, φ)_{d} we have F(f)(π) = 0 for all π ∈ Ĝ with κ(π) ≠ 0, i.e., Γ(V_{π′,−∞} ⊗ V_{φ})_{d} ≠ 0. Thus, by the results of Section 9, it remains to show that F(f)(π) = 0 for the unitary principal series representations π^{σ,−iλ}. Because of the equivalence π^{σ,iλ} ≅ π^{σ^{w},−iλ} we can assume λ ∈ 𝔞_{+}^{∗}. Clearly, F(f)(π^{σ,−iλ}) = 0 if [γ_{|M} : σ] = 0.

By Corollary 8.7 and Lemma 9.4 we have

F(f)(π^{σ,−iλ}) ∈ ^{Γ}C^{−∞}(∂X, V (σ̃_{iλ}, φ)) ⊗ L^{2}(∂X, V (σ_{−iλ}))(γ) .

For each σ ∈ M̂, with [γ_{|M} : σ] ≠ 0 let {T_{i}^{σ}}_{i=1}^{dim Hom_{M} (Vσ̃,Vγ̃)} be a basis of Hom_{M}(Vσ̃, Vγ̃). Let {v^{j}}_{j=1}^{dim(V_{γ})} be a basis of V_{γ}. We conclude that there are sections ϕ_{σ,iλ}^{ij} ∈ C^{−∞}(B, V_{B}(σ̃_{iλ}, φ)), i = 1, …, dim Hom_{M}(Vσ̃, Vγ̃), j = 1, …, dim(V_{γ}), such that

F(f)(π^{σ,−iλ}) = ∑_{i,j} ext(ϕ_{σ,iλ}^{ij}) ⊗ v_{T_{i}^{σ}}^{j} .

Using Lemma 9.1 we find

c_{π^{σ,−iλ}}(F(f)(π^{σ,−iλ})) = ∑_{i,j}⟨E(iλ, ϕ_{σ,iλ}^{ij}, T_{i}^{σ}), v^{j}⟩ .

We now evaluate the scalar product of f with some wave packet of the form W P_{σ′}(ψ ⊗ v_{T}), where ψ belongs to the space H_{0}^{σ̃′}(φ), v ∈ V_{γ}, and T ∈ Hom(V_{}_{σ̃′}, Vγ̃).

The map 𝔞_{+}^{∗} ∋ λ ↦ π^{σ,−iλ} ∈ Ĝ identifies 𝔞_{+}^{∗} with an open subset of Ĝ. Let dκ(σ, λ) be the restriction of the Plancherel measure dκ to this subset. Using (76), Proposition 10.8 we obtain

0 = (f, W P_{σ′}(ψ ⊗ v_{T}))_{L^{2}(Γ\G,φ)}
= ∫Ĝ(c_{π}(F(f)(π)), W P_{σ′}(ψ ⊗ v_{T})) dκ(π)
= ∑_{σ∈M̂,[γ_{|M}:σ]≠0} ∑_{i,j} ∫_{𝔞_{+}^{∗}}(⟨E(iλ, ϕ_{σ,iλ}^{ij}, T_{i}^{σ}), v^{j}⟩, ⟨E(ψ, T), v⟩) dκ(σ, λ)

= 2πω_{X} ∑_{i,j}(v_{j}, v)(T_{i}^{σ′}, T) ∫_{𝔞_{+}^{∗}}(ϕ_{σ′,iλ}^{ij}, ψ_{iλ})_{B} dκ(σ′, λ) .

Varying σ′, T and v we see that for all i, j and σ′

∫_{𝔞_{+}^{∗}}(ϕ_{σ′,iλ}^{ij}, ψ_{iλ})_{B} dκ(σ′, λ) = 0 .

Moreover, we are free to multiply ψ with a function h ∈ C_{c}^{∞}(𝔞_{+}^{∗}). Hence for all such h

∫_{𝔞_{+}^{∗}} h(λ)(ϕ_{σ′,iλ}^{ij}, ψ_{iλ})_{B} dκ(σ^{l}, λ) = 0 .

We conclude that (ϕ_{σ′,iλ}^{ij}, ψ_{iλ})_{B} = 0 for almost all λ (mod κ). We choose a countable dense set {ψ_{m}} ⊂ C^{∞}(B, V_{B}(σ̃′_{0}, φ)). Using a holomorphic trivialization of the family of bundles {V_{B}(σ̃′_{μ}, φ)}_{μ∈𝔞_{ℂ}^{∗}} we extend these sections to families 𝔞_{ℂ}^{∗} ∋ μ ↦ ψ_{m,μ} ∈ C^{∞}(B, V_{B}(σ̃′_{μ}, φ)). We form B_{m} := {λ|(ϕ_{σ′,iλ}^{ij}, ψ_{m,iλ}) ≠ 0} ⊂ 𝔞_{+}^{∗}. Then κ(B_{m}) = 0. Moreover let U := ⋃_{m} B_{m}. Then κ(U) = 0, and we have (ϕ_{σ′,iλ}^{ij}, ψ_{m,iλ}) = 0 for all λ ∈ 𝔞_{+}^{∗} \ U and all m. Thus ϕ_{σ′,iλ}^{ij} = 0 for λ ∈ 𝔞_{+}^{∗} \ U. Hence F(f)(π) = 0 for almost all π (mod κ). Therefore f = 0. This proves that the image of H_{ac}(φ)⊕H_{cusp}(φ)⊕H_{res}(φ)⊕H_{U}(φ) is all of L^{2}(Γ\G, φ) and hence the theorem. □

Let now γ be a finite-dimensional unitary representation of K. We want to draw consequences of Theorem 11.1 for the spectral decomposition of the algebra Z containing the Casimir operator Ω_{G} and acting by unbounded operators on L^{2}(Y, V_{Y} (γ, φ)). Let D(G, γ) be the algebra of G-invariant differential operators on V (γ). This algebra is a finitely generated module over Z, where according to our convention the homomorphism of Z to D(G, γ) is induced by the left regular representation of G. The algebra D(G, γ) might be noncommutative, even if γ is irreducible (but then X = ℍH^{n} or 𝕆H^{2}).

As before, we identify C^{∞}(Y, V_{Y} (γ, φ)) with [C^{∞}(Γ\G, φ)⊗V_{γ}]^{K}. This identification provides an isomorphism of [U(𝔤)⊗_{U(𝔨)}End(V_{γ})]^{K} with D(G, γ) , where the action of U(𝔤) on C^{∞}(Γ\G, φ) is induced by the right regular representation. In particular, the decomposition of L^{2}(Γ\G, φ) with respect to G induces a decomposition of L^{2}(Y, V_{Y} (γ, φ)) with respect to D(G, γ).

If π is any admissible representation of G, then the finite-dimensional space Hom_{K}(V_{π}, V_{γ}) has a natural structure of a D(G, γ)-module. In fact, the action of U(𝔤)⊗End(V_{γ}) on Hom(V_{π}, V_{γ}) induces an action of D(G, γ) ≅ [U(𝔤) ⊗_{U(𝔨)} End(V_{γ})]^{K} on Hom_{K}(V_{π}, V_{γ}). Note that if π is irreducible, then Hom_{K}(V_{π}, V_{γ}) is an irreducible D(G, γ)-module. Moreover, if γ is irreducible, then the functor π ↦ Hom_{K}(V_{π}, V_{γ}) provides a one-to-one correspondence between equivalence classes of irreducible G-representations (strictly speaking of irreducible (𝔤, K)-modules) containing γ and of irreducible D(G, γ)-modules (see [55], 3.5.4.). The induced action of Z on Hom_{K}(V_{π}, V_{γ}) is given by the infinitesimal character of π.

We define the Hilbert space and D(G, γ)-module

H_{cusp}^{γ}(φ) := ⨁_{π∈Ĝ_{d}} ^{Γ}(V_{π,−∞} ⊗ V_{φ})_{d} ⊗ Hom_{K}(V_{π}, V_{γ}) .

This sum is finite since every irreducible subrepresentation of γ only occurs in finitely many discrete series representations. There is a natural unitary D(G, γ)-equivariant map of H_{cusp}^{γ}(φ) into L^{2}(Y, V_{Y} (γ, φ)) which is given on each summand by

Γ(V_{π,−∞} ⊗ V_{φ})_{d} ⊗ Hom_{K}(V_{π}, V_{γ}) ∋ ϕ ⊗ t ↦ t(π(g^{−1}) ⊗ id)ϕ) ∈ [L^{2}(Γ\G, φ) ⊗ V_{γ}]^{K} . (77)

By the Frobenius reciprocity Hom_{M}(V_{σ}, V_{γ}) ≅ Hom_{K}(V_{π^{σ,λ}}, V_{γ}) the space Hom_{M}(V_{σ}, V_{γ}) carries a structure of a D(G, γ)-module depending on λ ∈ 𝔞_{ℂ}^{∗}. We denote this module by Hom_{M}(V_{σ}, V_{γ})_{λ}. For Re(λ) > 0 we consider I^{σ,λ} as a submodule of V_{π^{σ,λ}}. We define the D(G, γ)- module \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ} to be the quotient of Hom_{M}(V_{σ}, V_{γ})_{λ} which corresponds to the quotient Hom_{K}(I^{σ,λ}, V_{γ}) of Hom_{K}(V_{π^{σ,λ}}, V_{γ}) by Frobenius reciprocity. Using Lemma 5.5, 1., one can check that \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ} is the quotient of Hom_{M}(V_{σ}, V_{γ})_{λ} by the subspace which is annihilated by multiplication by c_{γ}(λ).

Recall the definition of the scalar product (T_{1}, T_{2})id_{V_{σ}} := T_{2}^{∗}T_{1} on Hom_{M}(V_{σ}, V_{γ}). For λ ∈ i𝔞^{∗} this scalar product is equal to the scalar product induced by Frobenius reciprocity Hom_{M}(V_{σ}, V_{γ}) ≅ Hom_{K}(L^{2}(∂X, V (σ, λ)), V_{γ}) rescaled by dim(V_{σ})^{−1}. Note that if σ ∈ M̂_{2}, then there is an orthogonal decomposition

Hom_{M}(V_{σ}, V_{γ})_{0} = Hom_{M}(V_{σ}, V_{γ})_{0}^{+} ⊕ Hom_{M}(V_{σ}, V_{γ})_{0}^{−}

corresponding to the decomposition π^{σ,0} = π^{σ,+} ⊕ π^{σ,−}.

If Re(λ) > 0 and π̄^{σ,λ} is unitary, then we define the scalar product on \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ} by

([T_{1}], [T_{2}]) := \frac{dim(V_{σ})}{c_{σ}(λ)}(c_{γ}(λ)T_{1}, T_{2}^{w}) ,

where T_{i} ∈ Hom_{M}(V_{σ}, V_{γ}) are representatives of [T_{i}] ∈ \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ}. Note that this scalar product coincides with the scalar product induced by \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ} ≅ Hom_{K}(I^{σ,λ}, V_{γ}).

We define the Hilbert spaces and D(G, γ)-modules

H_{res}^{γ}(φ) := ⨁_{\{σ∈M̂ | [γ_{|M}:σ]≠0 , p_{σ}(0)=0\}} ⨁_{λ∈P S_{res}(σ,φ)\\{0\}} E_{Λ}(σ_{λ}, φ) ⊗ \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ}

0H_{U}^{γ}(φ) := ⨁_{\{σ∈M̂ | [γ_{|M}:σ]≠0 , p_{σ}(0)=0\}} ⨁_{λ∈P S_{U} (σ,φ)\\{0\}} U_{Λ}(σ_{λ}, φ) ⊗ \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ}

1H_{U}^{γ}(φ) := ⨁_{{σ∈M̂_{1} | [γ_{|M}:σ]≠0 , 0∈P S_{U} (σ,φ)}} U_{Λ}(σ_{0}, φ) ⊗ Hom_{M}(V_{σ}, V_{γ})_{0}

2H_{U}^{γ}(φ) := ⨁_{{σ∈M̂_{2} | [γ_{|M}:σ]≠0 , 0∈P S_{U} (σ,φ)}} U_{Λ}^{+}(σ_{0}, φ) ⊗ Hom_{M}(V_{σ}, V_{γ})_{0}^{+} ⊕ U_{Λ}^{−}(σ_{0}, φ) ⊗ Hom_{M}(V_{σ}, V_{γ})_{0}^{−}

3H_{U}^{γ}(φ) := ⨁_{{σ∈M̂_{4} , [γ_{|M}:σ]≠0 | 0∈P S_{U} (σ,φ)}} U_{Λ}(σ_{0}, φ) ⊗ Hom_{M}(V_{σ}, V_{γ})_{0} .

All these sums are finite. We further define

H_{U}^{γ}(φ) := ⨁_{i=0}^{3} _{i}H_{U}^{γ}(φ) .

The matrix coefficient maps defined on H_{res}(φ) ⊕ H_{U}(φ) induces a unitary map of D(G, γ)- modules from H_{res}^{γ}(φ) ⊕ H_{U}^{γ}(φ) to L^{2}(Y, V_{Y} (γ, φ)) which is given by the Poisson transform on each summand (see Lemma 9.1). In particular, if λ > 0, then the Poisson transform factors over (E_{Λ}(σ_{λ}, φ) ⊕ U_{Λ}(σ_{λ}, φ)) ⊗ \overline{Hom_{M}(V_{σ}, V_{γ})}_{λ}. By (62) the image under the Poisson transform of H_{res}^{γ}(φ) consists exactly of the residues of Eisenstein series.

Last not least we introduce the absolute continuous part as the finite sum

H_{ac}^{γ}(φ) := ⨁_{σ∈M̂ , [γ_{|M}:σ]≠0} H_{ac}^{σ,γ}(φ) ,

where H_{ac}^{σ,γ} is the direct integral of Hilbert spaces and D(G, γ)-modules

H_{ac}^{σ,γ}(φ) := ∫_{𝔞_{+}^{∗}}^{⊕} L^{2}(B, V_{B}(σ, iλ)) ⊗ Hom_{M}(V_{σ}, V_{γ})_{iλ} 2πω_{X}p_{σ}(iλ) dλ .

By Proposition 10.8 the wave packet transforms extend to isometric D(G, γ)-equivariant embeddings of H_{ac}^{γ}(φ) into L^{2}(Y, V_{Y} (γ, φ)). Note that D(G, γ) acts on H_{ac}^{σ,γ}(φ) as an algebra of unbounded operators via multiplication by End(Hom_{M}(V_{σ}, V_{γ}))-valued polynomials on 𝔞^{∗}.

Now the following theorem is an immediate consequence of Theorem 11.1.

Theorem 11.2 The maps (77), the Poisson transforms and the wave packets of Eisenstein series combine to a unitary equivalence of D(G, γ)-representations

H_{ac}^{γ}(φ) ⊕ H_{cusp}^{γ}(φ) ⊕ H_{res}^{γ}(φ) ⊕ H_{U}^{γ}(φ) ≅ L^{2}(Y, V_{Y} (γ, φ)) .

It gives rise to a corresponding decomposition

L^{2}(Y, V_{Y} (γ, φ)) = L^{2}(Y, V_{Y} (γ, φ))_{ac} ⊕ L^{2}(Y, V_{Y} (γ, φ))_{d} ,

where the discrete subspace

L^{2}(Y, V_{Y} (γ, φ))_{d} := L^{2}(Y, V_{Y} (γ, φ))_{cusp} ⊕ L^{2}(Y, V_{Y} (γ, φ))_{res} ⊕ L^{2}(Y, V_{Y} (γ, φ))_{U}

is the sum of the cuspidal, the residual, and the ”stable” part.

The algebra Z has pure point spectrum on L^{2}(Y, V_{Y} (γ, φ))_{d}, whereas its spectrum on L^{2}(Y, V_{Y} (γ, φ))_{ac} consists of finitely many branches of absolute continuous spectrum of infinite multiplicity. L^{2}(Y, V_{Y} (γ, φ))_{cusp} is a finite sum of infinite-dimensional eigenspaces. The remaining part of the discrete subspace is finite-dimensional. If δ_{Γ} < 0, then it is empty. □

We conclude this paper by some comments on Theorem 11.2.

• It is clear from Corollary 8.7 that integration against Eisenstein series gives a map

C(Y, V_{Y} (γ, φ)) → H_{ac}^{γ}(φ)

which we call Eisenstein-Fourier transform. It is a left-inverse of the wave packet transform, and it would be interesting to investigate its image.

• The decomposition of L^{2}(Y, V_{Y} (γ, φ)) given by Theorem 11.2 is finer than the spectral decomposition with respect to the Casimir operator. In particular, the Casimir operator can have eigenvalues embedded in the continuous spectrum. This is a kind of accident, because these embedded eigenvalues can be separated from the continuous spectrum by additional operators belonging to D(G, γ) with one possible exception. Namely an eigenspace arising from ⊕_{i=1}^{3}_{i}H_{U}^{γ}(φ) contributes an eigenvalue lying at the bottom of one branch of the absolute continuous spectrum of Ω_{G}.

• So far we have not presented any example where the space L^{2}(Y, V_{Y} (γ, φ))_{U} is non-trivial. Here is one. It also sheds some light on the previous remark. Let Γ ⊂ SL(2, ℝ) be a cocompact Fuchsian group. Consider Γ ⊂ SL(2, ℂ) in the standard way. The limit set consists of the equator of the sphere S^{2}. If we interpret the equator as a 1-current, then it is not difficult to see that it defines an element of U_{Λ}(σ_{0}, 1), where σ = σ^{2} ⊕ σ^{2,w} is the representation of M ≅ U(1) corresponding to 1-forms. The corresponding square integrable 1-form on Y is harmonic, i.e., it contributes to the L^{2}-cohomology of Y (compare Mazzeo-Phillips [40]).

• Even in the case δ_{Γ} > 0 it can happen that L^{2}(Y, V_{Y} (γ, φ))_{d} = 0. For instance if Y is odd-dimensional, then Corollary 7.9 implies that the Dirac operator acting on spinors has pure absolute continuous spectrum (−∞, ∞).

• Using the meromorphic continuation of the Eisenstein series to all of 𝔞_{ℂ}^{∗} and Theorem 11.2 one can show that the resolvent kernel of (Ω − z)^{−1} on L^{2}(Y, V_{Y} (γ, φ)) extends meromorphically to a finite-sheeted branched cover of ℂ.

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