Scattering theory for geometrically finite groups

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

November 28, 2007

Contents

1 Introduction and statement of the results 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Notion of geometrical finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Twists and bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Description of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Some machinery 13 2.1 Limits of bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Holomorphic vector bundles over ℂ . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Pure cusps 24 3.1 Geometry of cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Schwartz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Asymptotics for the trivial group . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 The spaces B_{U,k}(σ_{λ}, φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 The push-down for Schwartz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6 Push down for cusps of smaller rank . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Push-down for cusps of full rank and for general σ . . . . . . . . . . . . . . . . . 54 3.8 Compatibility with embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.9 Extension and restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.10 The scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

∗NFW I–Mathematik, Universität Regensburg, D-93040 Regensburg, Germany, E-mail:

ulrich.bunke@mathematik.uni-regensburg.de

†Université du Luxembourg, UR en Mathématiques, 162 a, avenue de la Faïencerie, L-1511 Luxembourg, Grand-Duchy of Luxembourg, E-mail: martin.olbrich@uni.lu

4 The general case 80 4.1 The space B_{Γ}(σ_{λ}, φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Push-down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Extension, restriction, and the scattering matrix . . . . . . . . . . . . . . . . . . 90 4.4 Vanishing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Meromorphic continuations and general σ . . . . . . . . . . . . . . . . . . . . . . 96

1 Introduction and statement of the results

1.1 Introduction

1.1.1 Scattering theory is a collection of methods and results for studying the continuous spectrum of operators. Classical fields of application of scattering theory are the propagation of waves in mechanics, electro-dynamics and quantum mechanics. Often one considers a model that is close to some ideal case in which the wave equation can be solved explicitely. Scattering theory provides the tools for comparison. 1.1.2 In many cases it is possible to separate the time and space variables. In this case one can apply methods from stationary scattering theory. This branch of scattering theory investigates the generalized eigenfunctions contributing to the continuous spectrum. In good situations these eigenfunctions come in families which extend meromorphically. Often they are studied via a meromorphic continuation of the distribution kernel of the resolvent of the spacial part of the operator. 1.1.3 The problem of stationary scattering theory that is considered in the present paper has mainly an internal mathematical motivation. The model situation is a Laplace-type operator on a globally symmetric space of negative curvature. Due to the presence of a large symmetry group it is possible to get an essentially complete description of the generalized eigenfunctions.

The real problem is to understand the generalized eigenfunctions on associated locally symmetric spaces. The special case of quotients of the globally symmetric space by arithmetic subgroup groups has many applications in number theory and arithmetic. While arithmetic quotients have finite volume the main emphasis in the present work is on spaces of infinite volume. The question is, how far the interplay between global and local symmetries of the problem can be used in order to get a complete picture.

1.1.4 There is lot of literature on the stationary scattering theory for the function Laplacian on spaces which become close to the global symmetric space asymptotically at infinity. The major part is devoted to the asymptotically hyperbolic case. The main results are meromorphic continuation of the resolvent kernel, the scattering matrix and the parametrization of the relevant generalized eigenfunctions. Most of this work is based on a fine study of the resolvent kernel. We refer to [Gui05b] for one of the latest works and the discussion of the literature therein. 1.1.5 In the more rigid case of a locally symmetric space one can obtain better results. In this case one can approach stationary scattering theory by pushing the analysis to the boundary. Using this method we obtained in [BO00] the spectral decomposition of all locally invariant differential operators on vector bundles in the convex-cocompact case. We refer to this paper for a review of the literature about the convex-cocompact case. 1.1.6 The goal of the present paper is to generalize this method to geometrically finite spaces. These are locally symmetric spaces of negative curvature which at infinity look like the symmetric space or a cusp. From the point of view of analysis the part at infinity which can be compared with the globally symmetric space is easy with the results of [BO00] at hand. The complications are due to the presence of cusps, in particular of those of non-maximal rank. 1.1.7 Let G be a real simple linear connected Lie group of real rank one. It covers the connected component of the group of isometries of the associated symmetric space X. We consider a geometrically finite group Γ ⊂ G which determines the locally symmetric space Γ\X. For technical reasons we exclude the exceptional symmetric space X = H𝕆^{2} from our considerations.

The classical problems of meromorphic continuation of the Eisenstein series, the scattering matrix, and their functional equations was previously adressed by Guillopé [Gu92] in the special case that X is the hyperbolic plane Hℝ^{2} and by Froese/Hislop/Perry [FHP91b] for X = Hℝ^{3} and spherical Eisenstein series. Cusps of non-maximal rank in the three-dimensional hyperbolic case have been investigated first in [FHP91a] via the resolvent. For the meromorphic continuation of the resolvent on Γ\Hℝ^{n} see [Per99],[Gui06]. All these papers are restricted to the case of so-called rational cusps. 1.1.8 The main result of the present paper is, in the geometrically finite case, the meromorphic continuation of the Eisenstein series, i.e. of the families of generalized eigensections for all locally invariant differential operators on bundles, see Corollary 1.16 below. Note that this result also includes the case of irrational cusps.

At the moment we are not able to obtain results, which are as complete as in the convex cocompact case. In particular, we are still quite far from showing a spectral decomposition. 1.1.9 The class of geometrically finite discrete subgroups Γ ⊂ G subsumes cocompact, convexcocompact groups, and subgroups of finite covolume as well as various kinds of combinations of these cases with cusps of non-maximal rank. The paper can be considered as a continuation of [BO00], where the special case of a convex-cocompact group Γ is considered. We will frequently refer to this paper for notations and conventions as well as for technical results. 1.1.10 In the present paper the analysis takes place on the boundary ∂X of the symmetric space. The group Γ acts on vector bundles over this boundary. As in [BO00] the main players of the geometric scattering theory are the push-down π_{∗}^{Γ} (average of sections over Γ), the extension ext^{Γ} (the adjoint ot the push-down), the restriction res^{Γ}, and the scattering matrix S^{Γ}. Our task is to construct these as families of maps between suitable function/distribution spaces depending meromorphically on the spectral parameter. Due to the presence of cusps of non-maximal rank the detailed description of these spaces turns out to be quite difficult.

1.2 Notion of geometrical finiteness

1.2.1 In the present paper the symmetric space X belongs to one of the series of hyperbolic spaces Hℝ^{n}, Hℂ^{n}, and Hℍ^{n} (real, complex, and quaternionic) of real dimension n, 2n, 4n. We exclude the exceptional symmetric space of rank one, the Cayley hyperbolic plane H𝕆^{2}, for technical reasons since we are going to employ the fact that X belongs to a series in several places. 1.2.2 By G we denote a real simple linear connected Lie group of real rank one covering the connected component of the group of isometries of X. By ∂X we denote the geodesic boundary of X. The union X̄ := X ∪∂X has the structure of a compact G-manifold with boundary. While X parametrizes the set of maximal compact subgroups the boundary ∂X parametrizes the set of parabolic subgroups of G. If P ⊂ G is a parabolic subgroup, then we denote by ∞_{P} ∈ ∂X the unique fixed point of P. Let Ω_{P} := ∂X \ ∞_{P}. 1.2.3 Let P ⊂ G be parabolic, and N ⊂ P be its nil-radical. Then we can write P as an extension

0 → N → P →^{l} L → 0 .  (1)

Here L is a reductive Lie group which is canonically isomorphic to a product MA of a compact group M and a group A ≅ ℝ_{+}^{∗}. 1.2.4 Let Γ ⊂ G be a torsion-free discrete subgroup.

Definition 1.1 A parabolic subgroup P ⊂ G is called Γ-cuspidal if U_{P} := Γ ∩ P is an infinite subgroup such that l(U_{P}) ⊂ L is precompact, i.e. l(U_{P}) ⊂ M.

Let p denote the Γ-conjugacy class of the Γ-cuspidal parabolic subgroup P. We call such classes cusps and say that the cusp p has full rank (as opposed to smaller rank) if U_{P} \Ω_{P} is compact for one (and hence for any) P ∈ p. For each cusp p and P ∈ p we form the manifold with boundary Ȳ_{U_{P}} := U_{P} \(X ∪ Ω_{P}). 1.2.5 The boundary ∂X of X has a Γ-equivariant decomposition ∂X = Ω_{Γ} ∪ Λ_{Γ} into a limit set Λ_{Γ} and a domain of discontinuity Ω_{Γ} ([EO73], Prop. 8.5). Let Ȳ_{Γ} denote the manifold with boundary Ȳ_{Γ} := Γ\(X ∪ Ω_{Γ}).

Definition 1.2 The group Γ is called geometrically finite if the following conditions hold:

1. The set P_{Γ} of Γ-conjugacy classes of Γ-cuspidal parabolic subgroups is finite.

2. There is a bijection end(Ȳ_{Γ}) →^{∼} P_{Γ}, where end(Ȳ_{Γ}) denotes the set of ends of the manifold with boundary Ȳ_{Γ}.

3. For all p ∈ P_{Γ} and P ∈ p there is a representative Ȳ_{p} of the end corresponding to p and an isometric embedding e_{P} : Ȳ_{p} → Ȳ_{U_{P}} such that its image e_{p}(Ȳ_{p}) represents the end of Ȳ_{U_{P}}.

By Bowditch [Bow95], Corollary 6.3, this definition is equivalent to the slightly weaker definition [Bow95], F1 (which can be derived from 1.2 by replacing “set P_{Γ} of Γ-conjugacy classes of Γ-cuspidal parabolic subgroups” by “set end(Ȳ_{Γ}) of ends of Ȳ_{Γ}” in 1., “bijection” by “map c” in 2. and “all p ∈ P_{Γ}” by “all p ∈ P_{Γ} in the range of c” in 3.). 1.2.6 Let Y_{Γ} := Γ\X denote the locally symmetric space associated to Γ. By Ȳ_{0} we denote the compact subset Ȳ_{Γ} \ ⋃_{p∈P_{Γ}} Ȳ_{p} of Ȳ_{Γ}. The boundary of Ȳ_{Γ} is the manifold B_{Γ} := Γ\Ω_{Γ}. Let P_{Γ}^{<} ⊂ P_{Γ} denote the subset of cusps of smaller rank. The decompositions of Ȳ_{Γ} into a compact piece and its ends induces a decomposition B_{Γ} = B_{0} ∪ ⋃_{p∈P_{Γ}^{<}} B_{p}, where B_{p} := B ∩ Ȳ_{p}, p ∈ P_{Γ}^{<} ∪ {0}. 1.2.7 Set P_{Γ}^{max} := P_{Γ} \ P_{Γ}^{<}. It will be convenient to fix a set P̃ of parabolic subgroups representing the elements of P_{Γ}. It comes with a partition P̃ = P̃^{<} ∪ P̃^{max}. 1.2.8 Our analysis will require an additional assumption on the cusps. By Lemma 3.5 this assumption is automatically satisfied in the cases Hℝ^{n}, Hℂ^{n}, but by Lemma 3.6 it is non-trivial in the case Hℍ^{n}. We are now going to describe this assumption in detail. Let P ⊂ G be parabolic. A Langlands decomposition P = MAN is the same as a split s : MA → P of the extension (1), where we identify M and A with their images s(M) and s(A).

Let p be a cusp of Γ and P ∈ p. In Subsection 3.1 we will construct a Langlands decomposition P = MAN, a discrete subgroup V ⊂ N, and a homomorphism m : N_{V} → M from the Zariski closure N_{V} of V in N such that the group U^{0} := {m(v)v|v ∈ V } is a subgroup of U_{P} (see Def. 1.1) of finite index.

Definition 1.3 The cusp p is called regular if the Langlands decomposition of P above can be adjusted such that N_{V} is invariant with respect to conjugation by A. We call such a Langlands decomposition adapted.

Assumption 1.4 In the remainder of the present paper we assume that Γ is geometrically finite and that all its cusps are regular.

1.3 Twists and bundles

1.3.1 We fix a parabolic subgroup P of G and write ∂X = G/P. We further fix a Langlands decomposition P = MAN and let 𝔞 denote the Lie algebra of A. Let 𝔫 denote the Lie algebra of N, and let α ∈ 𝔞^{∗} denote the short root of 𝔞 on 𝔫. We define ρ ∈ 𝔞^{∗} by ρ(H) := \frac{1}{2}tr(ad(H)_{|𝔫}), H ∈ 𝔞. 1.3.2 If λ ∈ 𝔞_{ℂ}^{∗} and (σ, V_{σ}) is a representation of M, then we define the representation (σ_{λ}, V_{σ_{λ}}) of P by V_{σ_{λ}} := V_{σ} and σ_{λ}(man) := a^{ρ−λ}σ(m). By 1 we denote the trivial one-dimensional representation of M. 1.3.3 If (θ, V_{θ}) is a representation of P, then we define the G-homogeneous bundle V (θ) := G ×_{P} V_{θ} and denote by π^{θ} the representation of G on spaces of sections of V (θ). 1.3.4 If (φ, V_{φ}) is a finite-dimensional representation of G (or Γ), then we denote by V (θ, φ) the tensor product of V (θ) with the trivial bundle ∂X × V_{φ}, and by π^{θ,φ} the representation of G (or Γ) on spaces of sections of V (θ, ϕ). Note that we can identify

C^{∞}(∂X, V (θ, φ)) ≅ C^{∞}(∂X, V (θ)) ⊗ V_{φ}, π^{θ,φ} = π^{θ} ⊗ φ .

1.3.5 The representation φ of Γ is called twist. Our analysis requires further assumptions on the twist going under the name“admissible”. Let p be a cusp of Γ, P ∈ p, and MAN be an adapted Langlands decomposition of P (see Definition 1.3). We define M_{U} := \overline{l(U_{P})}. Then P_{U} := M_{U}N_{V} ⊂ P is a subgroup containing U_{P}. Let (φ, V_{φ}) be a twist.

Definition 1.5 The twist (φ, V_{φ}) is called admissible at the cups p if its restriction to U_{P} extends to a continuous representation of AP_{U} such that A acts algebraically by semisimple endomorphisms. A twist is called admissible if it is admissible at all cusps of Γ.

Note that the algebraic functions on A are linear combinations of A ↦ a^{nα}, n ∈ ℤ (see 1.3.1 for the definition of α). 1.3.6 If Γ has cusps, then the condition of admissibility excludes most unitary representations of Γ. Examples of admissible twists are restrictions of finite-dimensional representations of G to Γ. As explained in Subsection 2.3 twists are used as a technical device.

1.3.7 If (γ, V_{γ}) is a representation of K, then we can form the homogeneous bundle V (γ) := G ×_{K} V_{γ} over X. For a twist φ let V (γ, φ) denote the product V (γ) ⊗ V_{φ}. Furthermore let V_{Γ}(γ, φ) := Γ\V (γ, φ) denote the corresponding bundle over Y.

1.4 Exponents

1.4.1 The main numerical invariant associated to a discrete subgroup Γ ⊂ G is its exponent δ_{Γ} ∈ 𝔞^{∗}. Note that each g ∈ G can be written as g = ka_{g}h ∈ KA_{+}K, where a_{g} is uniquely determined. Here A_{+} := {a ∈ A | a^{α} ≥ 1}.

Definition 1.6 The exponent δ_{Γ} ∈ 𝔞^{∗} is defined as the infimum of the set

{ν ∈ 𝔞^{∗} | ∑_{g∈Γ} a_{g}^{−ν−ρ} < ∞} .

If follows from the discreteness of Γ that δ_{Γ} ≤ ρ. 1.4.2 The critical exponent δ_{Γ} has been extensively studied, in particular by Patterson [Pat76], Sullivan [Sul79], [Sul84], and Corlette [Cor90], Corlette and Iozzi [CI99]. From these papers we know that δ_{Γ} ∈ [−ρ, ρ], if Γ is infinite. Moreover, we have δ_{Γ} = ρ if and only if Γ has finite covolume. If Λ_{Γ} contains at least 2 points, then 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. If Y = Γ\X is an infinite volume quotient of 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 [Cor90],[CI99]. 1.4.3 Let φ be a twist.

Definition 1.7 We define the exponent δ_{φ} ∈ 𝔞^{∗} of φ as the infimum of the set

{ν ∈ 𝔞 | sup_{g∈Γ} a_{g}^{−ν}∥φ(g)∥ < ∞} ,

where ∥.∥ denotes any norm on End(V_{φ}).

1.5 Description of the main results

1.5.1 Let Γ ⊂ G be a discrete, torsion-free, geometrically finite subgroup such that all its cusps are regular. Furthermore let φ be an admissible twist. Let B_{Γ} := Γ\Ω_{Γ} and V_{B_{Γ}}(σ_{λ}, φ) := Γ\V (σ_{λ}, φ). Under the name push-down we subsume several constructions related to the average of elements of C^{∞}(∂X, V (σ_{λ}, φ)) with respect to Γ. It is a crucial matter to construct a space B_{Γ}(σ_{λ}, φ) which contains the result of the average. If λ ∈ 𝔞_{ℂ}^{∗} varies, then these spaces assemble as a projective limit of locally trivial bundles of Fréchet spaces in the sense of Subsection 2.1. Therefore we can speak of meromorphic families of continuous maps from and to these spaces.

1.5.2 We define the bundle B_{Γ}(σ_{λ}, φ) as a direct sum

B_{Γ}(σ_{λ}, φ) := B_{Γ}(σ_{λ}, φ)_{1} ⊕ R_{Γ}(σ_{λ}, φ)_{max} ,

where the first component satisfies

C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) ⊂ B_{Γ}(σ_{λ}, φ)_{1} ⊂ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

(equality occurs if all cusps have full rank). The second component is associated to the cusps of full rank and is finite-dimensional. 1.5.3 The elements of B_{Γ}(σ_{λ}, φ)_{1} are characterized as smooth sections of V_{B_{Γ}}(σ_{λ}, φ) with a certain asymptotic expansion near the cusps. We postpone the detailed description of the asymptotics until Subsection 3.4. The bundle R_{Γ}(σ_{λ}, φ)_{max} has a further decomposition

R_{Γ}(σ_{λ}, φ)_{max} = ⨁_{P ∈P̃^{max}} R_{U_{P}} (σ_{λ}, φ) .

Recall from 1.2.7 that P̃^{max} is in bijection with the set of cusps of full rank. In order to define R_{U_{P}} (σ_{λ}, φ) we first consider the sheaf E_{∞_{P}} (σ̃, φ̃) on 𝔞_{ℂ}^{∗} of holomorphic families ϕ_{ν}, ν ∈ 𝔞_{ℂ}^{∗}, of U_{P} -invariant distribution sections of V (σ̃_{ν}, φ̃) supported in ∞_{P}. Here σ̃, φ̃ are the dual representations to σ, φ. The sheaf E_{∞_{P}} (σ̃, φ̃) is the sheaf of holomorphic sections of a trivial finite-dimensional holomorphic vector bundle over 𝔞_{ℂ}^{∗}, and we denote by E_{∞_{P}} (σ̃_{λ}, φ̃) the fibre of this bundle over λ ∈ 𝔞_{ℂ}^{∗} (see Lemma 3.40 for an alternative description of the space E_{∞_{P}} (σ̃_{λ}, φ̃) and 3.7.2 for the finite-dimensionality). Then we define

R_{U_{P}} (σ_{λ}, φ) := E_{∞_{P}} (σ̃_{−λ}, φ̃)^{∗} .

1.5.4 We can now state the definition of the push-down.

Definition 1.8 For f ∈ C^{∞}(∂X, V (σ_{λ}, φ)) we define the push-down π_{∗}^{Γ}(f) ∈ B_{Γ}(σ_{λ}, φ) as the direct sum of π_{∗}^{Γ}(f)_{1} ∈ B_{Γ}(σ_{λ}, φ)_{1} and π_{∗}^{Γ}(f)_{2} ∈ R_{Γ}(σ_{λ}, φ)_{max}. Here π_{∗}^{Γ}(f)_{1} is given by π_{∗}^{Γ}(f)_{1} := ∑_{g∈Γ} π^{σ_{λ},φ}(g)f_{|Ω_{Γ}} (if the sum converges). The component π_{∗}^{Γ}(f)_{2,P} ∈ R_{U_{P}} (σ_{λ}, φ) for P ∈ P̃^{max} is defined by the condition that

⟨ϕ, π_{∗}^{Γ}(f)_{2,P} ⟩ = ∑_{[g]∈Γ/U_{P}} ⟨π^{σ̃_{−λ},φ̃}(g)ϕ, f⟩

for all ϕ ∈ E_{∞_{P}} (σ̃_{−λ}, φ̃) (if the sum converges).

1.5.5 In 1.8 the push-down is defined if certain sums converge. The first part of the following theorem gives a range of λ ∈ 𝔞_{ℂ}^{∗} for which these convergence conditions are satisfied. The second part asserts its meromorphic continuation to all of 𝔞_{ℂ}^{∗}.

Theorem 1.9 If f ∈ C^{∞}(∂X, V (σ_{λ}, φ)), then the push-down π_{∗}^{Γ}(f) ∈ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) converges for Re(λ) < −δ_{Γ} − δ_{φ}. The push-down induces a meromorphic family on 𝔞_{ℂ}^{∗} of continuous maps

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

with finite-dimensional singularities.

We will first prove the part of the theorem asserting the convergence of the push-down for Re(λ) < −δ_{Γ} − δ_{φ}. Then we consider the adjoint of the push-down, the extension ext^{Γ}, and obtain the meromorphic continuation of the latter (see Theorem 1.11). The remainder of Theorem 1.9 then follows from this result by duality. 1.5.6 By D_{Γ}(σ_{λ}, φ) we denote the topological dual space of B_{Γ}(σ̃_{−λ}, φ̃). It is a dual Fréchet and Montel space. By the latter property it is reflexive so that

D_{Γ}(σ_{λ}, φ)′ ≅ B_{Γ}(σ̃_{−λ}, φ̃) .

Varying λ ∈ 𝔞_{ℂ}^{∗} the spaces D_{Γ}(σ_{λ}, φ) form a direct limit of locally trivial holomorphic bundles of dual Fréchet spaces in the sense of Subsection 2.1.

Definition 1.10 For Re(λ) > δ_{Γ} + δ_{φ} the extension

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

is defined to be the adjoint of π_{∗}^{Γ} : C^{∞}(∂X, V (σ̃_{−λ}, φ̃)) → B_{Γ}(σ̃_{−λ}, φ̃).

Theorem 1.11 The extension induces a meromorphic family of continuous maps on 𝔞_{ℂ}^{∗}

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

with finite-dimensional singularities and values in ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)).

1.5.7 The meromorphic continuation of the extension is the main goal of the present paper. We obtain the continuation in close connection with the meromorphic continuation of the scattering matrix which will be defined below. In order to define this scattering matrix we need the restriction map res^{Γ} which is a left-inverse of the extension. Due to the presence of cusps the definition of the restriction map is more complicated than in [BO00].

1.5.8 We first recall the definition of res^{Γ} in the case of convex-cocompact groups Γ (i.e. in the case without cusps, compare Sec. 4 of [BO00]). In this case

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

is given by the composition of the restriction of distributions to Ω_{Γ} and the identification ΓC^{−∞}(Ω_{Γ}, V (σ_{λ}, φ)) ≅ C^{−∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)). For the purpose of the present paper we employ a different description. 1.5.9 Let us still assume that Γ is convex-cocompact. We choose a cut-off function χ^{Γ} ∈ C_{c}^{∞}(Ω_{Γ}) such that ∑_{g∈Γ} g^{∗}χ^{Γ} = 1. Then we define

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

by π^{∗}(f) = χ^{Γ}f (in order to understand the right-hand side properly one must identify sections of bundles over B_{Γ} with Γ-equivariant sections on Ω_{Γ}). We define

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

to be the adjoint of π^{∗}. Then res^{Γ} coincides with the restriction of r̃es to the subspace ΓC^{−∞}(∂X, V (σ_{λ}, φ)). While r̃es depends on the choice of χ^{Γ}, the restriction res^{Γ} is independent of choices. 1.5.10 If Γ has cusps of smaller rank, then we can not assume that χ^{Γ} has compact support. The definition of π^{∗} breaks down. Our way arround that problem is as follows. For each k ∈ ℕ_{0} we consider the Banach spaces C^{k}(∂X, V (σ̃_{−λ}, φ̃)) and define a meromorphic family of maps π_{k}^{∗} : B_{Γ}(σ̃_{−λ}, φ̃) → C^{k}(∂X, V (σ̃_{−λ}, φ̃)). The map π_{k}^{∗} is a right-inverse of the push-down π_{∗}^{Γ}, and its definition is similar in spirit to that of π^{∗} above, but the details are more complicated since we have to take into account the asymptotic expansions of the elements of B_{Γ}(σ̃_{−λ}, φ̃) near cusps. One problem is that we can only deal with a finite number (depending on k) of these terms at a time in order to define π_{k}^{∗}. The situation is similar to the problem of the construction of smooth functions with given Taylor series at a given point. It is impossible to construct a continuous map from the space of formal power series to smooth functions which is right-inverse to the surjective map which takes the Taylor series. Back to the definition of the restriction, let C^{−k}(∂X, V (σ_{λ}, φ)) denote the distributions of order k, i.e. the topological dual of C^{k}(∂X, V (σ̃_{−λ}, φ̃)), and define res_{k}^{Γ} : C^{−k}(∂X, V (σ_{λ}, φ)) → D_{Γ}(σ_{λ}, φ) as the adjoint of π_{k}^{∗}. The collection of maps res_{k}^{Γ} play the role of the map r̃es above. These maps depend on choices and are, in particular, not compatible if we vary k. 1.5.11 Nevertheless, we have the following uniqueness property. If f_{ν} ∈ ^{Γ}C^{−∞}(∂X, V (σ_{ν}, φ)), ν ∈ 𝔞_{ℂ}^{∗}, is a meromorphic family, and W ⊂ 𝔞_{ℂ}^{∗} is compact, then for sufficiently large k the meromorphic family res_{k}^{Γ}(f_{ν}), ν ∈ W, is well-defined. In fact, due to the compactness of W the order of f_{ν} as a distribution is uniformly bounded for ν ∈ W. In 4.3.9 we will see that res_{k}^{Γ}(f_{ν}) is independent of the choices. 1.5.12 In general the restriction maps res_{k}^{Γ} may have poles. Moreover, not every element of ΓC^{−∞}(∂X, V (σ_{λ}, φ)) can be written as evaluation of a holomorphic family f_{ν} ∈ ^{Γ}C^{−∞}(∂X, V (σ_{ν}, φ)) defined near λ. But for generic λ ∈ 𝔞_{ℂ}^{∗} the restriction map res_{k}^{Γ} is regular and every element f ∈ ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) extends to a family (in the present paper we will prove a weaker statement only, which suffices for our purposes). In this case res_{k}^{Γ}(f) is well-defined independent of the choice of (the sufficiently large) k, and we can omit the subscript k and write res^{Γ}(f) for res_{k}^{Γ}(f). We have the identity

res^{Γ} ∘ ext^{Γ} = id_{D_{Γ}(σ_{λ},φ)} ,

which holds true for generic λ ∈ 𝔞_{ℂ}^{∗} or as an identity of maps defined on meromorphic families (see Lemma 4.17). 1.5.13 If f ∈ ^{Γ}C^{−k}(∂X, V (σ_{λ}, φ)), k ∈ ℕ_{0}, then f_{|Ω_{Γ}} is a Γ-invariant distribution on Ω_{Γ}, hence can be considered as an element of C^{−∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)). If

i : D_{Γ}(σ_{λ}, φ) → C^{−∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

is the natural map (adjoint to the inclusion C_{c}^{∞}(B, V_{B}(σ̃_{−λ}, φ̃)) ↪ B_{Γ}(σ̃_{−λ}, φ̃)), then i ∘ res^{Γ}(f) is defined (even though res^{Γ} may have a pole at λ) and coincides with f_{|Ω_{Γ}}. In particular, the condition i ∘ res^{Γ}(f) = 0 is equivalent to supp(f) ⊂ Λ_{Γ}.

If Γ has cusps, then i is not injective. In this case the condition supp(f) ∈ Λ_{Γ} does not imply that res^{Γ}(f) = 0 (provided the latter is defined). 1.5.14 Vanishing of res^{Γ}(f) is a rather strong condition and should play the role of the condition supp(f) ⊂ Λ_{Γ} in the convex-cocompact case which was successfully exploited in [BO00] and [BO99]. One application of this condition is the following result which is employed in proving the functional equation of the scattering matrix in the domain of convergence.

Theorem 1.12 (Corollary 4.21) Assume that Re(λ) > max ({δ_{Γ}} ∪ {ρ_{U_{P}} | P ∈ P̃}) + δ_{φ}. If f_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (1_{μ}, φ)) is a germ of a meromorphic family at λ, then ext^{Γ} ∘ res^{Γ}(f_{μ}) = f_{μ}.

Further applications of the condition res^{Γ}(f) = 0 are contained in [BO02] (non-existence of such f for Re(λ) large) and [BOa] (estimates of the regularity of f). In fact, the proof of Theorem 1.12 is based on the main result of [BO02]. 1.5.15 We now turn to the scattering matrix. We assume that σ is either a Weyl-invariant irreducible representation of M or of the form σ′ ⊕ (σ′)^{w}, where σ′ is irreducible and not Weylinvariant, and (σ′)^{w}(m) := σ′(w^{−1}mw) is the Weyl-conjugate representation of σ′. Here w ∈ N_{K}(A) is a representative of the non-trivial element of the Weyl group W = N_{K}(A)/M ≅ ℤ/2ℤ.

1.5.16 Our starting point is the scattering matrix

S_{λ}^{{1}} = J_{λ} : C^{−∞}(∂X, V (σ_{λ}, φ)) → C^{−∞}(∂X, V (σ_{−λ}, φ))

associated to the trivial group {1}. In representation theory it firms under the name Knapp-Stein intertwining operator. Here we employ a suitably normalized version as in the paper [BO00], Sec. 5, to which we refer for further details. Let I_{𝔞} ⊂ 𝔞^{∗} be the ℤ-module spanned by α if 2α is a root of (𝔤, 𝔞), and by \frac{1}{2}α otherwise. The operators J_{λ} form a meromorphic family of continuous maps with singularities in I_{𝔞} and satisfy the functional equation J_{λ} ∘ J_{−λ} = id. 1.5.17 If Γ is non-trivial, then we obtain the scattering matrix S_{λ}^{Γ} from J_{λ} using restriction and extension.

Definition 1.13 For Re(λ) > δ_{Γ} + δ_{φ} we define the scattering matrix

S_{λ}^{Γ} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(σ_{−λ}, φ)

as the meromorphic family of continuous maps

S_{λ}^{Γ} := res^{Γ} ∘ J_{λ} ∘ ext^{Γ} .

Theorem 1.14 The scattering matrix has a meromorphic continuation to all of 𝔞_{ℂ}^{∗}. It satisfies the functional equation S_{λ}^{Γ} ∘ S_{−λ}^{Γ} = id and the relation J_{λ} ∘ ext^{Γ} = ext^{Γ} ∘ S_{λ}^{Γ}.

As noted above this theorem is proved in a multistep procedure which involves the meromorphic continuation of ext^{Γ} at the same time. The basic ideas are adapted from [BO00]. The non-compactness of B_{Γ} due to the presence of cusps of non-maximal rank is responsible for the various complications which have to be resolved on the way. 1.5.18 The following application to the Eisenstein series is an easy consequence of the preceding results and can be derived exactly as in [BO00], Cor. 10.2. Let γ be a finite-dimensional representation of K and T ∈ Hom_{M}(V_{σ}, V_{γ}). Then we have a holomorphic family of G-equivariant maps

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

which are called Poisson transformations. We refer to [BO00], Def. 4.8, for a definition of the Poisson transformation and to [BO00], Sec. 6, for a discussion of its properties.

Definition 1.15 If λ ∈ 𝔞_{ℂ}^{∗}, f ∈ D_{Γ}(σ_{λ}, φ), ext^{Γ}(f) is regular, and T ∈ Hom_{M}(V_{σ}, V_{γ}), then we define the Eisenstein series E(λ, f, T) ∈ C^{∞}(Y, V_{Y} (γ, φ)) by P_{λ}^{T} ∘ ext^{Γ}(f).

In the special case that σ, γ, and φ are trivial, T is the identity, Re(λ) > δ_{Γ}, and f = δ_{b} is the delta-distribution located at b ∈ B_{Γ}, this definition coincides with the classical definition of the Eisenstein series as the Γ-average of the Poisson kernel, i.e. the integal kernel of P_{λ}^{id}.

Let c_{σ} and c_{γ} be the c-functions introduced in [BO00], Sec. 5. We further employ the notation (c_{γ}(λ)T)^{w} as introduced in the same section of [BO00].

Corollary 1.16 The Eisenstein series forms a meromorphic family of continuous maps with finite-dimensional singularities

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

It satisfies the functional equation

E(λ, S_{−λ}f, T) = E(−λ, f, (\frac{c_{γ}(λ)}{c_{σ}(λ)}T)^{w}) .

2 Some machinery

2.1 Limits of bundles

2.1.1 In the present paper a main issue is the construction of families of topological vector spaces V_{λ}, λ ∈ ℂ, and the study of holomorphic (meromorphic) families of vectors f_{λ} ∈ V_{λ} or holomorphic (meromorphic) families of homomorphisms h_{λ} ∈ Hom(V_{λ}, W_{λ}) between such families of spaces. In order to make the notion of a holomorphic family precise we must relate the spaces V_{λ} for neighbouring λ in some holomorphic manner. One way to do this is to equip the family of spaces with the structure of a locally trivial holomorphic bundle of topological vector spaces. If we fix two local trivializations U_{0} × W_{0}, U_{1} × W_{1} of the family V_{λ} with non-trivial overlap, then the transition function is a holomorphic map from U_{0} ∩ U_{1} to Hom(W_{0}, W_{1}) with values in the subspace of isomorphisms. 2.1.2 All topological vector spaces in the present paper will be locally convex. Spaces of homomorphisms between topological vector spaces will always be equipped with the topology of uniform convergence on bounded sets. We refer to [BO99], Sec.2.2 for more details.

If we have two holomorphic families h_{λ} ∈ Hom(W_{0}, W_{1}) and g_{λ} ∈ Hom(W_{1}, W_{2}), then we can consider the composition g_{λ} ∘ h_{λ} ∈ Hom(W_{0}, W_{2}). This composition is again holomorphic (see [Glö02]). If the evaluation W_{0} → W′′_{0} is continuous, then the adjoint h′_{λ} ∈ Hom(W′_{1}, W′_{0}) is holomorphic, too (see again [Glö02]).

In order to show that the composition of two holomorphic families of maps is again holomorphic in [BO99], Sec.2.2, we assumed that W_{0} is a Montel space. The discussion of [Glö02] shows that this assumption can be omitted.

2.1.3 In this paper we do not show that the spaces B_{Γ}(σ_{λ}, φ) (D_{Γ}(σ_{λ}, φ)) form locally trivial bundles of Fréchet (dual Fréchet) spaces. These families of spaces arise instead as projective (symbol lim) and inductive limits (symbol colim) of locally trivial bundles, respectively. E.g. the bundle B_{U}(σ_{λ}, φ) comes as a limit limB_{Γ,k}(σ_{λ}, φ) of trivial bundles B_{Γ,k}(σ_{λ}, φ), k ∈ ℕ_{0}. The connecting maps (in the example B_{Γ,k}(σ_{λ}, φ) → B_{Γ,k+1}(σ_{λ}, φ)) of the diagrams are holomorphic families of continuous maps which are injective and have dense range (resp. are injective). But they are not compatible with the trivializations. 2.1.4 We must extend the notions of a holomorphic or meromorphic family of continuous of maps between such limits of locally trivial holomorphic bundles. Let

⋯ ⊂ E_{n+1} ⊂ E_{n} ⊂ …,
⋯ ⊂ F_{n+1} ⊂ F_{n} ⊂ …

be decreasing families of locally trivial bundles of Fréchet spaces defined over some open subset U ⊂ ℂ such that all inclusion maps are holomorphic. Let E := limE_{n}, F := limF_{n}.

Definition 2.1 A family of maps ϕ_{z} : E_{z} → F_{z}, z ∈ U, is called holomorphic (meromorphic) if for each x ∈ U and n ∈ ℕ_{0} there exists a neighbourhood U_{x,n} ⊂ U of x, m(n) ∈ ℕ, and a holomorphic (meromorphic) bundle map Φ_{n} : (E_{m(n)})_{|U_{x,n}} → (F_{n})_{|U_{x,n}} such that ϕ_{y} is the restriction of Φ_{n}(y) to E_{y} for all y ∈ U_{n,x}.

The composition of two holomorphic families is again a holomorphic family (compare [BO00], Subsection 2.2). 2.1.5 We now consider the dual situation. Let

⋯ → E′_{n} → E′_{n+1} → …,
⋯ → F′_{n} → F′_{n+1} → …

be direct systems of holomorphic locally trivial bundles of dual Fréchet spaces defined over some open subset U ⊂ ℂ. Let E′ := colimE′_{n}, F′ := colimF′_{n}.

Definition 2.2 A family of maps ϕ_{z} : F′_{z} → E′_{z}, z ∈ U, is called holomorphic (meromorphic) if for each x ∈ U and n ∈ ℕ_{0} there exists a neighbourhood U_{x,n} ⊂ U of x, m(n) ∈ ℕ, and a holomorphic (meromorphic) bundle map Φ′_{n} : (F′_{n})_{|U_{x,n}} → (E′_{m(n)})_{|U_{x,n}} such that for each y ∈ U_{x,n} the restriction of ϕ′_{y} to (F′_{n})_{y} is the composition of (Φ′_{n})_{y} with the natural map (E′_{m(n)})_{y} → E′_{y} .

Again the composition of two such families is a holomorphic (meromorphic) family. The adjoint of a holomorphic family is again a holomorphic (meromorphic) family.

2.2 Embedding

2.2.1 The symmetric spaces X considered in the present paper belong to a series of symmetric spaces. Let X^{n} denote the n’th space of the series. We will use the superscript ^{n} as a decoration of symbols for other objects in order to indicate that they are associated to X^{n}. For a number of arguments we need that δ_{Γ}^{n} is sufficiently negative. Note that δ_{Γ}^{n} → −∞ as n → ∞. E.g., we will first obtain a meromorphic continuation of ext^{Γ,n+k} for sufficiently large k, and then use the propositions below in order to conclude that ext^{Γ,n} has a meromorphic continuation, too. 2.2.2 The main point of this subsection is to explain that the concept of embedding is compatible with the function and distribution spaces introduced in 1.5.2 and 1.5.6. Furthermore we need compatibility with the push-down and extension maps. Note that the concept of embedding is only applied to the spherical case σ = 1. 2.2.3 We have embeddings X^{n} ↪ X^{n+1} and i : ∂X^{n} ↪ ∂X^{n+1}. In order to have compatible embeddings G^{n} ↪ G^{n+1} of the groups we assume at this point that G^{n} is one of

{Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)} .

If we realize the last three groups of the list using 𝔽-valued matrices, 𝔽 ∈ {ℝ, ℂ, ℍ}, then the embedding G^{n} ↪ G^{n+1} is the usual embedding into the left upper corner. We have compatible Iwasawa decompositions such that A = A^{n} = A^{n+1}, and in particular P^{n} ↪ P^{n+1}. Let ζ be α/2, α, and 2α in the cases X = Hℝ^{n}, X = Hℂ^{n}, and X = Hℍ^{n}, respectively. Then ζ = ρ^{n+1} − ρ^{n}, and we have (V_{1_{λ}^{n+1}})_{|P^{n}} = V_{1_{λ−ζ}^{n}}, and hence V (1_{λ}^{n+1}, φ)_{|∂X^{n}} = V (1_{λ−ζ}^{n}, φ). 2.2.4 Let

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

be the restriction of sections. It is a Γ-equivariant and surjective continuous linear map.

Proposition 2.3 There exists a meromorphic family of maps

i_{Γ}^{∗} : B_{Γ}(1_{λ}^{n+1}, φ) → B_{Γ}(1_{λ−ζ}^{n}, φ)

such that the following diagram commutes:

\begin{matrix}C^{∞}(∂X, V (1_{λ}^{n+1}, φ)) & →^{i^{∗}} & C^{∞}(∂X^{n}, V (1_{λ−ζ}^{n}, φ)) \\ ↓ π_{∗}^{Γ,n+1} & & ↓ π_{∗}^{Γ,n} \\ B_{Γ}(1_{λ}^{n+1}, φ) & →^{i_{Γ}^{∗}} & B_{Γ}(1_{λ−ζ}^{n}, φ) \end{matrix} .

If Γ does not have cusps of full rank, then i_{Γ}^{∗} is holomorphic.

This proposition will be proved in various stages. First we consider a version for the spaces associated to the cusps, see Lemma 3.38. The global result is stated in Lemma 4.12.

2.2.5 In the following we explain the construction of i_{Γ}^{∗}. If all cusps of Γ as a subgroup of G^{n} have smaller rank, then i_{Γ}^{∗} is induced by the ususal restriction i_{Γ}^{∗} : C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ}^{n+1}, φ)) → C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ−ζ}^{n}, φ)), and both π_{∗}^{Γ,n} and π_{∗}^{Γ,n+1} are given as the average of sections over Γ. The relation i_{Γ}^{∗} ∘ π_{∗}^{Γ,n+1} = π_{∗}^{Γ,n} ∘ i^{∗} is obvious in the domain of convergence Re(λ) > δ_{Γ}^{n+1} + δ_{φ}, and follows by meromorphic continuation for all λ ∈ 𝔞_{ℂ}^{∗}. We postpone the proof of the fact that i_{Γ}^{∗} really maps B_{Γ}(1_{λ}^{n+1}, φ) to B_{Γ}(1_{λ−ζ}^{n}, φ) until these spaces are defined. 2.2.6 If Γ has cusps of full rank (when considered as a subgroup of G^{n}), then i_{Γ}^{∗} is the sum of two maps (i_{Γ}^{∗})_{1} : B_{Γ}(1_{λ}^{n+1}, φ) → B_{Γ}(1_{λ−ζ}^{n}, φ)_{1} and (i_{Γ}^{∗})_{2} : B_{Γ}(1_{λ}^{n+1}, φ) → R_{Γ}(1_{λ−ζ}^{n}, φ)_{max}. The map (i_{Γ}^{∗})_{1} is the restriction of smooth sections, and we can apply the argument above in order to show commutativity of the part of the commutative diagram involving (π_{∗}^{Γ,n})_{1} and (i_{Γ}^{∗})_{1}. 2.2.7 We define (i_{Γ}^{∗})_{2} such that the corresponding diagram is commutative. We choose k ∈ ℕ_{0} so large that E_{∞_{P^{n}}} (1_{−λ+ζ}^{n}, φ̃) ⊂ C^{−k}(∂X^{n}, V (1_{−λ+ζ}^{n}, φ̃)) for all P^{n} ∈ P̃^{max,n}. For f ∈ B_{Γ}(1_{λ}^{n+1}, φ) the component i_{Γ}^{∗}(f)_{2,P^{n}} ∈ R^{U_{P^{n}}} (1_{λ−ζ}^{n}, φ) is characterized by

⟨ϕ, i_{Γ}^{∗}(f)_{2,P^{n}}⟩ = ⟨res_{k}^{U_{P^{n+1}},n+1} ∘ i_{∗}(ϕ), T_{P^{n+1}}(f)⟩

for all ϕ ∈ E_{∞_{P^{n}}} (1_{−λ+ζ}^{n}, φ̃), where P^{n+1} ⊂ G^{n+1} is the unique parabolic subgroup containing P^{n}, and

T_{P^{n+1}} : C^{∞}(B_{Γ}^{n+1}, V_{B_{Γ}}(1_{λ}^{n+1}, φ)) → C^{∞}(B_{U_{P^{n+1}}}^{n+1} , V_{B_{U_{P^{n+1}}}^{n+1}} (1_{λ}^{n+1}, φ))

is the natural map defined using the map e_{P^{n+1}} and a cut-off function χ_{p} ∈ C^{∞}(B_{Γ}) which is supported in B_{p} and one on a neighbourhood of infinity of B_{p}. This definition is independent of the choice of k. In order to show that (i_{Γ}^{∗})_{2} ∘ π_{∗}^{Γ,n+1} = (π_{∗}^{Γ,n})_{2} ∘ i^{∗} we compute in the domain of convergence for generic λ (so that the restrictions are defined)

⟨ϕ, (i_{Γ}^{∗})_{2} ∘ π_{∗}^{Γ,n+1}(f)⟩ = ⟨res_{k}^{U_{P^{n+1}},n+1} ∘ i_{∗}(ϕ), T_{P^{n+1}}^{∗} ∘ π_{∗}^{Γ,n+1}(f)⟩
= ∑_{g∈Γ^{P^{n+1}}}⟨res_{k}^{U_{P^{n+1}},n+1} ∘ i_{∗}(ϕ), χ_{P^{n+1}} ∘ π_{∗}^{U_{P^{n+1}},n+1}(π^{1_{λ}^{n+1},φ}(g)f)⟩
=^{(∗)} ∑_{g∈Γ^{P^{n+1}}}⟨i_{∗}(ϕ), π^{1_{λ}^{n+1},φ}(g)f⟩
= ∑_{g∈Γ^{P^{n+1}}}⟨π^{1_{−λ+ζ}^{n+1},φ̃}(g^{−1})ϕ, i^{∗}(f)⟩
= ⟨ϕ, (π_{∗}^{Γ,n})_{2} ∘ i^{∗}(f)⟩ ,

where Γ^{P^{n+1}} denotes any system of representatives of U_{P^{n+1}}\Γ, and χ_{P^{n+1}} := e_{P^{n+1}}^{∗}χ_{p} (see Def. 1.2 for the map e_{p}). In order to get the equality marked by (∗) we use that

(1 − χ_{P^{n+1}}) ∘ res_{k}^{U_{P^{n+1}},n+1} ∘ i_{∗}(ϕ) = 0

and

ext^{U_{P^{n+1}},n+1} ∘ res_{k}^{U_{P^{n+1}},n+1} ∘ i_{∗}(ϕ) = i_{∗}(ϕ) .
□

2.2.8 Let i_{∗} and i_{∗}^{Γ} denote the dual maps to i^{∗} and i_{Γ}^{∗}. Dualizing Prop. 2.3 we obtain the following corollary.

Corollary 2.4 We have a commutative diagram of meromorphic families of maps

\begin{matrix}C^{−∞}(∂X, V (1_{λ}^{n}, φ)) & →^{i_{∗}} & C^{−∞}(∂X^{n+1}, V (1_{λ−ζ}^{n+1}, φ)) \\ ↑ ext_{∗}^{Γ,n} & & ↑ ext_{∗}^{Γ,n+1} \\ D_{Γ}(1_{λ}^{n}, φ) & →^{i_{∗}^{Γ}} & D_{Γ}(1_{λ−ζ}^{n+1}, φ) \end{matrix} .

2.2.9

Lemma 2.5 There exists holomorphic families of continuous maps

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

and

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

such that j^{∗} ∘ i_{∗} = id, i^{∗} ∘ j_{∗} = id.

Proof. We choose a tubular neighbourhood T : 𝔽 × ∂X^{n} ↪ ∂X^{n+1} of ∂X^{n} such that T ({0} × ∂X^{n}) = ∂X^{n}. Furthermore we choose a cut-off function χ ∈ C_{c}^{∞}(𝔽) such that χ(0) = 1. Then we define the map t : C^{∞}(∂X^{n}, V (1_{ρ^{n}}^{n}, φ̃)) → C^{∞}(∂X^{n+1}, V (1_{ρ^{n+1}}^{n+1}, φ̃)) setting (tf)(T (u, x)) := χ(u)f(x) and extending this function by zero outside the tubular neighbourhood. Here we use the canonical identifications C^{∞}(∂X^{n}, V (1_{ρ^{n}}^{n}, φ̃)) = C^{∞}(∂X^{n}) ⊗ Vφ̃ and C^{∞}(∂X^{n+1}, V (1_{ρ^{n+1}}^{n+1}, φ)) = C^{∞}(∂X^{n+1}) ⊗ Vφ̃.

We choose a positive section s_{n+1} ∈ C^{∞}(∂X^{n+1}, V (1_{ρ^{n+1}+α}^{n+1})) and put i^{∗}s_{n+1} =: s_{n} ∈ C^{∞}(∂X^{n}, V (1_{ρ^{n}+α}^{n})). Then we define Φ_{ρ^{n}} : C^{∞}(∂X^{n}, V (1_{−λ}^{n}, φ̃)) → C^{∞}(∂X^{n}, V (1_{ρ^{n}}^{n}, φ̃)) and Φ_{ρ^{n+1}} : C^{∞}(∂X^{n+1}, V (1_{−λ+ζ}^{n+1}, φ̃)) → C^{∞}(∂X^{n+1}, V (1_{ρ^{n+1}}^{n+1}, φ̃)) as multiplication by s_{n}^{(ρ^{n}+λ)/α} and s_{n+1}^{(ρ^{n+1}+λ−ζ)/α}, respectively. We define j_{∗} := Φ_{ρ^{n+1}}^{−1} ∘ t ∘ Φ_{ρ^{n}}, and j^{∗} as the adjoint of j_{∗}. It is now easy to check that j_{∗} and j^{∗} have the required properties. □

2.2.10 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 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}.

2.3 Twisting

2.3.1 Twisting is an important technical device of the present paper. We explain in 2.3.11 and 2.3.12 how this concept is used. But first we must introduce some notation. 2.3.2 If (π, V_{π}) is a finite-dimensional representation of G, then we can form the bundles V (σ_{λ} ⊗ π, φ) and V (σ_{λ}, π ⊗ φ). There is an isomorphism T : V (σ_{λ} ⊗ π, φ) →^{∼} V (σ_{λ}, π ⊗ φ), which is given on the level of sections by T : C^{∞}(∂X, V (σ_{λ} ⊗ π, φ)) →^{∼} C^{∞}(∂X, V (σ_{λ}, π ⊗ φ)), T (f)(g) := π(g)f(g). 2.3.3 Given an irreducible representation σ of M and μ_{0} ∈ 𝔞^{∗} there exists a finite-dimensional representation (π_{σ,μ}, V_{π_{σ,μ}}) of G with highest A-weight μ ≥ μ_{0} such that V_{σ} = V_{π_{σ,μ}}(μ) as representations of M. Here V_{π_{σ,μ}}(μ) denotes the subspace of V_{π_{σ,μ}} on which A acts with weight μ. Note that V_{π_{σ,μ}}(−μ) = V_{σ^{w}} as representations of M. 2.3.4 If σ is a representation of M of the form σ′⊕(σ′)^{w}, where σ′ irreducible and not equivalent to its Weyl conjugate, then given μ_{0} ∈ 𝔞^{∗} there exist (π_{σ′,μ}, V_{π_{σ′,μ}}) and (π_{(σ′)^{w},μ}, V_{π_{(σ′)^{w},μ}}) for suitable μ ≥ μ_{0}. In this case we set π_{σ,μ} := π_{σ′,μ} ⊕ π_{(σ′)^{w},μ}. In connection with twisting without further notice we will always assume that σ is Weyl-invariant and either irreducible or of the form σ′ ⊕ (σ′)^{w}, where σ′ is irreducible and not Weyl invariant. Note that π_{σ̃,μ} = π̃_{σ,μ}. 2.3.5 We have an embedding of P -modules V_{σ_{λ}} ↪ V_{1_{λ+μ}}⊗V_{π_{σ,μ}} and a corresponding embedding of Γ-equivariant bundles V (σ_{λ}, φ) ↪ V (1_{λ+μ} ⊗ π_{σ,μ}, φ). Composing this embedding with the isomorphism T (introduced in 2.3.2) we obtain the Γ-equivariant embedding

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

Similarly we have a projection of P -modules V_{1_{λ−μ}} ⊗ V_{π_{σ,μ}} → V_{σ_{λ}} and a corresponding projection of Γ-equivariant bundles V (1_{λ−μ} ⊗ π_{σ,μ}, φ) → V (σ_{λ}, φ). Composing this projection with the inverse of T we obtain the Γ-equivariant projection

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

2.3.6 Since i_{σ,μ} and p_{σ,μ} are induced by homomorphisms of Γ-equivariant bundles restriction to Γ-equivariant sections over Ω_{Γ} provides the embedding

i_{σ,μ}^{Γ} : C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) ↪ C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ+μ}, π_{σ,μ} ⊗ φ))

and the projection

p_{σ,μ}^{Γ} : C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ−μ}, π_{σ,μ} ⊗ φ)) → C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) .

In Lemma 4.5 will show that i_{σ,μ}^{Γ} (resp. p_{σ,μ}^{Γ}) maps B_{Γ}(σ_{λ}, φ)_{1} (resp. B_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ)_{1}) to B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)_{1} (resp. B_{Γ}(σ_{λ}, φ)_{1}). Hence we can consider the adjoint of

p_{σ̃,μ}^{Γ} : B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)_{1} → B_{Γ}(σ̃_{−λ}, φ̃)_{1}

which will be denoted by by

(i_{σ,μ}^{Γ})_{1} : D_{Γ}(σ_{λ}, φ)_{1} → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)_{1} .

In a similar manner the adjoint of

i_{σ̃,μ}^{Γ} : B_{Γ}(σ̃_{−λ}, φ̃)_{1} → B_{Γ}(1_{−λ+μ}, π_{σ̃,μ} ⊗ φ̃)_{1}

will be a map

(p_{σ,μ}^{Γ})_{1} : D_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ)_{1} → D_{Γ}(σ_{λ}, φ)_{1} .

2.3.7 In the presence of cusps of full rank we must take into account their contribution to the function spaces (see 1.5.2). It is clear that i_{σ,μ} (resp. p_{σ,μ}) maps E_{∞_{P}} (σ_{λ}, φ) (resp. E_{∞_{P}} (1_{λ−μ}, π_{σ,μ} ⊗ φ)) to E_{∞_{P}} (1_{λ+μ}, π_{σ,μ} ⊗ φ) (resp. E_{∞_{P}} (σ_{λ}, φ)). We get second components

(i_{σ,μ}^{Γ})_{2} : R_{Γ}(σ_{λ}, φ)_{max} → R_{Γ}(1_{λ+μ}, π_{σ,μ}φ)_{max}
(p_{σ,μ}^{Γ})_{2} : R_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ)_{max} → R_{Γ}(σ_{λ}, φ)_{max}

Combining all these maps we obtain maps

i_{σ,μ}^{Γ} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)
p_{σ,μ}^{Γ} : D_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → D_{Γ}(σ_{λ}, φ)

and dually corresponding maps between spaces of functions. 2.3.8 Using the isomorphism T we transfer the action π^{1_{−λ−μ}⊗π_{σ̃,μ},idφ̃} of Z(𝔤) to

C^{∞}(∂X, V (1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)) .

Since this action commutes with Γ and is implemented by local operators we obtain an action of Z(𝔤) on C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)) as well. We will show that B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)_{1} is a Z(𝔤)-invariant subspace of C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)). By duality we obtain an action of Z(𝔤) on D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)_{1}. Since Z(𝔤) also acts on E_{∞_{P}} (1_{λ+μ}, π_{σ,μ} ⊗ φ), P ∈ P̃^{max}, the space D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) has the structure of a Z(𝔤)-module.

2.3.9 Let Ω ∈ Z(𝔤) denote the Casimir operator. Let {σ^{i}|i ∈ I_{ν}}, be the set of Weyl-invariant representations of M (irreducible or sum of two non-Weyl invariant irreducible) occuring in the restriction of V_{π_{σ,μ}}(ν) to M, where we set I_{μ} := {1}. Let ν_{i}(λ) := π^{σ_{λ+ν}^{i}}(Ω) ∈ ℂ be the eigenvalue of the Casimir operator on the principal series representation π^{σ_{λ+ν}^{i}}. We set I := ⋃_{ν≠μ} I_{ν} and define the meromorphic function

𝔞_{ℂ}^{∗} ∋ λ ↦ Π(λ) := 1 − ∏_{i∈I} \frac{Ω − ν_{i}(λ)}{ν_{1}(λ) − ν_{i}(λ)} ∈ Z(𝔤) .

This function has a finite number of poles all contained in the subset I_{𝔞} ⊂ 𝔞^{∗}. Furthermore, we define

Z(λ) := π^{1_{λ+μ}⊗π_{σ,μ},id_{φ}}(Π(λ)) .

If λ ∉ I_{𝔞} (or more precisely, if Π(λ) is regular), then Z(λ) is a projection. 2.3.10 For λ ∈ I_{𝔞} we will show in Lemma 4.15 that

i_{σ,μ}^{Γ} : D_{Γ}(σ_{λ}, φ) → ker{Z(λ) : D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)} =: ker_{Γ}(Z(λ))

is an isomorphism. We then define the continuous map

j_{σ,μ}^{Γ} : D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) ⟶^{1−Z(λ)} ker_{Γ}(Z(λ)) ⟶^{(i_{σ,μ}^{Γ})^{−1}} D_{Γ}(σ_{λ}, φ) .

In Lemma 4.15 we will furthermore show that as a function of λ the maps j_{σ,μ}^{Γ} form a meromorphic family of continuous maps which is holomorphic on 𝔞_{ℂ}^{∗} \ I_{𝔞}. If Γ is trivial, then we omit the superscript and write j_{σ,μ} for j_{σ,μ}^{{1}}. 2.3.11 We now explain by examples how the concept of twisting is applied. Assume that we have obtained a meromorphic continuation of the extension ext^{Γ} : D_{Γ}(1_{λ}, φ) → C^{−∞}(∂X, V (1_{λ}, φ)) to some half-plane W = {Re(λ) > λ_{0}} for all admissible twists φ. Then we can obtain a meromorphic continuation of ext^{Γ} : D_{Γ}(σ_{λ}, φ) → C^{−∞}(∂X, V (σ_{λ}, φ)) to some larger half plane W − μ_{0}, μ_{0} ≥ 0, and for all σ as follows. We choose (π_{σ,μ}, V_{π_{σ,μ}}) for some μ ≥ μ_{0}. Then we can define a meromorphic family ẽxt^{Γ} : D_{Γ}(σ_{λ}, φ) → C^{−∞}(∂X, V (σ_{λ}, φ)) for λ ∈ W − μ_{0} by the diagram

\begin{matrix}D_{Γ}(σ_{λ}, φ) & →^{i_{σ,μ}^{Γ}} & D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)) \\ ↓ ẽxt^{Γ} & & ↓ ext^{Γ} \\ C^{−∞}(∂X, V (σ_{λ}, φ)) & ←^{j_{σ,μ}} & C^{−∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \end{matrix} .

We then show that ext^{Γ} = ẽxt^{Γ} for all λ with sufficiently large real part. Thus ẽxt^{Γ} provides a meromorphic continuation of ext^{Γ}.

2.3.12 Assume that we have defined the scattering matrix S_{λ}^{Γ} : D_{Γ}(1_{λ}, φ) → D_{Γ}(1_{−λ}, φ) in the spherical case for all admissible twists φ. Then we could define the scattering matrix S_{λ}^{Γ} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(σ_{−λ}, φ) for all σ using the diagram

D_{Γ}(σ_{λ}, φ) →^{i_{σ,μ}^{Γ}} D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) ↓ S_{λ}^{Γ}  ↓ S_{λ+μ}^{Γ} D_{Γ}(σ_{−λ}, φ) ←^{p_{σ,μ}^{Γ}} D_{Γ}(1_{−λ−μ}, π_{σ,μ} ⊗ φ) .  (2)

For generic λ ∈ 𝔞_{ℂ}^{∗} we have a well-defined map res^{Γ} : ^{Γ}C^{−∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ), and we can define res^{Γ} : ^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) → D_{Γ}(σ_{λ}, φ) using the diagram

\begin{matrix}^{Γ}C^{−∞}(∂X, V (σ_{λ}, φ)) & →^{i_{σ,μ}} & ^{Γ}C^{−∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \\ ↓ res^{Γ} & & ↓ res^{Γ} \\ D_{Γ}(σ_{λ}, φ) & ←^{j_{σ,μ}^{Γ}} & D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) \end{matrix} .

We then check that S_{λ}^{Γ} defined by (2) coincides with res^{Γ} ∘ J_{λ} ∘ ext^{Γ}.

2.4 Holomorphic vector bundles over ℂ

2.4.1 It is a general fact that each finite-dimensional holomorphic vector bundle over ℂ or a half-plane is trivial. Let U ⊂ ℂ be open and E be a finite-dimensional holomorphic vector bundle over U. Furthermore let U × V → U be a trivial bundle of Fréchet spaces over U. By E and V we denote the corresponding sheaves of holomorphic sections. Let ϕ : E → V be a meromorphic family of linear maps. The assertions of the following lemma are well-known in the case where V is finite-dimensional.

Lemma 2.6 1. There exists a unique finite-dimensional subbundle F ⊂ U × V such that ϕ factors over a meromorphic family of maps ψ : E → F which is surjective for generic z ∈ U.

2. If we have meromorphic families of bundle maps Z_{E} : E → E and Z_{V} : V → V such that ϕ ∘ Z_{E} = Z_{V} ∘ ϕ, then Z_{V} restricts to a meromorphic family of bundle maps of F.

3. Furthermore, if U = ℂ or a half-plane, then there exists a meromorphic right-inverse η : F → E such that ϕ ∘ η = id_{F}.

Proof. The proof of this lemma will occupy the remainder of the present subsection. The main point which we will explain in detail is the reduction to the finite-dimensional case.

2.4.2 We first show that F exists. Let ϕ have singularities in the discrete subset A ⊂ U, and let n_{z}, z ∈ A, denote the order of the corresponding singularity. We consider the divisor D := ∑_{z∈A} n_{z}z, the associated line bundle L(D), and its sheaf of sections L(D). Let V(D) be the sheaf of sections of V ⊗ L(D). Then ϕ induces a holomorphic map ϕ_{D} : E → V ⊗ L(D). Let W be the quotient sheaf

E →^{ϕ_{D}} V(D) → W → 0

and denote by T or(W) its torsion subsheaf. We define F(D) as the kernel

0 → F(D) → V(D) → W/T or(W) → 0 .

There is a natural factorization of ϕ_{D} over ψ_{D} : E → F(D). 2.4.3 We claim that F(D) is a coherent sheaf. Let x ∈ U. If v ∈ V(D)_{x}, then let lp(v) ∈ V denote the leading part of v at x. Note that lp(v) = 0 implies that v = 0 since a Laurent series must start somewhere.

We define the subspace Z ⊂ V as the set of all leading parts lp(ϕ_{D}(e)), e ∈ E_{x}. 2.4.4 We now show that dim(Z) < ∞. We choose a connected neighbourhood U_{x} ⊂ U of x and a holomorphic trivialization E_{|U_{x}} = U_{x} × E_{x}. Let ϕ = ∑_{n≥m} ϕ_{n}z^{n} denote the Laurent expansion of ϕ in this trivialization, where ϕ_{n} ∈ Hom(E_{x}, V). Then using that dim E_{x} < ∞ we have

Z = ϕ_{m}(E_{x}) + ϕ_{m+1}(ker ϕ_{m}) + ϕ_{m+2}(ker ϕ_{m} ∩ ker ϕ_{m+1}) + ⋯ + ϕ_{m+k}( ⋂_{i=0}^{k−1} ker ϕ_{m+i}) ,

where k ∈ ℕ_{0} is sufficiently large such that ⋂_{i=0}^{k−1} ker ϕ_{m+i} = ⋂_{i=0}^{l−1} ker ϕ_{m+i} for all l ≥ k. This proves that dim(Z) < ∞. 2.4.5 We choose some closed subspace Y ⊂ V of finite codimension such that Z ∩ Y = {0}. We consider the composition \overline{ϕ}_{D} : E →^{ϕ} V → V/Y. We form the quotient of sheaves

E →^{\overline{ϕ}_{D}} V(D)/Y(D) → W̃ → 0 ,

and we define F̃(D) as the kernel

0 → F̃(D) → V(D)/Y(D) → W̃/T or(W̃) → 0 .

2.4.6 Consider the following commutative diagram of sheaves on U_{x}:

0  0  0
↓  ↓  ↓
0 →  0  →  Y(D)  →^{≅} Y(D) → 0
↓  ↓  ↓
0 → E/ ker ϕ_{D} →  V(D)  → W → 0
↓≅  ↓  ↓
0 → E/ ker \overline{ϕ}_{D} → V(D)/Y(D) → W̃ → 0
↓  ↓  ↓
0  0  0

The two lower rows are exact by construction. We now show that the left column is exact. We have a sequence

ker(ϕ_{D}) ⊂ ker(\overline{ϕ_{D}}) ⊂ E

of inclusioins of torsion-free sheaves. In particular, ker(ϕ_{D}) and ker(\overline{ϕ_{D}}) are sheaves of holomorphic sections of vector bundles on U_{x}. If we show that the inclusion induces an isomorphism

ker(ϕ_{D})_{x} ≅ ker(\overline{ϕ_{D}})_{x} ,  (3)

then we conclude an isomorphism of sheaves ker(ϕ_{D}) ≅ ker(\overline{ϕ_{D}}) after shrinking U_{x}, if necessary. To see (3) consider h ∈ ker(\overline{ϕ}_{D})_{x}. Then lp(ϕ_{D}(h)) ∈ Y. Since on the other hand lp(ϕ_{D}(h)) ∈ Z we conclude that lp(ϕ_{D}(h)) = 0, hence h ∈ ker(ϕ_{D})_{x}.

We now conclude that the last column is exact, too. 2.4.7 We now consider the diagram

\begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & 0 & → & T orW & →^{≅} & T orW̃ & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & Y(D) & → & W & → & W̃ & → & 0 \\ & & ↓≅ & & ↓ & & ↓ \\ 0 & → & Y(D) & → & W/T orW & → & W̃/T orW̃ & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned} .

Since Y(D) is torsion-free we have Y(D)∩T orW = 0 This implies the isomorphism in the upper row. It follows that all rows and columns of this diagram are exact.

2.4.8 We now consider the following diagram of sheaves on U_{x}:

0  0  0
↓  ↓  ↓
0 → 0 →  Y(D)  →^{≅} Y(D) → 0
↓  ↓  ↓
0 → F(D) →  V(D)  → W/T orW → 0
↓≅  ↓  ↓
0 → F̃(D) → V(D)/Y(D) → W̃/T orW̃ → 0
↓  ↓  ↓
0  0  0

The rows and the middle column are exact. We just have shown that the right column is exact. 2.4.9 We conclude that the natural map F(D) → F̃(D) is an isomorphism of sheaves. This proves the claim 2.4.3 since F̃(D) is obviously coherent. 2.4.10 Since F(D) is torsion-free it is the sheaf of sections of a holomorphic vector bundle F (D) (here we use the fact that the base space is smooth and one-dimensional). By construction F(D)/ψ_{D}(E) is torsion and therefore ψ_{D,x} : E_{x} → F (D)_{x} is surjective for generic x ∈ U. We define F := F (D) ⊗ L(−D) and let ψ : E → F be the corresponding meromorphic family of maps. The uniqueness part and assertion 2. are left to the reader. 2.4.11 We now construct the meromorphic right-inverse η. Note that we can assume that E and F are trivial. We fix trivializations and a constant Hermitian metric on E. Let z ∈ ℂ be such that ϕ : E_{z} → F_{z} is regular and surjective. Let P be the orthogonal projection onto the orthogonal complement of ker ϕ_{z}. We extend P constantly over U and let P (E) be the range of P. The composition ϕ ∘ P : P (E) → F is invertible at z and therefore has a meromorphic family of inverses η : F → P (E) ⊂ E. □

2.4.12

Definition 2.7 The bundle F constructed in Lemma 2.6 is called the image bundle of ϕ.

3 Pure cusps

3.1 Geometry of cusps

3.1.1 In this subsection we analyze the geometry of cusps. First we recall the following theorem of Auslander.

Theorem 3.1 ([Aus61], [Apa97]) Let N be a connected, simply-connected nilpotent Lie group and M be a compact group of automorphisms of N. Furthermore let U ⊂ N ⋊ M be a discrete subgroup and U^{∗} := \overline{NU}_{0} ∩ U = p_{|U}^{−1}(\overline{p(U)}_{0}), where p : N ⋊ M → M is the projection, ()_{0} stands for connected component of the identity, and ”¯.” means the closure of the set in the argument. Then

1. U^{∗} ⊂ U is a normal subgroup of finite index.

2. There exists b ∈ N and a connected subgroup N_{V} ⊂ N such that (U^{∗})^{b} := bU^{∗}b^{−1} acts effectively and cocompactly by translations on N_{V}, i.e. there is lattice V^{∗} ⊂ N_{V} and an isomorphism θ : (U^{∗})^{b} → V^{∗} such that u.x = θ(u)x, x ∈ N_{V}, where (N ⋊ M) × N ∋ (a, x) ↦ a.x ∈ N denotes the natural action of N ⋊ M on N.

3. The element b ∈ N of 2. can be choosen such that U^{b} leaves the space N_{V} invariant.

4. M_{U^{0}} := \overline{p(U^{∗})} ⊂ M is a torus.

The third assertion is due to Apanasov [Apa97]. 3.1.2 We now apply Theorem 3.1 to a discrete torsion-free subgroup U ⊂ P of a parabolic subgroup P ⊂ G such that P is U-cuspidal. We have the exact sequence

0 → N → P →^{l} L → 0 ,  (4)

where N is the nil-radical of P. The group L decomposes as L = M × A, and we denote by l_{M} : P → M the composition of l with the projection from L to M. Let E := l^{−1}(M × {1}). If we choose a split s_{1} : M → E of the sequence 0 → N → E →^{l_{M}} M → 0, then m ∈ M acts by the automorphism s_{1}(m) on N, and we have an isomorphism T_{s_{1}} : E → N ⋊_{s_{1}} M. Since P is U-cuspidal we have U ⊂ E. Applying Theorem 3.1 to T_{s_{1}}(U) we obtain a subgroup T_{s_{1}}(U)^{∗} ⊂ T_{s_{1}}(U) of finite index, b ∈ N, a connected subgroup N_{V}, a lattice V^{∗} ⊂ N_{V}, and an isomorphism θ : bT_{s_{1}}(U)^{∗}b^{−1} → V^{∗} such that bT_{s_{1}}(U)b^{−1} leaves N_{V} invariant and acts by translations on N_{V} via θ. 3.1.3 If we replace the split s_{1} and N_{V} by b^{−1}N_{V} b and s := b^{−1}s_{1}b, then T_{s}(U) itself leaves N_{V} invariant, and T_{s}(U)^{∗} acts by translations on N_{V}.

From now on we use the split s in order to identify M with the subgroup s(M) ⊂ E ⊂ P, to write E as the product NM, and to indentify T_{s}(U) with U. 3.1.4 Any element u ∈ U^{∗} (u ∈ U) can be written as n_{u}m_{u} ∈ NM such that n_{u} ∈ N_{V} and m_{u} centralizes (normalizes) N_{V}.

We have V^{∗} := {n_{u}|u ∈ U^{∗}} ⊂ N_{V}. The map V^{∗} ∋ n_{u} ↦ m_{u} ∈ M defines a homomorphism m^{∗} : V^{∗} → M_{U^{0}}.

Lemma 3.2 There exists a subgroup V ⊂ V^{∗} of finite index such that the restriction m := m_{|V}^{∗} extends to a homomorphism m : N_{V} → M_{U^{0}}.

Proof. Let π : V^{∗} → N_{V} /[N_{V} , N_{V}] be the projection. Then π(V^{∗}) ⊂ N_{V} /[N_{V} , N_{V}] is a lattice (see e.g. [Rag72], proof of Thm. 2.10). Let g_{1}, … g_{n} ∈ V^{∗}, n = dim(N_{V} /[N_{V} , N_{V}]) be such that π(g_{1}), …, π(g_{n}) generate the lattice π(V^{∗}). We define V := ⟨g_{1}, …, g_{n}⟩. Furthermore we put X_{i} := log(g_{i}) ∈ 𝔫_{V} and consider π(X_{i}) ∈ 𝔫_{V} /[𝔫_{V} , 𝔫_{V}]. Then π(X_{i}) is a basis of 𝔫_{V} /[𝔫_{V} , 𝔫_{V}], and {X_{1}, …, X_{n}} generate the Lie algebra 𝔫_{V}. We conclude that N_{V} is the smallest connected group containing V, and by [Rag72], Ch.2, V is a lattice in N_{V}. Therefore V ⊂ V^{∗} has finite index.

We claim that V ∩ [N_{V} , N_{V}] = [V, V]. Let g ∈ V ∩ [N_{V} , N_{V}]. Then there are finite sequences i_{k} ∈ {1, …, n} and e_{k} ∈ {1, −1}, k = 1, … r, such that g = g_{i_{1}}^{e_{1}} … g_{i_{r}}^{e_{r}}. Applying π we obtain 1 = π(g_{i_{1}})^{e_{1}} … π(g_{i_{r}})^{e_{r}}. Using the fact that N_{V} /[N_{V} , N_{V}] is free abelian we conclude that there is a permutation σ ∈ S_{r} such that 1 = g_{i_{σ(1)}}^{e_{σ(1)}} … g_{i_{σ(r)}}^{e_{σ(r)}}. We conclude that 1 = g modulo [V, V], i.e. g ∈ [V, V]. This proves the claim.

We first define the derivative dm : 𝔫_{V} → 𝔪_{U^{0}} of m as the composition

𝔫_{V} →^{π} 𝔫_{V} /[𝔫_{V} , 𝔫_{V}] →^{q} 𝔪_{U^{0}} ,

where q is given by q(π(X_{i})) := Z_{i}, i = 1, … n, where Z_{i} ∈ 𝔪_{N_{V}} is any element satisfying exp(Z_{i}) = m(g_{i}). Integrating the derivative we obtain a representation m̃ : N_{V} → M_{U^{0}} such that m(g_{i}) = m̃(g_{i}), i = 1 …, n. Since M_{U^{0}} is a torus we see that m must factor over [V, V] and conclude that m = m̃. This finishes the proof of the lemma. □

3.1.5 We consider the subgroup of finite index U^{1} := {vm(v)|v ∈ V } ⊂ U.

Definition 3.3 We define U^{0} to be the largest normal subgroup of U such that U^{0} ⊂ U^{1} ⊂ U, i.e.

U^{0} := ⋂_{u∈U}(U^{1})^{u} .

U^{0} ⊂ U has finite index, too. The torus M_{U^{0}} coincides with \overline{m(N_{V})}.

Definition 3.4 We set P_{U^{0}} := N_{V} M_{U^{0}}. Furthermore we define M_{U} := \overline{l_{M}(U)} and P_{U} := N_{V} M_{U}.

Then U ⊂ P_{U}, M_{U^{0}} ⊂ M_{U} has finite index, and M_{U} normalizes N_{V}.

3.1.6 We have an exact sequence

0 → E → P →^{l_{A}} A → 0 ,

where the decomposition E = NM is already fixed by the split s. The split extends uniquely to a split s : MA → P of (4) and induces a split of the sequence above. In particular we obtain a Langlands decomposition P = NAM, where we identify A with its image by s. The split furthermore defines an action of A on N by automorphisms commuting with the action of M. 3.1.7 We call the cusp associated to U ⊂ P regular if we can choose s such that N_{V} is Ainvariant (compare Definition 1.3).

Lemma 3.5 If X is a real or complex hyperbolic space, then every cusp is regular.

Proof. If X is real-hyperbolic, then for any split s the group A acts on 𝔫 as multiplication by scalars. Any subspace, in particular 𝔫_{V}, is invariant with respect to A.

We now consider the case that X = Hℂ^{n}. Assume that we have chosen a split s. Then we can decompose 𝔫 = 𝔫_{α} ⊕ 𝔫_{2α} with respect to the action of A such that a ∈ A acts on 𝔫_{iα} as multiplication by a^{iα}. Here 𝔫_{α} is a symplectic vector space of dimension 2(n − 2), where the symplectic form with values in the one-dimensional space 𝔫_{2α} is given by the commutator. Taking M_{U^{0}}-invariants we obtain a decomposition 𝔫^{M_{U^{0}}} = 𝔫_{1}⊕𝔫_{2}, where 𝔫_{1} ⊂ 𝔫_{α} is a symplectic subspace ([GS77], Prop. 4.2.1). If 𝔫_{1} = {0}, then 𝔫_{V} = 𝔫_{2} is invariant with respect to A, and we are done. We now assume that 𝔫_{1} ≠ {0}. We have two cases. If 𝔫_{2} ⊂ 𝔫_{V}, then 𝔫_{V} = 𝔫_{V} ∩𝔫_{1} ⊕𝔫_{2}, and this space is invariant with respect to A. Thus we assume that 𝔫_{2} ⊄ 𝔫_{V}. Since dim(𝔫_{2}) = 1 we then have 𝔫_{2} ∩ 𝔫_{V} = {0}. Let p_{i} : 𝔫_{V} → 𝔫_{i} denote the projections. Then p_{1} : 𝔫_{V} → 𝔫_{1} is injective. Let λ : 𝔫_{1} → 𝔫_{2} be a linear extension of p_{2} ∘ p_{1}^{−1} : p_{1}(𝔫_{V}) → 𝔫_{2}. Since the symplectic form is non-degenerate there exists a unique Y ∈ 𝔫_{1} such that λ(X) = [Y, X] for all X ∈ 𝔫_{1}.

Let h := exp(−Y) ∈ N^{M_{U^{0}}} . We claim that 𝔫_{V}^{h} ⊂ 𝔫_{1}. Indeed, if X ∈ 𝔫_{V}, then p_{2}(X^{h}) = p_{2}(X − [Y, X]) = λ(p_{1}(X)) − [Y, X] = [Y, X] − [Y, X] = 0.

If we replace the split s : M × A → P by hsh^{−1}, then 𝔫_{V} gets replaced by 𝔫_{V}^{h}. Thus by an appropriate choice of the split we can assume that 𝔫_{V} ⊂ 𝔫_{1}, and we are done. □

3.1.8 The next lemma shows that regularity of cusps is a proper restriction in the case of Hℍ^{n}.

Lemma 3.6 If X = Hℍ^{n}, n ≥ 2, then there exist non-regular cusps.

Proof. It suffices to provide an example for n = 2. We can identify 𝔫 = ℍ ⊕ Im_{ℍ}(ℍ) (Im_{ℍ}(ℍ) denoting the imaginary quaternions), and the commutator of X, Y ∈ ℍ is given by [X, Y] = X̄Y − Ȳ X ∈ Im_{ℍ}(ℍ). Implicitly we have fixed some split s × t such that a ∈ A acts on ℍ as multiplication by a^{α} and on Im_{ℍ}(ℍ) by a^{2α}. Let 1, I, J, K be a base of the copy of ℍ and i, j, k be the base of Im_{ℍ}(ℍ). Then we consider 𝔫_{V} = span_{ℝ}{1, I + j, i} and let U ⊂ N_{V} be any lattice. It is not possible to conjugate this subspace into an A-invariant one. □

3.1.9 Assume that U ⊂ P defines a regular cusp.

Definition 3.7 We define ρ_{U} ∈ 𝔞^{∗} by ρ_{U}(H) := \frac{1}{2}tr(ad(H)_{|𝔫_{V}}), where 𝔫_{V} is the Lie algebra of N_{V}. Furthermore, we set ρ^{U} := ρ − ρ_{U}.

3.2 Schwartz spaces

3.2.1 In this subsection we start the description of our function spaces. Here we introduce the Schwartz spaces S_{U}(σ_{λ}, φ). We will show that these spaces form trivial holomophic bundles of Fréchet spaces for λ ∈ 𝔞_{ℂ}^{∗}, and that they are compatible with twisting. The Schwartz spaces are basic building blocks of the function spaces B_{U}(σ_{λ}, φ) which we will define later (see Subsection 3.4) . 3.2.2 We fix a representative w ∈ K of the non-trivial element of the Weyl group W (𝔤, 𝔞) such that w^{2} = 1. Let ∞_{P} ∈ ∂X be the fixed point of P and 0_{P} := w∞_{P}. The subset Ω_{U} = Ω_{P} = ∂X \ ∞_{P} is the N-orbit of 0_{P} (see Subsection 1.2 for notation). Indeed, the map N ∋ x ↦ xw∞_{P} ∈ Ω_{U} is a diffeomorphism from N to Ω_{U}. 3.2.3 Let (σ, V_{σ}) be a finite-dimensional unitary representation of M. Given λ ∈ 𝔞_{ℂ}^{∗} we form the representation (σ_{λ}, V_{σ_{λ}}) of P (see 1.3.2). Furthermore, let (φ, V_{φ}) be an admissible twist (see Definition 1.5). 3.2.4 The action of A on U(𝔫) induces a grading. We choose a basis {A_{i}} of U(𝔫) such that the subset {A_{i} | deg(A_{i}) ≤ dα} spans U(𝔫)^{≤dα}. Furthermore, we choose a norm |.| on V_{φ}. For any d ∈ ℕ_{0}, k ∈ ℝ, and compact subset W ⊂ N \ N_{V} (see Theorem 3.1 for the definition of N_{V}) we define the seminorm q_{W,d,k} on C_{c}^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)) by

q_{W,d,k}(f) := sup_{{i| deg(A_{i})≤dα}} sup_{x∈W} sup_{a∈A_{+}} a^{kα−2(λ−ρ^{U})}|φ(a)^{−1}f(x^{a}A_{i}w)| .

Here we consider a section of V_{B_{U}} (σ_{λ}, φ) as a U-invariant function on G \ P with values in V_{σ_{λ}} ⊗ V_{φ} satisfying the corresponding invariance conditions with respect to the right P -action, and where u ∈ U acts by (π^{σ_{λ},φ}(u)f)(xw) = φ(u)f(u^{−1}xw). 3.2.5 We fix W and another compact subset W′ ⊂ N such that W′ ∪W^{A_{+}} projects surjectively onto B_{U}. We add an arbitrary C^{d}-norm over W′ to q_{W,d,k} in order to obtain a norm ∥.∥_{k,d}. We first define S_{U,k,d}(σ_{λ}, φ) to be the Banach space closure of C_{c}^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)) with respect to ∥.∥_{k,d}. This Banach space is independent (up to equivalent norms) of the choices of W, W′, |.|, and the base of U(𝔫).

Definition 3.8 We define S_{U,k}(σ_{λ}, φ) to be the intersection of the spaces S_{U,k,d}(σ_{λ}, φ) for all d ∈ ℕ_{0}. The space S_{U,k}(σ_{λ}, φ) is a Fréchet space which is topologized by the countable set of norms ∥.∥_{k,d}, d ∈ ℕ_{0}.

The space S_{U,k}(σ_{λ}, φ) is a space of smooth sections with a fixed growth rate at infinity of B_{U} measured by k ∈ ℕ_{0}. 3.2.6 We now show that the family of spaces {S_{U,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} forms a trivial holomorphic bundle of Fréchet spaces. To this end we construct a certain holomorphic family of functions s^{z}, z ∈ ℂ. In the proof of Lemma 3.10 we use multiplication by these functions in order to identify the Schwartz spaces for different λ ∈ 𝔞_{ℂ}^{∗}. 3.2.7

Lemma 3.9 There exists a positive P_{U}-invariant function s ∈ C^{∞}(N) such that for any compact subset W ⊂ N \ N_{V} there is a_{0} ∈ A with s(x^{a}) = a^{2α}s(x) for all a ≥ a_{0} and x ∈ W.

Proof. Let N̄ := N^{w} and consider the decomposition

G \ wP = N̄MAN , g = n̄(g)m(x)a(x)n(x) .

Note that a(xw) satisfies a(x^{a}w) = a^{2}a(xw) for all a ∈ A and x ∈ N \ {1} (see (6) below). We consider the function N \ {1} ∋ x ↦ a(xw)^{α} as a positive function s_{1} ∈ C^{∞}(N \ {1}). Let s_{2} ∈ C^{∞}(N) be any positive function. Using a partition of unity we glue s_{1} and s_{2} to obtain a positive function s_{3} ∈ C^{∞}(N) which coincides with s_{1} outside of a compact subset W_{1} of N. We now choose z ∈ ℝ such z < −ρ_{U}/α and define s_{4} as the average

s_{4}(x) := ∫_{P_{U}} s_{3}^{z}(y.x)dy

(see Definition 3.4 for P_{U} and Theorem 3.1, (2) for the action (y, x) ↦ y.x). The integral converges (compare Lemma 3.16) and defines a positive P_{U}-invariant smooth function on N. If W ⊂ N \ N_{V} is compact, then there is a^{0} ∈ A such that (N_{V} W)^{aM_{U}} ∩ W_{1} = ∅ for all a ≥ a_{0}. If a ≥ a_{0} and x ∈ W, then we have

s_{4}(x^{a}) = ∫_{P_{U}} s_{3}^{z}(y.x^{a})dy
= ∫_{M_{U}} ∫_{N_{V}} s_{3}^{z}(vx^{au})dvdu
= a^{2ρ_{U}} ∫_{M_{U}} ∫_{N_{V}} s_{3}^{z}((vx)^{au})dvdu
= a^{2(zα+ρ_{U})}s_{4}(x) .

We define s as the α/(zα + ρ_{U})’th power of s_{4}. □

3.2.8 We now consider the function s constructed in Lemma 3.9 as a section s ∈ C^{∞}(Ω_{P} , V (1_{ρ+α})) by defining s(xw) := s(x).

If λ ∈ 𝔞^{∗}, then the bundle V (1_{λ}) is a complex line bundle which can be written as the complexification of a trivial G-equivariant real line bundle V (1_{λ})^{ℝ}. It makes sense to speak of a positive section of V (1_{λ})^{ℝ}. A section of V (1_{λ}) is called positive if it is a positive section of the real subbundle V (1_{λ})^{ℝ}. In particular, the section s of V (1_{ρ+α}) constructed above is positive. 3.2.9 For each z ∈ ℂ we can form s^{z}, which is a section of V (1_{ρ+zα}). There is a natural identification V (σ_{λ}) ⊗ V (1_{ρ+zα}) ≅ V (σ_{λ+zα}). Multiplication by s^{z} identifies C^{∞}(Ω_{P} , V (σ_{λ}, φ)) with C^{∞}(Ω_{P} , V (σ_{λ+zα}, φ)).

Lemma 3.10 For each d ∈ ℕ_{0} and z ∈ ℂ multiplication by s^{z} defines a continuous map from S_{U,k,d}(σ_{λ}, φ) to S_{U,k,d}(σ_{λ+zα}, φ). The family {S_{U,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} is a trivial holomorphic bundle of Fréchet spaces.

Proof. Since s is U-invariant and non-vanishing multiplication by s^{z} is an isomorphism of C^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)) with C^{∞}(B_{U}, V_{B_{U}} (σ_{λ+zα}, φ)) and of the subspaces of sections with compact support. It suffices to show that there is a constant C ∈ ℝ such that for all f ∈ C_{c}^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)) we have ∥s^{z}f∥_{k,d} ≤ C∥f∥_{k,d}. This follows from the Leibniz rule and the following estimate. Let A ∈ U(𝔫) be homogeneous of degree dα. Then for any compact subset W ⊂ N \ N_{V} there is a constant C ∈ ℝ and a_{0} ∈ A such that for x ∈ W and a ≥ a_{0}

|s^{z}(x^{a}Aw)| = |s^{z}((xA^{a^{−1}})^{a}w)|
= a^{(2z−d)α}|s^{z}(xAw)|
≤ Ca^{(2z−d)α} .

For any λ_{0} we now define the trivialization Φ_{λ_{0}} : ⋃_{λ∈𝔞_{ℂ}^{∗}} S_{U,k}(σ_{λ}, φ) → S_{U,k}(σ_{λ_{0}}, φ) × 𝔞_{ℂ}^{∗} such that the restriction of Φ_{λ_{0}} to the fibre S_{U,k}(σ_{λ}, φ) is multiplication by s^{(λ_{0}−λ)/α}. The transition map Φ_{λ_{0}} ∘ Φ_{λ_{1}}^{−1} is multiplication by s^{(λ_{0}−λ_{1})/α} and thus independent of λ. In particular, it is a holomorphic family of continuous maps. □

3.2.10 The Schwartz space is the space of smooth rapidly decaying sections on B_{U}.

Definition 3.11 We define the Schwartz space S_{U}(σ_{λ}, φ) as the intersection of the spaces S_{U,k}(σ_{λ}, φ) over all k ∈ ℕ_{0}.

3.2.11 Since the trivializations Φ_{λ} are compatible with the inclusions S_{U,k′}(σ_{λ}, φ) ↪ S_{U,k}(σ_{λ}, φ), k′ ≥ k, we obtain the following corollary.

Corollary 3.12 The family {S_{U}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} is a trivial holomorphic vector bundle of Fréchet spaces.

3.2.12 Note that the inclusions S_{U,k+1}(σ_{λ}, φ) ↪ S_{U,k}(σ_{λ}, φ) are compact. The proof of this fact is a simple application of the Lemma of Arzela-Ascoli. The compactness of these inclusions imply the following fact.

Corollary 3.13 The Schwartz space S_{U}(σ_{λ}, φ) is a Montel space. In particular, it is reflexive.

3.2.13 If the cusp associated to U ⊂ P has full rank, then we have for all k ∈ ℕ_{0}

S_{U,k}(σ_{λ}, φ) = S_{U}(σ_{λ}, φ) = C^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)) .

3.2.14 Next we show that the Schwartz spaces are compatible with twisting. Let σ be a Weyl invariant representation of M and (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in 2.3.3. See 2.3.9 for the definition of Z(λ) and Π(λ).

Lemma 3.14

1. The Schwartz space S_{U}(σ_{λ}, φ) coincides with the closed subspace of ^{U}C^{∞}(∂X, V (σ_{λ}, φ)) of sections which vanish at ∞_{P} of infinite order.

2. We have holomorphic families of continuous maps

{i_{σ,μ}^{U}} : S_{U}(σ_{λ}, φ) → S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)
{p_{σ,μ}^{U}} : S_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → S_{U}(σ_{λ}, φ) .

3. S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) is a Z(𝔤)-module.

4. If λ ∉ I_{𝔞}, then {i_{σ,μ}^{U}} maps S_{U}(σ_{λ}, φ) isomorphically onto

ker{Z(λ) : S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)} ,

and the restriction of {p_{σ,μ}^{U}} to

ker{Z̃(λ) : S_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → S_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ)}

is an isomorphism onto S_{U}(σ_{λ}, φ) (here Z̃(λ) is the adjoint of π^{1_{−λ+μ}⊗π̃_{σ,μ},id_{φ}}(Π(λ))).

5. The composition

{j_{σ,μ}^{U}} : S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) →^{1−Z(λ)} ker(Z(λ)) →^{({i_{σ,μ}^{U}})^{−1}} S_{U}(σ_{λ}, φ)

which is intitially defined for λ ∉ I_{𝔞} extends to a meromorphic family of continuous maps. Similarly, the composition

{q_{σ,μ}^{U}} : S_{U}(σ_{λ}, φ) →^{({p_{σ,μ}^{U}})^{−1}} ker(Z̃(λ)) → S_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ)

extends to a meromorphic family of continuous maps.

Proof. We leave the proof of 1. at this place to the interested reader. Alternatively the assertion can be considered as an immediate consequence of the material of Subsection 3.3. 3.2.15 2. follows from 1. and the fact that {i_{σ,μ}^{U}} and {p_{σ,μ}^{U}} are induced by an inclusion

i_{σ,μ} : V (σ_{λ}, φ) → V (1_{λ+μ}, π_{σ,μ} ⊗ φ)

and a projection

p_{σ,μ} : V (1_{λ−μ}, π_{σ,μ} ⊗ φ) → V (σ_{λ}, φ)

of bundles. Another argument for the assertion about {i_{σ,μ}^{U}} would be to check that {i_{σ,μ}^{U}} maps S_{U,k,d}(σ_{λ}, φ) continuously to S_{U,k′,d}(1_{λ+μ}, π_{σ,μ} ⊗ φ), k′ = k + μ/α, for all k, d ∈ ℕ_{0} by comparing the norms explicitly. A similar argument works for {p_{σ,μ}^{U}} as well. 3.2.16 In order to see 3. note that the action of cZ(𝔤) is implemented by differential operators. Using the fact that the action π^{1_{λ+μ}⊗π_{σ,μ},id_{φ}} of Z(𝔤) commutes with the action π^{1_{λ+μ},π_{σ,μ}⊗φ} of P_{U}A which is used in order to characterize the growth of elements of S_{U,k}(1_{λ+μ}, π_{σ,μ} ⊗ φ) one could alternatively check that this space is a Z(𝔤)-module. 3.2.17 We now prove the first assertion of 4. and leave the second to the reader since the argument is similar. The representation V_{1_{λ+μ}} ⊗ V_{π_{σ,μ}} of P fits into an exact sequence

0 → V_{σ_{λ}} →^{i_{σ,μ}} V_{1_{λ+μ}} ⊗ V_{π_{σ,μ}} → W → 0 ,

which induces a corresponding exact sequence of bundles

0 → V (σ_{λ}, φ) →^{i_{σ,μ}} V (1_{λ+μ}, π_{σ,μ} ⊗ φ) → V (w, φ) → 0 ,  (5)

where w denotes the representation of P on W. Using 1. we obtain an exact sequence of sections which vanish of infinite order at ∞_{P}. Going over to U-invariant sections we obtain the exact sequence

0 → S_{U}(σ_{λ}, φ) →^{{i_{σ,μ}^{U}}} S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) →^{p} S_{U}(w, φ) .

The action of Z(𝔤) commutes with p. The operator Π(λ) is designed such that π^{w,φ}(Π(λ)) = 1 and π^{σ_{λ},φ}(Π(λ)) = 0 (note that λ ∉ I_{𝔞}). Since {i_{σ,μ}^{U}} is a morphism of Z(𝔤)-modules we see that {i_{σ,μ}^{U}} maps to ker(Z(λ)). In order to show that it is onto take f ∈ ker(Z(λ)). Then

p(f) = p((1 − Z(λ))f) = (1 − π^{w,φ}(Π(λ)))p(f) = 0 ,

and f is in the range of {i_{σ,μ}^{U}}. 3.2.18 Finally we show the first assertion of 5. and leave the second to the reader since the argument is similar, again. We choose a holomorphic family of splits j : V (1_{λ+μ}, π_{σ,μ} ⊗ φ) → V (σ_{λ}, φ) of the sequence of bundles (5) (not necessarily U-invariant). The split j induces a holomorphic family of maps J : C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) → C^{∞}(∂X, V (σ_{λ}, φ)). The map J is compatible with the subspaces of sections vanishing of infinite order at ∞_{P}. We define the meromorphic family of continuous maps j̃_{σ,μ}^{U} as the restricition of J ∘(1−Z(λ)) to S_{U}(1_{λ+μ}, π_{σ,μ}⊗ φ). Since

j̃_{σ,μ}^{U} ∘ i_{σ,μ}^{U} = J ∘ (1 − Z(λ)) ∘ i_{σ,μ}^{U} = id

and the restriction of j̃_{σ,μ}^{U} to ker(1 − Z(λ)) vanishes we see by 4. that j̃_{σ,μ}^{U} maps to S_{U}(σ_{λ}, φ), and that it coincides with {j_{σ,μ}^{U}} for λ ∉ I_{𝔞}. □

3.3 Asymptotics for the trivial group

3.3.1 We consider the decomposition G \ wP = N̄MAN, g = n̄(g)m(g)a(g)n(g). There is a unique diffeomorphism F : N\{1} → N̄\{1} such that nw ∈ F (n)MAN. Indeed, F (n) = n̄(nw). One can check that

a(n^{ma}w) = a^{2}a(nw)  (6)
m(n^{a}w) = m(nw)
F (n^{ma}) = F (n)^{ma} .

3.3.2 Let (φ, V_{φ}) be an admissible twist for U. For simplicity we normalize the restriction of φ to A such that the lowest A-weights of all its irreducible coponents are zero. Let l_{φ} ∈ 𝔞^{∗} be the highest weight of φ. 3.3.3 In the present subsection we describe the space C^{∞}(∂X, V (σ_{λ}, φ)) =: B_{{1}}(σ_{λ}, φ) as an extension

0 → S_{{1}}(σ_{λ}, φ) → B_{{1}}(σ_{λ}, φ) → R_{{1}}(σ_{λ}, φ) → 0 .

The Schwartz space S_{{1}}(σ_{λ}, φ) coincides with the subspace of C^{∞}(∂X, V (σ_{λ}, φ)) of all section that vanish at ∞_{P} of infinite order. Note that N̄ is diffeomorphic to ∂X \ 0_{P} by n̄ ↦ n̄∞_{P} (recall that 0_{P} = w∞_{P}). Using a theorem of E. Borel to the effect that each formal power series can be realized as a Taylor series of a smooth function we obtain an exact sequence

0 → S_{{1}}(σ_{λ}, φ) → C^{∞}(∂X, V (σ_{λ}, φ)) →^{T S} Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ}) → 0 ,  (7)

where T S is given by T S(f)(A) := f(Ae). 3.3.4 The space Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ}) admits an action of AM_{U} by (u.f)(A) := (σ_{λ}(u) ⊗ φ(u))f(A^{u^{−1}}). With respect to the action of A it can be decomposed as

Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ}) = ∏_{n∈ℕ_{0}} Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ})^{n} ,

where A acts on the M_{U}-module Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ})^{n} with weight ρ − λ + αn. 3.3.5 By definition a function p : N̄ → V_{σ_{λ}} ⊗ V_{φ} is a polynomial iff p ∘ exp : 𝔫̄ → V_{σ_{λ}} ⊗ V_{φ} is a polynomial. Such a function p is called homogeneous of degree n, iff

(σ_{λ}(a) ⊗ φ(a))p(n̄^{a^{−1}}) = a^{ρ−λ+nα}p(n̄) .

3.3.6 The space Hom(U(𝔫̄), V_{σ_{λ}} ⊗ V_{φ})^{n} can be identified AM_{U}-equivariantly with the space of polynomials Pol(N̄, V_{σ_{λ}} ⊗ V_{φ})^{n} on N̄ with values in V_{σ_{λ}} ⊗ V_{φ} being homogeneous of degree n. Here a polynomial p ∈ Pol(N̄, V_{σ_{λ}} ⊗V_{φ})^{n} corresponds to the map U(𝔫̄) ∋ A ↦ p(Ae) ∈ V_{σ_{λ}} ⊗V_{φ}. 3.3.7 If a homogeneous polynomial p ∈ Pol(N̄, V_{σ_{λ}} ⊗ V_{φ})^{n} is considered as a section f_{p} of V (σ_{λ}, φ) defined over ∂X \ {0_{P} }, then we have for x ∈ N

f_{p}(xw) = f_{p}(F (x)m(xw)a(xw)n(xw)) = σ(m(xw))^{−1}a(xw)^{λ−ρ}f_{p}(F (x)) .

In particular, for x ∈ N, x ≠ 1, and a ∈ A we have

φ(a)^{−1}f_{p}(x^{a}w) = φ(a)^{−1}a(x^{a}w)^{λ−ρ}σ(m(x^{a}w))^{−1}f_{p}(F (x^{a}))
= a^{2(λ−ρ)−nα}a(xw)^{λ−ρ}σ(m(xw))^{−1}f_{p}(F (x)) .

Definition 3.15 We define A_{{1}}(σ_{λ}, φ)^{n} ⊂ C^{∞}(∂X \ {∞_{P} , 0_{P} }, V (σ_{λ}, φ)) as the subspace of sections f satisfying

φ(a)^{−1}f(x^{a}w) = a^{2(λ−ρ)−nα}f(xw)  (8)

for all x ∈ N \ {1}, a ∈ A. We define the subspace R_{{1}}(σ_{λ}, φ)^{n} ⊂ A_{{1}}(σ_{λ}, φ)^{n} as the subspace spanned by the sections f_{p} corresponding to p ∈ Pol(N̄, V_{φ})^{n}. We further define R_{{1},k}(σ_{λ}, φ) := ⨁_{n=0}^{k} R_{{1}}(σ_{λ}, φ)^{n} and R_{{1}}(σ_{λ}, φ) := ∏_{n=0}^{∞} R_{{1}}(σ_{λ}, φ)^{n}.

3.3.8 The spaces A_{{1}}(σ_{λ}, φ)^{n} for the various λ can be identified using multiplication by suitable powers of the function xw ↦ a(xw)^{α} and therefore form a trivial vector bundle of Fréchet spaces over 𝔞_{ℂ}^{∗}. The spaces R_{{1}}(σ_{λ}, φ)^{n} form trivial finite-dimensional subbundles trivialized by the identification with the trivial bundles 𝔞_{ℂ}^{∗} × Pol(N̄, V_{σ} ⊗ V_{φ})^{n} → 𝔞_{ℂ}^{∗}. Thus R_{{1},k}(σ_{λ}, φ) has the structure of a trivial holomorphic vector bundle.

3.3.9 We fix a smooth cut-off function χ ∈ C^{∞}(A) such that χ(a) = 0 in a neighbourhood of A_{−} and χ(a) = 1 if a^{α} > 2. Multiplication by the function

Ω_{P} ∋ x 0_{P} ↦ χ(a(xw))

defines inclusions L : R_{{1}}(σ_{λ}, φ)^{n} → C^{∞}(B_{{1}}, V_{{1}}(σ_{λ}, φ)) for each n, and summing these maps up we obtain inclusions

L : R_{{1},k}(σ_{λ}, φ) → C^{∞}(B_{{1}}, V_{B_{{1}}}(σ_{λ}, φ)) .

In the present paper the symbol L is used for various maps of this kind. It will always be clear from the context which version of L is meant. 3.3.10 If f ∈ S_{{1},k}(σ_{λ}, φ), then

lim_{a→∞}a^{kα−2(λ−ρ)}φ(a)^{−1}f(x^{a}w) = 0

uniformly for x in compact subsets of N \ {1}. Thus L(R_{{1},k}(σ_{λ}, φ)) intersects S_{{1},k}(σ_{λ}, φ) trivially. We define

B_{{1},k}(σ_{λ}, φ) := S_{{1},k}(σ_{λ}, φ) ⊕ L(R_{{1},k}(σ_{λ}, φ)) .

This space fits into the exact sequence

0 → S_{{1},k}(σ_{λ}, φ) → B_{{1},k}(σ_{λ}, φ) →^{AS} R_{{1},k}(σ_{λ}, φ) → 0 ,  (9)

where AS takes the finite asymptotic expansion. To AS applies the same remark as as above for L. This symbol appears in various versions which will be denoted by the same symbol, and it will be clear from the context which version is meant. 3.3.11 The sequence (9) is split by L and defines B_{{1},k}(σ_{λ}, φ) as a topological vector space and the family of spaces {B_{{1},k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} as a trivial bundle of Fréchet spaces. Moreover, we have continuous and compact inclusions B_{{1},k+1}(σ_{λ}, φ) ↪ B_{{1},k}(σ_{λ}, φ). The intersection of these spaces over all k ∈ ℕ_{0} is the Fréchet and Montel space B_{{1}}(σ_{λ}, φ), which concides with C^{∞}(∂X, V (σ_{λ}, φ)) because of exactness of (7). Note that the sequence

0 → S_{{1}}(σ_{λ}, φ) → B_{{1}}(σ_{λ}, φ) → R_{{1}}(σ_{λ}, φ) → 0

does not admit any continuous split. 3.3.12 Though this is clearly possible we will not attempt to trivialize the family of spaces B_{{1}}(σ_{λ}, φ). But we keep in mind that B_{{1}}(σ_{λ}, φ) is an intersection of trivial holomorphic bundles (see 2.1.4). Below we will obtain a similar description of the spaces B_{U,k}(σ_{λ}, φ).

3.4 The spaces B_{U,k}(σ_{λ}, φ)

3.4.1 In the present subsection σ is any finite-dimensional representation of M, and φ denotes a twist. Below we define the spaces B_{U}(σ_{λ}, φ). In particular, we construct the spaces R_{U}(σ_{λ}, φ) which describe the asymptotic behaviour of elements of B_{U}(σ_{λ}, φ). For simplicity we assume that the twist φ is normalized as in 3.3.2. 3.4.2 We assume that U ⊂ P defines a cusp of smaller rank. The case of cusps of full rank will be discussed in Subsection 3.7. We want to define a map π_{∗}^{P_{U}} : A_{{1}}(σ_{λ}, φ)^{n} → C^{∞}(Ω_{P} \ N_{V} {0_{P} }, V (σ_{λ}, φ)) by

π_{∗}^{P_{U}} (f) := ∫_{P_{U}} π^{σ_{λ},φ}(u)fdu .

If x ∉ N_{V}, then the integrand (π^{σ_{λ},φ}(u)f)(xw) is well-defined for all u ∈ P_{U}. We will see below that this integral converges for Re(λ) < ρ^{U} − l_{φ}/2. Let S(N_{V}) := {x ∈ N_{V} |a(xw) = 1} (see 3.3.2 for a definition of l_{φ} and Definition 3.7 for ρ^{U}). 3.4.3 Employing the assumption that N_{V} is invariant under conjugation by A (regularity of the cusp, Definition 1.3) we can consider polar coordinates S(N_{V}) × A ∋ (ξ, a) ↦ ξ^{a} ∈ N_{V} of N_{V} \ {1}.

Lemma 3.16 There is a measure dξ on S(N_{V}) such that the Haar measure dx of N_{V} is given by a^{2ρ_{U}} dadξ.

Proof. There is a family of measures dξ(a) on S(N_{V}) such that for any f ∈ C_{c}(N_{V}) we have

∫_{N_{V}} f(x)dx = ∫_{A} ∫_{S(N_{V} )} f(ξ^{a})dξ(a)a^{2ρ_{U}} da .

We compute for any b ∈ A

∫_{N_{V}} f(x^{b^{−1}})dx = b^{2ρ_{U}} ∫_{N_{V}} f(x)dx
= b^{2ρ_{U}} ∫_{A} ∫_{S(N_{V} )} f(ξ^{a})dξ(a)a^{2ρ_{U}} da ,
∫_{N_{V}} f(x^{b^{−1}})dx = ∫_{A} ∫_{S(N_{V} )} f(ξ^{ab^{−1}})dξ(a)a^{2ρ_{U}} da
= b^{2ρ_{U}} ∫_{A} ∫_{S(N_{V} )} f(ξ^{a})dξ(ab)a^{2ρ_{U}} da .

We conclude that dξ(ab) = dξ(a) for all b ∈ A and hence dξ(a) = dξ(1) =: dξ. □

3.4.4 Recall the the Definition 3.4 of P_{U} = N_{V} M_{U}. We introduce the abbreviation

f_{0} := ∫_{M_{U}} π^{σ_{λ},φ}(t)fdt .

Then we can write for x ∉ N_{V} and using the homogeneity (8)

π_{∗}^{P_{U}} (f)(xw) = ∫_{A} ∫_{S(N_{V} )} φ(ξ^{a})^{−1}f_{0}(ξ^{a}xw)a^{2ρ_{U}} dξda  (10)
= ∫_{A} ∫_{S(N_{V} )} φ(a)φ(ξ)^{−1}φ(a)^{−1}f_{0}(aξa^{−1}xw)a^{2ρ_{U}} dξda
= ∫_{A} ∫_{S(N_{V} )} φ(a)φ(ξ)^{−1}φ(a)^{−1}f_{0}(aξx^{a^{−1}}a^{−1}w)a^{2ρ_{U}} dξda
= ∫_{A} ∫_{S(N_{V} )} φ(a)φ(ξ)^{−1}f_{0}(ξx^{a^{−1}}w)a^{2(λ−ρ^{U})−nα}dξda .  (11)

We decompose the outer integral into the integrals over A_{+} and A_{−} and obtain π_{+}^{P_{U}} (f), π_{−}^{P_{U}} (f) such that π_{∗}^{P_{U}} (f) = π_{+}^{P_{U}} (f) + π_{−}^{P_{U}} (f). It is clear from (10) that π_{−}^{P_{U}} (f) converges for all λ ∈ 𝔞_{ℂ}^{∗}. We conclude from (11) that π_{+}^{P_{U}} (f) converges for Re(λ) < ρ^{U} + (nα − l_{φ})/2.

In this domain of convergence we compute for b ∈ A

φ(b)^{−1}π_{∗}^{P_{U}} (f)(x^{b}w) = b^{2(λ−ρ^{U})−nα}π_{∗}^{P_{U}} (f)(xw) .

3.4.5 Motivated by this calculation we make the following definition.

Definition 3.17 We define A_{P_{U}} (σ_{λ}, φ)^{n} to be the space of all P_{U}-invariant f ∈ C^{∞}(Ω_{P} \ N_{V} 0_{P} , V (σ_{λ}, φ)) satisfying

φ(a)^{−1}f(x^{a}w) = a^{2(λ−ρ^{U})−nα}f(xw)

for all a ∈ A, x ∈ N \ N_{V}.

3.4.6 Lemma 3.18 The family of spaces {A_{P_{U}} (σ_{λ}, φ)^{n}}_{λ∈𝔞_{ℂ}^{∗}} forms a trivial holomorphic family of Fréchet and Montel spaces. Proof. For some z ∈ ℝ with z < −ρ_{U}/α we define the function

s_{1}(x) := ∫_{P_{U}} a(y.xw)^{zα}dy .

We further define s as the α/(zα + ρ_{U})’th power of s_{1}. Now we consider the function s as a section s ∈ C^{∞}(Ω_{P} \ N_{V} 0_{P} , V (1_{ρ+α})) defining s(xw) := s(x). Multiplication by s^{μ} defines an continuous isomorphism of A_{P_{U}} (σ_{λ}, φ)^{n} with A_{P_{U}} (σ_{λ+μ}, φ)^{n}. We employ these isomorphisms in order to define a holomorphic trivialization of the bundle. □

3.4.7 For Re(λ) < ρ^{U} + (nα − l_{φ})/2 we have defined above a holomorphic family of continuous maps

π_{∗}^{P_{U}} : A_{{1}}(σ_{λ}, φ)^{n} → A_{P_{U}} (σ_{λ}, φ)^{n} .

Lemma 3.19 The family π_{∗}^{P_{U}} extends meromorphically to all of 𝔞_{ℂ}^{∗} with at most first order poles in the set ρ^{U} + \frac{nα−l_{φ}}{2} + \frac{1}{2}ℕ_{0}. The residues are finite-dimensional.

Proof. Recall the decomposition π_{∗}^{P_{U}} = π_{+}^{P_{U}} + π_{−}^{P_{U}} introduced in 3.4.4. We have already seen that π_{−}^{P_{U}} has a holomorphic continuation. It suffices to show that π_{+}^{P_{U}} has a meromorphic continuation.

Consider f ∈ A_{{1}}(σ_{λ}, φ)^{n}. Let

F (x) := ∫_{S(N_{V} )} φ(ξ)^{−1}f_{0}(ξxw)dξ

(see 3.4.4 for the definition of f_{0}). This function is smooth in a small neighbourhood of 1 ∈ N and on N \ N_{V}. Using the Taylor formula for each r ∈ ℕ_{0} we can write F (x^{a}) = ∑_{q=0}^{r} F_{q}(x)a^{qα} + a^{(r+1)α}R_{r}(x, a), where F_{q} is a homogeneous polynomial of degree qα on N and R_{r}(x, a) is uniformly bounded as a → 0. We write π_{+}^{P_{U}} (f)(xw) := ∫_{A_{+}} φ(a)F (x^{a^{−1}})a^{2(λ−ρ^{U})−nα}da and insert the expansion for F in order to obtain

π_{+}^{P_{U}} (f)(xw) := J_{r}^{1}(f)(xw) + J_{r}^{2}(f)(xw) ,

where

J_{r}^{1}(f)(xw) := ∑_{q=0}^{r} ∫_{A_{+}} φ(a)a^{2(λ−ρ^{U})−(q+n)α}daF_{q}(x)
J_{r}^{2}(f)(xw) := ∫_{A_{+}} φ(a)R_{r}(x, a^{−1})a^{2(λ−ρ^{U})−(r+1+n)α}da .

The integral J_{r}^{2} converges for Re(λ) < ρ^{U} + (n + r + 1)α/2 − l_{φ}/2. In order to evaluate J_{r}^{1} we introduce the operator B := (d/da)_{|a=1}φ(a) ∈ End(V_{φ}). In order to define this derivative we embed A into the multiplicative group of ℝ. Since φ is assumed to be algebraic as a representation of A the eigenvalues of B are integral. We have

J_{r}^{1}(f)(xw) = − ∑_{q=0}^{r}(B + 2(λ − ρ^{U}) − (n + q)α)^{−1}F_{q}(x) .

In particular we see that J_{r}^{1} has a meromorphic continuation to all of 𝔞_{ℂ}^{∗}. Since we can choose r arbitrarily large we obtain a meromorphic continuation of π_{+}^{P_{U}} to all of 𝔞_{ℂ}^{∗}. □

3.4.8 Recall the Definition 3.15 of the subspace of polynomials R_{{1}}(σ_{λ}, φ)^{n} ⊂ A_{1}(σ_{λ}, φ)^{n}. We consider the following diagram

\xymatrix{R_{\{1\}}(σ_{λ}, φ)^{n} \ar[d] \ar[r] & A_{1}(σ_{λ}, φ)^{n} \ar[d]^{π_{∗}^{P_{U}}} \\ ? \ar[r] & A_{P_{U}} (σ_{λ}, φ)^{n}} .

We can complete the diagram using the concept of an image bundle Definition 2.7.

Definition 3.20 We define R_{U}(σ_{λ}, φ)^{n} ⊂ A_{P_{U}} (σ_{λ}, φ)^{n} as the image bundle of the restriction of π_{∗}^{P_{U}} to R_{{1}}(σ_{λ}, φ)^{n}. We define the meromorphic family of maps

[π_{∗}^{U}] : R_{{1}}(σ_{λ}, φ)^{n} → R_{U}(σ_{λ}, φ)^{n}

to be induced by π_{∗}^{P_{U}}. Furthermore, we define R_{U,k}(σ_{λ}, φ) := ⨁_{n=0}^{k} R_{U}(σ_{λ}, φ)^{n} and let

[π_{∗}^{U}] : R_{{1},k}(σ_{λ}, φ) → R_{U,k}(σ_{λ}, φ)

denote the corresponding meromorphic family of maps.

Similar to the usage of AS and L the symbol [π_{∗}^{Γ}] denotes various versions of the push-down on the level of asymptotic terms. It will be clear from the context wich version is meant. 3.4.9 Using the second assertion of Lemma 2.6 we choose once and for all meromorphic families of right-inverses

[Q] : R_{U}(σ_{λ}, φ)^{n} → R_{{1}}(σ_{λ}, φ)^{n}

of [π_{∗}^{U}]. 3.4.10 Let χ ∈ C^{∞}(B_{U}) be a cut-off function such that 1 − χ has compact support, and which vanishes in a neighbourhood of U\(P_{U}0_{P}). We can assume that χ is P_{U}-invariant (otherwise we replace it by the average xw ↦ ∫_{U\P_{U}} χ(u^{−1}xw)du). Multiplication by χ defines an inclusion L : R_{U,k}(σ_{λ}, φ) → C^{∞}(B_{U}, V_{B_{U}} (σ_{λ}, φ)). As in Subsection 3.3 we see that L(R_{U,k}(σ_{λ}, φ)) ∩ S_{U,k}(σ_{λ}, φ) = {0}. 3.4.11

Definition 3.21 We define the Fréchet space

B_{U,k}(σ_{λ}, φ) := S_{U,k}(σ_{λ}, φ) ⊕ L(R_{U,k}(σ_{λ}, φ)) .

Furthermore we define the Fréchet and Montel space B_{U}(σ_{λ}, φ) to be the intersection of the spaces B_{U,k}(σ_{λ}, φ) for all k ∈ ℕ.

Note that B_{U,k}(σ_{λ}, φ) fits into the split exact sequence

0 → S_{U,k}(σ_{λ}, φ) → B_{U,k}(σ_{λ}, φ) →^{AS} R_{U,k}(σ_{λ}, φ) → 0 ,  (12)

where AS takes the finite asymptotic expansion. Since the spaces S_{U,k}(σ_{λ}, φ) and R_{U,k}(σ_{λ}, φ) form trivial holomorphic bundles we can employ the split L of the sequence in order to equip the family of spaces {B_{U,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} with the structure of a trivial holomorphic bundle over 𝔞_{ℂ}^{∗}. The space B_{U}(σ_{λ}, φ) fits into the exact sequence

0 → S_{U}(σ_{λ}, φ) → B_{U}(σ_{λ}, φ) → R_{U}(σ_{λ}, φ) → 0  (13)

which does not admit any continuous split. The spaces B_{U}(σ_{λ}, φ) form a projective limit of locally trivial holomorphic bundles in the sense of 2.1.4. 3.4.12 We now show compatibility of the spaces B_{U}(σ_{λ}, φ) with twisting. We assume that σ is Weyl invariant as in 2.3.4. Let (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in 2.3.3. Recall the notation p_{σ,μ} and i_{σ,μ} from 2.3.5.

Lemma 3.22 1. For each n ∈ ℕ_{0} we have the following commutative diagrams

\begin{matrix}R_{\{1\}}(σ_{λ}, φ)^{n} & →^{i_{σ,μ}^{n}} & R_{\{1\}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} \\ ↓ [π_{∗}^{U}] & & ↓ [π_{∗}^{U}] \\ R_{\{U\}}(σ_{λ}, φ)^{n} & →^{[i_{σ,μ}^{U}]} & R_{\{U\}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} \end{matrix}  (14)

\begin{matrix}R_{\{1\}}(σ_{λ}, φ)^{n} & ←^{p_{σ,μ}^{n}} & R_{\{1\}}(1_{λ−μ}, π_{σ,μ} ⊗ φ)^{m} \\ ↓ [π_{∗}^{U}] & & ↓ [π_{∗}^{U}] \\ R_{\{U\}}(σ_{λ}, φ)^{n} & ←^{[p_{σ,μ}^{U}]} & R_{\{U\}}(1_{λ−μ}, π_{σ,μ} ⊗ φ)^{m} \end{matrix} ,

where m := n + μ/α.

2. We have holomorphic families of continuous maps

i_{σ,μ}^{U} : B_{U}(σ_{λ}, φ) → B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)

and

p_{σ,μ}^{U} : B_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → B_{U}(σ_{λ}, φ) .

3. B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) is a Z(𝔤)-module.

4. If λ ∉ I_{𝔞}, then i_{σ,μ}^{U} maps B_{U}(σ_{λ}, φ) isomorphically onto

ker{Z(λ) : B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)} ,

and the restriction of p_{σ,μ}^{U} to

ker{Z̃(λ) : B_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → B_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ)}

is an isomorphism onto B_{U}(σ_{λ}, φ) (here Z̃(λ) is the adjoint of π^{1_{−λ+μ}⊗π̃_{σ,μ},idφ̃}(Π(λ))).

5. The composition

j_{σ,μ}^{U} : B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) →^{1−Z(λ)} ker_{U}(Z(λ)) →^{(i_{σ,μ}^{U})^{−1}} B_{U}(σ_{λ}, φ)

which is initially defined for λ ∉ I_{𝔞} extends to a meromorphic family of continuous maps. Similarly, the composition

q_{σ,μ}^{U} : B_{U}(σ_{λ}, φ) →^{(p_{σ,μ}^{U})^{−1}} ker(Z̃(λ)) → B_{U}(1_{λ−μ}, π_{σ,μ} ⊗ φ)

extends to a meromorphic family of continuous maps.

Proof. 3.4.13 We consider the first diagram of 1. and leave the second to the reader since the argument is similar. First one checks that the inclusion i_{σ,μ} induces an inclusion

i_{σ,μ}^{P_{U} ,n} : A_{P_{U}} (σ_{λ}, φ)^{n} → A_{P_{U}} (1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m}

for all n ∈ ℕ_{0}, m := n+μ/α. Now we obtain diagram (14), where the map [i_{σ,μ}^{U}] is the restriction of i_{σ,μ}^{P_{U} ,n} which is well-defined by the naturality of the image bundle construction Lemma 2.6. 3.4.14 Assertion 2. follows from Lemma 3.14, 2. and 1. Indeed, i_{σ,μ}^{U} maps B_{U,k}(σ_{λ}, φ) to

{i_{σ,μ}^{U}}({S_{U,k}(σ_{λ}, φ)}) + L([i_{σ,μ}^{U}](R_{U,k}(σ_{λ}, φ))) ⊂ B_{U,k′}(1_{λ+μ}, π_{σ,μ} ⊗ φ) ,

where k′ = k + μ/α. The argument for the second assertion of 2. is similar. 3.4.15 In order to prove 3. we first show that the spaces R_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} are a Z(𝔤)- modules. Note that the representations π^{1_{λ+μ}⊗π_{σ,μ},id_{φ}} of Z(𝔤) and π^{1_{λ+μ},π_{σ,μ}⊗φ} of AP_{U} commute. Since we employ the latter representation in order to define homogeneities we see that the spaces A_{P_{U}} (1_{λ+μ}, π_{σ,μ}⊗φ)^{m} are Z(𝔤)-modules. The space R_{{1}}(1_{λ+μ}, π_{σ,μ}⊗φ) is clearly a Z(𝔤)-module since it is the quotient of C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) by the submodule S_{{1}}(1_{λ+μ}, π_{σ,μ} ⊗ φ) (Lemma 3.14, 3.). Thus the subspaces R_{{1}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} are Z(𝔤)-modules. We obtain the Z(𝔤)-module structure on R_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} from Lemma 2.6 since π_{∗}^{P_{U}} is Z(𝔤)-equivariant.

Assertion 3. now follows from Lemma 3.14, 3. and the fact that for any A ∈ Z(𝔤) the commutator [π^{1_{λ+μ}⊗π_{σ,μ},id_{φ}}(A), χ] is a differential operator with compactly supported coefficients on B_{U}.

3.4.16 We now prove the first assertion of 4. and leave the second to the reader. The exact sequence of bundles (5) induces an exact sequence

0 → R_{U}(σ_{λ}, φ)^{n} →^{[i_{σ,μ}^{U}]} R_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)^{m} →^{p} R_{U}(w, φ)^{k} ,

where m = n+μ/α and we employ an appropriate definition of homogeneity for the last space. If λ ∉ I_{𝔞}, then we have ker(Z(λ) : R_{U}(1_{λ+μ}, π_{σ,μ}⊗φ)^{m} → R_{U}(1_{λ+μ}, π_{σ,μ}⊗φ)^{m}) = ker(p). For these λ the map [i_{σ,μ}^{U}] identifies R_{U}(σ_{λ}, φ)^{n} with the kernel of Z(λ). If f ∈ B_{U}(1_{λ+μ}, π_{σ,μ} ⊗φ) satisfies Z(λ)(f) = 0, then Z(λ)AS(f) = 0. From what was shown above and (13) it follows that there exists g ∈ B_{U}(σ_{λ}, φ) such that [i_{σ,μ}^{U}] ∘ AS(g) = AS(f). Then f − i_{σ,μ}^{U}(g) ∈ S_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ), and by Lemma 3.14, 4. there exists h ∈ S_{U}(σ_{λ}, φ) such that {i_{σ,μ}^{U}}(h) = f − i_{σ,μ}^{U}(g). Thus f = i_{σ,μ}^{U}(h + g), and this finishes the proof of 4. 3.4.17 We now prove the first assertion of 5. and leave the second to the reader. We employ the notation introduced in the proof of Lemma 3.14, 5. For each n ∈ ℕ_{0} there is m ∈ ℕ_{0} such that J : S_{U,m}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → S_{U,n}(σ_{λ}, φ) is a holomorphic family of maps. We define the meromorphic family of continuous maps j̃_{σ,μ}^{U} : B_{U,m}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → B_{U,n}(σ_{λ}, φ) by

j̃_{σ,μ}^{U}(f) := L ∘ [i_{σ,μ}^{U}]^{−1} ∘ AS ∘ (1 − Z(λ))(f) + {j_{σ,μ}^{U}}(f − L ∘ AS ∘ (1 − Z(λ))(f)) .

Then j̃_{σ,μ}^{U} ∘ i_{σ,μ}^{U}(f) = f for any f ∈ B_{U}(σ_{λ}, φ). Since j̃_{σ,μ}^{U} vanishes on ker(1 − Z(λ)) we conclude that the restriction of j̃_{σ,μ}^{U} to B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) coincides with j_{σ,μ}^{U} for λ ∉ I_{𝔞}. This proves 5. □

3.5 The push-down for Schwartz spaces

3.5.1 In this subsection we show that the push-down induces a map between Schwartz spaces

{π_{∗}^{U}} : S_{{1},k}(1_{λ}, φ) → S_{U,k}(1_{λ}, φ) .

Note that we only consider the spherical case σ = 1.

Lemma 3.23 1. The push-down {π_{∗}^{U}} converges for Re(λ) < ρ^{U} + (kα − l_{φ})/2.

2. For k_{1} < k it induces a holomorphic family of maps {π_{∗}^{U}} : S_{{1},k}(1_{λ}, φ) → S_{U,k_{1}}(1_{λ}, φ).

Proof. 3.5.2 Let f ∈ C_{c}^{∞}(B_{{1}}, V_{B_{{1}}}(1_{λ}, φ)). Then {π_{∗}^{U}}(f) = ∑_{u∈U} π^{1_{λ},φ}(u)f converges and defines an element of C_{c}^{∞}(B_{U}, V_{B_{U}} (1_{λ}, φ)).

3.5.3 Let h ∈ ℕ_{0}, W ⊂ N \ N_{V} be compact, |.| be a φ(M_{U})-invariant norm on V_{φ}, and D ∈ U(𝔫)^{≤dα} (see 3.2.4 for notation). Then we consider the seminorm

q_{W,D,h}(f) := sup_{x∈W} sup_{a∈A_{+}} a^{hα−2(λ−ρ^{U})}|φ(a)^{−1}(f)(x^{a}Dw)| .

The first assertion of the Lemma will easily follow from estimates of the form

q_{W,D,k}({π_{∗}^{U}}f) < C∥f∥_{k,d} ,

and the second assertion will follow from the convergence of sums of the form

∑_{u∈U} q_{W,D,k_{1}}(π^{1_{λ},φ}(u)f) < C∥f∥_{k,d} .

3.5.4 If u ≠ 1, then we can write u^{−1} = m_{u}ξ_{u}^{b_{u}} for suitable ξ_{u} ∈ S(N_{V}), b_{u} ∈ A, and m_{u} ∈ M_{U} (see 3.4.2 for the notation S(N_{V})). There is a constant C ∈ R such that for all f ∈ C_{c}^{∞}(B_{{1}}, V_{B_{{1}}}(1_{λ}, φ)), x ∈ W, a ∈ A, 1 ≠ u ∈ U with b_{u} ≥ a we have the estimate

|φ(a)^{−1}π^{1_{λ},φ}(u)f(x^{a}Dw)| = |φ(a)^{−1}φ(ξ_{u}^{b_{u}})^{−1}f(m_{u}ξ_{u}^{b_{u}}x^{a}Dw)|
= |φ(a^{−1}b_{u})φ(ξ_{u})^{−1}φ(b_{u}^{−1})f(m_{u}(ξ_{u}x^{ab_{u}^{−1}})^{b_{u}}Dw)|
≤ C∥f∥_{d,k}|φ(a^{−1}b_{u})|b_{u}^{2(λ−ρ)−kα}

and for b_{u} ≤ a

|φ(a)^{−1}π^{1_{λ},φ}(u)f(x^{a}Dw)| = |φ(a)^{−1}φ(ξ_{u}^{b_{u}})^{−1}f(m_{u}ξ_{u}^{b_{u}}x^{a}Dw)|
= |φ(ξ_{u}^{a^{−1}b_{u}})^{−1}φ(a)^{−1}f(m_{u}(ξ^{b_{u}a^{−1}}x)^{a}Dw)|
≤ C∥f∥_{d,k}a^{2(λ−ρ)−kα} .

3.5.5 If Re(λ) < ρ^{U} + (kα − l_{φ})/2, then summing up these estimates over U and estimating the sum by an integral over P_{U} we obtain C_{1} ∈ ℝ such that all x ∈ W, a ∈ A^{+}, f ∈ C_{c}^{∞}(B_{{1}}, V_{B_{{1}}}(1_{λ}, φ))

|φ(a)^{−1}{π_{∗}^{U^{0}}}(f)(x^{a}Dw)| < C_{1}a^{2(λ−ρ^{U})−kα} .

This implies

q_{W,D,k}({π_{∗}^{U}}f) < C_{1}∥f∥_{k,d} .

3.5.6 Multiplying the two inequalities above by a^{k_{1}α−2(λ−ρ^{U})} and taking the supremum over x and a we obtain q_{W,D,k_{1}}(π^{1_{λ},φ}(u)f) < b_{u}^{−2ρ_{U} −(k−k_{1})α}. Summing this over U and estimating the sum over U by the integral over P_{U} we obtain

∑_{u∈U} q_{W,D,k_{1}}(π^{1_{λ},φ}(u)f) < C∥f∥_{k,d} .

□

3.5.7 Recall (Definition 3.11) that S_{{1}}(1_{λ}, φ) is the intersection of the spaces S_{{1},k}(1_{λ}, φ) for k ∈ ℕ_{0}. Hence Lemma 3.23 implies the following corollary.

Corollary 3.24 We have a holomorphic family of maps

{π_{∗}^{U}} : S_{{1}}(1_{λ}, φ) → S_{U}(1_{λ}, φ)

defined on all of 𝔞_{ℂ}^{∗}.

3.5.8 We now turn to the problem of constructing a right-inverse of the push-down for Schwartz spaces. Its adjoint plays an important role in the construction of the restriction map. Below we encounter the effect of a loss of regularity (−2ρ_{U}). This is one of the places where the presence of cusps makes the theory much more complicated in comparison with the convex cocompact case. 3.5.9

Lemma 3.25 For all k_{1} ∈ ℕ_{0} satisfying k_{1}α < kα − 2ρ_{U} there exists a holomorphic family of maps

{Q} : S_{U,k}(1_{λ}, φ) → S_{{1},k_{1}}(1_{λ}, φ)

such that {π_{∗}^{U}} ∘ {Q} is the natural inclusion S_{{1},k}(1_{λ}, φ) ↪ S_{{1},k_{1}}(1_{λ}, φ).

Proof. We define a holomorphic family {Q} of right-inverses of {π_{∗}^{U}}. We employ a cut-off function χ^{U} ∈ C^{∞}(Ω_{P}) with ∑_{u∈U} u^{∗}χ^{U} = 1, and such that χ^{U}(xD) is bounded for each D ∈ U(𝔫). Here we have identified Ω_{P} with N. Then χ^{U} considered as a function on N can be derived with respect to the left invariant differential operator D.

To see that such a function exists we equip Ω_{P} ≅ N with a MN-invariant Riemannian metric such that U acts isometrically. Note that the Riemannian manifold U\N admits a lower bound of the injectivity radius.

For f ∈ S_{U,k}(1_{λ}, φ) we define {Q}(f) to be the lift of f to Ω_{P} multiplied by χ^{U}. It is then easy to see that {Q} : S_{U,k}(1_{λ}, φ) → S_{{1},k_{1}}(1_{λ}, φ) for all k_{1} ∈ ℕ_{0} satisfying k_{1}α < kα − 2ρ_{U}. Therefore we obtain a holomorphic family of maps {Q} : S_{U}(1_{λ}, φ) → S_{{1}}(1_{λ}, φ). By definition {π_{∗}^{U}} ∘ {Q} = id. □

3.6 Push down for cusps of smaller rank

3.6.1 We assume that the cusp associated to U ⊂ P does not have full rank. In order to construct the push-down it remains to extend {π_{∗}^{U}} (constructed in Lemma 3.23) for each k ∈ ℕ from S_{{1},k}(1_{λ}, φ) to B_{{1},k}(1_{λ}, φ) such that the following diagram is commutative:

0 → S_{{1},k}(1_{λ}, φ) → B_{{1},k}(1_{λ}, φ) →^{AS} R_{{1},k}(1_{λ}, φ) → 0
↓ {π_{∗}^{U}} ↓ π_{∗}^{U} ↓ [π_{∗}^{U}]
0 → S_{U,k}(1_{λ}, φ) → B_{U,k}(1_{λ}, φ) →^{AS} R_{U,k}(1_{λ}, φ) → 0 .

The main technical result in this direction is the following Proposition 3.26. 3.6.2 Note that R_{{1}}(1_{λ}, φ)^{n} is a finite-dimensional representation of the torus M_{U^{0}} (see 3.1.5 for the explanation of M_{U^{0}}). Let X (M_{U^{0}}) denote the set of characters of M_{U^{0}}. For any θ ∈ X (M_{U^{0}}) let R_{{1}}(1_{λ}, φ)_{θ}^{n} be the subspace of all f ∈ R_{{1}}(1_{λ}, φ)^{n} such that

π^{1_{λ},φ}(t)f = θ(t)f .

Then we have a finite decomposition

R_{{1}}(1_{λ}, φ)^{n} = ⨁_{θ∈X (M_{U^{0}})} R_{{1}}(1_{λ}, φ)_{θ}^{n} .

3.6.3 Recall the construction of L from 3.4.10. Furthermore see Definition 3.20 for [π_{∗}^{U^{0}}], Definition 3.7 for ρ^{U} and 3.3.2 for l_{φ}.

Proposition 3.26 The composition

π_{∗}^{U^{0}} ∘ L : R_{{1}}(1_{λ}, φ)_{θ}^{n} → B_{U^{0}}(1_{λ}, φ)

converges for Re(λ) < ρ^{U} + (nα − l_{φ})/2 and has a meromorphic continuation to all of 𝔞_{ℂ}^{∗} such that AS ∘ π_{∗}^{U^{0}} ∘ L = [π_{∗}^{U^{0}}].

Proof. 3.6.4 We decompose the push-down π_{∗}^{U^{0}} into two intermediate maps. The first map is the push-down π_{∗}^{[U^{0},U^{0}]} ∘ L : R_{{1}}(1_{λ}, φ)_{θ}^{n} → B_{[U^{0},U^{0}]}(1_{λ}, φ) with respect to the commutator group [U^{0}, U^{0}], and the second is a relative push-down π^{U^{0}/[U^{0},U^{0}]}. Note that M_{U^{0}} centralizes [U^{0}, U^{0}] (see 3.1.4) and therefore acts on the sequence

0 → S_{[U^{0},U^{0}]}(1_{λ}, φ) → B_{[U^{0},U^{0}]}(1_{λ}, φ) → R_{[U^{0},U^{0}]}(1_{λ}, φ) → 0 .

The representation of M_{U^{0}} on R_{[U^{0},U^{0}]}(1_{λ}, φ) is compatible with the decomposition ⨁_{n=0}^{∞} R_{[U^{0},U^{0}]}(1_{λ}, φ)^{n}. 3.6.5 Let R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} be the subspace of all f ∈ R_{[U^{0},U^{0}]}(1_{λ}, φ)^{n} satisfying

π^{1_{λ},φ}(t)f = θ(t)f .

Then we have a further finite decomposition

R_{[U^{0},U^{0}]}(1_{λ}, φ)^{n} = ⨁_{θ∈X (M_{U^{0}})} R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} .

3.6.6 We will first show Lemma 3.27 The push-down over [U^{0}, U^{0}] defines a meromorphic family of maps

π^{[U^{0},U^{0}]} : R_{{1}}(1_{λ}, φ)_{θ}^{n} → S_{[U^{0},U^{0}]}(1_{λ}, φ) ⊕ L(R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n})

such that

AS ∘ π^{[U^{0},U^{0}]} ∘ L = [π^{[U^{0},U^{0}]}] .

It is clear that the average {π_{∗}^{U^{0}/[U^{0},U^{0}]}} : S_{[U^{0},U^{0}]}(1_{λ}, φ) → S_{U^{0}}(1_{λ}, φ) over U^{0}/[U^{0}, U^{0}] converges and depends holomorphically on λ. A simple way to see this formally is to write

{π_{∗}^{U^{0}/[U^{0},U^{0}]}} = {π_{∗}^{U^{0}}} ∘ {Q} ∘ π_{∗}^{[U^{0},U^{0}]}

using the split {Q} constructed in Lemma 3.25 and Corollary 3.24.

Proposition 3.26 now immediately follows from Lemma 3.28 The composition

π_{∗}^{U^{0}/[U^{0},U^{0}]} ∘ L : R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} → B_{U^{0}}(1_{λ}, φ)

defines a meromorphic family of maps such that

AS ∘ π_{∗}^{U^{0}/[U^{0},U^{0}]} ∘ L ∘ [π_{∗}^{[U^{0},U^{0}]}] = [π_{∗}^{U^{0}}] .

3.6.7 We now start with the proof of Lemma 3.27. Since M_{U^{0}} is abelian the homomorphism m : N_{V} → M_{U^{0}} vanishes on [N_{V} , N_{V}]. Therefore [U^{0}, U^{0}] is a discrete subgroup of the center of N. If f ∈ R_{{1}}(1_{λ}, φ)^{n}, then for all d ∈ ℕ_{0} we have ∥L(f)∥_{n,d} < ∞. Therefore the same arguments as in the proof of Lemma 3.23 show that the push-down π_{∗}^{[U^{0},U^{0}]} ∘ L converges for Re(λ) < ρ^{[U^{0},U^{0}]} + (nα − l_{φ})/2. 3.6.8 We consider the abelian group Z := P_{[U^{0},U^{0}]} = [N_{V} , N_{V}]. If we are given an unitary character ϑ ∈ X (Z), then we consider the push-down π_{∗}^{Z,ϑ}, which associates to f ∈ C^{∞}(∂X, V (1_{λ}, φ)) the average

π_{∗}^{Z,ϑ}(f) := ∫_{Z} ϑ(z)^{−1}π^{1_{λ},φ}(z)(f_{|Ω})dz

provided the integral converges. 3.6.9 Observe that [U^{0}, U^{0}] is a cocompact discrete subgroup of Z. We normalize the Haar measure of Z such that vol([U^{0}, U^{0}]\Z) = 1. Let X (Z, [U^{0}, U^{0}]) denote the set of all unitary characters of Z which are trivial on [U^{0}, U^{0}]. We identify X (Z) with ı𝔷^{∗} and X (Z, [U^{0}, U^{0}]) with a lattice in ı𝔷^{∗}, where 𝔷 denotes the Lie algebra of Z. Our approach is based on the Poisson summation formula which states that

π_{∗}^{[U^{0},U^{0}]}(f) = ∑_{ϑ∈X (Z,[U^{0},U^{0}])} π_{∗}^{Z,ϑ}(f)

in the domain of convergence.

3.6.10 We choose an euclidean structure on 𝔷. This structure induces norms on 𝔷 and 𝔷^{∗}. Furthermore we use the euclidean structure in order to define a Laplace operator Δ ∈ U(𝔷) which is normalized such that if ϑ ∈ X (Z), then ϑ(Δ^{l}x) = |ϑ|^{2l}ϑ(x). Let COP : U(𝔷) → U(𝔷) ⊗ U(𝔷) denote the coproduct on U(𝔷). There are A_{l,j}, B_{l,j}, C_{l,j} ∈ U(𝔷) such that

(COP ⊗ 1) ∘ COP (Δ^{l}) = ∑_{j} A_{l,j} ⊗ B_{l,j} ⊗ C_{l,j} .

Let D ↦ D̃ be the canonical antiautomorphism of U(𝔷). 3.6.11 Fix l ∈ ℕ and let f ∈ R_{{1}}(1_{λ}, φ)_{θ}^{n}. Recall that L(f)(xw) = χ(a(xw))f(xw), where χ ∈ C^{∞}(A) was some cut-off function which vanishes on a neighbourhood of A_{−} and is equal to one near ∞. By S(Z) we denote the unit-sphere in Z. For ξ ∈ S(Z) and a ∈ A we compute

π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)(.w)
= π^{1_{λ},φ}(a)π^{1_{λ},φ}((Δ^{l})^{a^{−1}}ξ)π^{1_{λ},φ}(a^{−1})χ(a(.w))f(.w)
= a^{ρ−λ−4lα}π^{1_{λ},φ}(a)π^{1_{λ},φ}(Δ^{l}ξ)φ(a^{−1})χ(a(a.a^{−1}w))f(a.a^{−1}w)
= a^{λ−ρ−(n+4l)α}π^{1_{λ},φ}(a)π^{1_{λ},φ}(Δ^{l})φ(ξ)χ(a^{2}a(ξ^{−1}.w))f(ξ^{−1}.w)
= a^{λ−ρ−(n+4l)α} ∑_{j} π^{1_{λ},φ}(a)φ(A_{l,j}ξ)χ(a^{2}a(ξ^{−1}B̃_{l,j}.w))f(ξ^{−1}C̃_{l,j}.w)
= a^{2(λ−ρ)−(n+4l)α} ∑_{j} φ(a)φ(A_{l,j}ξ)χ(a^{2}a(ξ^{−1}B̃_{l,j}a^{−1}.aw))f(ξ^{−1}C̃_{l,j}a^{−1}.aw) . (15)

3.6.12 If W ⊂ N is any compact subset and D ∈ U(𝔫), then there is a constant C ∈ ℝ such that

|π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)(yDw)| ≤ Ca^{2(λ−ρ)−(n+4l)α+l_{φ}}

for all ξ ∈ S(Z) and a ∈ A_{+}.

If Re(λ) is sufficiently small, then by partial integration for ϑ ∈ X (Z, [U^{0}, U^{0}]), ϑ ≠ 0

π_{∗}^{Z,ϑ}(L(f)) = ∫_{Z} ϑ(x)^{−1}π^{1_{λ},φ}(x)L(f)dx
= \frac{1}{|ϑ|^{2l}} ∫_{Z} ϑ(Δ^{l}x)^{−1}π^{1_{λ},φ}(x)L(f)dx
= \frac{1}{|ϑ|^{2l}} ∫_{Z} ϑ(x)^{−1}π^{1_{λ},φ}(Δ^{l}x)L(f)dx
= \frac{1}{|ϑ|^{2l}} ∫_{S(Z)} ∫_{A} ϑ(ξ^{a})^{−1}π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)a^{2ρ_{[U^{0},U^{0}]}}dadξ .

By the estimate above the integral converges locally uniformly on {Re(λ) < ρ^{[U^{0},U^{0}]} + (n + 4l)α/2 − l_{φ}/2}, and for any compact subset of this region, D ∈ U(𝔫), and compact subset W ⊂ N there is a constant C_{1} such that

|π_{∗}^{Z,ϑ}(L(f))(yDw)| ≤ \frac{C_{1}}{|ϑ|^{2l}}

for all y ∈ W. 3.6.13 Choosing 2l > dim(Z) we see that the sum

∑_{0≠ϑ∈X (Z,[U^{0},U^{0}])} π_{∗}^{Z,ϑ}(L(f))

converges in C^{∞}(Ω_{P} , V (1_{λ}, φ)). 3.6.14 Refining the estimates above we now show that this sum in fact converges in the space of rapidly decreasing functions. Let W_{1} ⊂ N \ Z be a compact subset and D ∈ U(𝔫). It immediately follows from (15), that there is a constant C_{2} such that for all ξ ∈ S(Z), a, b ∈ A_{+}, with a ≥ b and y ∈ W_{1}

|φ(b)^{−1}(π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f))(y^{b}Dw)| ≤ C_{2}a^{2(λ−ρ)−(n+4l)α}|φ(ab^{−1})| .

Let COP (D) = ∑_{h} D_{h}^{l} ⊗D_{h}^{r} denote the coproduct of D, where here COP : U(𝔫) → U(𝔫)⊗U(𝔫). We find a constant C_{3} ∈ ℝ such that for all ξ ∈ S(Z), a, b ∈ A_{+} with a ≤ b and y ∈ W_{1}

|φ(b)^{−1}(π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f))(y^{b}Dw)|

= a^{2(λ−ρ)−(n+4l)α}| ∑_{j,h} φ(ab^{−1})φ(A_{l,j}ξ)χ(a^{2}a(ξ^{−1}B̃_{l,j}a^{−1}y^{b}D_{h}^{l}aw))f(ξ^{−1}C̃_{l,j}a^{−1}y^{b}D_{h}^{r}aw)|
= a^{−4lα}b^{2(λ−ρ)−nα}| ∑_{j,h} φ((A_{l,j}ξ)^{ab^{−1}})χ(b^{2}a((ξ^{−1}B̃_{l,j})^{b^{−1}a}y(D_{h}^{l})^{b^{−1}}w))f((ξ^{−1}C̃_{l,j})^{b^{−1}a}y(D_{h}^{r})^{b^{−1}}w)|
≤ C_{3}b^{2(λ−ρ)−(n+4l)α} .

Using this estimate we see that

φ(b)^{−1} ∑_{0≠ϑ∈X (Z,[U^{0},U^{0}])} π_{∗}^{Z,ϑ}(L(f))(y^{b}Dw)

can be estimated by Cb^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α}, where C can be choosen uniformly for y ∈ W_{1} and λ in compact subsets of {Re(λ) < ρ^{[U^{0},U^{0}]} + (n + 4l)α/2 − l_{φ}/2}. Since we can choose l arbitrary large we obtain a holomorphic continuation of the sum above to all of 𝔞_{ℂ}^{∗}. Moreover we see that this sum is rapidly decreasing with respect to b. 3.6.15 We now consider π_{∗}^{Z,0}(L(f)). If Re(λ) is sufficiently small then we can write

π_{∗}^{Z,0}(L(f)) = ∫_{A} ∫_{S(Z)} π^{1_{λ},φ}(ξ^{a})L(f)dξa^{2ρ_{[U^{0},U^{0}]}}da .

We again employ the relation

π^{1_{λ},φ}(ξ^{a})L(f)(yw)
= a^{2(λ−ρ)−nα}φ(a)φ(ξ)χ(a^{2}a(ξ^{−1}a^{−1}yaw))f(ξ^{−1}a^{−1}yaw) .

Note that

F (a, y) := ∫_{S(Z)} φ(ξ)χ(a^{2}a(ξ^{−1}yw))f(ξ^{−1}yw)dξ

is a smooth function of y near y = 0 which is independent of a for large a. Let

F (a, y^{a^{−1}}) = ∑_{q=0}^{r} F_{q}(y)a^{−qα} + a^{−(r+1)α}R_{r}(a, y)

be the asymptotic expansion for large a ∈ A obtained from the Taylor series of F (a, y) at y = 0. The remainder R_{r}(a, y) remains bounded as a → ∞. We write

π_{∗}^{Z,1}(L(f))(yw) = I_{+}(y) + I_{−}(y) ,

where

I_{+}(y) := ∫_{A_{+}} a^{2(λ−ρ^{[U^{0},U^{0}]})−nα}φ(a)F (a, y^{a^{−1}})da
I_{−}(y) := ∫_{A_{−}} ∫_{S(Z)} π^{1_{λ},φ}(ξ^{a})L(f)(yw)dξa^{2ρ_{[U^{0},U^{0}]}}da .

The integral I_{−} converges for all λ ∈ 𝔞_{ℂ}^{∗} and defines a holomorphic family of smooth functions. We write I_{+}(y) := J_{r}^{1}(y) + J_{r}^{2}(y), where

J_{r}^{1}(y) := ∫_{A_{+}} a^{2(λ−ρ^{[U^{0},U^{0}]})−nα}φ(a) ∑_{q=0}^{r} F_{q}(y)a^{−qα}da
J_{r}^{2}(y) := ∫_{A_{+}} a^{2(λ−ρ^{[U^{0},U^{0}]})−nα}φ(a)a^{−(r+1)α}R_{r}(a, y)da .

The integral J_{r}^{2} converges for Re(λ) < ρ^{[U^{0},U^{0}]} + (n + r + 1)α/2 − l_{φ}/2 and defines a smooth function in y. The integral J_{r}^{1} can be evaluated:

J_{r}^{1}(y) = − ∑_{q=0}^{r}(B + 2(λ − ρ^{[U^{0},U^{0}]}) − (n + q)α)^{−1}F_{q}(y) ,

where B := \frac{d}{da}_{|a=1}φ(a) ∈ End(V_{φ}) (compare with the proof of Lemma 3.19). It obviously defines a meromorphic family of smooth functions. Since we can choose r arbitrary large we obtain a meromorphic continuation of π_{∗}^{Z,0}(L(f)) to all of 𝔞_{ℂ}^{∗}. Note that if y ∈ N \ Z and b ∈ A is sufficiently large, then we have π_{∗}^{Z,0}(L(f))(y^{b}w) = π_{∗}^{P_{[U^{0},U^{0}]}}(f)(y^{b}w), where π_{∗}^{P_{[U^{0},U^{0}]}} was discussed in Subsection 3.4. We conclude that π_{∗}^{Z,0}(L(f)) ∈ S_{[U^{0},U^{0}]}(1_{λ}, φ)⊕L(R_{[U^{0},U^{0}]}(1_{λ}, φ)^{n}).

We have shown that

π_{∗}^{[U^{0},U^{0}]} ∘ L : R_{{1}}(1_{λ}, φ)^{n} → S_{[U^{0},U^{0}]}(1_{λ}, φ) ⊕ L(R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n}) ⊂ B_{[U^{0},U^{0}]}(1_{λ}, φ)

has a meromorphic continuation to all of 𝔞_{ℂ}^{∗} such that AS ∘π_{∗}^{[U^{0},U^{0}]} ∘L = [π_{∗}^{[U^{0},U^{0}]}]. This finishes the proof of the lemma. □

3.6.16 We now prove Lemma 3.28 in a similar manner. Since M_{U^{0}} centralizes N_{V} we can form the direct product P_{1} := P_{U^{0}}/Z = M_{U^{0}}T where T := N_{V} /Z (the group Z was defined in 3.6.8). By construction of U^{0} there is a lattice V_{1} ⊂ T and a homomorphism m : T → M_{U^{0}} such that U^{0}/[U^{0}, U^{0}] = {m(v)v|v ∈ V_{1}} ⊂ P_{1}. For f ∈ R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} (this space is defined in 3.6.5) we have

π^{1_{λ},φ}(m(v)v)L(f)(yw) = θ^{m}(v)φ(ṽ)f(ṽ^{−1}yw) ,

where ṽ ∈ N is any lift of v ∈ T, and θ^{m} = θ ∘ m ∈ X (T). 3.6.17 We normalize the Haar measure on T such that vol(V_{1}\T) = 1. Given a unitary character ϑ ∈ X (T), we consider the push-down π_{∗}^{T,ϑ}, which associates to f ∈ C^{∞}(B_{Z}, V_{B_{Z}}(1_{λ}, φ)) the average

π_{∗}^{T,ϑ}(f) := ∫_{T} ϑ(z)^{−1}π^{1_{λ},φ}(z)fdz

provided the integral converges. Note that we define L : R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} → B_{[U^{0},U^{0}]}(1_{λ}, φ) using a Z-invariant cut-off function (recall that Z = P_{[U^{0},U^{0}]}). It follows from the proof of Lemma 3.27 that all f ∈ R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n} are Z-invariant. Therefore L(f) ∈ C^{∞}(B_{Z}, V_{B_{Z}}(θ, φ)) and π_{∗}^{T,ϑ}(L(f)) is well-defined (up to convergence). 3.6.18 We fix an isomorphism of abelian groups T ≅ ℝ^{dim(T)} preserving the Haar measure such that we obtain an euclidean structure on T and its Lie algebra 𝔱. As before let X (T, V_{1}) denote the set of all unitary characters of T which are trivial on V_{1}. We identify X (T) with ı𝔱^{∗} and X (T, V_{1}) with a lattice in ı𝔱^{∗}. If f ∈ R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n}, then by the Poisson summation formula

π_{∗}^{U^{0}/[U^{0},U^{0}]}(L(f)) = ∑_{ϑ∈X (T,V_{1})} π_{∗}^{T,ϑ−θ^{m}}(L(f))

in the domain of convergence. 3.6.19 Note that if θ^{m} ∈ X (T, V_{1}), then θ = 0 by the construction of M_{U^{0}}. If θ ≠ 0, then will show that ∑_{ϑ∈X (T,V_{1})} π_{∗}^{T,ϑ−θ^{m}}(L(f)) gives rise to a rapidly decreasing section for all λ ∈ 𝔞_{ℂ}^{∗}. If θ = 0, then again ∑_{0≠ϑ∈X (T,V_{1})} π_{∗}^{T,ϑ−θ^{m}}(L(f)) is rapidly decreasing. The remaining term π_{∗}^{T,0}(L(f)) contributes to S_{U^{0}}(1_{λ}, φ) ⊕ L(R_{U^{0}}(1_{λ}, φ)_{θ}^{n}) and depends meromorphically on λ. 3.6.20 Observe that A normalizes Z. This is a consequence of the assumption that the cusp is regular, and that the Langlands decomposition of P is adapted (Definition 1.3). It is in fact the reason for making this assumption. We see that A acts on T. In particular 𝔱^{∗} decomposes into two eigenspaces 𝔱 = 𝔱_{1}^{∗} ⊕ 𝔱_{2}^{∗} (we write ϑ = ϑ_{1} ⊕ ϑ_{2} for the corresponding decomposition of ϑ ∈ 𝔱^{∗}) with respect to A such that A acts on 𝔱_{i}^{∗} by a^{iα}, i = 1, 2. We define the A-homogeneous ”norm” on 𝔱^{∗} by |ϑ| := (|ϑ_{1}|^{4} + |ϑ_{2}|^{2})^{1/2}. Furthermore we use the euclidean structure on 𝔱_{i} in order to define Laplace operators Δ_{i} ∈ U(𝔱_{i}) and the A-homogeneous operator Δ := Δ_{1}^{2} + Δ_{2}. We fix the normalizations such that for ϑ ∈ X (T) we have ϑ(Δ^{l}x) = |ϑ|^{2l}ϑ(x). Let ρ_{U^{0}/[U^{0},U^{0}]} ∈ 𝔞^{∗} be such that A acts on Λ^{max}𝔱 by the character a^{2ρ_{U^{0}/[U^{0},U^{0}]}}. Note that ρ_{U^{0}/[U^{0},U^{0}]} + ρ_{[U^{0},U^{0}]} = ρ_{U}.

3.6.21 Let COP : U(𝔱) → U(𝔱) ⊗ U(𝔱) denote the coproduct on U(𝔱). There are A_{l,j}, B_{l,j}, C_{l,j} ∈ U(𝔱) such that

(COP ⊗ 1) ∘ COP (Δ^{l}) = ∑_{j} A_{l,j} ⊗ B_{l,j} ⊗ C_{l,j} .

Let D ↦ D̃ be the canonical antiautomorphism of U(𝔱).

Recall that L(f)(xw) = χ(xw)f(xw), where χ is Z-invariant. By S(T) we denote the unitsphere in T. For ξ ∈ S(T) and a ∈ A we compute

π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)(.w)
= π^{1_{λ},φ}(a)π^{1_{λ},φ}((Δ^{l})^{a^{−1}}ξ)π^{1_{λ},φ}(a^{−1})χf(.w)
= a^{ρ−λ−4lα}π^{1_{λ},φ}(a)π^{1_{λ},φ}(Δ^{l}ξ)φ(a^{−1})χ(a.a^{−1}w)f(a.a^{−1}w)
= a^{λ+ρ−2ρ^{[U^{0},U^{0}]}−(n+4l)α}π^{1_{λ},φ}(a)π^{1_{λ},φ}(Δ^{l})φ(ξ)χ(aξ^{−1}.a^{−1}w)f(ξ^{−1}.w)
= a^{λ+ρ−2ρ^{[U^{0},U^{0}]}−(n+4l)α} ∑_{j} π^{1_{λ},φ}(a)φ(A_{l,j}ξ)χ(aξ^{−1}B̃_{l,j}.a^{−1}w)f(ξ^{−1}C̃_{l,j}.w)
= a^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α} ∑_{j} φ(a)φ(A_{l,j}ξ)χ(aξ^{−1}B̃_{l,j}a^{−1}.w)f(ξ^{−1}C̃_{l,j}a^{−1}.aw) . (16)

If W ⊂ N is any compact subset and D ∈ U(𝔫), then there is a constant C ∈ ℝ such that

|π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)(yDw)| ≤ Ca^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α+l_{φ}}

for all ξ ∈ S(T) and a ∈ A_{+}. 3.6.22 If Re(λ) is sufficiently small, then by partial integration for ϑ ∈ X (T, V_{1}), ϑ − θ^{m} ≠ 0

π_{∗}^{Z,ϑ−θ^{m}}(L(f)) = ∫_{Z}(ϑ − θ^{m})^{−1}(x)π^{1_{λ},φ}(x)L(f)dx
= \frac{1}{|ϑ − θ^{m}|^{2l}} ∫_{T} (ϑ − θ^{m})^{−1}(Δ^{l}x)π^{1_{λ},φ}(x)L(f)dx
= \frac{1}{|ϑ − θ^{m}|^{2l}} ∫_{T} (ϑ − θ^{m})^{−1}(x)π^{1_{λ},φ}(Δ^{l}x)L(f)dx
= \frac{1}{|ϑ − θ^{m}|^{2l}} ∫_{S(T)} ∫_{A}(ϑ − θ^{m})^{−1}(ξ^{a})π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f)a^{2ρ_{U^{0}/[U^{0},U^{0}]}}dadξ .

By the estimate above the integral converges locally uniformly on {Re(λ) < ρ^{U} + (n + 4l)α/2 − l_{φ}/2}, and for any compact subset of this region, D ∈ U(𝔫), and compact subset W ⊂ N there is a constant C_{1} such that

|π_{∗}^{T,ϑ−θ^{m}}(L(f))(yDw)| ≤ \frac{C_{1}}{|ϑ − θ^{m}|^{2l}}

for all y ∈ W.

3.6.23 Choosing 2l > dim(T) we see that the sum

∑_{ϑ∈X (T,V_{1}),ϑ−θ^{m}≠0} π_{∗}^{T,ϑ−θ^{m}}(L(f))

converges in C^{∞}(Ω, V (1_{λ}, φ)). 3.6.24 Refining the estimates above we now show that this sum in fact converges in the space of rapidly decreasing functions. Let W_{1} ⊂ N \ N_{V} be a compact subset and D ∈ U(𝔫). It immediately follows from (16), that there is a constant C_{2} such that for all ξ ∈ S(T), a, b ∈ A_{+}, with a ≥ b and y ∈ W_{1}

|φ(b)^{−1}(π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f))(y^{b}Dw)| ≤ C_{2}a^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α}|φ(ab^{−1})| .

Let COP (D) = ∑_{h} D_{h}^{l} ⊗ D_{h}^{r} denote the coproduct of D, where here COP : U(𝔫) → U(𝔫) ⊗ U(𝔫).

We find a constant C_{3} such that for all ξ ∈ S(T), a, b ∈ A_{+}, with a ≤ b and y ∈ W_{1}

|φ(b)^{−1}(π^{1_{λ},φ}(Δ^{l}ξ^{a})L(f))(y^{b}Dw)|

= a^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α}| ∑_{j,h} φ(ab^{−1})φ(A_{l,j}ξ)χ(aξ^{−1}B̃_{l,j}a^{−1}y^{b}D_{h}^{l}w)f(ξ^{−1}C̃_{l,j}a^{−1}y^{b}D_{h}^{r}aw)
= a^{−4lα}b^{2(λ−ρ^{[U^{0},U^{0}]})−nα}| ∑_{j,h} φ((A_{l,j}ξ)^{ab^{−1}})χ(aξ^{−1}B̃_{l,j}a^{−1}y^{b}D_{h}^{l}w)f((ξ^{−1}C̃_{l,j})^{b^{−1}a}y(D_{h}^{r})^{b^{−1}}w)
≤ C_{3}b^{2(λ−ρ^{[U^{0},U^{0}]})−(n+4l)α} .

Using this estimate we see that

φ(b)^{−1} ∑_{ϑ∈X (T,V_{1}),ϑ−θ^{m}≠0} π_{∗}^{T,ϑ−θ^{m}}(L(f))(y^{b}Dw)

can be estimated by Cb^{2(λ−ρ^{U})−(n+4l)α}, where C can be choosen uniformly for y ∈ W_{1} and λ in compact subsets of {Re(λ) < ρ^{U} + (n + 4l)α/2 − l_{φ}/2}. Since we can choose l arbitrary large we obtain a holomorphic continuation of the sum above to all of 𝔞_{ℂ}^{∗} and that it is rapidly decreasing with respect to b. 3.6.25 Let now θ = 0 and consider π_{∗}^{T,0}(L(f)). We write for Re(λ) sufficiently small

π_{∗}^{T,0}(L(f)) = ∫_{A} ∫_{S(T)} π^{1_{λ},φ}(ξ^{a})L(f)dξa^{2ρ_{U^{0}/[U^{0},U^{0}]}}da .

We again employ

π^{1_{λ},φ}(ξ^{a})L(f)(yw)
= a^{2(λ−ρ^{[U^{0},U^{0}]})−nα}φ(a)φ(ξ)χ(aξ^{−1}y^{a^{−1}}a^{−1}w)f(ξ^{−1}a^{−1}yaw) .

Note that

F (a, y) := ∫_{S(T)} φ(ξ)χ(aξ^{−1}ya^{−1}w)f(ξ^{−1}yw)dξ

is a smooth function of y near y = 0 which is independent of a for large a. Let

F (a, y^{a^{−1}}) = ∑_{q=0}^{r} F_{q}(y)a^{−qα} + a^{−(r+1)α}R_{r}(a, y)

be the asymptotic expansion for large a ∈ A obtained from the Taylor series of F (a, y) at y = 0. The remainder R_{r}(a, y) remains bounded as a → ∞. We write

π_{∗}^{T,0}(L(f))(yw) = I_{+}(y) + I_{−}(y) ,

where

I_{+}(y) = ∫_{A_{+}} a^{2(λ−ρ^{U})−nα}φ(a)F (a, y^{a^{−1}})da
I_{−}(y) = ∫_{A_{−}} ∫_{S(T)} π^{1_{λ},φ}(ξ^{a})L(f)(yw)dξa^{2ρ_{U^{0}/[U^{0},U^{0}]}}da .

The integral I_{−} converges for all λ ∈ 𝔞_{ℂ}^{∗} and defines a holomorphic family of smooth functions. We write I_{+}(y) := J_{r}^{1}(y) + J_{r}^{2}(y), where

J_{r}^{1}(y) := ∫_{A_{+}} a^{2(λ−ρ^{U})−nα}φ(a) ∑_{q=0}^{r} F_{q}(y)a^{−qα}da
J_{r}^{2}(y) := ∫_{A_{+}} a^{2(λ−ρ^{U})−nα}φ(a)a^{−(r+1)α}R_{r}(a, y)da .

The integral J_{r}^{2} converges for Re(λ) < ρ^{U} + (n + r + 1)α/2 − l_{φ}/2 and defines a smooth function in y. The integral J_{r}^{1} can be evaluated:

J_{r}^{1}(y) = ∑_{q=0}^{r}(B + 2(λ − ρ^{U}) − (n + q)α)^{−1}F_{q}(y) .

It obviously defines a meromorphic family of smooth functions. Since we can choose r arbitrary large we obtain a meromorphic continuation of π_{∗}^{U,1}(L(f)) to all of 𝔞_{ℂ}^{∗}. 3.6.26 Let now f ∈ R_{{1}}(1_{λ}, φ)_{0}^{n}. If y ∈ N \ T and b ∈ A is sufficiently large, then we have

π_{∗}^{T,0} ∘ L ∘ [π_{∗}^{U^{0}/[U^{0},U^{0}]}](f)(y^{b}) = π_{∗}^{T,0} ∘ π_{∗}^{Z,0}(f)(y^{b})
= π_{∗}^{P_{U^{0}}} (f)(y^{b})
= [π_{∗}^{U^{0}}](f)(y^{b}) .

We now have shown that if f ∈ R_{[U^{0},U^{0}]}(1_{λ}, φ)_{θ}^{n}, then π_{∗}^{U^{0}/[U^{0},U^{0}]}(f) ∈ S_{U^{0}}(1_{λ}, φ)⊕L(R_{U^{0}}(1_{λ}, φ)^{n}), and that π_{∗}^{U^{0}/[U^{0},U^{0}]} depends meromorphically on λ. □

3.6.27 The finite group U/U^{0} acts on S_{U^{0},k}(1_{λ}, φ) and R_{U^{0}}(1_{λ}, φ)^{n}. We can define {π^{U/U^{0}}} : S_{U^{0},k}(1_{λ}, φ) → S_{U,k}(1_{λ}, φ), π^{U/U^{0}} : B_{U^{0}}(1_{λ}, φ) → B_{U}(1_{λ}, φ), and [π_{∗}^{U/U^{0}}] : R_{U^{0}}(1_{λ}, φ)^{n} → R_{U}(1_{λ}, φ)^{n}. Using π_{∗}^{U} := π_{∗}^{U/U^{0}} ∘ π_{∗}^{U^{0}} and Proposition 3.26 (and its proof) we obtain

Proposition 3.29 The composition

π_{∗}^{U} ∘ L : R_{{1}}(1_{λ}, φ)_{θ}^{n} → B_{U}(1_{λ}, φ)

converges for Re(λ) < ρ^{U} + (nα − l_{φ})/2 and has a meromorphic continuation to all of 𝔞_{ℂ}^{∗} such that AS ∘ π_{∗}^{U} ∘ L = [π_{∗}^{U}]. If θ ≠ 0, then it is in fact a holomorphic family of maps π_{∗}^{U} ∘ L : R_{{1}}(1_{λ}, φ)_{θ}^{n} → S_{U}(1_{λ}, φ). If θ = 0, then π_{∗}^{U} ∘ L has at most first order poles in the set ρ^{U} + \frac{nα−l_{φ}}{2} + \frac{1}{2}ℕ_{0}.

3.6.28 Combining Proposition 3.29 with Lemma 3.23 we obtain

Corollary 3.30 Assume that the cusp associated to U ⊂ P does not have full rank. Then for any k_{1} < k the push-down π_{∗}^{U} : B_{{1},k}(1_{λ}, φ) → B_{U,k_{1}}(1_{λ}, φ) forms a meromorphic family of continuous maps with finite-dimensional singularities defined on {Re(λ) < ρ^{U} + (kα − l_{φ})/2}. It fits into the following commutative diagram

0 → S_{{1},k}(1_{λ}, φ) → B_{{1},k}(1_{λ}, φ) →^{AS} R_{{1},k}(1_{λ}, φ) → 0
↓ {π_{∗}^{U}} ↓ π_{∗}^{U} ↓ [π_{∗}^{U}]
0 → S_{U,k}(1_{λ}, φ) → B_{U,k}(1_{λ}, φ) →^{AS} R_{U,k}(1_{λ}, φ) → 0 .

Moreover we have a meromorphic family of maps π_{∗}^{U} : B_{{1}}(1_{λ}, φ) → B_{U}(1_{λ}, φ) with finitedimensional singularities and defined on all of 𝔞_{ℂ}^{∗} such that

0 → S_{{1}}(1_{λ}, φ) → B_{{1}}(1_{λ}, φ) →^{AS} R_{{1}}(1_{λ}, φ) → 0
↓ {π_{∗}^{U}} ↓ π_{∗}^{U} ↓ [π_{∗}^{U}]
0 → S_{U}(1_{λ}, φ) → B_{U}(1_{λ}, φ) →^{AS} R_{U}(1_{λ}, φ) → 0 .

is commutative.

3.7 Push-down for cusps of full rank and for general σ

3.7.1 In this subsection (σ, V_{σ}) denotes a Weyl-invariant representation of M as in 2.3.4.

Let C^{−∞}(∞_{P} , V (σ_{λ}, φ)) ⊂ C^{−∞}(∂X, V (σ_{λ}, φ)) be the space of distributions which are supported on ∞_{P}. This space can be identified P_{U}-equivariantly with the tensor product of a generalized Verma module by V_{φ}

(U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ}

such that (X ⊗s⊗v) ∈ (U(𝔤)⊗_{U(𝔭)}V_{σ_{λ+2ρ}})⊗V_{φ} maps f ∈ C^{∞}(∂X, V (σ̃_{−λ}, φ̃)) to (s⊗v)(f(Xe)). We can further identify C^{−∞}(∞_{P} , V (σ_{λ}, φ)) with R_{{1}}(σ̃_{−λ}, φ̃)^{∗}.

3.7.2 We now assume that the cusp associated to U ⊂ P has full rank. In this case N is the Zariski closure of U^{0}. The space (U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ} carries an algebraic representation of N. Therefore

U[(U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ}] ⊂ ^{U^{0}}[(U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ}]
= ^{N}[(U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ}]

is finite-dimensional since the space of highest weight vectors of the 𝔤-module U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}} is finite-dimensional [Jan79]. 3.7.3 Let E_{∞_{P}} (σ, φ) denote the sheaf of holomorphic families f_{ν} ∈ ^{U}C^{−∞}(∞_{P} , V (σ_{ν}, φ)). Since E_{∞_{P}} (σ, φ) is torsion-free it is the space of sections of a unique holomorphic vector bundle E_{∞_{P}} (σ, φ) over 𝔞_{ℂ}^{∗}. By E_{∞_{P}} (σ_{λ}, φ) we denote the fibre of E_{∞_{P}} (σ, φ) at λ ∈ 𝔞_{ℂ}^{∗}. We will discuss this bundle in detail in Lemma 3.40. 3.7.4 We now define the function space B_{U}(σ_{λ}, φ) in the case of a cusp of full rank.

Definition 3.31 We define R_{U}(σ_{λ}, φ) := E_{∞_{P}} (σ̃_{−λ}, φ̃)^{∗}. Furthermore we set

B_{U}(σ_{λ}, φ) := S_{U}(σ_{λ}, φ) ⊕ R_{U}(σ_{λ}, φ) .

Let AS : B_{U}(σ_{λ}, φ) → R_{U}(σ_{λ}, φ) be the projection and L : R_{U}(σ_{λ}, φ) → B_{U}(σ_{λ}, φ) be the inclusion.

These families of spaces form trivial holomorphic bundles of Fréchet and Montel spaces over 𝔞_{ℂ}^{∗}. 3.7.5

Definition 3.32 We define

[ext^{U}] : E_{∞_{P}} (σ̃_{−λ}, φ̃) ↪ R_{{1}}(σ̃_{−λ}, φ̃)^{∗}

as the natural inclusion. We define the push-down

[π_{∗}^{U}] : R_{{1}}(σ_{λ}, φ) → R_{U}(σ_{λ}, φ)

to be the adjoint of [ext^{U}]. Furthermore we define

(π_{∗}^{U})_{1} : C^{∞}(∂X, V (σ_{λ}, φ)) → S_{U}(σ_{λ}, φ)

by (π_{∗}^{U})_{1}(f) := ∑_{u∈U} π^{σ_{λ},φ}(u)(f_{|Ω}) (convergence provided). Finally we set

π_{∗}^{U} := (π_{∗}^{U})_{1} ⊕ [π_{∗}^{U}] : C^{∞}(∂X, V (σ_{λ}, φ)) → B_{U}(σ_{λ}, φ) .

Note that [π_{∗}^{U}] is a holomorphic family of surjective maps. Once and for all we fix a right-inverse [Q] of [π_{∗}^{U}]. We leave it to the interested reader to show that the statements of Lemma 3.22 hold true for cusps of full rank as well.

3.7.6

Lemma 3.33 The push-down (π_{∗}^{U})_{1} : C^{∞}(∂X, V (1_{λ}, φ)) → S_{U}(1_{λ}, φ) converges for Re(λ) < −l_{φ}/2 and has a meromorphic continuation to all of 𝔞_{ℂ}^{∗} with finite-dimensional singularities.

Proof. One checks that the corresponding parts of the proofs of Lemma 3.23 and Proposition 3.26 apply to the case of cusps of full rank as well. □

3.7.7 Note that in Subsection 3.6 we have considered the push-down in the spherical case σ = 1. We now deal with the general case using the concept of twisting. We assume that U ⊂ P defines a cusp of smaller rank and consider a Weyl invariant σ (see 2.3.4). We show the existence of a meromorphic family of push-down maps

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

using twisting. We employ a finite-dimensional representation (π_{σ,μ}, V_{π_{σ,μ}}) of G as in 2.3.3. Note that

π_{∗}^{U} : C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) → B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ)

is Z(𝔤)-equivariant. Therefore we can make the following definition:

Definition 3.34 If λ ∉ I_{𝔞}, then using Lemma 3.22, 4., we define the push-down

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

by the following coummutative diagram:

\begin{aligned}0 & → & C^{∞}(∂X, V (σ_{λ}, φ)) & →^{i_{σ,μ}} & C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) & →^{Z(λ)} & C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \\ & & ↓ π_{∗}^{U} & & ↓ π_{∗}^{U} & & ↓ π_{∗}^{U} \\ 0 & → & B_{U}(σ_{λ}, φ) & →^{i_{σ,μ}^{U}} & B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) & →^{Z(λ)} & B_{U}(1_{λ+μ}, π_{σ,μ} ⊗ φ) \end{aligned} .

3.7.8 It is clear that in the domain of convergence Re(λ) ≪ 0 this definition coincides with Definition 1.8. In order to see that π_{∗}^{U} extends to a meromorphic family note that i_{σ,μ}^{U} admits a meromorphic family of left inverses j_{σ,μ}^{U} (Lemma 3.22, 5.), and that we can express the push-down for σ through the spherical push-down by j_{σ,μ}^{U} ∘ π_{∗}^{U} ∘ i_{σ,μ}^{U}. 3.7.9 We claim that π_{∗}^{U} has finite-dimensional singularities. Note that given ν ∈ 𝔞^{∗} there exists k ∈ ℕ_{0} such that the restriction of π_{∗}^{U} to S_{{1},k}(σ_{λ}, φ) ∩ B_{{1}}(σ_{λ}, φ) converges for all Re(λ) < ν. The rank of the singularities of π_{∗}^{U} for Re(λ) < ν is bounded by the codimension of S_{{1},k}(σ_{λ}, φ) ∩ B_{{1}}(σ_{λ}, φ) in B_{{1}}(σ_{λ}, φ) which is finite. This proves the claim.

3.7.10 The following corollary is a consequence of the discussion above and Corollary 3.30.

Corollary 3.35 Let U ⊂ P define a regular cusp and φ be an admissible twist. Then the push-down is a meromorphic family of continuous maps π_{∗}^{U} : B_{{1}}(σ_{λ}, φ) → B_{U}(σ_{λ}, φ) with finite-dimensional singularities such that

0 → S_{{1}}(σ_{λ}, φ) → B_{{1}}(σ_{λ}, φ) →^{AS} R_{{1}}(σ_{λ}, φ) → 0
↓ {π_{∗}^{U}} ↓ π_{∗}^{U} ↓ [π_{∗}^{U}]
0 → S_{U}(σ_{λ}, φ) → B_{U}(σ_{λ}, φ) →^{AS} R_{U}(σ_{λ}, φ) → 0 .

is commutative.

3.8 Compatibility with embedding

3.8.1 In this subsection we assume that G^{n} belongs to the list

{Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)} .

First assume that U ⊂ P defines a cusp of smaller rank. Then we have a commutative diagram

\begin{matrix}A_{\{1\}}(1_{λ}^{n+1}, φ)^{m} & →^{i^{∗}} & A_{\{1\}}(1_{λ−ζ}^{n}, φ)^{m} \\ ↓ π_{∗}^{P_{U}^{n+1}} & & ↓ π_{∗}^{P_{U}^{n}} \\ A_{P_{U}} (1_{λ}^{n+1}, φ)^{m} & →^{i_{U}^{∗}} & A_{P_{U}} (1_{λ−ζ}^{n}, φ)^{m} \end{matrix} .

In fact, commutativity is obvious in the domain of convergence. 3.8.2 By the definition of the spaces R_{{1}}(1_{λ}^{n+1}, φ), R_{{1}}(1_{λ−ζ}^{n}, φ) as the spaces of asymptotics of smooth sections and the fact that that i^{∗} maps smooth sections to smooth sections we obtain

U-equivariant maps R_{{1}}(1_{λ}^{n+1}, φ)^{m} →^{i^{∗}} R_{{1}}(1_{λ−ζ}^{n}, φ)^{m}, m ∈ ℕ_{0}.

It follows from the naturality of the image bundle Lemma 2.6 that we have a commutative diagram

\begin{matrix}R_{\{1\}}(1_{λ}^{n+1}, φ)^{m} & →^{i^{∗}} & R_{\{1\}}(1_{λ−ζ}^{n}, φ)^{m} \\ ↓ [π_{∗}^{U,n+1}] & & ↓ [π_{∗}^{U,n}] \\ R_{U}(1_{λ}^{n+1}, φ)^{m} & →^{i_{U}^{∗}} & R_{U}(1_{λ−ζ}^{n}, φ)^{m} \end{matrix} .

3.8.3 On the level of Schwartz spaces we have for all k ∈ ℕ_{0}

\begin{matrix}S_{\{1\},k}(1_{λ}^{n+1}, φ) & →^{i^{∗}} & S_{\{1\},k}(1_{λ−ζ}^{n}, φ) \\ ↓ \{π_{∗}^{U,n+1}\} & & ↓ \{π_{∗}^{U,n}\} \\ S_{U,k}(1_{λ}^{n+1}, φ) & →^{i_{U}^{∗}} & S_{U,k}(1_{λ−ζ}^{n}, φ) \end{matrix} .

In order to see that i^{∗} and i_{U}^{∗} (initially defined on spaces of smooth sections) induce maps between Schwartz spaces one checks that these maps are bounded with respect to the norms ∥.∥_{k,d} introduced in 3.2.5. Commutativity of the diagram is clear.

3.8.4 We now easily obtain the diagram

\begin{matrix}B_{\{1\},k}(1_{λ}^{n+1}, φ) & →^{i^{∗}} & B_{\{1\},k}(1_{λ−ζ}^{n}, φ) \\ ↓ π_{∗}^{U,n+1} & & ↓ π_{∗}^{U,n} \\ B_{U,k}(1_{λ}^{n+1}, φ) & →^{i_{U}^{∗}} & B_{U,k}(1_{λ−ζ}^{n}, φ) \end{matrix} .

It gives Proposition 2.3 in the case of a pure cusp of lower rank. 3.8.5 We now assume that U ⊂ P^{n} defines a cusp of full rank. We are going to construct a map i_{U}^{∗} : B_{U}(1_{λ}^{n+1}, φ) → B_{U}(1_{λ−ζ}^{n}, φ) as the direct sum of

(i_{U}^{∗})_{1} : B_{U}(1_{λ}^{n+1}, φ) → S_{U}(1_{λ}^{n}, φ)

and

(i_{U}^{∗})_{2} : B_{U}(1_{λ}^{n+1}, φ) → R_{U}(1_{λ−ζ}^{n}, φ) .

While (i_{U}^{∗})_{1} is just the restriction of sections the definition of (i_{U}^{∗})_{2} is more complicated. In order to verify its properties we need some results of Subsection 3.9. 3.8.6 In the following we discuss (i_{U}^{∗})_{2}. This map comes as a meromorphic family and will essentially be fixed by the condition

π_{∗}^{U,n} ∘ i^{∗} = i_{U}^{∗} ∘ π_{∗}^{U,n+1} .  (17)

The details are as follows. 3.8.7 Since E_{∞_{P}} (1_{−λ+ζ}^{n}, φ̃) (see 3.7.3 for notation) is finite-dimensional it consists of distributions of uniformly bounded order. Further, since these distributions are supported in ∞_{P} we can choose k ∈ ℕ be such that E_{∞_{P}} (1_{−μ+ζ}^{n}, φ̃) pairs trivially with the space S_{{1},k−2ρ_{U}} (1_{μ−ζ}^{n}, φ) for all μ in some compact neighbourhood of λ. Thus we have an inclusion E_{∞_{P}} (1_{−μ+ζ}^{n}, φ̃) ⊂ R_{{1},k}(1_{μ−ζ}^{n}, φ)^{∗} (see Definition 3.15 for notation and use the dual of (9)).

Note that the target of (i_{U}^{∗})_{2} is the dual of E_{∞_{P}} (1_{−λ+ζ}^{n}, φ̃). We define

(i_{U}^{∗})_{2} : B_{U}(1_{λ}^{n+1}, φ) → R_{U}(1_{λ−ζ}^{n}, φ)

by the condition that

⟨ϕ, (i_{U}^{∗})_{2}(f)⟩ = ⟨i_{∗}(ϕ), L ∘ [Q] ∘ AS(f)⟩  (18)

for all ϕ ∈ E_{∞_{P}} (1_{−λ+ζ}^{n}, φ̃), where i_{∗} : C^{−∞}(∂X^{n}, V (1_{−λ+ζ}^{n}, φ̃)) → C^{−∞}(∂X^{n+1}, V (1_{−λ}^{n+1}, φ̃)) is the natural inclusion adjoint to i^{∗} (see 2.2.4), L is the split of (9), [Q] is defined in 3.4.9, and AS(f) ∈ R_{{1},k}(1_{λ}^{n}, φ). This formula defines (i_{U}^{∗})_{2}(f) for generic λ ∈ 𝔞_{ℂ}^{∗} where [Q] is regular. The meromorphic family is given by

(i_{U}^{∗})_{2} := (i_{∗})_{|E_{∞_{P}} (1_{−λ+ζ}^{n},φ̃)}^{∗} ∘ L ∘ [Q] ∘ AS .

3.8.8 Lemma 3.36 The definition of (i_{U}^{∗})_{2} is independent of the choice of k, the split L and [Q]. Proof. Let L′ ∘ [Q′] ∘ AS′ be defined with different choices (assume that k′ ≥ k). Let f_{μ} ∈ B_{U}(1_{μ}^{n+1}, φ) be the germ of a holomorphic family near λ, and consider a family ϕ_{μ} ∈ E_{∞_{P}} (1_{−μ+ζ}^{n}, φ̃). Then we have

AS ∘ π_{∗}^{U,n+1}(L′ ∘ [Q′] ∘ AS′(f) − L ∘ [Q] ∘ AS(f))
=^{Cor.3.30} [π_{∗}^{U,n+1}] ∘ AS(L′ ∘ [Q′] ∘ AS′(f) − L ∘ [Q] ∘ AS(f))
= [π_{∗}^{U,n+1}]([Q′] ∘ AS′(f) − [Q] ∘ L ∘ AS(f))
= AS′(f) − AS(f) .

We define (see Lemma 3.25 for {Q})

Δ := L′ ∘ [Q′] ∘ AS′(f) + {Q} ∘ L(AS′(f) − AS(f))
− L ∘ [Q] ∘ AS(f) .

Then by construction we have

π_{∗}^{U,n+1}(Δ) = 0 .  (19)

Using results which we will prove (independently of the present stuff) in Subsection 3.9 we show that ⟨i_{∗}(ϕ), Δ⟩ = 0. Proposition 3.58 states an equality of two spaces defined in 3.9.14 and 3.9.16:

Ext_{U}(1_{−μ}^{n+1}, φ̃) = E_{U}(1_{−μ}^{n+1}, φ̃) .

On the one hand,

E_{U}(1_{−μ}^{n+1}, φ̃) ⊂ ^{U}C^{−∞}(∂X^{n+1}, V (1_{−μ}^{n+1}, φ̃))

is the space of evaluations at μ of holomorphic families of U-invariant distributions. In particular we have i_{∗}(ϕ_{μ}) ∈ E_{U}(1_{−μ}^{n+1}, φ̃). On the other hand, for generic μ the space Ext_{U}(1_{−μ}^{n+1}, φ̃) is contained in the annihilator of the kernel of π_{∗}^{U,n+1}. Using (19) we obtain ⟨i_{∗}(ϕ_{μ}), Δ_{μ}⟩ = 0.

By our choice of k we have ⟨i_{∗}(ϕ_{μ}), g⟩ = 0 for every g ∈ S_{{1},k−2ρ_{U}} (1_{μ}^{n+1}, φ) and μ near λ since g vanishes at ∞_{P} with order larger than k. Note that L(AS′(f_{μ})−AS(f_{μ})) ∈ S_{U,k}(1_{μ}^{n+1}, φ). Since {Q} ∘ L(AS′(f_{μ}) − AS(f_{μ})) ∈ S_{{1},k−2ρ_{U}} (1_{μ}^{n+1}, φ) it follows that

⟨i_{∗}(ϕ_{μ}), {Q} ∘ L(AS′(f_{μ}) − AS(f_{μ}))⟩ = 0

for generic μ near λ. Finally we conclude for these μ that

⟨i_{∗}(ϕ_{μ}), L′ ∘ [Q]′ ∘ AS′(f_{μ})⟩
= ⟨i_{∗}(ϕ_{μ}), L′ ∘ [Q′] ∘ AS′(f_{μ}) + {Q} ∘ L(AS′(f_{μ}) − AS(f_{μ}))⟩
= ⟨i_{∗}(ϕ_{μ}), L ∘ [Q] ∘ AS(f_{μ})⟩ .

□

3.8.9

Lemma 3.37 The equality (17) holds true.

Proof. Let f ∈ B_{{1}}(1_{λ}^{n+1}, φ). Then we have

π_{∗}^{U,n+1}(f − L ∘ [Q] ∘ AS ∘ π_{∗}^{U,n+1}(f)
−{Q}(π_{∗}^{U,n+1}(f) − π_{∗}^{U,n+1} ∘ L ∘ [Q] ∘ AS ∘ π_{∗}^{U,n+1}(f)) = 0 .  (20)

We have seen in the proof of Lemma 3.36 that i_{∗}(ϕ) annihilates the kernel of π_{∗}^{U,n+1} (for generic λ, where all the maps are regular). We combine this fact with (20) in order to derive Equality (21) below. For all ϕ ∈ E_{∞_{P}} (1_{−λ+ζ}^{n}, φ̃) we have

⟨ϕ, AS ∘ i_{U}^{∗} ∘ π_{∗}^{U,n+1}(f)⟩ =^{def} ⟨ϕ, (i_{U}^{∗})_{2} ∘ π_{∗}^{U,n+1}(f)⟩
=^{(18)} ⟨i_{∗}(ϕ), L ∘ [Q] ∘ AS ∘ π_{∗}^{U,n+1}(f)⟩
= ⟨i_{∗}(ϕ), L ∘ [Q] ∘ AS ∘ π_{∗}^{U,n+1}(f)⟩
+{Q}(π_{∗}^{U,n+1}(f) − π_{∗}^{U,n+1} ∘ L ∘ [Q] ∘ AS ∘ π_{∗}^{U,n+1}(f))⟩
= ⟨i_{∗}(ϕ), f⟩  (21)
= ⟨ϕ, i^{∗}(f)⟩
= ⟨ϕ, [π_{∗}^{U,n}] ∘ AS ∘ i^{∗}(f)⟩ .

Since clearly {π_{∗}^{U,n}} ∘ i^{∗} = (i_{U}^{∗})_{1} ∘ {π_{∗}^{U,n+1}} we conclude the required indentity (17). □

3.8.10 Let us combine the results of the present subsection into one statement. Let now U ⊂ P^{n} define a cusp of arbitrary rank. Note that (i_{U}^{∗})_{2} and therefore i_{U}^{∗} may have poles in the case of a cusp of full rank. The following Proposition settles Proposition 2.3 in the case of pure cusps.

Proposition 3.38 We have the following commutative diagram (to be understood as an identity of meromorphic families if i_{U}^{∗} has poles)

\begin{matrix}C^{∞}(∂X, V (1_{λ}^{n+1}, φ)) & →^{i^{∗}} & C^{∞}(∂X^{n}, V (1_{λ−ζ}^{n}, φ)) \\ ↓ π_{∗}^{U,n+1} & & ↓ π_{∗}^{U,n} \\ B_{U}(1_{λ}^{n+1}, φ) & →^{i_{U}^{∗}} & B_{U}(1_{λ−ζ}^{n}, φ) \end{matrix} .  (22)

3.9 Extension and restriction

3.9.1 In the present subsection we assume for simplicity that the twist φ is normalized such that all its highest A-weights are zero. This differs from the convention adopted in 3.3.2. Our present convention has the effect that the dual φ̃ has the normalization adopted in 3.3.2.

Let l_{φ} be the highest weight of φ̃, i.e. l_{φ} := lφ̃. 3.9.2 The space C^{−∞}(∞_{P} , V (σ_{λ}, φ)) carries an action π^{σ_{λ},φ} of P_{U}A. Using the isomorphism of P_{U}A-modules

C^{−∞}(∞_{P} , V (σ_{λ}, φ)) ≅ (U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ}

and of M_{U}A-modules (compare 3.3.4)

(U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}) ⊗ V_{φ} ≅ U(𝔫̄) ⊗ V_{σ_{λ+2ρ}} ⊗ V_{φ}

given by the PBW-theorem we see that A acts semisimply.

Let C^{−∞}(∞_{P} , V (σ, φ))^{n} denote the subspace on which A acts with weight −λ − ρ − nα. Then we have

C^{−∞}(∞_{P} , V (σ_{λ}, φ))^{n} = (R_{{1}}(σ̃_{−λ}, φ̃)^{n})^{∗} .

3.9.3 Since the action of P_{U} is algebraic and U ⊂ P_{U} is Zariski dense we have

UC^{−∞}(∞_{P} , V (σ_{λ}, φ)) = ^{P_{U}} C^{−∞}(∞_{P} , V (σ_{λ}, φ)) .

Since A normalizes P_{U} we conclude that ^{U}C^{−∞}(∞_{P} , V (σ_{λ}, φ)) is an A-invariant subspace. In particular, we obtain a decomposition

UC^{−∞}(∞_{P} , V (σ_{λ}, φ)) = ⨁_{n=0}^{∞}^{U}C^{−∞}(∞_{P} , V (σ_{λ}, φ))^{n} = ⨁_{n=0}^{∞}^{U}(R_{{1}}(σ̃_{−λ}, φ̃)^{n})^{∗} .

3.9.4 Let E_{∞_{P}} (σ, φ)^{n} be the sheaf of holomorphic families f_{ν} ∈ ^{U}(R_{{1}}(σ̃_{−ν}, φ̃)^{n})^{∗}. This sheaf is torsion-free, and it is therefore the sheaf of holomorphic sections of a holomorphic vector bundle E_{∞_{P}} (σ, φ)^{n}. By E_{∞_{P}} (σ_{λ}, φ)^{n} we denote its fibre at λ. For each n ∈ ℕ_{0} we define the space Q̄_{(}σ_{λ}, φ)^{n} by the exact sequence

0 → E_{∞_{P}} (σ_{λ}, φ)^{n} → ^{U}(R_{{1}}(σ̃_{−λ}, φ̃)^{n})^{∗} → Q̄_{∞_{P}} (σ_{λ}, φ)^{n} → 0 .

Furthermore let E_{∞_{P}} (σ_{λ}, φ) := ⨁_{n=0}^{∞} E_{∞_{P}} (σ_{λ}, φ)^{n} and Q̄_{∞_{P}} (σ_{λ}, φ) := ⨁_{n=0}^{∞} Q̄_{∞_{P}} (σ_{λ}, φ)^{n}.

Definition 3.39 We call the elements of E_{∞_{P}} (σ_{λ}, φ)^{n} deformable. An element of U(R_{{1}}(σ̃_{−λ}, φ̃)^{n})^{∗} is called undeformable, if it represents a nontrivial class in Q̄_{∞_{P}} (σ_{λ}, φ)^{n}.

3.9.5 Let 𝔱 ⊂ 𝔪 be a Cartan subalgebra of 𝔪. Then 𝔥 := 𝔱 ⊕ 𝔞 is a Cartan subalgebra of 𝔤. By Δ^{+}(𝔤, 𝔥) we denote a positive root system which is compatible with the orientation of 𝔞. By Δ^{+}(𝔪, 𝔥) ⊂ Δ^{+}(𝔤, 𝔥) we denote the subsystem of roots of 𝔪. For each σ ∈ M̂ we define

𝔞^{∗} ∋ d(σ) := −ρ + max\{\frac{⟨μ_{σ}, ε⟩}{⟨α, ε⟩} | ε ∈ Δ^{+}(𝔤, 𝔥) \ Δ^{+}(𝔪, 𝔥)\}α ,  (23)

where μ_{σ} is the highest weight of σ, and ⟨., .⟩ is any Weyl-invariant scalar product on 𝔥. There is a natural action of P_{U}A on the space Pol(N, V_{σ_{−λ}^{w}} ⊗ V_{φ}) of polynomials on N with values in V_{σ_{−λ}^{w}} ⊗ V_{φ} (compare 3.3.6) given by

(man.f)(x) = (σ_{−λ}^{w} ⊗ φ)(man)f((n^{−1}x)^{m^{−1}a^{−1}}), m ∈ M_{U}, a ∈ A, n ∈ N_{V} .

3.9.6

Lemma 3.40 1. There is a holomorphic family of A-equivariant maps

j_{λ} : E_{∞_{P}} (σ_{λ}, φ)→ ^{P_{U}} Pol(N, V_{σ_{−λ}^{w}} ⊗ V_{φ}) .

2. If Re(λ) > d(σ) or λ ∉ I_{𝔞}, then j_{λ} is an isomorphism and Q̄_{∞_{P}} (σ_{λ}, φ) = 0.

Proof. 3.9.7 Let

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

be the unnormalized Knapp-Stein intertwining operator (compare [BO00], Sec. 5, (15)). In order to fix the conventions we recall its definition. The restriction of Ĵ_{λ}^{w} to smooth sections is given for Re(λ) < 0 by

(Ĵ_{λ}^{w})f(g) = ∫N̄ f(gwn̄)dn̄ .

For the rest of parameters it is defined by meromorphic continuation, and it extends by continuity to distributions. 3.9.8 By

j_{λ} : C^{−∞}(∞_{P} , V (σ_{λ})) → Pol(N, V_{σ_{−λ}^{w}})

we denote the off-diagonal part of the Knapp-Stein intertwining operator. Here we identify Pol(N, V_{σ_{−λ}^{w}}) with a subspace of C^{∞}(Ω_{P} , V (σ_{−λ}^{w})) such that p ∈ Pol(N, V_{σ_{−λ}^{w}}) corresponds to f_{p} ∈ C^{∞}(Ω_{P} , V (σ_{−λ}^{w})) with f_{p}(xw) = p(x).

The off-diagonal part of Ĵ_{λ}^{w} maps to polynomials since it is P -equivariant and the elements of C^{−∞}(∞_{P} , V (σ_{λ})) are P -finite. Alternatively, using the identification

C^{−∞}(∞_{P} , V (σ_{λ})) ≅ U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}} ,

we can write

j_{λ}(X ⊗ v)(n) = π^{σ_{−λ}^{w}}(X)f_{1_{v}}(nw) ,

where 1_{v} ∈ Pol(N, V_{σ_{−λ}^{w}}) is the constant polynomial with value v ∈ V_{σ_{−λ}^{w}}. 3.9.9 The map j_{λ} is in fact 𝔤-equivariant, where the action π^{σ_{−λ}^{w}} of 𝔤 on Pol(N, V_{σ_{−λ}^{w}}) is induced by the embedding Pol(N, V_{σ_{−λ}^{w}}) ⊂ C^{∞}(Ω_{P} , V (σ_{−λ}^{w})). Therefore ker(j_{λ}) is a 𝔤-submodule of the Verma module U(𝔤) ⊗_{U(𝔭)} V_{σ_{λ+2ρ}}. By [Jan79], Satz 1.17, this Verma module is irreducible for Re(λ) > d(σ). If λ ∉ I_{𝔞}, then j_{λ} is injective by [BO00], Lemma 6.7. 3.9.10 Since

dim Pol(N, V_{σ_{−λ}^{w}})^{n} = dim C^{−∞}(∞_{P} , V (σ_{λ}))^{n} = dim(U(𝔫̄) ⊗ V_{σ_{λ}})^{n} ,

we conclude that j_{λ} is in fact an isomorphism. After tensoring with V_{φ} and taking U-invariants we obtain an isomorphism

j_{λ} : ^{U}(R_{{1}}(σ̃_{−λ}, φ̃)^{n})^{∗} ≅ ^{P_{U}} Pol(N, V_{σ_{−λ}^{w}} ⊗ V_{φ})^{n} .

Since j_{λ} and its inverse depend holomorphically on λ we have

P_{U} Pol(N, V_{σ_{−λ}^{w}} ⊗ V_{φ})^{n} ≅ E_{∞_{P}} (σ_{λ}, φ)^{n}

for λ ∈ 𝔞_{ℂ}^{∗} with Re(λ) > d(σ) or λ ∉ I_{𝔞}. This proves the lemma. □

3.9.11 Recall the definition of the function spaces B_{U,k}(σ̃_{−λ}, φ̃) given in 3.21.

Definition 3.41 We define D_{U,k}(σ_{λ}, φ) to be the dual space to B_{U,k}(σ̃_{−λ}, φ̃). Furthermore let D_{U}(σ_{λ}, φ) := ⋃_{k∈ℕ_{0}} D_{U,k}(σ_{λ}, φ).

As a consequence of the corresponding properties of the family of function spaces the family of spaces D_{U,k}(σ_{λ}, φ), λ ∈ 𝔞_{ℂ}^{∗}, forms local trivial holomorphic bundles of dual Fréchet spaces. Furthermore, the spaces D_{U}(σ_{λ}, φ) are Montel and form a direct limit of locally trivial holomorphic bundles. 3.9.12 Recall the definition of the push-down Definition 1.8. Its meromorphic continuation was finally established in Corollary 3.35.

Definition 3.42 We define the extension map

ext^{U} : D_{U}(σ_{λ}, φ) → D_{{1}}(σ_{λ}, φ)

as the adjoint of the push-down

π_{∗}^{U} : B_{{1}}(σ̃_{−λ}, φ̃) → B_{U}(σ̃_{−λ}, φ̃) .

It follows from the corresponding properties of the push-down that the extension maps form a meromorphic family of continuous maps with finite-dimensional singularities. 3.9.13 In the remainder of the present subsection we discuss the case σ = 1. Assume that the cusp associated to U ⊂ P has lower rank. For k ∈ ℕ_{0} and Re(λ) > −ρ^{U} + (l_{φ} − kα)/2 such that

π_{∗}^{U} : B_{{1},k}(1_{−λ}, φ̃) → B_{U,k}(1_{−λ}, φ̃)

is regular (compare Corollary 3.30) we also have a map

ext^{U} : D_{U,k}(1_{λ}, φ) → D_{{1},k}(1_{λ}, φ)

defined as the adjoint π_{∗}^{U}. If k_{1} > k, then ext^{U} : D_{U,k}(1_{λ}, φ) → D_{{1},k_{1}}(1_{λ}, φ) is a meromorphic family of continuous maps with finite-dimensional singularities defined for Re(λ) > −ρ^{U} + (l_{φ} − kα)/2. 3.9.14 Definition 3.43 We define

Ext_{U}(1_{λ}, φ) ⊂ D_{{1}}(1_{λ}, φ)

to be the subspace of all f ∈ D_{{1}}(1_{λ}, φ) of the form ext^{U}(h)_{λ}, where h_{μ} ∈ D_{U}(1_{μ}, φ) is a meromorphic family defined near λ such that μ ↦ ext^{U}(h)_{μ} is regular at μ = λ. In a similar manner we define Ext_{U,k}(1_{λ}, φ) to by requiring in addition that h_{μ} ∈ D_{U,k}(1_{μ}, φ). The subspaces Ext_{U,k}(1_{λ}, φ) ⊂ Ext_{U}(1_{λ}, φ) are defined for Re(λ) > −ρ^{U} + (l_{φ} − kα)/2. The space Ext_{U}(1_{λ}, φ) plays the role of the range of the extension. 3.9.15 It is clear that

Ext_{U}(1_{λ}, φ) ⊂ ^{U}D_{{1}}(1_{λ}, φ) .

In order to describe to which extent the space ^{U}D_{{1}}(1_{λ}, φ) is exhausted by Ext_{U}(1_{λ}, φ) we define Q_{U}(1_{λ}, φ) to be the following quotient:

0 → Ext_{U}(1_{λ}, φ) → ^{U}D_{{1}}(1_{λ}, φ) → Q_{U}(1_{λ}, φ) → 0 .

The elements in the space Q_{U}(1_{λ}, φ) turn out to be somewhat uncontrollable. We therefore take much effort to show that this space is trivial under certain conditions. 3.9.16 We now define the space of deformable U-invariant distributions (compare Def. 3.39 for a similar definition with an additional support condition).

Definition 3.44 We define the subspace

E_{U}(1_{λ}, φ) ⊂ ^{U}D_{{1}}(1_{λ}, φ)

as the space of evaluations of germs at λ of holomorphic families f_{ν} ∈ ^{U}D_{{1}}(1_{ν}, φ). In a similar manner we define E_{U,k}(1_{λ}, φ) ⊂ ^{U}D_{{1},k}(1_{λ}, φ) as the subspace of evaluations of families with the additional property that f_{ν} ∈ ^{U}D_{{1},k}(1_{ν}, φ).

3.9.17 We define the space Q̄_{U}(1_{λ}, φ) as the quotient

0 → E_{U}(1_{λ}, φ) → ^{U}D_{{1}}(1_{λ}, φ) → Q̄_{U}(1_{λ}, φ) → 0 .

Definition 3.45 An element of ^{U}D_{{1}}(1_{λ}, φ) which represents a non-trivial class in Q̄_{U}(1_{λ}, φ) is called undeformable. (compare Definition 3.39). 3.9.18 It follows immediately from the definitions that

Ext_{U}(1_{λ}, φ) ⊂ E_{U}(1_{λ}, φ) .

Hence we have a surjection

Q_{U}(1_{λ}, φ) → Q̄_{U}(1_{λ}, φ) .

One of the goals of the present subsection is to show that

Ext_{U}(1_{λ}, φ) = E_{U}(1_{λ}, φ) .

Furthermore, we want to show that for many (generic) λ ∈ 𝔞_{ℂ}^{∗} every element of ^{U}D_{{1}}(1_{λ}, φ) is deformable, i.e. Q_{U}(1_{λ}, φ) ≅ 0. The results are stated in Proposition 3.58. 3.9.19 Now we come to the definition of a left-inverse of ext^{U}: the restriction map res^{U}. We fix k ∈ ℕ_{0}. Recall the construction of the meromorphic family of right-inverses of [π_{∗}^{U}]

[Q] : R_{U,k}(1_{λ}, φ̃) → R_{{1},k}(1_{λ}, φ̃)

from 3.4.9. Let k_{1} ∈ ℕ_{0} be such that k_{1}α < kα − 2ρ_{U}, and let

{Q} : S_{U,k}(1_{λ}, φ) → S_{{1},k_{1}}(1_{λ}, φ)

be as in Lemma 3.25. We define a meromorphic family of maps

Q : B_{U,k}(1_{λ}, φ̃) → B_{{1},k_{1}}(1_{λ}, φ̃)

by

Q(h) := {Q} (h − π_{∗}^{U} ∘ L ∘ [Q] ∘ AS(h)) + L ∘ [Q] ∘ AS(h)

Then one easily checks that π_{∗}^{U} ∘ Q is just the inclusion B_{U,k}(1_{λ}, φ̃) → B_{U,k_{1}}(1_{λ}, φ̃).

Definition 3.46 We define the meromorphic family of restriction maps

res^{U} : D_{{1},k_{1}}(1_{λ}, φ) → D_{U,k}(1_{λ}, φ)

as the adjoint of Q.

Note that res^{U} depends on choices (we do not indicate these choices in the notation for the restriction map). In particular, these maps are not compatible if we change k and k_{1}.

3.9.20 Note that the composition res^{U} ∘ ext^{U} coincides with the inclusion

D_{U,k_{1}}(1_{λ}, φ) ↪ D_{U,k}(1_{λ}, φ) .

Recall the Definition 3.43 of Ext_{U,k_{1}}(1_{λ}, φ).

Lemma 3.47 Let k, k_{1} ∈ ℕ_{0} and λ ∈ 𝔞_{ℂ}^{∗} be such that res^{U} : D_{{1},k_{1}}(1_{λ}, φ) → D_{U,k}(1_{λ}, φ) is defined. Furthermore we assume that π_{∗}^{U} : B_{{1}}(1_{−λ}, φ̃) → B_{U}(1_{−λ}, φ̃) is regular. Then the restriction of res^{U} to Ext_{U,k_{1}}(1_{λ}, φ) is independent of choices.

Proof. This follows from res^{U} ∘ ext^{U}(f) = f and the fact that ext^{U} is regular at λ. □

3.9.21 Assume that λ ∈ 𝔞_{ℂ}^{∗} is such that

π_{∗}^{U} : B(1_{−λ}, φ̃) → B_{U}(1_{−λ}, φ̃)

and

[Q] : R_{U}(1_{−λ}, φ̃)^{n} → R_{{1}}(1_{−λ}, φ̃)^{n}

are regular for all n ∈ ℕ_{0}. We call λ ∈ 𝔞_{ℂ}^{∗} satisfying these conditions admissible. 3.9.22 Note that R_{{1}}(1_{−λ}, φ̃)^{∗} is the space of those distribution sections of V (1_{λ}, φ) that are supported at the point ∞_{P}. We define

Ext_{∞_{P}} (1_{λ}, φ) := R_{{1}}(1_{−λ}, φ̃)^{∗} ∩ Ext_{U}(1_{λ}, φ) .

Recall the definition 3.9.15 of the space Q_{U}(1_{λ}, φ). The point of the following lemma is that every element of Q_{U}(1_{λ}, φ) can be represented by an invariant distribution supported in ∞_{P}. Note that

[ext^{U}] : R_{U}(1_{−λ}, φ̃)^{∗} → ^{U}R_{{1}}(1_{−λ}, φ̃)^{∗}

generates a subspace of Ext_{∞_{P}} (1_{λ}, φ), where [ext^{U}] is the restriction of ext^{U} to R_{U}(1_{−λ}, φ̃)^{∗}, or equivalently, the adjoint of [π_{∗}^{U}] (see Def. 3.20).

Lemma 3.48 1. The family of maps [ext^{U}] generates all of Ext_{∞_{P}} (1_{λ}, φ).

2. If λ ∈ 𝔞_{ℂ}^{∗} is admissible in the sense of 3.9.21, then there is an exact sequence

0 → Ext_{∞_{P}} (1_{λ}, φ) → ^{U}R_{{1}}(1_{−λ}, φ̃)^{∗} → Q_{U}(1_{λ}, φ) → 0

of semisimple A-modules.

Proof. Recall that

UR_{{1}}(1_{−λ}, φ̃)^{∗} = ⨁_{n∈ℕ_{0}} ^{U}(R_{{1}}(1_{−λ}, φ̃)^{n})^{∗}

is a weight-decomposition of the A-module. Moreover, we have

[ext^{U}] : (R_{U}(1_{−λ}, φ̃)^{n})^{∗} → (R_{{1}}(1_{−λ}, φ̃)^{n})^{∗} .

If f ∈ Ext_{∞_{P}} (1_{λ}, φ) is represented by ext^{U}(h) for some meromorphic family h_{μ} ∈ D_{U}(1_{μ}, φ), then {res^{U}}{ext^{U}(h)} = {h} vanishes at λ, where {h} denotes the restriction of h to S_{{1}}(1_{−λ}, φ̃), and {res^{U}} := {Q}^{∗}. Therefore all non-positive Laurent-coefficients of the expansion of h at λ belong to R_{U}(1_{−λ}, φ̃)^{∗}. Thus we can choose the family h such that h_{μ} ∈ R_{U}(1_{−μ}, φ̃)^{∗}, and such that [ext^{U}](h)_{λ} = f. This shows that [ext^{U}] generates Ext_{∞_{P}} (1_{λ}, φ), and that A acts semisimply on Ext_{∞_{P}} (1_{λ}, φ). 3.9.23 It remains to show that any element of Q_{U}(1_{λ}, φ) can be represented by some element of ^{U}R_{{1}}(1_{−λ}, φ̃)^{∗}. Let f ∈ ^{U}D_{{1}}(1_{λ}, φ). Then there is k ∈ ℕ_{0} such that f ∈ ^{U}D_{{1},k}(1_{λ}, φ). We choose k_{1} such that

{res^{U}} : S_{{1},k}(1_{−λ}, φ̃)^{∗} → S_{U,k_{1}}(1_{−λ}, φ̃)^{∗}

is defined. We then put h := {res^{U}}{f}. Let

T : S_{U,k_{1}}(1_{−λ}, φ̃)^{∗} → D_{U}(1_{λ}, φ)

be the split induced by the dual split L. We form f − ext^{U} ∘ T (h). This difference represents the same element in Q_{U}(1_{λ}, φ) as f, but its restriction to S_{{1}}(1_{−λ}, φ̃) vanishes. Indeed, for g in S_{{1}}(1_{−λ}, φ̃) we have

⟨ext^{U} ∘ T (h), g⟩ = ⟨f, {Q} ∘ {π_{∗}^{U}}(g)⟩ = ⟨f, g⟩ .
□

3.9.24

Proposition 3.49 Let n ≥ 0. If λ ∈ 𝔞^{∗} is sufficiently large, then the inclusion

Ext_{∞_{P}} (1_{λ}, φ)^{n} ↪ ^{U}(R_{{1}}(1_{−λ}, φ̃)^{n})^{∗}

is an isomorphism.

Proof. If U defines a cusp of full rank, then the proposition is a direct consequence of Lemma 3.40 and the definition of [ext^{U}] (Definition 3.32). Thus we may and will assume in the following that U defines a cusp of smaller rank. Note that both spaces appearing in the proposition are finite-dimensional. The proposition is an immediate consequence of the following lemma.

Lemma 3.50 If λ ∈ 𝔞^{∗} is sufficiently large, then we have the inequality

dim Ext_{∞_{P}} (1_{λ}, φ)^{n} ≥ dim^{U}(R_{{1}}(1_{−λ}, φ̃)^{n})^{∗} .

Proof. By Lemma 3.40,2, if λ ∈ 𝔞^{∗} is sufficiently large we have dim^{U}(R_{{1}}(1_{−λ}, φ̃)^{n})^{∗} = dim^{P_{U}} Pol(N, V_{1_{−λ}} ⊗ V_{φ})^{n}, where the action of AP_{U} on a polynomial is given by

(p f)(n) := φ(p)f(n^{p^{−1}}) .

The degree-n subspace of the polynomial maps is characterized by

a f = a^{−nα}f , a ∈ A .  (24)

We now consider

[π_{∗}^{U}] : R_{1}(1_{−λ}, φ̃)^{n} → R_{U}(1_{−λ}, φ̃)^{n}

which is regular for Re(λ) sufficiently large. Its adjoint is

[ext^{U}] : (R_{U}(1_{−λ}, φ̃)^{∗})^{n} → Ext_{∞_{P}} (1_{λ}, φ)^{n} ,

and we have dim im[π_{∗}^{U}] = dim im[ext^{U}]. 3.9.25 Therefore, Lemma 3.50 follows directly from the following lemma.

Lemma 3.51 If λ ∈ 𝔞^{∗} is sufficiently large, then we have

dim im[π_{∗}^{U}] ≥ dim^{P_{U}} Pol(N, V_{1_{−λ}} ⊗ V_{φ})^{n} .

Proof. Let ⟨., .⟩ be some non-degenerate invariant bilinear form on 𝔤. We define the linear subspace

𝔫̄^{U} := {Y ∈ 𝔫̄ | ⟨[Y, H], X⟩ = 0 ∀X ∈ 𝔫_{V} }  (25)

and the submanifold N̄^{U} := exp(𝔫̄^{U}) ⊂ N̄. We furthermore choose a Cartan involution θ of 𝔤 compatible with 𝔞.

Lemma 3.52 1. The submanifold N̄^{U} is AM_{U}-invariant.

2. The multiplication map N̄_{V} × N̄^{U} → N̄ is a diffeomorphism, where N̄_{V} := N_{V}^{θ}.

3. The composition

N̄ →^{∼} N̄_{V} × N̄^{U} →^{pr} N̄^{U}

is an AM_{U}-equivariant polynomial map.

Proof. The subspace 𝔫̄^{U} is AM_{U}-invariant since 𝔫_{V} is AM_{U}-invariant. The first assertion now follows from the AM_{U}-equivariance of the exponential map

By the A-invariance of N̄^{U}, N̄_{V} and the equivariance of the multiplication, it suffices to show that the multiplication map is a diffeomorphism near (1, 1). Infinitesimally it is given by the map 𝔫̄_{V} × 𝔫̄^{U} → 𝔫̄, (Y_{V} , Y^{U}) ↦ Ȳ_{V} + Y^{U}. This map is an isomorphism. In fact, the dimensions of the domain and the target coincide, and 𝔫̄_{V} ∩ 𝔫̄^{U} = {0}.

The last assertion follows from the diagram

\xymatrix{𝔫̄ \ar[d]^{exp} & 𝔫̄_{V} × 𝔫̄^{U} \ar[l]_{q} \ar[d]^{(exp,exp)} \ar[r]^{pr} & 𝔫̄^{U} \ar[d]^{exp} \\ N̄ & N̄_{V} × N̄^{U} \ar[l]_{mult} \ar[r]^{pr} & N̄^{U}}

and the fact that q is an AM_{U}-equivariant polynomial map with a polynomial inverse. □

3.9.26 Note that N̄_{V} AM_{U} acts on Pol(N̄, V_{1_{−λ}} ⊗ Vφ̃) by

n̄_{V} am f(n̄) = φ̃(am)f((n̄_{V}^{−1}n̄)^{(am)^{−1}}) .

We now consider the space

I(λ)^{n} := ^{N̄_{V} M_{U}} Pol(N̄, V_{1_{−λ}} ⊗ Vφ̃)^{n} ⊂ Pol(N̄, V_{1_{−λ}} ⊗ Vφ̃)^{n} ,

where the degree n-subspace is distinguished by the condition

a f = a^{nα}f .  (26)

We have a degree-preserving inclusion

I(λ)^{n} ↪ R_{1}(1_{−λ}, φ̃)^{n} .

Furthermore, it follows from Lemma 3.52 that the restriction to N̄^{U} induces an isomorphism

I(λ)^{n} →^{∼} ^{M_{U}} Pol(N̄^{U}, 1_{−λ} ⊗ Vφ̃)^{n} .

Lemma 3.51 now follows from the following two assertions.

Lemma 3.53 dim I(λ)^{n} = dim^{P_{U}} Pol(N, V_{1_{−λ}} ⊗ V_{φ})^{n}

Lemma 3.54 If λ ∈ 𝔞^{∗} is sufficiently large, then the restriction of [π_{∗}^{U}] to I(λ)^{n} is injective.

3.9.27 We first show Lemma 3.53. We define N^{U} := (N̄^{U})^{θ} ⊂ N. Then we have an Aequivariant diffeomorphism N_{V} × N^{U} →^{mult} N, and the projection N →^{∼} N_{V} × N^{U} →^{pr} N^{U} is an AM_{U}-equivariant polynomial map. These facts follow from Lemma 3.52 by an application of the Cartan involution θ. We conclude that the restriction to N^{U} induces an isomorphism

P_{U} Pol(N, V_{1_{−λ}} ⊗ V_{φ})^{n} →^{∼} ^{M_{U}} Pol(N^{U}, V_{1_{−λ}} ⊗ V_{φ})^{n} .

Note that there is a canonical AM_{U}-equivariant isomorphism between S((𝔫^{U})^{∗}) ⊗ V_{φ} and Pol(N^{U}, V_{1_{−λ}} ⊗ V_{φ}). Here S(.) stands for the symmetric algebra. Similarly, we have

S((𝔫̄^{U})^{∗}) ⊗ Vφ̃ ≅ Pol(N̄^{U}, V_{1_{−λ}} ⊗ Vφ̃) .

The natural pairing between 𝔫̄^{U} and 𝔫^{U} := (𝔫̄^{U})^{θ} via the G-invariant form ⟨., .⟩ induces a nondegenerate pairing between the two symmetric algebras above. We conclude that the spaces M_{U} Pol(N^{U}, V_{1_{−λ}} ⊗ V_{φ})^{n} and ^{M_{U}} Pol(N̄^{U}, V_{1_{−λ}} ⊗ Vφ̃)^{n} ≅ I(λ)^{n} are each others duals (the pairing is degree-preserving in view of the characterizations (24) and (26)). Lemma 3.53 now follows. 3.9.28 We now start with the proof of Lemma 3.54. First of all note that the push-down (see Def. 3.20)

[π_{∗}^{U}] : I(λ)^{n} → R_{U}(1_{−λ}, φ̃)^{n} ⊂ A_{U}(1_{−λ}, φ̃)^{n}

is given by a convergent integral

[π_{∗}^{U}](f)(n̄) = ∫_{N_{V}} φ(n_{V})^{−1}f(n̄(n_{V} n̄))a(n_{V} n̄)^{−λ−ρ}dn_{V} , n̄ ∈ N̄ \ {1} .

Here we have employed again the Bruhat decomposition g = n̄(g)m(g)a(g)n(g). The vector space Vφ̃ has a filtration induced by the action of A that is preserved by M_{U}N_{V}. The induced action of N_{V} on the associated graded vector space Gr(Vφ̃) is trivial. The filtration of Vφ̃ induces filtrations on I(λ)^{n} and A_{U}(1_{−λ}, φ̃)^{n}. From the integral representation of [π_{∗}^{U}] we see that that this map preserves the filtrations and induces a map Gr[π_{U}^{∗}] : GrI(λ)^{n} → GrA_{U}(1_{−λ}, φ̃)^{n}. Since injectivity of the associated graded map implies injectivity of a filtration preserving map, Lemma 3.54 is a consequence of

Lemma 3.55 If λ ∈ 𝔞^{∗} is sufficiently large, then

Gr[π_{∗}^{U}] : GrI(λ)^{n} → GrA_{U}(1_{−λ}, φ̃)^{n}

is injective.

3.9.29 Note that

Gr[π_{∗}^{U}](f)(n̄) = ∫_{N_{V}} f(n̄(n_{V} n̄))a(n_{V} n̄)^{−λ−ρ}dn_{V} , n̄ ∈ N̄ \ {1} .

We define

c_{λ}(n̄) := Gr[π_{∗}^{U}](1)(n̄) = ∫_{N_{V}} a(n_{V} n̄)^{−λ−ρ}dn_{V} , n̄ ∈ N̄ \ {1} .

We will show the following lemma.

Lemma 3.56 For n̄^{U} ∈ N̄^{U} \ exp(𝔫̄_{−2α}) we have

\frac{1}{c_{λ}(n̄^{U})}Gr[π_{∗}^{U}](f)(n̄^{U}) = f(n̄^{U}) + O(|λ|^{−1}) .

3.9.30 Let us first show that Lemma 3.56 implies 3.55. The natural identification V_{1_{−λ}} ≅ ℂ induces identifications Pol(N̄^{U}, 1_{−λ} ⊗ Vφ̃)^{n} ≅ Pol(N̄^{U}, Vφ̃)^{n} for all λ.

Note that 𝔫̄^{U} ∩ 𝔫̄_{−α} is non-trivial. Otherwise we would have 𝔫_{V} ∩ 𝔫_{α} = 𝔫_{α} and therefore 𝔫_{V} = 𝔫, i.e., U would define a cusp of full rank. Hence we can choose a finite sequence of base points n̄_{i}^{U} ∈ N̄^{U} \ exp(𝔫̄_{−2α}) and vectors v_{i} ∈ V_{φ}, i = 1, …, r := dim^{M_{U}} Pol(N̄^{U}, Vφ̃)^{n} such that the following map is an isomorphism:

Φ : ^{M_{U}} Pol(N̄^{U}, Vφ̃)^{n} → ℂ^{r} , f ↦ (⟨v_{1}, f(n_{1}^{U})⟩, …, ⟨v_{r}, f(n_{r}^{U})⟩) .

We now consider the composition

A(λ) := Φ ∘ \frac{1}{c_{λ}} Gr[π_{∗}^{U}] ∘ Φ^{−1} : ℂ^{n} → ℂ^{n} .

Lemma 3.56 implies that

A(λ) = 1 + O(|λ|^{−1}) .

In particular, A is injective, if λ ∈ 𝔞 is sufficiently large. This implies the assertion of Lemma 3.55. 3.9.31 We now show Lemma 3.56. Fix n̄^{U} ∈ N̄^{U} \ exp(𝔫̄_{−2α}). We set

Ψ(n_{V}) := log a(n_{V} n̄^{U}) , g(n_{V}) := f(n̄(n_{V} n̄^{U}))

Then we can write

∫_{N_{V}} f(n̄(n_{V} n̄^{U}))a(n_{V} n̄^{U})^{−λ−ρ}dn_{V} = ∫_{N_{V}} e^{(−λ−ρ)Ψ(n_{V} )}g(n_{V})dn_{V} .

Lemma 3.57 The function Ψ(n_{V}) has a unique non-degenerate absolute minimum Ψ(1) = 0 at n_{V} = 1. Furthermore, there exists a compact neighbourhood K ⊂ N_{V} of 1 such that α(Ψ(n_{V})) ≥ 1 for n_{V} ∉ K.

We first show that Lemma 3.57 implies 3.56. We split the integral as ∫_{N_{V}} = ∫_{K} + ∫_{N_{V} \K}. Note that n̄(N_{V} n̄^{U}) ⊂ N̄ is a pre-compact subset. It follows that g is smooth and uniformly bounded. We approximate the first summand by a Gaussian integral at the minimum of Ψ and get

∫_{K} e^{(−λ−ρ)(Ψ(n_{V} ))}g(n_{V})dn_{V} = ∫_{K} e^{(−λ−ρ)(Ψ(n_{V} ))}dn_{V} (g(0) + O(λ^{−1})) .

Furthermore, ∫_{K} e^{(−λ−ρ)(Ψ(n_{V} ))}dn_{V} decreases at most as |λ|^{− dim N_{V} /2}. We claim that the contributions ∫_{N_{V} \K} decrease exponentially so that these parts of the integrals can only contribute exponentially small error terms. This claim implies Lemma 3.56.

In order to see the claim we write

∫_{N_{V} \K} e^{(−λ−ρ)(Ψ(n_{V} ))}dn_{V} = min_{n_{V} ∈N_{V} \K} e^{(−λ+α)(Ψ(n_{V} ))} ∫_{N_{V} \K} e^{(−α−ρ)(Ψ(n_{V} ))}dn_{V} .

The integral on the right-hand side converges, and

min_{n_{V} ∈N_{V} \K} e^{(−λ+α)(Ψ(n_{V} ))} ≤ e^{−c|λ|}

for a suitable constant c > 0. This finishes the proof of Lemma 3.56 under the assumption of Lemma 3.57. 3.9.32 We now show Lemma 3.57. In order to compute a(n_{V} n̄^{U}) we may assume that G is the subgroup of GL(n + 1, 𝔽) that preserves the 𝔽-valued Hermitian scalar product

v̄_{0}w_{n} + v̄_{n}w_{0} + v̄_{1}w_{1} + ⋯ + v̄_{n−1}w_{n−1}  (27)

on the right 𝔽-vector space 𝔽^{n+1}. We choose

A := \{( \begin{matrix}a & 0 & 0 \\ 0 & 1_{n−1×n−1} & 0 \\ 0 & 0 & a^{−1} \end{matrix} ) | a ∈ ℝ^{+}\}

and get

N := \{( \begin{matrix}1 & w & p − \frac{∥w∥^{2}}{2} \\ 0 & 1 & −w̄^{t} \\ 0 & 0 & 1 \end{matrix} ) | w ∈ 𝔽^{n−1} , p ∈ im 𝔽\} .

The parametrization of N given here is via the exponential map, if we identify 𝔫 = 𝔫_{α} ⊕ 𝔫_{2α} ≅ 𝔽^{n−1} ⊕ Im 𝔽. Furthermore,

N̄ := \{( \begin{matrix}1 & 0 & 0 \\ −v̄^{t} & 1 & 0 \\ q − \frac{∥v∥^{2}}{2} & v & 1 \end{matrix} ) | v ∈ 𝔽^{n−1} , q ∈ Im 𝔽\} .

Since they are A-invariant, the subspaces 𝔫_{V} ⊂ 𝔫 and 𝔫̄^{U} ⊂ 𝔫̄ are given in this identification as W ⊕ P ⊂ 𝔫 and V ⊕ Q ⊂ 𝔫̄ for real subspaces W, V ⊂ 𝔽^{n−1} and P, Q ⊂ Im 𝔽. We consider the invariant form ⟨A, B⟩ := Re TrAB on 𝔤. Given W ⊕ P, the space V ⊕ Q is characterized by ⟨ad(V ⊕ W)(H), W ⊕ H⟩ = 0 (see (25)). Explicitly,

H = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ⟨ad(v + q)(H), w + p⟩ = −2Re(w̄v^{t} + qp) .

Therefore we have

V = W^{⊥} , Q = P^{⊥}  (28)

with respect to the natural euclidean pairings on 𝔽^{n−1} and Im 𝔽. The vector e_{0} := (1, …, 0) ∈ 𝔽^{n+1} is the highest weight vector of the standard representation of G with weight α. Similarly, e_{n} := (0, …, 0, 1) is the lowest weight vector with weight −α. Let g = n̄man ∈ N̄MAN. If ⟨., .⟩ denotes the 𝔽-valued scalar product (27), then

⟨ge_{0}, e_{n}⟩ = ⟨m̄ane_{0}, e_{n}⟩ = ⟨mane_{0}, n̄^{−1}e_{n}⟩ = a^{α}⟨me_{0}, e_{n}⟩ .

The subspace e_{0}𝔽 ≅ 𝔽 is invariant under the group M. In particular, we have a homomorphism ϑ : M → 𝔽^{∗} such that me_{0} = e_{0}ϑ(m). It now follows that ⟨me_{0}, e_{n}⟩ = ϑ̄(m). Since M is compact, we have ∥ϑ(m)∥ = 1 and therefore a^{α} = ∥⟨ge_{0}, e_{n}⟩∥. If we parametrize (w, p) = n_{V} ∈ N_{V} and (v, q) = n̄^{U} ∈ N̄^{U} as above, then we get

a(n_{V} n^{U})^{2α} = ∥1 − wv̄^{t} + (p − \frac{∥w∥^{2}}{2} )(q − \frac{∥v∥^{2}}{2} )∥^{2} .  (29)

We fix (v, q) ∈ V ⊕ Q with v ≠ 0. We must show that the right-hand side has a unique absolute minimum 1 at (w, p) = (0, 0), and that this minimum is non-degenerate. Using (28) we get

Re(1 − wv̄^{t} + (p − \frac{∥w∥^{2}}{2} )(q − \frac{∥v∥^{2}}{2} )) = 1 + \frac{∥v∥^{2}∥w∥^{2}}{4}
Im(1 − wv̄^{t} + (p − \frac{∥w∥^{2}}{2} )(q − \frac{∥v∥^{2}}{2} )) = pq − wv̄^{t} − \frac{∥w∥^{2}}{2} q − \frac{∥v∥^{2}}{2} p .

First of all, a(n_{V} n^{U})^{2α} ≥ 1 and a(1)^{2α} = 1. Moreover, if a(n_{V} n̄^{U})^{2α} = 1, then w = 0. Since Re(\overline{pq}p) = Req̄∥p∥^{2}) = 0 we have pq ⊥ \frac{∥v∥^{2}}{2} p. Hence, the equality a(n_{V} n̄^{U}) = 1 implies in addition to w = 0 that also p = 0. We thus have shown that a(n_{V} n̄^{U})^{2α} takes its unique absolute minimum at 1. Next we show that it is non-degenerate. The Hessian h(p, w) is the part of the polynomial (29) which is quadratic in (w, p). It can be written as

h(w, p) := ∥w∥^{2} \frac{∥v∥^{2}}{2} + ∥p(q − \frac{∥v∥^{2}}{2} ) − wv̄^{t}∥^{2} .

If h(w, p) = 0, then from the first summand and v ≠ 0 we get w = 0 and p(q − \frac{∥v∥^{2}}{2} ) = 0. Since (q − \frac{∥v∥^{2}}{2} ) ≠ 0 we conclude that p = 0. The last assertion of Lemma 3.57 follows from lim_{n_{V} →∞}a(n_{V} n̄^{U})^{2α} = ∞, which is easy to check using the explicit formula (29). This finishes the proof of Lemma 3.57.

We now have also finished the proof of Proposition 3.49. □

3.9.33

Proposition 3.58 1. We have Ext_{U}(1_{λ}, φ) = E_{U}(1_{λ}, φ) for all λ ∈ 𝔞_{ℂ}^{∗}.

2. If λ ∉ I_{𝔞} or Re(λ) > −ρ, then Q_{U}(1_{λ}, φ) = 0.

Proof. 3.9.34 As in3.9.4 let E_{∞_{P}} (1, φ)^{n} be the torsion-free coherent sheaf on 𝔞_{ℂ}^{∗} of holomorphic families of U-invariant distributions supported on ∞_{P}, on which A acts by multiplication by the function λ ↦ a^{λ−ρ−nα}. By Ext_{∞_{P}} (1, φ)^{n} we denote the subsheaf generated by the restriction of [ext^{U}] to the homogeneous part (R_{U}(1_{−.}, φ̃)^{n})^{∗}.

The space Ext_{∞_{P}} (1_{λ}, φ)^{n} is the geometric fibre of Ext_{∞_{P}} (1, φ)^{n} for the generic set of λ ∈ 𝔞_{ℂ}^{∗}, where [ext^{U}] is regular. 3.9.35 The sheaf E_{∞_{P}} (1, φ)^{n} is the sheaf of sections of a finite-dimensional trivial holomorphic vector bundle E_{∞_{P}} (1, φ)^{n} → 𝔞_{ℂ}^{∗}. The torsion-free subsheaf Ext_{∞_{P}} (1, φ)^{n} corresponds to a bundle Ext_{∞_{P}} (1, φ)^{n}.

This discussion shows the following: if the inclusion Ext_{∞_{P}} (1_{λ}, φ)^{n} ↪ E_{∞_{P}} (1_{λ}, φ)^{n} is surjective at one point λ ∈ 𝔞_{ℂ}^{∗}, then it is surjective for generic λ, i.e. outside a discrete set. However, we know from Proposition 3.49 that this inclusion is surjective for many λ, hence it is so generically. 3.9.36 In view of Lemma 3.48, 2. we have

E_{∞_{P}} (1_{λ}, φ)^{n}/Ext_{∞_{P}} (1_{λ}, φ)^{n} ≅ ker (Q_{U}(1_{λ}, φ)^{n} → Q̄_{∞_{P}} (1_{λ}, φ)^{n}) .  (30)

By 3.9.35 the quotient on the left hand side is trivial generically. The same is true for Q̄_{∞_{P}} (1_{λ}, φ)^{n} by Lemma 3.40. We conclude that Q_{U}(1_{λ}, φ)^{n} is trivial outside a discrete set. 3.9.37 We now prove the first assertion of Prop. 3.58. We know that Ext_{U}(1_{λ}, φ) ⊂ E_{U}(1_{λ}, φ). Let now f ∈ E_{U}(1_{λ}, φ) be given as the value at λ of a meromorphic family f_{μ} ∈ ^{U}D_{{1},k}(1_{μ}, φ) for some sufficiently large k. Let res^{U} be the meromorphic family of continuous maps res^{U} : UD_{{1},k}(1_{μ}, φ) :→ D_{U,k_{1}}(1_{μ}, φ) for suitable k_{1} ∈ ℕ (see Definition 3.46). By 3.9.36 for generic μ we can write f_{μ} = ext^{U}g_{μ} for some g_{μ} ∈ D_{U,k}(1_{μ}, φ). We conclude ext^{U} ∘ res^{U}(f_{μ}) = ext^{U} ∘ res^{U} ∘ ext^{U}(g_{μ}) = ext^{U}(g_{μ}) = f_{μ} by Lemma 3.47. Thus

ext^{U} ∘ res^{U}(f_{μ}) = f_{μ} for all μ .  (31)

We conclude that f ∈ Ext_{U}(1_{λ}, φ). 3.9.38 We now turn to the second assertion. The first assertion implies that the left hand side of (30) is trivial for all λ. Therefore the map Q_{U}(1_{λ}, φ)^{n} → Q̄_{∞_{P}} (1_{λ}, φ)^{n} is always injective. We now apply Lemma 3.40. □

3.9.39 The argument that proofs Equation (31) also shows the following.

Lemma 3.59 The composition ext^{U} ∘ res^{U} is the identity on Ext_{U}(1_{λ}, φ).

3.10 The scattering matrix

3.10.1 Recall that the family of intertwining operators (see 3.9.7 for the unnormalized version and [BO00] for normalizations) forms a meromorphic family of operators. It therefore maps (holomorphic) families of invariant sections to (meromorphic) families. It follows that J_{λ} maps E_{U}(1_{λ}, φ) to E_{U}(1_{−λ}, φ) if λ ∈ 𝔞_{ℂ}^{∗} is such that J_{λ} is regular (e.g. λ ∉ I_{𝔞}, see 1.5.16). By Proposition 3.58 we get a mapping

J_{λ} : Ext_{U}(1_{λ}, φ) → Ext_{U}(1_{−λ}, φ) .  (32)

Moreover, for given k ∈ ℕ_{0} we have

J_{λ} : Ext_{U,k}(1_{λ}, φ) → Ext_{U,k_{1}}(1_{−λ}, φ)

if k_{1} ∈ ℕ_{0} is sufficiently large. 3.10.2 We can now define the scattering matrix

S_{λ}^{U} : D_{U}(1_{λ}, φ) → D_{U}(1_{−λ}, φ) .

Fix k ∈ ℕ_{0} and a compact subset W ⊂ 𝔞^{∗}. Then we choose k_{0} > k and k_{1}, k_{2} such that

J_{λ} : Ext_{U,k_{0}}(1_{λ}, φ) → Ext_{U,k_{1}}(1_{−λ}, φ)

for generic λ (e.g. non-integral) with Re(λ) ∈ W and

res^{U} : D_{{1},k_{1}}(1_{λ}, φ) → D_{U,k_{2}}(1_{λ}, φ)

is defined (Definition 3.46) as a meromorphic family on W. Then we consider the composition

S_{λ}^{U} := res^{U} ∘ J_{λ} ∘ ext^{U} : D_{U,k}(1_{λ}, φ) → D_{U,k_{2}}(1_{−λ}, φ) .

By definition S_{λ}^{U} is a meromorphic family of continuous maps. Since

J_{λ} ∘ ext^{U} : D_{U}(1_{λ}, φ) → Ext_{U}(1_{−λ}, φ)

Lemma 3.47 now implies that S_{λ}^{U} is well-defined independently of the choices made for res^{U}. 3.10.3 Letting k tend to infinity and W run over a sequence of compact subsets exhausting 𝔞^{∗} we are arrive at the following definition.

Definition 3.60 We define the scattering matrix as the meromorphic family of continuous maps

S_{λ}^{U} : D_{U}(1_{λ}, φ) → D_{U}(1_{−λ}, φ) ,

which is given by the composition

S_{λ}^{U}(f) := res^{U} ∘ J_{λ} ∘ ext^{U}(f)

whenever the constituents are regular.

3.10.4 If −λ is admissible in the sense of 3.9.21 (with φ replaced by φ̃), then the push-down

π_{∗}^{U} : B_{{1}}(1_{λ}, φ̃) → B_{U}(1_{λ}, φ̃)

is regular and admits a right-inverse. Hence it induces an isomorphism

B_{U}(1_{λ}, φ̃) ≅ B_{{1}}(1_{λ}, φ̃)/ ker π_{∗}^{U} .

If λ ∉ I_{𝔞}, then J_{λ} is regular. We claim that it maps ker π_{∗}^{U} to the kernel of π_{U}^{∗} at −λ. Let f ∈ ker π_{∗}^{U} and ϕ ∈ D_{U}(1_{λ}, φ). By (32) there is an ψ ∈ D_{U}(1_{−λ}, φ) such that J_{λ} ∘ ext^{U}(ϕ) = ext^{U}(ψ). We find

⟨π_{∗}^{U} ∘ J_{λ}(f), ϕ⟩ = ⟨f, J_{λ} ∘ ext^{U}(ϕ) = ⟨f, ext^{U}(ψ)⟩ = ⟨π_{∗}^{U}(f), ψ⟩ = 0 .

The claim follows. Therefore, if ±λ is admissible and non-integral, then the operator J_{λ} descends to a map

S̃_{λ}^{U} : B_{U}(1_{λ}, φ̃) → B_{U}(1_{−λ}, φ̃) .  (33)

On the other hand, we have a meromorphic family of maps given by the adjoint of the scattering matrix

tS_{λ}^{U} : B_{U}(1_{λ}, φ̃) → B_{U}(1_{−λ}, φ̃) .

Lemma 3.61 If ±λ is non-integral and admissible, then we have S̃_{λ}^{U} = ^{t}S_{λ}^{U}.

Proof. Recall from Lemma 3.59 that ext^{U} ∘res^{U} is the identity on Ext_{U}(1_{−λ}, φ). For non-integral λ we have Ext_{U}(1_{−λ}, φ) = ^{U}C^{−∞}(∂X, V (1_{−λ}, φ)) by Proposition 3.58, 2.

The assumptions on λ imply that S̃_{λ}^{U} is defined, that π_{∗}^{U} : B_{{1}}(1_{λ}, φ̃) → B_{U}(1_{λ}, φ̃), ext^{U} : D_{U}(1_{λ}, φ) → D_{{1}}(1_{λ}, φ), and J_{λ} are regular. Let f ∈ B_{U}(1_{λ}, φ̃) be given by π_{∗}^{U}(F), F ∈ B_{{1}}(1_{λ}, φ). Furthermore let ϕ ∈ D_{U}(1_{λ}, φ). Then we compute

⟨S̃_{λ}^{U}(f), ϕ⟩ = ⟨π_{∗}^{U} ∘ J_{λ}(F), ϕ⟩
= ⟨F, J_{λ} ∘ ext^{U}(ϕ)⟩
= ⟨F, ext^{U} ∘ res^{U} ∘ J_{λ} ∘ ext^{U}(ϕ)⟩
= ⟨f, S_{λ}^{U}(ϕ)⟩
= ⟨^{t}S_{λ}^{U}(f), ϕ⟩ .
□

3.10.5 Note that the normalization of φ in the present subsection differs from that in 3.3.2, since we want φ̃ to be normalized as required there. As a consquence the space R_{U}(1_{λ}, φ) contains summands with negative index. If we put k_{φ} := l_{φ}/α, then we have

R_{U}(1_{λ}, φ) = ∏_{n=−k_{φ}}^{∞} R_{U}(1_{λ}, φ)^{n} , B_{U}(1_{λ}, φ) = ⋂_{k≥−k_{φ}} B_{U,k}(1_{λ}, φ) .

3.10.6

Lemma 3.62 Assume that U defines a cusp of smaller rank. If l_{φ} < 2ρ^{U}, then there is a natural non-degenerate pairing between B_{U,−k_{φ}}(1_{λ}, φ) and B_{U,0}(1_{−λ}, φ̃) given by integration over B_{U}. We obtain an inclusion

B_{U}(1_{λ}, φ) ⊂ B_{U,−k_{φ}}(1_{λ}, φ) ↪ D_{U}(1_{λ}, φ) .

Proof. 3.10.7 Let f ∈ B_{U,−k_{φ}}(1_{λ}, φ) and ϕ ∈ B_{U,0}(1_{−λ}, φ̃). Let W ⊂ N \ N_{V} be any compact subset. Then there exists a constant C ∈ ℝ such that for all ξ ∈ W and a ∈ A_{+} we have

|⟨f(ξ^{a}w), ϕ(ξ^{a}w)⟩| = |⟨φ(a)φ(a)^{−1}f(ξ^{a}w), ϕ(ξ^{a}w)⟩|
= |⟨φ(a)^{−1}f(ξ^{a}w), φ̃(a)^{−1}ϕ(ξ^{a}w)⟩|
≤ C∥f∥_{−k_{φ},0}∥ϕ∥_{0,0}a^{l_{φ}−4ρ^{U}}  (34)

(see 3.8 for the definition of the norms).

3.10.8 Let F ⊂ N_{V} be any compact fundamental domain for the lattice V ⊂ N_{V}. We can now write

∫_{U\N} |⟨f(xw), ϕ(xw)⟩|dx
= \frac{1}{[U : U^{0}]} ∫_{N_{V} \N} ∫_{F} |⟨f(yvw), ϕ(yvw)⟩|dvdy
= \frac{1}{[U : U^{0}]} ∫_{A} ∫_{S(N_{V} \N)} ∫_{F} |⟨f(ξ^{a}vw), ϕ(ξ^{a}vw)⟩|dvdξa^{2ρ^{U}} da
= \frac{1}{[U : U^{0}]} ∫_{A} ∫_{S(N_{V} \N)} ∫_{F} |⟨f((ξv^{a^{−1}})^{a}w), ϕ((ξv^{a^{−1}})^{a}w)⟩|dvdξa^{2ρ^{U}} da
= I_{+} + I_{−} ,

where I_{±} are the integrals over A_{±}, respectively. Inserting (34) we obtain

∫_{S(N_{V} \N)} ∫_{F} |⟨f((ξv^{a^{−1}})^{a}w), ϕ((ξv^{a^{−1}})^{a}w)⟩|dvdξ ≤ C_{1}∥f∥_{−k_{φ},0}∥ϕ∥_{0,0}a^{l_{φ}−4ρ^{U}}

and therefore

I_{+} ≤ C_{2}∥f∥_{−k_{φ},0}∥ϕ∥_{0,0} ∫_{A_{+}} a^{l_{φ}−2ρ^{U}} da .

The integral on the right-hand side converges since l_{φ} − 2ρ^{U} < 0 by assumption. Since I_{−} can clearly be estimated by C_{3}∥f∥_{−k_{φ},0}∥ϕ∥_{0,0}, we have shown the lemma. □

3.10.9 Using the inclusion (Lemma 3.62)

B_{U}(1_{λ}, φ) ↪ D_{U}(1_{λ}, φ)

we can consider the restriction of the scattering matrix

(S_{λ}^{U})_{|B_{U} (1_{λ},φ)} : B_{U}(1_{λ}, φ) → D_{U}(1_{−λ}, φ) .

If f ∈ B_{U}(1_{λ}, φ), then the restriction of the distribution ext^{U}(f) to Ω_{P} is smooth. Since J_{λ} is pseudo-local it it follows that J_{λ} ∘ ext(f) is smooth on Ω_{P}. Hence the restriction of the distribution S_{λ}^{U}(f) to the Schwartz space S_{U}(1_{λ}, φ̃) is given by integration against a smooth section. The following lemma asserts that S_{λ}^{U}(f) = ^{t}S_{λ}^{U}(f) ∈ B_{U}(1_{λ}, φ) considered as an element of D_{U}(1_{−λ}, φ) via Lemma 3.62. 3.10.10

Lemma 3.63 Assume that U defines a cusp of smaller rank and that l_{φ} < 2ρ^{U}. Then we have an equality of meromorphic families

(S_{λ}^{U})_{|B_{U} (1_{λ},φ)} = ^{t}S_{λ}^{U} .  (35)

Proof. By meromorphic continuation it suffices to show the equality (35) for non-integral λ in the open subset {|Re(λ)| < ρ^{U} − l_{φ}/2} ⊂ 𝔞_{ℂ}^{∗} such that −λ is admissible in the sense of 3.9.21. Then the push-down converges at ±λ. Let f ∈ B_{{1}}(1_{λ}, φ̃). We consider an element ϕ ∈ S_{{1}}(1_{λ}, φ).

We view S̃_{λ}^{U} ∘ π_{∗}^{U}(f) (see (33)) as an element of D_{U}(1_{−λ}, φ̃) and compute

⟨S̃_{λ}^{U} ∘ π_{∗}^{U}(f), π_{∗}^{U}(ϕ)⟩_{B_{U}} = ⟨π_{∗}^{U} ∘ J_{λ}(f), π_{∗}^{U}(ϕ)⟩_{B_{U}}
= ⟨π_{∗}^{U} ∘ J_{λ}(f), ϕ⟩_{∂X}
= ∑_{u∈U}⟨π^{1_{−λ},φ̃}(u)J_{λ}(f), ϕ⟩_{∂X}
= ∑_{u∈U}⟨f, π^{1_{−λ},φ}(u)J_{λ}(ϕ)⟩_{∂X}
= ⟨f, π_{∗}^{U} ∘ J_{λ}(ϕ)⟩_{∂X}
= ⟨π_{∗}^{U}(f), π_{∗}^{U} ∘ J_{λ}(ϕ)⟩_{B_{U}}
= ⟨ext^{U} ∘ π_{∗}^{U}(f), J_{λ}(ϕ)⟩_{∂X}
= ⟨J_{λ} ∘ ext^{U} ∘ π_{∗}^{U}(f), ϕ⟩_{∂X}
= ⟨ext^{U} ∘ res^{U} ∘ J_{λ} ∘ ext^{U} ∘ π_{∗}^{U}(f), ϕ⟩_{∂X}
= ⟨S_{λ}^{U} ∘ π_{∗}^{U}(f), π_{∗}^{U}(ϕ)⟩_{B_{U}} .

Since the π_{∗}^{U}(ϕ), ϕ ∈ S_{{1}}(1_{λ}, φ) (resp. π_{∗}^{U}(f) ∈ B_{U}(1_{λ}, φ̃), f ∈ B_{{1}}(1_{λ}, φ̃)), exhaust S_{U}(1_{λ}, φ) (resp. B_{U}(1_{λ}, φ̃)) we conclude that

S̃_{λ}^{U}(h) = S_{λ}^{U}(h)

as smooth sections for all h ∈ B_{U}(1_{λ}, φ̃) and hence

tS_{λ}^{U} = (S_{λ}^{U})_{|B_{U} (1_{λ},φ)}

by Lemma 3.61. □

3.10.11 Next we show that the scattering matrix is off-diagonally smoothing. Let χ, χ̃ be smooth cut-off functions on B_{U} with compact support such that χ̃χ = χ.

Lemma 3.64 Assume that U defines a cusp of smaller rank and that l_{φ} < 2ρ^{U}. Then we have a meromorphic family of continuous maps

(1 − χ̃) ∘ S_{λ}^{U} ∘ χ : D_{U}(1_{λ}, φ) → B_{U}(1_{−λ}, φ) .

Proof. We are going employ the fact that J_{λ} is off-diagonally smoothing. Let χ^{U} ∈ C^{∞}(Ω_{P}) be a cut-off function such that ∑_{u∈U} u^{∗}χ^{U} ≡ 1, and such that the restriction of the projection Ω_{P} → B_{U} to supp(χ^{U}) is proper. Then there exists a cut-off function χ̄ on ∂X such that χ̄χχ^{U} = 0 and χ̄(1 − χ̃) = 1 − χ̃. Finally we choose a compactly supported cut-off function χ̂ on B_{U} such that χ^{U} χ̄χ̂ = 0 and χ̂χ = χ.

Let ϕ ∈ S_{U}(1_{λ}, φ̃) be any test function and f ∈ D_{U}(1_{λ}, φ). We approximate χf by a sequence of smooth functions f_{α} with compact support. We can in addition assume that suppf_{α} ⊂ {χ̂ = 1} for all α. Then we compute

⟨(1 − χ̃)S_{λ}^{U}(χf), ϕ⟩_{B_{U}}  =  lim_{α}⟨(1 − χ̃)S_{λ}^{U}f_{α}, ϕ⟩_{B_{U}}
=^{Lemma 3.63} lim_{α}⟨(1 − χ̃)^{t}S_{λ}^{U}f_{α}, ϕ⟩_{B_{U}}
=  lim_{α}⟨f_{α}, S_{λ}^{U}((1 − χ̃)ϕ)⟩_{B_{U}}
=  lim_{α}⟨f_{α}, {res^{U}} ∘ J_{λ} ∘ ext^{U}((1 − χ̃)ϕ)⟩_{B_{U}}
=  lim_{α}⟨χ^{U}f_{α}, J_{λ} ∘ ext^{U}((1 − χ̃)ϕ)⟩_{∂X}
=  lim_{α}⟨χ^{U} χ̂f_{α}, J_{λ} ∘ χ̄ext^{U}((1 − χ̃)ϕ)⟩_{∂X}
=  lim_{α}⟨π_{∗}^{U}(χ̄J_{λ}(χ^{U} χ̂f_{α})), (1 − χ̃)ϕ⟩_{B_{U}}
=^{(∗)}  ⟨π_{∗}^{U} (χ̄J_{λ}(χ^{U}χf) − χ̃π_{∗}^{U}(χ̄J_{λ}(χχ^{U}f))) , ϕ⟩_{B_{U}} .

In order to see the equality marked by (∗) note that that χ̄J_{λ}χ̂χ^{U} is a continuous map from D_{U}(1_{λ}, φ) to B_{{1}}(1_{−λ}, φ) (since J_{λ} is off-diagonal smoothing) and that lim_{α}χ^{U} χ̂f_{α} = χ^{U}χf. The assertion of the Lemma follows from the identity proved above:

(1 − χ̃) ∘ S_{λ}^{U} ∘ χ(.) = π_{∗}^{U} (χ̄J_{λ}(χχ^{U}.) − χ̃π_{∗}^{U}(χ̄J_{λ}(χχ^{U}.))) .
□

4 The general case

4.1 The space B_{Γ}(σ_{λ}, φ)

4.1.1 Let Γ ⊂ G be a torsion-free geometrically finite discrete subgroup (see Definition 1.2) such that all its cusps are regular (Definition 1.3). Furthermore let (φ, V_{φ}) be an admissible twist (Definition 1.5). We are going to employ the notation introduced in Subsection 1.2. 4.1.2 On Ȳ we choose a partition of unity {χ_{p}}_{p∈P_{Γ}∪{0}} such that χ_{p} ∈ C^{∞}(Ȳ) and supp(χ_{p}) ⊂ Ȳ_{p} for all p. Restriction of these functions to B_{Γ} gives a partition of unity {χ_{p}}_{p∈P_{Γ}^{<}∪{0}} on B_{Γ} such that supp(χ_{p}) ⊂ B_{p}. Here we denote the restriction of χ_{p} to the boundary by the same symbol χ_{p}. In a similar manner for P ∈ p ∈ P_{Γ}^{<} we let e_{P} : B_{p} → B_{U_{P}} denote the map defined as restriction of the map e_{P} : Ȳ_{p} → Ȳ_{U_{P}}. Let χ_{P} ∈ C^{∞}(Ȳ_{U_{P}}) be the cut-off function which is supported on the range of e_{P} and satisfies χ_{p} = e_{P}^{∗} χ_{P}. 4.1.3 Using cut-off with χ_{p} and the map e_{P} : B_{p} → B_{U_{P}} for P ∈ p ∈ P_{Γ}^{<} we define maps

T_{P} : C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) → C^{∞}(B_{U_{P}} , V_{B_{U_{P}}} (σ_{λ}, φ))
T^{P} : C^{∞}(B_{U_{P}} , V_{B_{U_{P}}} (σ_{λ}, φ)) → C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) .

4.1.4 We define the Schwartz space for Γ as the space of smooth sections which belong to the Schwartz space 3.11 near all cusps.

Definition 4.1 We define S_{Γ,k}(σ_{λ}, φ) to be the subspace of all f ∈ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) such that T_{P} (f) ∈ S_{U_{P} ,k}(σ_{λ}, φ) for all P ∈ p ∈ P_{Γ}^{<}. The inclusion

S_{Γ,k}(σ_{λ}, φ) ↪ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

and the maps

T_{P} : S_{Γ,k}(σ_{λ}, φ) → S_{U_{P} ,k}(σ_{λ}, φ)

induce on S_{Γ,k}(σ_{λ}, φ) the structure of a Fréchet space. Furthermore we define the Fréchet and Montel space

S_{Γ}(σ_{λ}, φ) := ⋂_{k∈ℕ_{0}} S_{Γ,k}(σ_{λ}, φ) .

4.1.5

Lemma 4.2 The families {S_{Γ,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} and {S_{Γ}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} form locally trivial holomorphic bundles.

Proof. Let Φ_{P,λ_{0}} denote the trivialization constructed in Lemma 3.10 which is given by multiplication by s_{P}^{(λ_{0}−λ)/α} (here we add the index P to the notation for the section s constructed in Lemma 3.9 in order to indicate its dependence on the parabolic subgroup). Let s_{p} := T^{P} (s_{P}), where P ∈ P̃^{<} represents p. Let s_{0} ∈ C^{∞}(B_{Γ}, V_{B_{Γ}}(1_{ρ+α}, φ)) be any positive section (compare 3.2.8). Then we form s_{Γ} := ∑_{p∈P_{Γ}} s_{p} + χ_{0}s_{0}. Multiplication by s_{Γ}^{(λ_{0}−λ)/α} defines isomorphisms

Φ_{Γ,λ_{0}} : S_{Γ,k}(σ_{λ}, φ) → S_{Γ,k}(σ_{λ_{0}}, φ)

and

Φ_{Γ,λ_{0}} : S_{Γ}(σ_{λ}, φ) → S_{Γ}(σ_{λ_{0}}, φ) .

We employ the maps Φ_{Γ,λ_{0}} in order to obtain the required local trivializations and to implement the holomorphic structures. □

4.1.6 Similarly to the case of the Schwartz space we define the space B_{Γ}(σ_{λ}, φ) as the space of smooth sections with the same asymptotic expansions near the cusps as the spaces as the space defined in 3.21. Recall the notation P̃^{max} from 1.2.7.

Definition 4.3 We define

B_{Γ,k}(σ_{λ}, φ) := B_{Γ,k}(σ_{λ}, φ)_{1} ⊕ ⨁_{P ∈P̃^{max}} R_{U_{P}} (σ_{λ}, φ) ,

where

B_{Γ,k}(σ_{λ}, φ)_{1} ⊂ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

is the subspace of all f such that T_{P} (f) ∈ B_{U_{P} ,k}(σ_{λ}, φ) for all P ∈ p ∈ P_{Γ}^{<}. The map

B_{Γ,k}(σ_{λ}, φ) ↪ C^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) ,

the maps

T_{P} : B_{Γ,k}(σ_{λ}, φ) → B_{U_{P} ,k}(σ_{λ}, φ) ,

P ∈ P̃^{<}, and the natural projections

AS_{P} : B_{Γ,k}(σ_{λ}, φ) → R_{U_{P}} (σ_{λ}, φ) ,

P ∈ P̃^{max}, equip B_{Γ,k}(σ_{λ}, φ) with the structure of a Fréchet space. We further define the Fréchet and Montel space

B_{Γ}(σ_{λ}, φ) := ⋂_{k∈ℕ_{0}} B_{Γ,k}(σ_{λ}, φ) .

4.1.7 We define

R_{Γ,k}(σ_{λ}, φ) := ⨁_{P ∈P̃} R_{U_{P} ,k}(σ_{λ}, φ) , R_{Γ}(σ_{λ}, φ) := ⨁_{P ∈P̃} R_{U_{P}} (σ_{λ}, φ) .

We have asymptotic term maps

AS_{P} : B_{Γ,k}(σ_{λ}, φ) → R_{U_{P} ,k}(σ_{λ}, φ) ,

AS_{P} := AS ∘ T_{P}, where AS was defined in 3.4.11 for pure cusps. The asymptotic term maps admit right-inverses

L_{P} : R_{U_{P} ,k}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ)

given by the natural inclusion for P ∈ P̃^{max}, and by L_{P} := T^{P} ∘ L if P ∈ P̃^{<}, where L was defined in 3.4.10 for pure cusps.

Let

AS : B_{Γ,k}(σ_{λ}, φ) → R_{Γ,k}(σ_{λ}, φ)

be induced by the maps AS_{P} and

L : R_{Γ,k}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ)

be given by the sum of the maps L_{P} for the various P ∈ P̃. 4.1.8

Lemma 4.4 The family {B_{Γ,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} forms a trivial holomorphic bundle of Fréchet spaces. We have a split (by L) exact sequence

0 → S_{Γ,k}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ) →^{AS} R_{Γ,k}(σ_{λ}, φ) → 0 .  (36)

Furthermore, the spaces {B_{Γ}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} form a limit of locally trivial holomorpic bundles in the sense of Subsection 2.1 and fit into the exact sequence (which does not admit any continuous split)

0 → S_{Γ}(σ_{λ}, φ) → B_{Γ}(σ_{λ}, φ) → R_{Γ}(σ_{λ}, φ) → 0 .

Proof. The assertions follow from Lemma 4.2 and the fact that the spaces R_{Γ,k}(σ_{λ}, φ) form locally trivial holomorphic vector bundles. □

4.1.9 We now show that the spaces B_{Γ}(σ_{λ}, φ) are compatible with twisting. Let (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in 2.3.3.

Lemma 4.5 1. We have holomorphic families of continuous maps (see 2.3.6 for notation)

i_{σ,μ}^{Γ} : B_{Γ}(σ_{λ}, φ) → B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)

and

p_{σ,μ}^{Γ} : B_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → B_{Γ}(σ_{λ}, φ) .

2. B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) is a Z(𝔤)-module.

3. If λ ∉ I_{𝔞}, then i_{σ,μ}^{Γ} maps B_{Γ}(σ_{λ}, φ) isomorphically onto

ker_{Γ}(Z(λ)) := ker{Z(λ) : B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)}

(see 2.3.9 for notation), and the restriction of p_{σ,μ}^{Γ} to

ker{Z̃(λ) : B_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ) → B_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ)}

is an isomorphism onto B_{Γ}(σ_{λ}, φ) (here Z̃(λ) is the adjoint of π^{1_{−λ+μ}⊗π̃_{σ,μ},idφ̃}(Π(λ))).

4. The composition

j_{σ,μ}^{Γ} : B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) →^{1−Z(λ)} ker_{Γ}(Z(λ)) →^{(i_{σ,μ}^{Γ})^{−1}} B_{Γ}(σ_{λ}, φ)

which is initially defined for λ ∉ I_{𝔞} extends to a meromorphic family of continuous maps. Similarly, the composition

q_{σ,μ}^{Γ} : B_{Γ}(σ_{λ}, φ) →^{(p_{σ,μ}^{Γ})^{−1}} ker_{Γ}(Z̃(λ)) → B_{Γ}(1_{λ−μ}, π_{σ,μ} ⊗ φ)

extends to a tame meromorphic family of continuous maps.

Proof. 4.1.10 In order to see 1. we employ Lemma 3.22 2., and the identities

i_{σ,μ}^{U_{P}} ∘ T_{P} = T_{P} ∘ i_{σ,μ}^{Γ} , p_{σ,μ}^{U_{P}} ∘ T_{P} = T_{P} ∘ p_{σ,μ}^{Γ} .

4.1.11 In order to see 2. we use Lemma 3.22, 3., and the fact that [π^{1_{λ+μ}⊗π_{σ,μ},id_{φ}}(A), χ_{p}] is a differential operator with compactly supported coefficients on B_{Γ} for any A ∈ Z(𝔤). 4.1.12 We show the first assertion of 3. and leave the second to the reader, since the argument is similar. Since i_{σ,μ}^{Γ} is Z(𝔤)-equivariant it maps to ker_{Γ}(Z(λ)) by construction of Z(λ). Let now f ∈ ker_{Γ}(Z(λ)). We claim that for any h ∈ C^{∞}(B_{Γ}) we have Z(λ)(hf) = 0. Let k be the order of the differential operator Z(λ) which is a projection. Then we have

Z(λ)(hf) = Z(λ)^{k+1}(hf)
= Z(λ)^{k}[Z(λ), h]f
= Z(λ)^{k−1}[Z(λ), [Z(λ), h]]f
…
= \underbrace{[Z(λ), …, [Z(λ), h] …]}_{k+1} f
= 0 .

This shows that claim.

We conclude that T_{P} (f) ∈ ker_{U}(Z(λ)) for all P ∈ P̃. By Lemma 3.22 we find g_{P} ∈ B_{U}(σ_{λ}, φ), P ∈ P̃, such that T_{P} (f) = i_{σ,μ}^{U_{P}}(g_{P}). Let

f_{0} := f − ∑_{P ∈P̃} i_{σ,μ}^{Γ} ∘ T^{P} (g_{P}) ∈ C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)) .

As in the proof of Lemma 3.14,4. we can find g_{0} ∈ C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ)) such that f_{0} = i_{σ,μ}^{Γ}(g_{0}). Thus f = i_{σ,μ}^{Γ} (∑_{P ∈P̃∪{0}} g_{P} ).

4.1.13 We show the first assertion of 4. and leave the second to the reader. As in the proof of Lemma 3.14, 5. we construct a holomorphic family

J : C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)) → C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

as the composition J := j ∘ (1 − Z(λ)), where

j : C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ+μ}, π_{σ,μ} ⊗ φ)) → C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(σ_{λ}, φ))

comes from a bundle homomorphism. Then we define

j̃_{σ,μ}^{Γ} := Jχ_{0}(1 − Z(λ)) + ∑_{P ∈P̃} T̃^{P} ∘ j_{σ,μ}^{U_{P}} ∘ T_{P} ∘ (1 − Z(λ)) .

Here T̃^{P} is defined as T^{P} but using a cut-off function χ̃_{p} ∈ C^{∞}(B_{Γ}), P ∈ p, satisfying supp(χ̃_{p}) ∈ B_{p} and χ̃_{p}χ_{p} = χ_{p} if P̃^{<}. If P ∈ P̃^{max}, then we set T̃^{P} := T^{P}. One checks that j̃_{σ,μ}^{Γ} ∘ i_{σ,μ}^{Γ} = id. Since j̃_{σ,μ}^{Γ} vanishes on ker(1 − Z(λ)) we conclude that it coincides with j_{σ,μ}^{Γ} for λ ∉ I_{𝔞}. We thus have shown that j_{σ,μ}^{Γ} extends to a meromorphic family of continuous maps. □

4.2 Push-down

4.2.1 For the first part of the present subsection we choose an Iwasawa decomposition G = KAN and a parabolic subgroup P = MAN, M ⊂ K. Then we write X = G/K and ∂X = G/P = K/M. 4.2.2 For f ∈ C^{∞}(∂X, V (σ_{λ}, φ)) we consider the push-down

π_{∗}^{Γ}(f) ∈ C^{∞}(B_{Γ}, B_{B_{Γ}}(σ_{λ}, φ)) ⊕ ⨁_{P ∈P̃^{max}} R_{U_{P}} (σ_{λ}, φ) .

It is given by the sum

(π_{∗}^{Γ}(f))_{1} := ∑_{g∈Γ} π^{σ_{λ},φ}(g)(f_{|Ω_{Γ}}) ∈ ^{Γ}C^{∞}(Ω_{Γ}, V (σ_{λ}, φ))

in the first component. Its second component

(π_{∗}^{Γ}(f))_{2} ∈ ⨁_{P ∈P̃^{max}} R_{U_{P}} (σ_{λ}, φ)

will be constructed below. The goal of the present section is to show that the push-down converges for Re(λ) < −δ_{Γ} − δ_{φ} and defines a holomorphic family of maps

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

4.2.3 In the following lemma we consider points of ∂X as subsets of K using the identification ∂X ≅ K/M.

Lemma 4.6 If W ⊂ Ω_{Γ} is compact, then Γ ∩ W MA_{+}K is finite.

Proof. The set W MA_{+}K is a precompact subset of X ∪ Ω_{Γ}. Since Γ acts properly discontinuously on X ∪ Ω_{Γ} the intersection of the orbit ΓK with W MA_{+}K is finite. □

4.2.4

Lemma 4.7 For Re(λ) < −δ_{Γ} − δ_{φ} the sum

∑_{g∈Γ} π^{σ_{λ},φ}(g)f_{|Ω_{Γ}}

converges in C^{∞}(Ω_{Γ}, V (σ_{λ}, φ)) and defines a holomorphic family of continuous maps

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

Proof. Using Lemma 4.6 we employ the same argument as in the proof of [BO00], Lemma 4.2. □

4.2.5 Next we study the behaviour of the sum ∑_{g∈Γ} π^{σ_{λ},φ}(g)f_{|Ω_{Γ}} near the cusps. Let P ∈ P̃. We choose a system Γ^{P} of representatives of U_{P} \Γ by taking in each class [g] ∈ U_{P} \Γ an element h ∈ [g] which minimizes dist_{X}(O, hO).

Lemma 4.8 There is a neighbourhood W ⊂ ∂X of ∞_{P} such that W MA_{+}K ∩ Γ^{P} is finite.

Proof. Let D(O, U_{P}) := {x ∈ X|dist_{X}(x, O) ≤ dist_{X}(gx, O) ∀g ∈ U_{P} } be the Dirichlet domain of U_{P}. If h ∈ Γ^{P}, then hO ∈ D(O, U_{P}). Let π : D(O, U_{P}) → Y_{U_{P}} denote the projection and consider E := π^{−1}(e_{P} (Y_{p})). Since e_{P} : Y_{p} → Y_{U_{P}} is an isometry we have ♯(E ∩ Γ^{P} O) ≤ 1. Now the closure in X̄ of D(O, U_{P}) \ E does not contain ∞_{P} and is therefore disjoint from a neighbourhood W ⊂ ∂X of ∞_{P}. Since W MA_{+}K ∩ (D(O, U_{P}) \ E) has a compact closure inside X we conclude that W MA_{+}K ∩ (D(O, U_{P}) \ E)K ∩ Γ^{P} is finite. Since Γ^{P} ⊂ D(O, U_{P})K we have shown the lemma. □

4.2.6 Let P ∈ P̃ and W_{P} be a neighbourhood of ∞_{P} as constructed in Lemma 4.8. Using Lemma 4.8 and the arguments of the proof of [BO00], Lemma 4.2, we show Lemma 4.9 For Re(λ) < −δ_{Γ} − δ_{φ} and f ∈ C^{∞}(∂X, V (σ_{λ}, φ)) the sum

π̄_{∗}^{P} (f) := ∑_{g∈Γ^{P}} (π^{σ_{λ},φ}(g)f)_{|W_{P} ∪Ω_{Γ}}

converges in C^{∞}(W_{P} ∪ Ω_{Γ}, V (σ_{λ}, φ)). For varying λ the maps π̄_{∗}^{P} form a holomorphic family of continuous maps.

□

4.2.7 We choose a cut-off function κ_{P} ∈ C_{c}^{∞}(W_{P}) such that κ_{P} is equal to one near ∞_{P}. Then κ_{P} π̄_{∗}^{P} (f)) ∈ C^{∞}(∂X, V (σ_{λ}, φ)), and we can write

(π_{∗}^{Γ}(f))_{1} = ((π_{∗}^{U_{P}} (κ_{P} π̄_{∗}^{P} (f)))_{1} + ∑_{u∈U_{P}} π^{σ_{λ},φ}(u)(1 − κ_{P})π̄_{∗}^{P} (f))_{|Ω_{Γ}} .

Here

(π_{∗}^{U_{P}} (κ_{P} π̄_{∗}^{P} (f)))_{1} = π_{∗}^{U_{P}} (κ_{P} π̄_{∗}^{P} (f))

if the cusp associated to U_{P} ⊂ P has smaller rank. The second sum is locally finite and defines a smooth section on Ω_{Γ}. Moreover we have (π̄_{∗}^{P} was defined in Lemma 4.9)

χ_{P} ∑_{u∈U_{P}} π^{σ_{λ},φ}(u)(1 − κ_{P})π̄_{∗}^{P} (f)_{|Ω_{Γ}} ∈ C_{c}^{∞}(B_{U_{P}} , V_{B_{U_{P}}} (σ_{λ}, φ)) .

4.2.8 We define

(π_{∗}^{Γ}(f))_{2} := ⨁_{P ∈P̃^{max}}[π_{∗}^{U_{P}}] ∘ AS(κ_{P} π̄_{∗}^{P} (f)) .

This definition is independent of the choice of κ_{P}. Finally we define π_{∗}^{Γ}(f) to be the sum (π_{∗}^{Γ}(f))_{1} ⊕ (π_{∗}^{Γ}(f))_{2}. 4.2.9 Lemma 4.10 For Re(λ) < −δ_{Γ} − δ_{φ} the push-down π_{∗}^{Γ} induces a holomorphic family of continuous maps

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

Proof. This follows from Lemmas 4.7, 4.9, the observation that

T_{P} (π_{∗}^{Γ}(f))_{1} ∈ χ_{P} π_{∗}^{U_{P}} (κ_{P} π̄_{∗}^{P} (f)) + C_{c}^{∞}(B_{U_{P}} , V_{B_{U_{P}}} (σ_{λ}, φ)) ⊂ B_{U_{P}} (σ_{λ}, φ)  (37)

for P ∈ P̃^{<}, and the definition of (π_{∗}^{Γ}(f))_{2}. □

4.2.10 Let χ^{Γ} ∈ C^{∞}(Ω_{Γ}) be a smooth cut-off function such that ∑_{g∈Γ} g^{∗}χ^{Γ} ≡ 1. We choose χ^{Γ} such that it coincides with χ^{U_{P}} in a neighbourhood of ∞_{P}, and such that the restriction of the projection Ω_{Γ} → B_{Γ} to supp(χ^{Γ}) is proper. Then multiplication by χ^{Γ} defines a holomorphic family of right-inverses of the push-down (see Lemma 3.25)

{Q} : S_{Γ}(σ_{λ}, φ) → C^{∞}(∂X, V (σ_{λ}, φ))

such that π_{∗}^{Γ} ∘ {Q} = id. 4.2.11 Occasionally we will need the partial push-down

π_{∗}^{Γ/U_{P}} : B_{U_{P} ,k}(σ_{λ}, φ) → B_{Γ,k}(σ_{λ}, φ)

which is defined as the average over Γ/U_{P} if the cusp associated to U_{P} ⊂ P has smaller rank. If the cusp associated to U_{P} ⊂ P has full rank, then we define

π_{∗}^{Γ/U_{P}} (f ⊕ v) := π_{∗}^{Γ}({Q}(f − (π_{∗}^{U_{P}} ∘ L ∘ [Q](v))_{1}) + L ∘ [Q](v)) ,

where

f ⊕ v ∈ C^{∞}(B_{U_{P}} , V_{B_{U_{P}}} (σ_{λ}, φ)) ⊕ R_{U_{P}} (σ_{λ}, φ) = B_{U_{P}} (σ_{λ}, φ) .

If P ∈ P̃^{<}, then we have

π_{∗}^{Γ/U_{P}} (f) := π_{∗}^{Γ}({Q}(f − π_{∗}^{U_{P}} ∘ L ∘ [Q] ∘ AS(f)) + L ∘ [Q] ∘ AS(f)) .

This formula together with (37) can be used to verify that π_{∗}^{Γ/U_{P}} has the required mapping properties. In particular we conclude that if P ∈ P̃^{<}, then the partial push-down converges for Re(λ) < −δ_{Γ} − δ_{φ} and depends holomorphically on λ. If P ∈ P̃^{max}, then the partial push-down is defined as a meromorphic family of continuous maps for Re(λ) < −δ_{Γ} − δ_{φ}. 4.2.12 We collect the following useful identities

π_{∗}^{Γ/U_{P}} ∘ π_{∗}^{U_{P}} = π_{∗}^{Γ}, π_{∗}^{Γ/U_{P}} ∘ T_{P} = χ_{p}, π_{∗}^{Γ/U_{P}} χ_{P} = T^{P} ,  (38)

where P ∈ P̃^{<} for the last two equations and multiplication by χ_{p} implicitly involves a projection onto the first component. If P ∈ P̃^{max}, then we have

π_{∗}^{Γ/U_{P}} ∘ AS_{P} = AS_{P} .  (39)

4.2.13 We define q_{σ̃,μ} := q_{σ̃,μ}^{{1}} as in Lemma 4.5, 4. (applied to the trivial group).

Lemma 4.11 We have the following commutative diagrams of meromorphic families of maps:

1.

\begin{matrix}C^{∞}(∂X, V (σ_{λ}, φ)) & →^{i_{σ,μ}} & C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) & →^{Z(λ)} & C^{∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \\ ↓ π_{∗}^{Γ} & & ↓ π_{∗}^{Γ} & & ↓ π_{∗}^{Γ} \\ B_{Γ}(σ_{λ}, φ) & →^{i_{σ,μ}^{Γ}} & B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) & →^{Z(λ)} & B_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) \end{matrix} .

2.

\begin{matrix}C^{∞}(∂X, V (σ̃_{−λ}, φ̃)) & →^{q_{σ̃,μ}} & C^{∞}(∂X, V (1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)) \\ ↓ π_{∗}^{Γ} & & ↓ π_{∗}^{Γ} \\ B_{Γ}(σ̃_{−λ}, φ̃) & ←^{p_{σ̃,μ}^{Γ}} & B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃) \end{matrix} .

Proof. The push-down is Z(𝔤)-equivariant. This implies commutativity of the right square of the diagram 1. The commutativity of the left square is obvious for the part involving (π_{∗}^{Γ})_{1}, and for (π_{∗}^{Γ})_{2} we invoke Definition 3.34.

For the second diagram consider the enlarged diagram

\xymatrix{C^{∞}(∂X, V (1_{−λ+μ′}, π_{σ̃,μ′} ⊗ φ̃)) \ar[d]^{π_{∗}^{Γ}} & C^{∞}(∂X, V (σ̃_{−λ}, φ̃)) \ar[r]^{q_{σ̃,μ}} \ar[l]^{i_{σ̃,μ′}} \ar[d]^{π_{∗}^{Γ}} & C^{∞}(∂X, V (1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃)) \ar[d]^{π_{∗}^{Γ}} \ar[l]^{p_{σ̃,μ}} \\ B_{Γ}(1_{−λ+μ′}, π_{σ̃,μ′} ⊗ φ̃) & B_{Γ}(σ̃_{−λ}, φ̃) \ar[l]^{i_{σ̃,μ′}^{Γ}} & B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃) \ar[l]^{p_{σ̃,μ}^{Γ}}} .

The left square commutes by the Definition 3.34 of the push-down in the middle. The outer rectangle commutes by the naturality of the definition of the push-down. □

4.2.14 Let G^{n} be one of {Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)}. Recall the definition of i_{Γ}^{∗} given in Subsection 2.2.

Lemma 4.12 We have the following commutative diagram:

\begin{matrix}C^{∞}(∂X^{n+m}, V (1_{λ}^{n+1}, φ)) & →^{i^{∗}} & C^{∞}(∂X^{n}, V (1_{λ−ζ}^{n}, φ)) \\ ↓ π_{∗}^{Γ,n+1} & & ↓ π_{∗}^{Γ,n} \\ B_{Γ}(1_{λ}^{n+1}, φ) & →^{i_{Γ}^{∗}} & B_{Γ}(1_{λ−ζ}^{n}, φ) \end{matrix} .

Proof. We use Proposition 3.38. □

4.3 Extension, restriction, and the scattering matrix

4.3.1 We now define the distribution spaces D_{Γ}(σ_{λ}, φ) associated to Γ. These spaces are the domain of the extension ext^{Γ} and the Eisenstein series (see Definition 1.15).

Definition 4.13 We define D_{Γ,k}(σ_{λ}, φ) to be the dual space to B_{Γ,k}(σ̃_{−λ}, φ̃). Furthermore let D_{Γ}(σ_{λ}, φ) := ⋃_{k∈ℕ_{0}} D_{Γ,k}(σ_{λ}, φ).

Lemma 4.4 has the following consequence: Corollary 4.14 The spaces {D_{Γ,k}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}}, form a trivial holomorphic bundle of dual Fréchet spaces and fit into the split exact sequence

0 → R_{Γ,k}(σ̃_{−λ}, φ̃)^{∗} →^{AS^{∗}} D_{Γ,k}(σ_{λ}, φ) → S_{Γ,k}(σ̃_{−λ}, φ̃)^{∗} → 0 .

The spaces {D_{Γ}(σ_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}}, are dual Fréchet and Montel spaces, and they form a direct limit of trivial bundles. Furthermore we have the exact sequence

0 → R_{Γ}(σ̃_{−λ}, φ̃)^{∗} →^{AS^{∗}} D_{Γ}(σ_{λ}, φ) → S_{Γ}(σ̃_{−λ}, φ̃)^{∗} → 0 .

Let (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in Subsection 2.3. 4.3.2 The following Lemma states that the distribution spaces are compatible with twisting.

Lemma 4.15 1. D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) is a Z(𝔤)-module.

2. There is a natural inclusion i_{σ,μ}^{Γ} : D_{Γ}(σ_{λ}, φ) → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) which identifies D_{Γ}(σ_{λ}, φ) with

ker_{Γ}(Z(λ)) := ker(Z(λ) : D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) → D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ))

for λ ∉ I_{𝔞}.

3. Moreover, the map

j_{σ,μ}^{Γ} : D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) ⟶^{1−Z(λ)} ker_{Γ}(Z(λ)) ⟶^{(i_{σ,μ}^{Γ})^{−1}} D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ)

extends to a tame meromorphic family of continuous maps.

Proof. We employ Lemma 4.5. The map i_{σ,μ}^{Γ} is defined as the adjoint of

p_{σ̃,μ}^{Γ} : B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃) → B_{Γ}(σ̃_{−λ}, φ̃) .

j_{σ,μ}^{Γ} is just the adjoint of

q_{σ̃,μ}^{Γ} : B_{Γ}(σ̃_{−λ}, φ̃) → B_{Γ}(1_{−λ−μ}, π_{σ̃,μ} ⊗ φ̃) .
□

4.3.3

Definition 4.16 For Re(λ) > δ_{Γ} + δ_{φ} we define the extension map

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

to be the adjoint of the push-down

π_{∗}^{Γ} : C^{∞}(∂X, V (σ̃_{−λ}, φ̃)) → B_{Γ}(σ̃_{−λ}, φ̃) .

The extension ext^{Γ} is a holomorphic family of continuous maps and has values in Γ-invariant distributions. We will also need the partial extensions ext^{Γ/U_{P}} : D_{Γ,k}(σ_{λ}, φ) → D_{U_{P} ,k}(σ_{λ}, φ), which are defined as the adjoints of π_{∗}^{Γ/U_{P}}. Taking the adjoint of the relations (38) and (39) we obtain

ext^{U_{P}} ∘ ext^{Γ/U_{P}} = ext^{Γ}, T_{P}^{∗} ∘ ext^{Γ/U_{P}} = χ_{p}, χ_{P} ext^{Γ/U_{P}} = (T^{P})^{∗}

(where P ∈ P̃^{<} for the last two, and multiplication by χ_{p} implicitly involves projection onto the first component), and

AS_{P}^{∗} ∘ ext^{Γ/U_{P}} = AS_{P}^{∗} ,

if P ∈ P̃^{max}. 4.3.4 The space D_{{1},k}(σ_{λ}, φ) defined in Def. 3.41 depends on the choice of a parabolic subgroup P. In the present subsection we write D_{P,k}(σ_{λ}, φ) for that space and let

D_{{1},k}(σ_{λ}, φ) := ⋂_{p∈P_{Γ},P ∈p} D_{P,k}(σ_{λ}, φ) .

Using the relation ext^{U_{P}} ∘ ext^{Γ/U_{P}} = ext^{Γ} we see that

ext^{Γ} : D_{Γ,k}(σ_{λ}, φ) → D_{{1},k}(σ_{λ}, φ) ,

and that it is holomorphic as a map ext^{Γ} : D_{Γ,k}(σ_{λ}, φ) → D_{{1},k_{1}}(σ_{λ}, φ) for any k_{1} > k (and, of course, for Re(λ) > δ_{Γ} + δ_{φ}). 4.3.5 From now on we consider the spherical case σ = 1. We introduce the restriction maps

res^{Γ} : D_{{1},k}(1_{λ}, φ) → D_{Γ,k_{1}}(1_{λ}, φ)

defined for all k and suitable k_{1} ≥ k depending on k. The map res^{Γ} is a refinement of the naive restriction

{res^{Γ}} : C^{−∞}(∂X, V (1_{λ}, φ)) → S_{Γ}(1_{−λ}, φ̃)^{∗}

which is given as the adjoint of {Q} (see 4.2.10). We define the meromorphic family of maps res^{Γ} by

res^{Γ}(f) := π_{∗}^{Γ}(χ^{Γ}χ_{0}f) + ∑_{P ∈P̃} T_{P}^{∗} ∘ res^{U_{P}} (f) ,  (40)

where we interprete T_{P}^{∗} as AS_{P}^{∗}, if P ∈ P̃^{max}. Here π_{∗}^{Γ} simply stands for the average over Γ. It is well-defined and holomorphic for all λ ∈ 𝔞_{ℂ}^{∗} since χ^{Γ}χ_{0}f ∈ C_{c}^{−∞}(Ω_{Γ}, V (1_{λ}, φ)) and thus π_{∗}^{Γ}(χ^{Γ}χ_{0}f) ∈ C_{c}^{−∞}(B_{Γ}, V_{B_{Γ}}(1_{λ}, φ)). Alternatively, one could write χ_{0}{res^{Γ}}(f) for π_{∗}^{Γ}(χ^{Γ}χ_{0}f). Like the maps res^{U_{P}} (see Definition 3.46) the restriction map res^{Γ} depends on choices. 4.3.6 A priori the composition res^{Γ} ∘ ext^{Γ} is meromorphic and depends on many choices. The following lemma shows that the situation is much better. It generalizes 3.9.20.

Lemma 4.17 The composition res^{Γ} ∘ ext^{Γ} is regular and coincides with the inclusion of

D_{Γ,k}(1_{λ}, φ) → D_{Γ,k_{1}}(1_{λ}, φ) .

Proof. Assume that λ ∈ 𝔞_{ℂ}^{∗} is such that res^{Γ} is regular. Note that we still assume Re(λ) > δ_{Γ} + δ_{φ} in order ensure convergence of ext^{Γ}. Let f ∈ D_{Γ,k}(1_{λ}, φ). It is easy to see that π_{∗}^{Γ}(χ^{Γ}χ_{0}ext^{Γ}(f)) = χ_{0}f. Furthermore, if P ∈ P̃^{<}, then

T_{P}^{∗} ∘ res^{U_{P}} ∘ ext^{Γ}(f) = T_{P}^{∗} ∘ res^{U_{P}} ∘ ext^{U_{P}} ∘ ext^{Γ/U_{P}} (f)
= T_{P}^{∗} ∘ ext^{Γ/U_{P}} (f)
= χ_{p}f .

If P ∈ P̃^{max}, then we have

AS_{P}^{∗} ∘ res^{U_{P}} ∘ ext^{Γ}(f) = AS_{P}^{∗} ∘ res^{U_{P}} ∘ ext^{U_{P}} ∘ ext^{Γ/U_{P}} (f)
= AS_{P}^{∗} ∘ ext^{Γ/U_{P}} (f)
= AS_{P} (f) .

Summing these equations over P̃ ∪ {0}, then we obtain the desired identity

res^{Γ} ∘ ext^{Γ}(f) = f .

Since this equation holds true for generic λ, the assertion of the lemma follows. □

4.3.7 We also have the following useful identity

T_{P}^{∗} ∘ res^{U_{P}} = χ_{p}res^{Γ}

for all P ∈ p ∈ P_{Γ}^{<}.

4.3.8 Consider P ∈ P̃ and recall the Definition 3.43 of Ext_{U_{P}} (1_{λ}, φ). If λ is admissible (for P) in the sense of 3.9.21, then on this space the restriction res^{U_{P}} is well-defined independent of choices.

We now say that λ ∈ 𝔞_{ℂ}^{∗} is admissible, if it is admissible for all P ∈ P̃. Furthermore we define

ΓC^{−∞}(∂X, V (1_{λ}, φ))^{0} := ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ)) ∩ ⋂_{P ∈P̃} Ext_{U_{P}} (1_{λ}, φ) .

It immediately follows from (40) that res^{Γ} is well-defined on this space independent of choices. 4.3.9 If we assume that λ ∉ I_{𝔞} or λ > −ρ, then

ΓC^{−∞}(∂X, V (1_{λ}, φ))^{0} = ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ))

by Prop. 3.58, 2. Thus res^{Γ} is pointwise well-defined at admissible λ satisfying these additional conditions. 4.3.10

Definition 4.18 For Re(λ) > δ_{Γ} + δ_{φ} we define the scattering matrix

S_{λ}^{Γ} : D_{Γ}(1_{λ}, φ) → D_{Γ}(1_{−λ}, φ)

as a meromorphic family of operators, which is given by

S_{λ}^{Γ} := res^{Γ} ∘ J_{λ} ∘ ext^{Γ}

provided −λ is non-integral and admissible in the sense of 4.3.8.

Using the scattering matrices for the cusps (Definition 3.60) we can rewrite the formula for S_{λ}^{Γ} as follows:

S_{λ}^{Γ} = π_{∗}^{Γ} ∘ χ^{Γ}χ_{0}J_{λ} ∘ ext^{Γ} + ∑_{P ∈P̃} T_{P}^{∗} ∘ S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}} .  (41)

Using this formula and the results of 3.10.2 we see that S_{λ}^{Γ} is a well-defined meromorphic family of continuous maps. 4.3.11 As in the case of pure cusps (Lemma 3.62) we obtain a natural inclusion

B_{Γ}(1_{λ}, φ) ↪ D_{Γ}(1_{λ}, φ)

provided that 2ρ^{U_{P}} > l_{φ_{|U_{P}}} for all P ∈ P̃^{<}. Note that l_{φ_{|U_{P}}} = 2δ_{φ_{|U_{P}}}.

Lemma 4.19 We assume that all cusps of Γ have smaller rank. Assume that δ_{φ_{|U_{P}}} < ρ^{U_{P}} for all P ∈ P̃. Then the restriction of the scattering matrix to B_{Γ}(1_{λ}, φ) induces a meromorphic family of continuous maps

(S_{λ}^{Γ})_{|B_{Γ}(1_{λ},φ)} : B_{Γ}(1_{λ}, φ) → B_{Γ}(1_{−λ}, φ) .

Proof. First note that if f ∈ B_{Γ}(1_{λ}, φ) (considered as distribution), then ext^{Γ}(f) is smooth on Ω_{Γ}. It follows that J_{λ} ∘ ext^{Γ}(f) is smooth on Ω_{Γ}, and hence S_{λ}^{Γ}(f) = res^{Γ} ∘ J_{λ} ∘ ext^{Γ}(f) is represented by a smooth function. We choose a cut-off function χ̃_{P} on B_{U_{P}} which is supported in e_{P} (B_{p}) such that χ̃_{P} χ_{P} = χ_{P}. In order to see that S_{λ}^{Γ}(f) ∈ B_{Γ}(1_{−λ}, φ) we employ Equation (41). Let P ∈ P̃. Then we have

T_{P} ∘ S_{λ}^{Γ}(f) = T_{P} (χ^{Γ}χ_{0}J_{λ} ∘ ext^{Γ}(f))  (42)
+T_{P} ∘ T_{P}^{∗} ∘ S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}} (f)
= T_{P} (χ^{Γ}χ_{0}J_{λ} ∘ ext^{Γ}(f))  (43)
+χ_{P}^{2} S_{λ}^{U_{P}} ∘ χ̃_{P} ext^{Γ/U_{P}} (f)  (44)
+χ_{P}^{2} S_{λ}^{U_{P}} ∘ (1 − χ̃_{P})ext^{Γ/U_{P}} (f) .  (45)

The term (43) belongs to C_{c}^{∞}(B_{Γ}, V_{B_{Γ}}(1_{λ}, φ)). We have

χ̃_{P} ext^{Γ/U_{P}} (f) = (T̃^{P})^{∗}(f) = T̃_{P} (f) ∈ B_{U_{P}} (1_{−λ}, φ) ,

where T̃_{P} , T̃^{P} are defined as T_{P} , T^{P}, but using χ̃_{P} instead of χ_{P}. We can now apply Lemma 3.63 to (44) and Lemma 3.64 to (45) in order to see that these terms belong to B_{Γ}(1_{−λ}, φ). □

4.4 Vanishing results

4.4.1 If λ ∈ 𝔞_{ℂ}^{∗} is admissible in the sense of 4.3.8 and Re(λ) > −ρ, then res^{Γ} is well-defined on ΓC^{−∞}(∂X, V (1_{λ}, φ)) (see 4.3.9). 4.4.2 Let f ∈ ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ)). The condition {res^{Γ}}(f) = 0 (see 4.3.5) is equivalent to the condition that the support of f as a distribution is contained in the limit set Λ_{Γ}.

In the case that Γ is convex cocompact we know from [BO00], Thm. 4.7 that the space

{f ∈ ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ))|{res^{Γ}}(f) = 0}

is trivial if Re(λ) > δ_{Γ} + δ_{φ}. Already the example of pure cusps shows that this is not true in the presence of cusps.

Note that in the convex-cocompact case {res^{Γ}} = res^{Γ}. It is at the heart of the matter that for geometrically finite groups Γ and Re(λ) large the stronger condition res^{Γ}(f) = 0 implies f = 0. In [BO02] we introduced a related condition “f is strongly supported on the limit set” in order to show such vanishing results.

4.4.3

Proposition 4.20 We assume that λ is admissible in the sense of 4.3.8 and satisfies

Re(λ) > max ({δ_{Γ}} ∪ {ρ_{U_{P}} | P ∈ P̃}) + δ_{φ} .  (46)

Let f ∈ ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ)). If res^{Γ}(f) = 0, then f = 0.

Proof. 4.4.4 Let λ be admissible and f ∈ ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ)). From the defining formula (40) for res^{Γ} we conclude that res^{Γ}(f) = 0 if and only if f is supported on Λ_{Γ} and res^{U_{p}}(f) ∈ C_{c}^{−∞}(B_{U_{P}} , V_{B_{U_{P}}} (1_{λ}, φ)) ⊂ D_{U_{P}} (1_{λ}, φ) for all P ∈ P̃. The latter condition implies that

res^{U_{P}} (f) = {res^{U_{P}} }(f)  (47)

for all P ∈ P̃. 4.4.5 If Re(λ) > −ρ^{U_{P}} and h ∈ C^{∞}(∂X, V (1_{−λ}, φ̃)), then Proposition 3.58, 2, Lemma 3.59, and Equation (47) imply

⟨f, h⟩ = ⟨ext^{U_{P}} ∘ res^{U_{P}} (f), h⟩ = ⟨{res}^{U_{P}} (f), π_{∗}^{U_{P}} (h)⟩

and therefore

⟨f, h⟩ = ∑_{u∈U_{P}} ⟨χ^{U_{P}} f, φ̃(u)h(u^{−1}.)⟩ .  (48)

Thus, if res^{Γ}(f) = 0, then f is supported on the limit set and satisfies (48) for all P ∈ P̃, h ∈ C^{∞}(∂X, V (1_{−λ}, φ̃)). These are precisely the defining conditions for being “strongly supported on the limit set” in the sense of [BO02]. The main result of [BO02] now states that an element f ∈ ^{Γ}C^{−∞}(∂X, V (1_{λ}, φ)) that is “strongly supported on the limit set” vanishes provided that (46) holds. The proposition follows. □

4.4.6

Corollary 4.21 Assume λ ∈ 𝔞_{ℂ}^{∗} satisfies (46). If f_{μ} ∈ ^{Γ}C^{−∞}(∂X, V (1_{μ}, φ)) is a germ of a meromorphic family at λ, then ext^{Γ} ∘ res^{Γ}(f_{μ}) = f_{μ}.

Proof. We apply res^{Γ}, and we use res^{Γ} ∘ ext^{Γ} = id and the injectivity of res^{Γ} for generic μ near λ proved in Proposition 4.20. □

.

4.5 Meromorphic continuations and general σ

4.5.1 Recall that we have defined ext^{Γ} for Re(λ) > δ_{Γ} + δ_{φ}. The scattering matrix S_{λ}^{Γ} was defined on the same range of λ for the trivial M-type 1.

In the present subsection we extend the definition of the scattering matrix to general Mtypes. Further we show that the extension and the scattering matrix have meromorphic continuations to all of 𝔞_{ℂ}^{∗}. This finishes the proof of theorems 1.9, 1.11, and 1.14. 4.5.2 We start with the meromorphic continuation of S_{λ}^{Γ} to a certain half-plane W ⊂ 𝔞_{ℂ}^{∗} in the spherical case. Under these conditions we can construct a meromorphic family of parametrices for S_{λ}^{Γ} with finite-dimensional singularities. In a second step we employ twisting in order to show meromorphy of the extension and the scattering matrix on all of 𝔞_{ℂ}^{∗} and for general σ. 4.5.3

Proposition 4.22 The extension

ext^{Γ} : D_{Γ}(1_{λ}, φ) → C^{−∞}(∂X, V (1_{λ}, φ))

and the scattering matrix

S_{λ}^{Γ} : D_{Γ}(1_{λ}, φ) → D_{Γ}(1_{−λ}, φ)

have meromorphic continuations to

W := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > −ρ + β} ,

where β = 0 for X = Hℝ^{n}, Hℂ^{n}, and β = 2α for X = Hℍ^{n}. The family ext^{Γ} has finitedimensional singularities.

Proof. 4.5.4 We first show the proposition under an additional assumption on Γ and φ.

Lemma 4.23 We assume that

δ_{Γ} + δ_{φ} < − max ({0} ∪ {ρ_{U_{P}} + δ_{φ} | P ∈ P̃}) .  (49)

Then the extension ext^{Γ} and the scattering matrix S_{λ}^{Γ} have a meromorphic continuation to the half-plane W. The family ext^{Γ} has finite-dimensional singularities.

Proof.

4.5.5 The additional assumption in particular implies that ρ^{U_{P}} > δ_{φ} ≥ 0 for all P ∈ P̃. Therefore all cusps of Γ have smaller rank, and there is an embedding

B_{Γ}(1_{λ}, φ) ↪ D_{Γ}(1_{λ}, φ)

(see 4.3.11). We consider the non-empty open subset

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

Based on Lemma 4.19 we first consider the restriction of the scattering matrix to B_{Γ}(1_{λ}, φ). For λ ∈ U we construct a parametrix (see 4.1.3 for T_{P} and 4.1.13 for T̃^{P})

Q_{−λ} : B_{Γ}(1_{−λ}, φ) → B_{Γ}(1_{λ}, φ)

of S_{λ}^{Γ} by

Q_{−λ}(f) := π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}f)) + ∑_{P ∈P̃} T_{P}^{∗} ∘ S_{−λ}^{U_{P}} ∘ (T̃^{P})^{∗}(f) .

Here χ̃_{0} is some cut-off function on B_{Γ} of compact support such that χ_{0}χ̃_{0} = χ_{0}, and T̃^{P} is defined in the same way as T^{P} but using a cut-off function χ̃_{p} with support on B_{p} such that χ_{p}χ̃_{p} = χ_{p}.

Note that Q_{−λ} has a continuous extension to a map

Q_{−λ} : D_{Γ}(1_{−λ}, φ) → D_{Γ}(1_{λ}, φ) .

This is clear for the second term ∑_{P ∈P̃} T_{P}^{∗} ∘S_{−λ}^{U_{P}} ∘(T̃^{P})^{∗} since the scattering matrices S_{−λ}^{U_{P}} extend to distributions. The map f ↦ χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}f) extends to a continuous map from D_{Γ}(1_{−λ}, φ) to C_{c}^{−∞}(Ω_{Γ}, V (1_{λ}, φ)). The push-down extends to a continuous map from C_{c}^{−∞}(Ω_{Γ}, V (1_{λ}, φ)) to C_{c}^{−∞}(B_{Γ}, V_{B_{Γ}}(1_{λ}, φ)). This implies that the first term π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}f)) extends to distributions, too. 4.5.6 We claim that for |Re(λ)| < ρ−β the scattering matrix S_{λ}^{U_{P}} has at most finite-dimensional singularities. To see the claim we write S_{λ}^{U_{P}} = res^{U_{P}} ∘J_{λ}∘ext^{U_{P}}. Since J_{λ} is regular and bijective for these λ, and ext^{U_{P}} has finite-dimensional singularities, the only term which may contribute infinite-dimensional singularities is res^{U_{P}}.

Since {res^{U_{P}} } is always regular the singular part of res^{U_{P}} has values in the space R_{U_{P}} (1_{λ}, φ̃)^{∗}. Now S_{λ}^{U_{P}} is the continuous extension of its restriction to the dense subspace B_{U_{P}} (1_{λ}, φ). Since S_{λ}^{U_{P}} maps B_{U_{P}} (1_{λ}, φ) to B_{U_{P}} (1_{−λ}, φ) the range of the singular part of S_{λ}^{U_{P}} does not contain non-trivial elements of R_{U_{P}} (1_{λ}, φ̃)^{∗}.

Let D_{U_{P}} (1_{λ}, φ)^{0} ⊂ D_{U_{P}} (1_{λ}, φ) be a closed subspace of finite codimension on which ext^{U_{P}} is regular and put B_{U_{P}} (1_{λ}, φ)^{0} := D_{U_{P}} (1_{λ}, φ)^{0} ∩ B_{U_{P}} (1_{λ}, φ). Then B_{U_{P}} (1_{λ}, φ)^{0} has finite codimension in B_{U_{P}} (1_{λ}, φ), and the closure of B_{U_{P}} (1_{λ}, φ)^{0} in D_{U_{P}} (1_{λ}, φ) is D_{U_{P}} (1_{λ}, φ)^{0}. Looking at Laurent expansions we conclude that the restriction of S_{λ}^{U_{P}} to B_{U_{P}} (1_{λ}, φ)^{0} and hence to D_{U_{P}} (1_{λ}, φ)^{0} is regular. This proves the claim.

We conclude that for ±λ ∈ U the family Q_{λ} has at most finite-dimensional singularities. 4.5.7 We define the remainder

R_{λ} := Q_{−λ} ∘ S_{λ}^{Γ} − id .

We are going to show that R_{λ} is a meromorphic family of smoothing operators with finitedimensional singularities. We start with

R_{λ}(f) = Q_{−λ} ∘ S_{λ}^{Γ}(f) − f
= π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}S_{λ}^{Γ}(f)))

+ ∑_{P ∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} (T̃^{P})^{∗}S_{λ}^{Γ}(f) − f

Next we insert the definition S_{λ}^{Γ} = res^{Γ} ∘ J_{λ} ∘ ext^{Γ} and the definition of res^{Γ} (40). We obtain

R_{λ}(f) = π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{λ} ∘ ext^{Γ}(f))))

+ ∑_{P ∈P̃} π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}T_{P}^{∗}res^{U_{P}} ∘ J_{λ} ∘ ext^{Γ}(f)))

+ ∑_{P ∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} (T̃^{P})^{∗}π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{λ} ∘ ext^{Γ}(f))

+ ∑_{P,Q∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} (T̃^{P})^{∗}T_{Q}^{∗}(res^{U_{Q}} ∘ J_{λ} ∘ ext^{Γ}(f)) − f .

Now we employ

χ^{Γ}χ̃_{0}π_{∗}^{Γ}χ^{Γ}χ_{0} = χ^{Γ}χ_{0}
χ^{Γ}χ̃_{0} ∑_{P ∈P̃} T_{P}^{∗} ∘ res^{U_{P}} = χ̃_{0}(1 − χ_{0})χ^{Γ}
∑_{Q∈P̃}(T̃^{P})^{∗}T_{Q}^{∗}res^{U_{Q}} + (T̃^{P})^{∗}π_{∗}^{Γ}χ^{Γ}χ_{0} = χ̃_{P} res^{U_{P}}
res^{U_{P}} ∘ J_{λ} ∘ ext^{Γ} = S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}}

in order to obtain

R_{λ}(f) = π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}(χ^{Γ}χ̃_{0}J_{λ} ∘ ext^{Γ}(f)))

+ ∑_{P ∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} (χ̃_{P} S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}} (f)) − f .

Using the functional equations of the intertwining operators and scattering matrices we can further write

R_{λ}(f) = π_{∗}^{Γ}(χ^{Γ}χ_{0}ext^{Γ}(f))

−π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}((1 − χ^{Γ}χ̃_{0})J_{λ} ∘ ext^{Γ}(f)))

+ ∑_{P ∈P̃} T_{P}^{∗}ext^{Γ/U_{P}} (f)

− ∑_{P ∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} ((1 − χ̃_{P})S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}} (f)) − f

= −π_{∗}^{Γ}(χ^{Γ}χ_{0}J_{−λ}((1 − χ^{Γ}χ̃_{0})J_{λ} ∘ ext^{Γ}(f)))

− ∑_{P ∈P̃} T_{P}^{∗}S_{−λ}^{U_{P}} ((1 − χ̃_{P})S_{λ}^{U_{P}} ∘ ext^{Γ/U_{P}} (f)) .

Note that χ^{Γ}χ_{0}(1 − χ^{Γ}χ̃_{0}) = 0, and that J_{−λ} is off-diagonal smoothing. Furthermore note that χ_{P} (1 − χ̃_{P}) = 0, and that S_{−λ}^{U_{P}} is off-diagonal smoothing by Lemma 3.64. We conclude that R_{λ} extends to a meromorphic family of continuous maps

R_{λ} : D_{Γ}(1_{λ}, φ) → B_{Γ}(1_{λ}, φ)

with finite-dimensional singularities. Since J_{0} = id and S_{0}^{U_{P}} = id we have R_{0} = 0. 4.5.8 We define the Banach space S_{Γ,0,0}(1_{λ}, φ) to be the subspace of all f ∈ C(B_{Γ}, V_{B_{Γ}}(1_{λ}, φ)) such that T_{P} (f) ∈ S_{U_{P} ,0,0}(1_{λ}, φ) for all P ∈ P̃ (see 3.2.5). Then we have compact inclusions

B_{Γ}(1_{λ}, φ) ↪ S_{Γ,0,0}(1_{λ}, φ) ↪ D_{Γ}(1_{λ}, φ)

(in order to see the second inclusion use Lemma 3.62). The family of Banach spaces {S_{Γ,0,0}(1_{λ}, φ)}_{λ∈𝔞_{ℂ}^{∗}} forms a trivial holomorphic bundle. 4.5.9 We now apply meromorphic Fredholm theory [RS78], Thm. VI.13, (or better its version for families of operators on Banach spaces) to the family

1 + R_{λ} : S_{Γ,0,0}(1_{λ}, φ) → S_{Γ,0,0}(1_{λ}, φ) .

As in [BO00], Lemma 3.6, we conclude that (1 + R_{λ})^{−1} exists as a meromorphic family of maps on S_{Γ,0,0}(1_{λ}, φ) of the form (1 + T_{λ}), where T_{λ} is a meromorphic family of continuous operators from D_{Γ}(1_{λ}, φ) to B_{Γ}(1_{λ}, φ) with finite-dimensional singularities. 4.5.10 We see that the inverse of the scattering matrix is given on U by

(S_{λ}^{Γ})^{−1} := (1 + R_{λ})^{−1}Q_{−λ} .

as a meromorphic family and with finite-dimensional singularities.

4.5.11 We have −U ∪ {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > δ_{Γ} + δ_{φ}} = W. By our assumption the set

W_{0} = {λ ∈ 𝔞_{ℂ}^{∗} | δ_{Γ} + δ_{φ} < Re(λ) < −δ_{φ} − max ({δ_{Γ}} ∪ {ρ_{U_{P}} | P ∈ P̃})}

is non-empty. By Corollary 4.21 we find for λ ∈ W_{0} the meromorphic identity

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

Thus defining S_{λ}^{Γ} := (S_{−λ}^{Γ})^{−1} for −λ ∈ U we obtain the continuation of S_{λ}^{Γ} to W. 4.5.12 We obtain the meromorphic continuation of ext^{Γ} to W with finite-dimensional singularities using the identity

ext^{Γ} = J_{−λ} ∘ ext^{Γ} ∘ S_{λ}^{Γ} .

See the proof of [BO00], Lemma 5.11, for the corresponding argument. □

4.5.13 The following lemma finishes the proof of Proposition 4.22.

Lemma 4.24 Lemma 4.23 holds true without the assumption (49).

Proof. We are going to employ the embedding trick. First we assume that G^{n} belongs to the list {Spin(1, n), SO(1, n)_{0}, SU(1, n), Sp(1, n)}. If we consider Γ ⊂ G^{n} as a subgroup of G^{n+m}, then for sufficiently large m the inequality (49) is satisfied. Indeed, all ingredients of (49) but δ_{Γ} are independent of m, while δ_{Γ}^{n+m} = δ_{Γ}^{n} − mζ, ζ = ρ^{n+1} − ρ^{n}.

It thus suffices to show that a meromorphic continuation of ext^{Γ,n+m} implies a meromorphic continuation of ext^{Γ,n}. Assume that we have a meromorphic continuation of ext^{Γ,n+m} to W^{n+m} := {λ ∈ 𝔞_{ℂ}^{∗} | Re(λ) > −ρ^{n+m} + β} with finite-dimensional singularities. Then we employ the diagram

C^{−∞}(∂X, V (1_{λ}^{n}, φ)) ←^{j^{∗}} C^{∞}(∂X^{n+m}, V (1_{λ−mζ}^{n+m}, φ))
↑ ext_{∗}^{Γ,n}  ↑ ext_{∗}^{Γ,n+m}
D_{Γ}(1_{λ}^{n}, φ) →^{i_{∗}^{Γ}} D_{Γ}(1_{λ−mζ}^{n+m}, φ)

which apriori holds in the domain of convergence Re(λ) > δ_{Γ}^{n} + δ_{φ}^{n} by Lemma 4.12. Note that the left-inverse j^{∗} of i^{∗} was constructed in Lemma 2.5 (or better its iterate for the m-fold embedding). We can define the meromorphic continuation to W^{n} = W^{n+m} + mζ of ext^{Γ,n} using the identity ext^{Γ,n} = j^{∗} ∘ ext^{Γ,n+m} ∘ i_{∗}^{Γ}.

Let now G be general. Then we can find a subgroup Γ^{0} ⊂ Γ of finite index to which we can apply the concept of embedding (see 2.2.10). In particular we obtain a meromorphic continuation of ext^{Γ^{0}} to W. If we identify D_{Γ}(1_{λ}, φ) with the subspace ^{Γ/Γ^{0}}D_{Γ^{0}}(1_{λ}, φ), then we obtain the meromorphic continuation of ext^{Γ} using the identity ext^{Γ} = ext_{|D_{Γ}(1_{λ},φ)}^{Γ^{0}}.

Using the meromorphic continuation of ext^{Γ,n} we then obtain the meromorphic continuation of the scattering matrix S_{λ}^{Γ} by S_{λ}^{Γ} = res^{Γ} ∘ J_{λ} ∘ ext^{Γ}. □

4.5.14 In Proposition 4.22 we have obtained (in the spherical case) a meromorphic extension of ext^{Γ} and the scattering matrix S_{λ}^{Γ} to a half space W ⊂ 𝔞_{ℂ}^{∗}. In the following argument, using twisting, we find the meromorphic continuation of these objects to all of 𝔞_{ℂ}^{∗}. 4.5.15 Let σ be a Weyl invariant representation of M, and (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in 2.3.3. The adjoint of the diagram 2. of Lemma 4.11 gives the following commutative diagram of meromorphic families of maps

\begin{matrix}D_{Γ}(σ_{λ}, φ) & →^{i_{σ,μ}^{Γ}} & D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) \\ ↓ ext^{Γ} & & ↓ ext^{Γ} \\ C^{−∞}(∂X, V (σ_{λ}, φ)) & ←^{j_{σ,μ}} & C^{−∞}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \end{matrix}  (50)

for Re(λ) > δ_{Γ} + δ_{φ}. Using Proposition 4.22 we conclude that

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

is meromorphic on the set W − μ. Since we can choose μ arbitrary large we obtain the meromorphic continutaion of ext^{Γ} to all of 𝔞_{ℂ}^{∗}.

From (50), injectivity of i_{σ,μ}, and Proposition 4.22 we conclude that ext^{Γ} has finite-dimensional singularities. This finishes the proof of Theorem 1.11. The adjoint of the extension is the pushdown. Therefore Theorem 1.9 follows from 1.11. 4.5.16 Let for a moment be σ = 1. Using the meromorphic continuation of the extension we obtain the meromorphic continuation of the scattering matrix to all of 𝔞_{ℂ}^{∗} by the formula S_{λ}^{Γ} = res^{Γ} ∘ J_{λ} ∘ ext^{Γ}. In order to proceed similarly for general σ we first have to define the restriction map in this situation.

Let now σ be a Weyl-invariant representation of M and (π_{σ,μ}, V_{π_{σ,μ}}) be a finite-dimensional representation of G as in 2.3.3. For k ∈ ℕ we define restriction maps

res^{Γ} : ^{Γ}C^{−k}(∂X, V (σ_{λ}, φ)) → D_{Γ}(σ_{λ}, φ)

using the diagrams

\begin{matrix}C^{−k}(∂X, V (σ_{λ}, φ)) & →^{i_{σ,μ}} & C^{−k}(∂X, V (1_{λ+μ}, π_{σ,μ} ⊗ φ)) \\ ↓ res^{Γ} & & ↓ res^{Γ} \\ D_{Γ}(σ_{λ}, φ) & ←^{j_{σ,μ}^{Γ}} & D_{Γ}(1_{λ+μ}, π_{σ,μ} ⊗ φ) \end{matrix} .

The right column depends on choices (including μ), and so does the left column. But it is easy to check that

res^{Γ} ∘ ext^{Γ} = id  (51)

4.5.17 Note that (50) implies the meromorphic identity ext^{Γ} ∘ j_{σ,μ}^{Γ} = j_{σ,μ} ∘ ext^{Γ}. Let f_{ν} ∈ ΓC^{−∞}(∂X, V (σ_{ν}, φ)) be a meromorphic family defined on all of 𝔞_{ℂ}^{∗}. We compute for ν ≫ 0 using Corollary 4.21 and any set of choices for res^{Γ}

ext^{Γ} ∘ res^{Γ}(f_{ν}) = ext^{Γ} ∘ j_{σ,μ}^{Γ} ∘ res^{Γ} ∘ i_{σ,μ}(f_{ν})
= j_{σ,μ} ∘ ext^{Γ} ∘ res^{Γ} ∘ i_{σ,μ}(f_{ν})
= j_{σ,μ} ∘ i_{σ,μ}(f_{ν})
= f_{ν} .  (52)

Formula (51) implies that ext^{Γ} is injective on families. Thus we can conclude from (52) that res^{Γ}(f_{ν}) is independent of choices. 4.5.18 Now we can define the scattering matrix for general σ by the same formula as in the spherical case

S_{λ}^{Γ} := res^{Γ} ∘ J_{λ} ∘ ext^{Γ} .

It is meromorphic on 𝔞_{ℂ}^{∗}.

In order to finish the proof of Theorem 1.14 it remains to show the functional equation. Using (52) we find

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

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