Colin Guillarmou

AND Sergiu Moroianu
email: moroianu@alum.mit.edu

Jinsung Park
email: jinsung@kias.re.kr

ETA INVARIANT AND SELBERG ZETA FUNCTION OF ODD TYPE OVER CONVEX CO-COMPACT HYPERBOLIC MANIFOLDS

Keywords: Mathematics Subject Classification. 58J52, 37C30, 11M36, 11F72

We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z o Γ,Σ (λ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ\H 2n+1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η(D) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ\H 2n+1 and that exp(πiη(D)) = Z o Γ,Σ (0), thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil-Petersson metric on Schottky space.

Introduction

The eta invariant is a measure of the asymmetry of the spectrum of self-adjoint elliptic operators which has been introduced by Atiyah, Patodi and Singer as the boundary term in the index formula for compact manifolds with boundary [START_REF] Atiyah | Spectral asymmetry and Riemannian geometry I[END_REF]. For an elliptic self-adjoint pseudodifferential operator D of positive order acting on a bundle over a closed manifold, it is defined as the value at s = 0 of the meromorphic function η(D, s) := Tr D(D 2 ) -s+1 2 = 1 Γ((s + 1)/2) ∞ 0 t s- 1 2 Tr De -tD 2 dt, which admits a meromorphic continuation from ℜ(s) ≫ 0 to s ∈ C and is regular at s = 0. In heuristic terms, η(D) := η(D, 0) computes the asymmetry Tr(D|D| -1 ). By applying Selberg's trace formula, Millson [START_REF] Millson | Closed geodesics and the η-invariant[END_REF] proved that for any (4m -1)-dimensional closed hyperbolic manifold X Γ := Γ\H 4m-1 , the eta invariant η(A) of the odd signature operator A on odd forms Λ odd = ⊕ 2m p=0 Λ 2p-1 can be expressed in terms of the geodesic flow on the unit sphere bundle of X Γ . Millson defined a Selberg zeta function of odd type by

(1.1) Z o Γ,Λ (λ) := exp   - γ∈P ∞ k=1 χ + (R(γ) k ) -χ -(R(γ) k ) |det(Id -P (γ) k )| 1 2
e -λkℓ(γ) k   where P denotes the set of primitive closed geodesics in X Γ , R(γ) ∈ SO(4m -2) is the holonomy along a geodesic γ, χ ± denotes the character associated to the two irreducible representations of SO(4m -2) corresponding to the ±i eigenspace of ⋆ acting on Λ 2m-1 , P (γ) is the linear Poincaré map along γ, and ℓ(γ) is the length of the closed geodesic γ. Then he showed that Z o Γ,Λ (λ) extends meromorphically to λ ∈ C, its only zeros and poles occur on the line ℜ(λ) = 0 with order given in terms of the multiplicity of the eigenvalues of A, and the following remarkable identity holds:

e πiη(A) = Z o Γ,Λ (0)
. The same result has been extended to compact locally symmetric manifolds of higher rank by Moscovici-Stanton [START_REF] Moscovici | Eta invariants of Dirac operators on locally symmetric manifolds[END_REF]. It is somehow believed that central values of Ruelle or Selberg type dynamical zeta functions have some kind of topological meaning and this identity, as well as Fried's identity [START_REF] Fried | Analytic torsion and closed geodesics on hyperbolic manifolds[END_REF], provide striking examples.

It is a natural question to try to extend this identity and to study the meromorphic extension and the zeros and poles of the zeta function Z o Γ,Λ (λ) on non-compact hyperbolic manifolds. The first step in this direction has been done by the third author in [START_REF] Park | Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps[END_REF] for cofinite hyperbolic quotients, where the functional equation satisfied by the Selberg zeta function of odd type holds with extra contributions from the cusps, in the guise of the determinant of the scattering matrix. In the present work, we carry out this program for convex co-compact manifolds, i.e., geometrically finite hyperbolic manifolds with infinite volume and no cusps. For particular 3-dimensional Schottky groups, our results have interesting connections to Teichmüller theory, as we explain below.

The proof of the meromorphic extension of any reasonable dynamical zeta function on cocompact hyperbolic manifolds is contained in the work of Fried [START_REF] Fried | The zeta functions of Ruelle and Selberg. I[END_REF] using transfer operator techniques in dynamics, but as explained by Patterson-Perry [START_REF] Patterson | The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein[END_REF], it extends to the convex co-compact setting in a natural way. However there is no general description of the zeros and poles, while we know in the co-compact and cofinite cases that these are related to spectral and topological data since the work of Selberg [START_REF] Selberg | Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series[END_REF]. There are now some rather recent works of Patterson-Perry [START_REF] Patterson | The divisor of Selberg's zeta function for Kleinian groups. Appendix A by Charles Epstein[END_REF] and Bunke-Olbrich [START_REF] Bunke | Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group[END_REF] which give a complete description of the zeros and poles of the original Selberg zeta function on convex co-compact hyperbolic (real) manifolds. The case of Selberg zeta functions attached to homogeneous vector bundles is not yet completely described.

In this paper we study mainly the Dirac operator acting on the spinor bundle Σ over a convex co-compact hyperbolic manifold X Γ := Γ\H 2n+1 . The basic quantity associated with Γ for this case is its exponent δ Γ defined to be the smallest number such that (1.2) γ∈Γ exp(-λr γ ) < ∞ for all λ > δ Γ . Here r γ denotes the hyperbolic distance d H 2n+1 (m, γm) for a fixed point m ∈ H d+1 . For λ > δ Γ -n, we define the Selberg zeta function of odd type Z o Γ,Σ (λ) associated to the spinor bundle Σ exactly like in (1.1) except that R(γ) denotes now the holonomy in the spinor bundle Σ along γ, and χ ± denotes the character of the two irreducible representations of Spin(2n) corresponding to the ±i eigenspaces of the Clifford multiplication cl(T γ ) with the tangent vector field T γ to γ. Like for the hyperbolic space H 2n+1 , the Dirac operator D acting on the spinor bundle Σ on a convex co-compact hyperbolic manifold X Γ has continuous spectrum the real line R, and one can define its resolvent for ℜ(λ) > 0 in two ways R + (λ) := (D + iλ) -1 , R -(λ) := (D -iλ) -1 as analytic families of bounded operators acting on L 2 (X Γ ; Σ). We then first show Theorem 1.1. The Selberg zeta function of odd type Z o Γ,Σ (λ) associated to the spinor bundle Σ on an odd dimensional spin convex co-compact hyperbolic manifold X Γ = Γ\H 2n+1 has a meromorphic extension to C and it is analytic in a neighborhood of the right half plane {ℜ(λ) ≥ 0}. The resolvents R ± (λ) of the Dirac operator have meromorphic continuation to λ ∈ C when considered as operators mapping C ∞ 0 (X Γ , Σ) to its dual C -∞ (X Γ , Σ * ), and the poles have finite rank polar part. A point λ 0 ∈ {ℜ(λ) < 0} is a zero or pole of Z o Γ,Σ (λ) if and only if the meromorphic extension of R + (λ) or of R -(λ) has a pole at λ 0 , in which case the order of λ 0 as a zero or pole of Z o Γ,Σ (λ) (with the positive sign convention for zeros) is given by rank Res λ0 R -(λ)rank Res λ0 R + (λ).

We stress that our approach is closer to that of Patterson-Perry than that of Bunke-Olbrich. In so far as analysis is concerned, we deal with a much more general geometric setting in arbitrary dimensions and we prove various results which were previously known for the Laplacian on functions. We consider asymptotically hyperbolic manifolds (AH in short). These are complete Riemannian manifolds (X, g) which compactify smoothly to compact manifolds with boundary X, whose metric near the boundary is of the form g = ḡ/x 2 where ḡ is a smooth metric on X and x is any boundary defining function of ∂ X in X, and finally such that |dx| ḡ = 1 at ∂ X, a condition which is equivalent to assuming that the curvature tends to -1 at the boundary. Convex co-compact hyperbolic manifolds are special cases of AH manifolds. Using the machinery of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], we show that the spectrum of D on AH manifolds is absolutely continuous and given by R, that the resolvents R ± (λ) defined above have meromorphic extensions to λ ∈ C, and we define the scattering operator S(λ) : C ∞ (∂ X, Σ) → C ∞ (∂ X, Σ) by considering asymptotic profiles of generalized eigenspinors on the continuous spectrum. The family S(λ) extends to a meromorphic family of elliptic pseudo-differential operators acting on the boundary with the same principal symbol as D h0 |D h0 | 2λ-1 (up to a multiplicative constant), where D h0 is the Dirac operator induced by the metric ḡ| T ∂ X . The scattering operator is a fundamental object in the analysis of Selberg zeta function for convex co-compact manifolds, and we study it thoroughly in this work. We also show in a follow-up note [START_REF] Guillarmou | Calderón projector and cobordism invariance for the Dirac operator[END_REF] that the construction and properties of the scattering operator have some nice applications, for instance the invertibility of S(λ) except at discrete λ's implies that the index of D + h0 vanishes (the so-called cobordism invariance of the index), and the operator 1 2 (Id -S(0)) is a complementary Calderón projector of the Dirac operator D corresponding to ḡ, providing a natural way of constructing the Calderón projector without extending D or doubling the manifold X.

For the second result, we prove that Millson's formula holds for the Dirac operator on odd dimensional spin convex co-compact hyperbolic manifolds, and also for the signature operator under some condition on δ Γ . Theorem 1.2. Let X Γ = Γ\H 2n+1 be an odd dimensional spin convex co-compact hyperbolic manifold. Then the function tr(De -tD 2 )(m) ∈ C ∞ (X Γ ) is in L 1 (X Γ ), where tr denotes the local trace on the spinor bundle. The eta invariant η(D) can be defined as a convergent integral by

(1.3) η(D) := 1 √ π ∞ 0 t -1 2 XΓ tr(De -tD 2 )(m) dv(m) dt,
and the following equality holds

(1.4) e πiη(D) = Z o Γ,Σ (0)
. If 2n + 1 = 4m -1 and the exponent of convergence of Poincaré series δ Γ is strictly less than n = 2m -1, then the eta invariant η(A) can also be defined replacing D by the odd signature operator A and Σ by the bundle of forms Λ odd = ⊕ 2m p=0 Λ 2p-1 in (1.3), moreover we also have e πiη(A) = Z o Γ,Λ (0).

The assumption about δ Γ for the equality e πiη(A) = Z o Γ,Λ (0) is rather a technical condition than a serious problem. Most of the analysis we do here for Dirac operator D goes through without significant difficulties to the signature operator A, but it appears to be slightly more involved essentially due to the fact that the continuous spectrum of A has two layers corresponding to closed and co-closed forms. The complete analysis for forms in all dimensions will be included elsewhere.

To conclude this introduction and to motivate the eta invariant η(A) of the odd signature operator A, we describe the particular case of Schottky 3-dimensional manifolds with δ Γ < 1, where the eta invariant η(A) can be considered as a function on the Schottky space S g . Here the Schottky space S g is the space of marked normalized Schottky groups with g generators. It is a complex manifold of dimension 3g -3, covering the Riemann moduli space M g and with universal cover the Teichmüller space T g . It describes the deformation space of the 3-dimensional hyperbolic Schottky manifolds X Γ = Γ\H 3 . Like T g , the Schottky space S g has a natural Kähler metric, the Weil-Petersson metric. In [START_REF] Takhtajan | On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces[END_REF][START_REF] Takhtajan | A local index theorem for families of ∂-operators on Riemann surfaces[END_REF], Takhtajan-Zograf constructed two Kähler potentials of the Weil-Petersson metric on S g , that is,

∂∂S = ∂∂ -12π log Det∆ det Im τ = 2i ω W P
where ∂ and ∂ are the (1, 0) and (0, 1) components of the de Rham differential d on S g respectively, and ω W P is the symplectic form of the Weil-Petersson metric; here S is the so-called classical Liouville action, Det∆ and τ denote the ζ-regularized determinant of the Laplacian ∆ of hyperbolic metric and the period matrix respectively over the Riemann surface corresponding to an inverse image in T g of a point in S g . Let us remark that Det∆ and det Im τ descend to well-defined functions on S g . We show that Theorem 1.3. The function F defined on S 0 g := {Γ ∈ S g ; δ Γ < 1} by

F := Det∆ det Im τ exp S 12π -iπη(A)
is holomorphic. In particular, the eta invariant η(A) is a pluriharmonic function on S 0 g . The condition δ Γ < 1 in Theorem 1.3 simplifies the proof at several stages. But, one can expect that a similar result still holds over the whole Schottky space S g . This extension problem will be discussed elsewhere.

2. The Dirac operator on real hyperbolic space 2.1. Dirac operators over hyperbolic spaces. The (d + 1)-dimensional real hyperbolic space is the manifold

H d+1 = x ∈ R d+2 | x 2 0 + x 2 1 + . . . + x 2 d -x 2 d+1 = -1,
x d+1 > 0 equipped with the metric of curvature -1. The orientation preserving isometries of H d+1 form the group SO 0 (d + 1, 1), which is the identity connected component of SO(d + 1, 1). The isotropy subgroup of the base point (0, . . . , 0, 1) is isomorphic to SO(d+ 1). Hence the real hyperbolic space H d+1 can be identified with the symmetric space SO 0 (d+1, 1)/SO(d+1). Since G = Spin(d+1, 1), K = Spin(d + 1) are double coverings of SO 0 (d + 1, 1), SO(d + 1) respectively, we see that SO 0 (d + 1, 1)/SO(d + 1) = G/K and we use the identification H d+1 ∼ = G/K for our purpose. We denote the Lie algebras of G, K by g = spin(d + 1, 1), k = spin(d + 1) respectively. The Cartan involution θ on g gives us the decomposition g = k ⊕ p where k, p are the 1, -1 eigenspaces of θ respectively. The subspace p can be identified with the tangent space T o (G/K) ∼ = g/k at o = eK ∈ G/K where e denotes the identity element in G. The invariant metric of curvature -1 over H d+1 is given by the normalized Cartan-Killing form (2.1)

X, Y := - 1 2d C(X, θY )
where the Killing form is defined by

C(X, Y ) = tr(ad X • ad Y ) for X, Y ∈ g.
Let a be a fixed maximal abelian subspace of p. Then the dimension of a is 1. Let M = Spin(d) be the centralizer of A = exp(a) in K with Lie algebra m. We put β to be the positive restricted root of (g, a). Let ρ denote the half sum of the positive roots of (g, a), that is, ρ = d 2 β. From now on, we use the identification

(2.2) a * C ∼ = C by λ β -→ λ.
Let n be the positive root space of β and N = exp(n) ⊂ G. The Iwasawa decomposition is given by G = KAN . Throughout this paper we use the following Haar measure on G, The spinor bundle Σ(H d+1 ) can be identified with the associated homogeneous vector bundle over H d+1 = G/K with the spin representation

τ d of K ∼ = Spin(d + 1) acting on V τ d = C 2 [d+1/2] , that is, (2.4) Σ(H d+1 ) = G × τ d V τ d -→ H d+1 = G/K.
Here points of

G × τ d V τ d are given by equivalence classes [g, v] of pairs (g, v) under (gk, v) ∼ (g, τ d (k)v). Hence the sections of G × τ d V τ d from G/K consist of functions f : G → V τ d with the K-equivariant condition, f (gk) = τ d (k) -1 f (g) for g ∈ G, k ∈ K. Recall that τ d is irreducible if d + 1 is odd, while it splits into 2 irreducible representations if d + 1 is even.
Let us denote by

∇ : C ∞ (H d+1 ; Σ(H d+1 )) -→ C ∞ (H d+1 ; T * (H d+1 ) ⊗ Σ(H d+1 ))
the covariant derivative induced by the lift of the Levi-Civita connection to the spinor bundle Σ(H d+1 ), and by cl :

T m (H d+1 ) → End Σ m (H d+1 ) the Clifford multiplication. Then the Dirac operator D H d+1 acting on C ∞ 0 (H d+1 ; Σ(H d+1 )) is defined by D H d+1 f (m) = d+1 j=1 cl(e j )∇ ej f (m) for f ∈ C ∞ 0 (H d+1 ; Σ(H d+1 ))
where (e j ) d+1 j=1 denotes an orthonormal frame of T m (H d+1 ). The Dirac operator D H d+1 is an essentially self-adjoint, elliptic and G-invariant differential operator of first order, and we use the same notation for its self adjoint extension to L 2 (H d+1 ; Σ(H d+1 )). It is well known that the spectrum of D H d+1 on L 2 (H d+1 ; Σ(H d+1 )) consists only of the absolutely continuous spectrum R (for instance, by Cor. 4.11 in [6]).

The resolvent on H

d+1 . Let us define the resolvent of D 2 in the half-plane {ℜ(λ) > 0} by R H d+1 (λ) := (D 2 H d+1 + λ 2 ) -1 which maps L 2 (H d+1 ; Σ(H d+1 )) to itself. Recall the hypergeometric function F (a, b, c, z) defined by F (a, b, c; z) = ∞ k=0 Γ(a + k)Γ(b + k)Γ(c) Γ(a)Γ(b)Γ(c + k) z k k! for |z| < 1.
Then we have from the work of Camporesi [6, Th. 6.2 and 6.3],

Proposition 2.1. [Camporesi] For ℜ(λ) > 0, the respective Schwartz kernels of R H d+1 (λ) and

D H d+1 R H d+1 (λ) are given by R H d+1 (λ; m, m ′ ) = 2 -(d+1) π -d+1 2 Γ( d+1 2 + λ)Γ(λ) Γ(2λ + 1) (cosh(r/2)) -d-2λ F d + 1 2 + λ, λ, 2λ + 1; cosh -2 (r/2) U (m, m ′ ), (2.5)
D H d+1 R H d+1 (λ; m, m ′ ) = -2 -(d+1) π -d+1 2 Γ( d+1 2 + λ)Γ(λ + 1) Γ(2λ + 1) (cosh(r/2)) -(d+1)-2λ sinh(r/2) F d + 1 2 + λ, λ + 1, 2λ + 1; cosh -2 (r/2) cl(v m,m ′ )U (m, m ′ ) (2.6) where r = d H d+1 (m, m ′ ) for m, m ′ ∈ H d+1 , v m,m
′ is the unit tangent vector at m to the geodesic from m ′ to m and U (m, m ′ ) is the parallel transport from m ′ to m along the geodesic between them. Moreover R H d+1 (λ) has an analytic continuation in C \ {0} with a simple pole at λ = 0 and D H d+1 R H d+1 (λ) admits an analytic continuation to λ ∈ C (thus with no pole), as distributions on

H d+1 × H d+1 . Remark. If one denotes R H d+1 (λ; m; m ′ ) = Q λ (r)U (m, m ′ ) where r = d H d+1 (m, m ′ ), the function Q λ (r) satisfies that Q λ (r) -Q -λ (r)
is smooth in r near r = 0. This can be checked using functional equations of hypergeometric functions but actually follows directly from elliptic regularity since Q λ (r) -Q -λ (r) (since the difference of resolvents too) solves an elliptic ODE. The kernel

DΠ(λ; m, m ′ ) of D(R H d+1 (λ) -R H d+1 (-λ)
) is then also smooth near the diagonal m = m ′ and following the proof of [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF]Th 6.3], we see that it can be written under the form

DΠ(λ; m, m ′ ) = - 1 2 sinh(r) ∂ cosh -2 (r/2) (cosh -2 ( r 2 )) -d 2 H λ (cosh -2 ( r 2 )) cosh( r 2 ) d+4 cl(v m,m ′ )U (m, m ′ )
where

H λ (cosh -2 (r/2)) := Q λ (r) -Q -λ (r) with H λ (u) smooth near u = 1.
Then we deduce that on the diagonal DΠ(λ; m, m) = 0.

2.3. Dirac operators over convex co-compact hyperbolic manifolds. Let Γ denote a convex co-compact torsion-free discrete subgroup of G = Spin(d + 1, 1) such that its co-volume Vol(Γ\G) = ∞. Hence X Γ := Γ\G/K is a (d + 1)-dimensional convex co-compact hyperbolic manifold of infinite volume, which is a spin manifold by construction. The boundary ∂H d+1 , which can be identified with K/M , admits a Γ-invariant decomposition into Ω(Γ) ∪ Λ(Γ) where Ω(Γ) = ∅ is open and Γ acts freely and cocompactly on H d+1 ∪ Ω(Γ). Hence X Γ can be compactified by adjoining the geodesic boundary Γ\Ω(Γ).

By the identification (2.4) of the spinor bundle Σ(H d+1 ) with the homogeneous vector bundle G × τ d V τ d , we can also identify the spinor bundle Σ(X Γ ) over X Γ with the locally homogeneous

vector bundle Γ\ G × τ d V τ d . Here Γ acts on G × τ d V τ d by γ[g, v] = [γg, v] for γ ∈ Γ.
We can also push down the Dirac operator D H d+1 to X Γ , which we denote by D. We also use the same notation for its unbounded self-adjoint extension in L 2 (X Γ ; Σ(X Γ )), that is,

D : L 2 (X Γ ; Σ(X Γ )) -→ L 2 (X Γ ; Σ(X Γ )).
By Corollary 3.4 below (cf. Cor. 7.9 and Th. 11.2 in [START_REF] Bunke | The spectrum of Kleinian manifolds[END_REF]), the Dirac operator D over L 2 (X Γ ; Σ(X Γ )) has no discrete spectrum and only absolutely continuous spectrum R.

Resolvent of Dirac operator on Asymptotically Hyperbolic manifolds

In this section, we analyze the resolvent R(λ) of D 2 on an asymptotically hyperbolic manifold (AH in short) of dimension (d + 1). An asymptotically hyperbolic manifold is a complete noncompact Riemannian manifold (X, g) which compactifies in a smooth manifold with boundary X and there is a diffeomorphism ψ (called product decomposition) from a collar neighbourhood [0, ǫ) x × ∂ X of the boundary to a neighbourhood of ∂ X in X so that

(3.1) ψ * g = dx 2 + h x
x 2 for some one-parameter family of metrics h x on the boundary ∂ X depending smoothly on x ∈ [0, ǫ). By abuse of notations, we will write x for ψ * x, and x is then a boundary defining function in X near ∂ X, satisfying |dx| x 2 g = 1. A boundary defining functions satisfying |dx| x 2 g = 1 near the boundary is called geodesic boundary defining function, and it yields a diffeomorphism ψ like in (3.1) by taking the flow of the gradient ∇ x 2 g x starting at the boundary. Following the terminology of [START_REF] Guillarmou | Meromorphic properties of the resolvent for asymptotically hyperbolic manifolds[END_REF], we shall say that (3.2) the metric is even if the Taylor expansion of h x at x = 0 contains only even powers of x.

This property does not depend on the choice of the diffeomorphism ψ but only on g, see [START_REF] Guillarmou | Meromorphic properties of the resolvent for asymptotically hyperbolic manifolds[END_REF]Lemma 2.1]. It is well known that convex co-compact quotients X Γ = Γ\H d+1 are even AH manifolds (see [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF]). Note that the metric h 0 is not canonical since it depends on the choice of ψ, but its conformal class [h 0 ] is canonical with respect to g.

3.1. 0-structures, spinor bundle and Dirac operator. Following the ideas of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] (and refer to this paper for more details), there is a natural structure associated to AH manifolds, this is encoded in the Lie algebra V 0 ( X) of smooth vector fields vanishing at the boundary, whose local basis over C ∞ ( X) is given near the boundary ∂ X by the vector fields (x∂ x , x∂ y1 , . . . , x∂ y d ) if (x, y 1 , . . . , y d ) is a local chart near a point p ∈ ∂ X and x is a smooth boundary defining function in X. The algebra is also the space of smooth section of a bundle 0 T X with local basis near p given by (x∂ x , x∂ y1 , . . . , x∂ y d ) and its dual space is denoted 0 T * X, with local basis (dx/x, dy 1 /x, . . . , dy d /x). The metric g is a smooth section of the bundle of positive definite symmetric form S 2 + ( 0 T * X) of 0 T * X.

Let us define ḡ := x 2 g where x is a boundary defining function appearing in (3.1). If ( X, ḡ) is orientable, there exists an SO(d + 1)-bundle o F ( X) → X over X, but also an SO(d + 1)-bundle 0 o F ( X) → X defined using the 0-tangent bundle 0 T X and the metric g smooth on it. If ( X, ḡ) admits a spin structure, then there exists a 0-spin structure on (X, g) in the sense that there is a Spin(d + 1)-bundle 0 s F ( X) → X which double covers 0 o F ( X) and is compatible with it in the usual sense. This corresponds to a rescaling of the spin structure related to ( X, ḡ). The 0-spinor bundle 0 Σ( X) can then be defined as a bundle associated to the Spin(d + 1) principal bundle 0 s F ( X), with fiber at p ∈ X 0 Σ p ( X) = 0 s F p × τ d V τ d . The vector field x∂ x := x∇ ḡ (x) in the collar neighbourhood is unit normal to all hypersurfaces {x = constant}. The 0-spinor bundle on X splits near the boundary under the form

0 Σ = 0 Σ + ⊕ 0 Σ -, where 0 Σ ± := ker(cl(x∂ x ) ∓ i),
note that this splitting is dependent of the choice of the geodesic boundary defining function x except at the boundary ∂ X where it yields an independent splitting of the spinor (since the one-jet of x∇ x 2 g x is independent of x at ∂ X). To avoid confusions later (and emphasize the fact that it is only depending on the conformal class (∂ X, [h 0 ])), we shall define cl(ν) the linear map on 0 Σ| ∂ X by cl(ν)ψ := cl(x∂ x )ψ. At the boundary, 0 Σ| ∂ X is diffeomorphic to the spinor bundle Σ(∂ X) on (∂ X, h 0 ), this is not canonical since it depends on h 0 and thus on the choice of x, however the splitting above is. Notice also that in even dimension d + 1 = 2m, the splitting 0 Σ + ⊕ 0 Σ -near the boundary is not the usual splitting of the spinor bundle into positive and negative spinors, i.e., into the ±1 eigenspaces of the involution ω := i m cl(e 1 ) . . . cl(e 2m ), where (e i ) i is any orthonormal oriented local basis of 0 T X. The Dirac operator near the boundary has the form

(3.3) D = x d 2 (cl(x∂ x )x∂ x + P ) x -d 2
, P ∈ Diff 1 0 ( X; 0 Σ) where P is a first order differential operator in tangential derivatives, which anticommutes with cl(x∂ x ) and such that P = xD h0 +O(x 2 ) where D h0 is the Dirac operator on ∂ X equipped with the metric h 0 . If the metric g is even, it is easy to see that locally near any point y ′ of the boundary, if (x∂ x , xY 1 , . . . , xY d ) is an orthonormal frame near y ′ and (x, y) are coordinates on [0, ǫ) × ∂ X there, then P is of the form

P = d i=1 P i (x 2 , y; ∇ ḡ xYi )
for some differential operators P i of order 1 and with smooth coefficients in (x 2 , y). This can be checked for instance by using the conformal change formula

D = x d 2 +1 Dx -d 2
where D is the Dirac operator for the metric ḡ = x 2 g which is smooth in the coordinates (x = x 2 , y) down to x = 0. From these properties, it is straightforward to check that if g is even, then for x geodesic boundary defining function fixed, D preserves the space A ± ⊂ C ∞ ( X; 0 Σ) of smooth spinors which have expansion at the boundary of the form

(3.4) σ ∼ x→0 ∞ j=0
x j ψ j , with ψ 2j ∈ Σ ± (∂ X) and ψ 2j+1 ∈ Σ ∓ (∂ X).

3.2. The stretched product. Following Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], we define the stretched product X × 0 X as the blow-up [ X × X, ∆ ∂ ] of X × X around the diagonal in the boundary ∆ ∂ := {(y, y) ∈ ∂ X × ∂ X}. The blow-up is a smooth manifold with codimension 2 corners, and 3 boundary hypersurfaces, the left boundary denoted lb, the right boundary denoted rb and the new face, called 'front face' and denoted ff, obtained from the blow-up. The blow-down map is denoted β : X × 0 X → X × X and maps int(lb) to ∂ X × X, int(rb) to X × ∂ X and ff to ∆ ∂ . The face ff is a bundle over ∆ ∂ ≃ ∂ X with fibers a quarter of d-dimensional sphere. Let us use the boundary defining function x in (3.1), which induces x := π * L x and x ′ := π * R x as boundary defining functions of X × X where π L , π R are the left and right projection X × X → X. The fibre ff p of the front face ff (with p = (y ′ , y ′ ) ∈ ∂ X × ∂ X) is, by definition of blow-up, given by the quotient

(3.5) ff p = N p (∆ ∂ , ∂ X × ∂ X) × (R + ∂ x ) × (R + ∂ x ′ ) \ {0} {(w, t, u) ∼ s(w ′ , t ′ , u ′ ), s > 0} where in general N (M, Y ) denotes the normal bundle of a submanifold M in a manifold Y . Since T y ′ ∂ X is canonically isomorphic to N p (∆ ∂ , ∂ X × ∂ X) by z ∈ T y ′ ∂ X → (z, -z) ∈ T p (∂ X × ∂ X), h 0 (y ′ ) induces a metric on N p (∆ ∂ , ∂ X × ∂ X).
Then ff p is clearly identified with the quarter of sphere

ff p ≃ {w + t∂ x + u∂ x ′ ∈ N p (∆ ∂ , ∂ X × ∂ X) × (R + ∂ x ) × (R + ∂ x ′ ), t 2 + u 2 + |w| 2 h0(y ′ ) = 1}.
In projective coordinates (s := t/u, z := w/u) ∈ (0, ∞) × R d , the interior of the front face fiber ff p is diffeomorphic to H d+1 . In the same way we define the blow-up

X × 0 ∂ X of X × ∂ X around ∆ ∂ and the blow-up ∂ X × 0 ∂ X of ∂ X × ∂ X around ∆ ∂ .
The first one is canonically diffeomorphic to the face rb of X × 0 X while the second one is canonically diffeomorphic to lb ∩ rb.

The manifold X × X carries the bundle

E = 0 Σ( X) ⊠ 0 Σ * ( X)
which on the diagonal is isomorphic to End( 0 Σ). This bundle lifts under β to a bundle over X × 0 X, still denoted by E, whose fiber at the front face ff is given by 0 Σ y ′ ( X) ⊠ 0 Σ * y ′ ( X) everywhere on the fiber ff p (here p = (y ′ , y ′ ) ∈ ∆ ∂ ) if 0 Σ y ′ ( X) is the fiber of 0 Σ( X) at the point y ′ ∈ ∂ X.

On a manifold with corners M with a smooth bundle E → M , let us denote by Ċ∞ (M ; E) the space of smooth section of E which vanish to all order at the (topological) boundary and let C -∞ (M ; E * ) be its dual, the elements of which are called extendible distributions. Then β * is an isomorphism between Ċ∞ ( X × X; E) and Ċ∞ ( X × 0 X; E) and also between their duals, meaning that distributions on X × X can be as well considered on the stretched product. In what follows, we consider the Schwartz kernel K A ∈ C -∞ ( X × X; E) of an operator A : Ċ∞ ( X; 0 Σ) → C -∞ ( X; 0 Σ) defined by Aψ, φ = K A , φ ⊠ ψ where ., . is the duality pairing using the volume density of the metric. By abuse of notations we will write A(m, m ′ ) for K A (m, m ′ ) and the bundle E at the diagonal will be identified to End( 0 Σ).

3.3.

Pseudo-differential operators. We define the space Ψ m,α,β 0 ( X; 0 Σ) for m ∈ R, α, β ∈ C as in [START_REF] Mazzeo | The Hodge cohomology on a conformally compact metric[END_REF][START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], and refer the reader to these references for more details. An operator A is in Ψ m,α,β 0 ( X; 0 Σ) if its Schwartz kernel K A lifts to X × 0 X to a distribution β * (K A ) which can be decomposed as a sum

K 1 A + K 2 A with K 1 A ∈ ρ α lb ρ β rb C ∞ ( X × 0 X; E) and K 2 A ∈ I m cl ( X × 0 X, ∆; E) where I m cl ( X × 0 X, ∆; E)
denotes the space of distribution on X × 0 X classically conormal to the lifted diagonal ∆ := β * ({(m, m) ∈ X × X}) of order m and vanishing to infinite order at the left and right boundaries lb ∪ rb.

3.4.

Microlocal structure of the resolvent on H d+1 . We want to describe the resolvent kernel as a conormal distribution on a compactification of H d+1 × H d+1 , in order to show later that a similar result holds for convex co-compact quotients and more generally asymptotically hyperbolic manifolds. Here we let H d+1 be the natural compactification of H d+1 , i.e., the unit ball in R d+1 .

D H d+1 on H d+1 is in the space Ψ -1,λ+ d 2 ,λ+ d 2 0 (H d+1 ; 0 Σ).
Proof. The proof is very similar to the case of the Laplacian on functions dealt with in [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], since we have cosh(r/2) -2 ∈ ρ lb ρ rb C ∞ (H d+1 × 0 H d+1 \ ∆) and is a smooth function of the distance, and since, by the remark after Lemma A.2, the lift of U (m, m) by β * is smooth on H d+1 × 0 H d+1 , combining the formulae (2.5), (2.6) proves our claim, except for the singularity at the diagonal ∆. The conormal diagonal singularity can be easily seen by applying the first step of the parametrix of Mazzeo-Melrose (we refer to the proof of Proposition 3.2 below) near the diagonal, indeed the construction shows that there exists

Q 0 ± (λ) ∈ Ψ -1,∞,∞ 0 (H d+1 ; 0 Σ) such that (D H d+1 ± iλ)(R H d+1 ± (λ) -Q 0 ± (λ)) and (R H d+1 ± (λ) -Q 0 ± (λ))(D H d+1 ± iλ)
have a smooth kernel in a neighbourhood of ∆ down to the front face ff, and so by 0-elliptic regularity

R H d+1 ± (λ) -Q 0 ± (λ) is smooth near ∆ down to ff.
Note that the resolvent can be also considered as a convolution kernel on H d+1 with a conormal singularity at the center 0 ∈ H d+1 .

3.5. The parametrix construction of Mazzeo-Melrose. We can construct the resolvent R ± (λ) := (D ± iλ) -1 through a pseudo-differential parametrix, following Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] or Mazzeo [START_REF] Mazzeo | The Hodge cohomology on a conformally compact metric[END_REF]. We will not give the full details since this is a straightforward application of the paper [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] and the analysis of the resolvent R H d+1 ± (λ) on the model space H d+1 . This will be done in 3 steps. If E, F are smooth bundles over X, we will say that a family of operator A(λ) : Ċ∞ ( X; E) → C -∞ ( X; F ) depending meromorphically on a parameter λ ∈ C is finite meromorphic if the polar part of A(λ) at any pole is a finite rank operator. Proposition 3.2. Let (X, g) be a spin asymptotically hyperbolic manifold and D be a Dirac operator over X.

Then the resolvent R ± (λ) = (D ± iλ) -1 extends from {ℜ(λ) > 0} to C \ (-N/2) as a finite meromorphic family of operators in Ψ -1,λ+ d 2 ,λ+ d 2 0 ( X; 0 Σ). Moreover R ± (λ) maps Ċ∞ ( X; 0 Σ) to x λ+ d 2 C ∞ ( X; 0 Σ) and for all σ ∈ Ċ∞ ( X; 0 Σ), we have [x -λ-d 2 R ± (λ)σ]| x=0 ∈ C ∞ (∂ X; 0 Σ ∓ ).
Proof. The proof goes along the lines of the construction of Mazzeo-Melrose [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF], but we also use arguments of Epstein-Melrose-Mendoza [START_REF] Epstein | emphResolvent of the Laplacian on strictly pseudoconvex domain[END_REF] which somehow simplify it. Since there is no real novelty, we do not give the full details but only the important steps and additional arguments to our case which are needed. First, we construct an operator

Q 0 ± (λ) ∈ Ψ -1,∞,∞ 0 ( X; 0 Σ) supported near the interior diagonal such that (D ± iλ)Q 0 ± (λ) = Id-K 0 ± (λ) with K 0 ± (λ) ∈ Ψ -∞,∞,∞ 0
( X; 0 Σ), thus a smooth kernel on X × 0 X and whose support actually does not intersect the right and left boundary. Note that this can be done thanks to the ellipticity of D and it can be chosen analytic in λ, moreover notice also that

Q 0 ± (λ)(D ± iλ) -Id ∈ Ψ -∞,∞,∞ 0
( X; 0 Σ) by standards arguments of pseudo-differential calculus. The error K 0 ± (λ) is a priori not compact on any weighted space x s L 2 (X; 0 Σ) so this parametrix is not sufficient for our purpose. To be compact on such a space, it would be enough to have vanishing of the error on the front face K 0 ± (λ)| ff = 0. Next we need to solve away the term at the front face ff, i.e., K 0 ± (λ)| ff . We can use the normal operator of D: the normal operator N y ′ (D) of D at y ′ ∈ ∂ X is an operator acting on the space

X y ′ := {z ∈ 0 T y ′ X, dx
x (z) > 0} obtained by freezing coefficients of D at y ′ ∈ ∂ X, when considered as polynomial in the 0-vector fields x∂ x , x∂ y . Here the spinor bundle over X y ′ is trivial, i.e., it is given by X y ′ × 0 Σ y ′ ( X) where 0 Σ y ′ ( X) is the fiber of 0 Σ at the boundary point y ′ ∈ ∂ X. The half space X y ′ equipped with the metric g frozen at y ′ (the metric here is considered as a symmetric tensor on the 0-cotangent space 0 T * X) is isometric to H d+1 and the operator N y ′ (D) corresponds to the Dirac operator on H d+1 using this isometry. Moreover this space is also canonically identified to the interior of the front face fibre ff p with basis point p = (y ′ , y ′ ) ∈ ∆ ∂ . One has from [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF] that the composition (D ±iλ)G ± for G ± ∈ Ψ -∞,α,β 0 ( X; 0 Σ) is in the calculus Ψ 0 ( X) and the restriction at the front face fiber ff p is given by

((D ± iλ)G ± )| ff p = N y ′ (D ± iλ).G ± | ff p
which is understood as the action of the differential operator N y ′ (D ± iλ) on the conormal distribution G ± | ff p on ff p ≃ X y ′ . Thus to solve away the error term at ff, it suffices to find an operator

Q 1 ± (λ) in the calculus such that N y ′ (D ± iλ).Q 1 ± (λ)| ff p = K 0 ± (λ)| ff p
for all y ′ ∈ ∂ X. This can be done smoothly in y ′ by taking

Q 1 ± (λ)| ff p := R X y ′ ± (λ)(K 0 ± (λ)| ff p ) where R X y ′ ± (λ) is the analytically extended resolvent of N y ′ (D ± iλ) ≃ (D H d+1 ± iλ) on X y ′ × 0 Σ y ′ ( X) ≃ H d+1 × 0 Σ(H d+1
), and then defining Q 1 ± (λ) to be a distribution on X × 0 X whose restriction to each fiber ff p is R

X y ′ ± (λ)(K 0 ± (λ)| ffp ). As we studied above, the resolvent R H d+1 ± (λ) is ana- lytic in λ and it maps Ċ∞ (H d+1 ; 0 Σ) to ρ λ+ d 2 C ∞ (H d+1 ; 0 Σ) if ρ is a boundary defining function of the compactification of H d+1 , moreover the leading asymptotic term is of the form ρ λ+ d 2 ψ ∓ for some ψ ∓ ∈ C ∞ (∂H d+1 ; 0 Σ ∓ ). Thus, the composition R X y ′ ± (λ)(K 0 ± (λ)| ffp ) is a conormal dis- tribution in the class (ρ lb ρ rb ) λ+ d 2 C ∞ (ff p ; End( 0 Σ y ′ )) and it is then possible to find Q 1 ± (λ) ∈ Ψ -∞,λ+ d 2 ,λ+ d 2 0
( X; 0 Σ) with the correct restriction at ff. Let P ± denote the canonical projection

P ± : 0 Σ(∂ X) → 0 Σ ± (∂ X).
The restriction of a conormal kernel in Ψ -∞,0,β 0 ( X; 0 Σ) at lb can be considered as a section C -∞ (∂ X × X; E) conormal to all boundary faces. From the mapping prop-

erty of R H d+1 ± (λ) just discussed, it is possible to choose Q 1 ± (λ) such that P ± [ρ -λ-d 2 lb Q 1 ± (λ)]| lb = 0, which will be important for the next step. Then we get (D ± iλ)(Q 0 ± (λ) + Q 1 ± (λ)) = Id -K 1 ± (λ) where K 1 ± (λ) ∈ ρ ff Ψ -∞,λ+ d 2 ,λ+ d 2 0
( X; 0 Σ). The final terms in the parametrix are those at the left boundary, solved away through the indicial equation for z ∈ C:

for all ψ ± ∈ C ∞ (∂ X; 0 Σ ± ) (D ± iλ)x d 2 +z (ψ + + ψ -) = i(z ± λ)x d 2 +z ψ + + i(-z ± λ)x d 2 +z ψ -+ O(x d 2 +z+1 ), (3.6)
which is an easy consequence of (3.3). Lifting D as acting on the left variable on the space X × 0 X, it satisfies the same type of indicial equation

: if G ∈ Ψ -∞,α+ d 2 ,β+ d 2 0 ( X; 0 Σ) for some α, β ∈ C, then (D ± iλ)G ∈ Ψ -∞,α+ d 2 ,β+ d 2 0
( X; 0 Σ) and the leading term at lb is

(3.7) [i(α ± λ)ρ α+ d 2 lb P + [(ρ -α-d 2 lb G)| lb ] + i(-α ± λ)ρ α+ d 2 lb P -[(ρ -α-d 2 lb G)| lb ]
where the restriction at lb is considered as a section C -∞ (∂ X × X; E) (conormal to all boundary faces). Then since for α = λ, the term (3.7) vanishes if P ± (ρ

-λ-d 2 lb G)| lb = 0, one clearly has K 1 ± (λ) ∈ ρ ff Ψ -∞,λ+ d 2 +1,λ+ d 2 0
( X; 0 Σ) thanks to the choice of Q 1 ± (λ) and now, since α = λ+j for j ∈ N is not solution of the indicial equation above when λ / ∈ -N/2, it is possible by induction and using Borel Lemma to construct a term

Q 2 ± (λ) ∈ ρ ff Ψ -∞,λ+ d 2 +1,λ+ d 2 0 ( X; 0 Σ), holomorphic in C \ (-N/2) such that (D ± iλ)Q 2 ± (λ) = K 1 ± (λ) -K 2 ± (λ) for some operator K 2 ± (λ) ∈ ρ ff Ψ -∞,∞,λ+ d 2 0
( X; 0 Σ). By [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]Prop. 3.29], the error term K 2 ± (λ) is now already compact on ρ z L 2 for all z ∈ [0, ∞) such that ℜ(λ) + z > 0. We can now improve the parametrix by using exactly the same arguments as in the proof of Theorem 14.5 of Epstein-Melrose-Mendoza [START_REF] Epstein | emphResolvent of the Laplacian on strictly pseudoconvex domain[END_REF] for complex asymptotically hyperbolic manifolds: take a kernel Q 3 ± (λ) which matches to infinite order on X × X with the formal Neumann series composition

(Q 0 ± (λ) + Q 1 ± (λ) + Q 2 ± (λ)) ∞ j=0 (K 2 ± (λ)) j ,
and the error term

K 3 ± (λ) will now be in ρ ∞ ff Ψ -∞,∞,λ+ d 2 0
( X; 0 Σ). The compositions above are still in the calculus by Mazzeo's composition theorem [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]Th. 3.15], but with a larger index set at the right boundary and front face than for 2 i=0 Q i ± (λ) (see the proof of Theorem 4.15 of [START_REF] Epstein | emphResolvent of the Laplacian on strictly pseudoconvex domain[END_REF] where this is explained in full details). Now fix λ 0 such that ℜ(λ 0 ) > 0 where R ± (λ 0 ) is bounded on L 2 (X). Then we use a standard argument, we can add a finite rank term [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]Th. 3.25], and so Fredholm theorem proves that R ±

Q 4 ± (λ) = Q 4 ± (λ 0 ) ∈ ρ ∞ ff Ψ -∞,∞,∞ 0 ( X; 0 Σ) to Q 3 ± (λ) in case (Id -K 3 ± (λ 0 )) has non empty null space in ρ z L 2 , so that Id -K 4 ± (λ 0 ) is invertible if K 4 ± (λ) := Id -(D ± iλ)(Q 3 ± (λ) + Q 4 ± (λ)). The operator Q ± (λ) := Q 3 ± (λ) + Q 4 ± (λ) is bounded from ρ z L 2 to ρ -z L 2 if ℜ(λ) + z > 0 by
(λ) = Q ± (λ)(Id -K 4 ± (λ)) -1 on the weighted space ρ z L 2 for z ∈ [0, ∞) such that ℜ(λ) + z > 0. Finally, writing (Id -K 4 ± (λ)) -1 = Id + T ± (λ), we see that T ± (λ) = K 4 ± (λ) + K 4 ± (λ)(Id -K 4 ± (λ)) -1 K 4 ± (λ). We claim that R ± (λ) ∈ Ψ -1,λ+ d 2 ,λ+ d 2 0
( X; 0 Σ): since there are no new arguments needed, we do not give details here and refer the reader to the proof of Theorem 4.15 in [START_REF] Epstein | emphResolvent of the Laplacian on strictly pseudoconvex domain[END_REF], we notice though that one of the points explained in [START_REF] Epstein | emphResolvent of the Laplacian on strictly pseudoconvex domain[END_REF] is to check that the additional exponents in the index sets obtained using the composition theorem of Mazzeo in the Neumann series of K 2 ± (λ) are actually absent in R ± (λ), this is based on the adjointness properties of the resolvent for what concerns the right boundary index set and on the properties of the normal operator for the front face index set (this however requires to add a last term to the parametrix). The mapping property of R ± (λ) acting on Ċ∞ ( X; 0 Σ) follows again from Mazzeo [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]Prop. 3.28], and the fact that for all σ ∈ Ċ∞ ( X; 0 Σ) we have

R ± (λ)σ = x λ+ d 2 ψ ∓ + O(x λ+ d 2 +1 ) for some ψ ∓ ∈ C ∞ (∂ X; 0 Σ ∓ ) is a straightforward consequence of the indicial equation (3.6).
Let us now discuss the nature of the spectrum of D. We start by an application of Green formula, usually called boundary pairing property (compare to [11,

Prop. 3.2]) Lemma 3.3. Let ℜ(λ) = 0 and σ i = x d 2 -λ σ - i + x d 2 +λ σ + i for i = 1, 2 and σ ± i ∈ C ∞ ( X; 0 Σ), then if (D 2 + λ 2 )σ i = r i ∈ Ċ∞ ( X; 0 Σ), one has X ( σ 1 , r 2 -r 1 , σ 2 )dv g = 2λ ∂ X σ - 1 | ∂ X , σ - 2 | ∂ X -σ + 1 | ∂ X , σ + 2 | ∂ X dv h0
where , denotes the scalar product with respect to g on X and to h 0 on ∂ X.

Proof. The proof of this Lemma is straightforward by using integration by parts in {x ≥ ǫ} and letting ǫ → 0.

As a corollary of the resolvent extension and this Lemma, we obtain the following Corollary 3.4. On a spin asymptotically hyperbolic manifold, the extended resolvent R ± (λ) = (D ± iλ) -1 is holomorphic on the imaginary line iR, consequently the spectrum of D is R and absolutely continuous.

Proof. In view of the meromorphy of R ± (λ), it clearly suffices from Stone's formula to prove that R ± (λ) has no pole on the imaginary line. Assume λ 0 is such a pole with order p, then the most singular coefficient of the Laurent expansion is a finite rank operator whose range is made of generalized eigenspinors σ solving (D ± iλ 0 )σ = 0 and σ ∈ x λ0+ d 2 C ∞ ( X; 0 Σ). In particular it satisfies (D 2 + λ 2 0 )σ = 0 and by applying Lemma 3.3 with

σ 1 = σ 2 = σ we see that (x -λ0-d 2 σ)| x=0 = 0 and so σ ∈ x λ0+ d 2 +1
C ∞ ( X; 0 Σ). Now from the indicial equation (3.6), this implies σ ∈ Ċ∞ ( X; 0 Σ) if λ 0 = 0. Then Mazzeo's unique continuation theorem [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds[END_REF] says that for a class of operators including D 2 , there is no eigenfunction vanishing to infinite order at the boundary except σ ≡ 0, we deduce that σ = 0 and thus by induction this shows that the polar part of Laurent expansion of R(λ) at λ 0 is 0. Now there remains the case λ 0 = 0. First from self-adjointness of D, we easily get

(3.8) λ||σ|| L 2 ≤ ||(D ± iλ)σ|| L 2
for all λ > 0 and σ in the L 2 -Sobolev space H 1 (X; 0 Σ) of order 1, and this implies that R ± (λ) has a pole of order at most 1 at λ = 0, i.e., one has R ± (λ) = A ± λ -1 + B ± (λ) for some B ± (λ) holomorphic (these can be considered as operators from Ċ∞ ( X; 0 Σ) to its dual). By (3.8), we also see by taking λ → 0 that ||A ± σ|| L 2 ≤ ||σ|| L 2 for all σ ∈ Ċ∞ ( X; 0 Σ) and so A ± is bounded on L 2 and also maps into ker(D) by (D ± iλ)R ± (λ) = Id. Now in view of the structure of the kernel of R ± (λ), it is not hard to check (e.g. see [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF]) that the elements in the range of A ± are harmonic spinors of the form

σ ∈ x d 2 C ∞ ( X; 0 Σ), which can only be L 2 if the leading asymptotic (x -d 2 σ)| x=0 = 0, i.e., if σ ∈ x d 2 +1 C ∞ ( X; 0 Σ).
Using again the indicial equation, (3.6), we deduce that σ ∈ Ċ∞ ( X; 0 Σ) and thus σ = 0 by Mazzeo's unique continuation theorem, so A ± = 0.

Another corollary of Proposition 3.2 is

Corollary 3.5. The resolvent R(λ) := (D 2 + λ 2 ) -1 extends meromorphically to λ ∈ C \ (-N/2) with poles of finite multiplicity, except at λ = 0 where it has a simple infinite rank pole with residue (2i

) -1 (R -(0) -R + (0)). Moreover R(λ) is an operator in Ψ -2,λ+ d 2 ,λ+ d 2 0
( X; 0 Σ) and for any σ ∈ Ċ∞ ( X; 0 Σ), one has

R(λ)σ ∈ x λ+ d 2 C ∞ ( X; 0 Σ).
Proof. : The extension and the structure of R(λ) are a consequence of Proposition 3.2 since

R(λ) = (2iλ) -1 (R -(λ) -R + (λ)).
As for the mapping property, this is a consequence of mapping properties of operators in Ψ * , * , * 0 ( X) in [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]. The question of the simple pole at λ = 0 is also clear since R ± (λ) are holomorphic at λ = 0. It remains to show that the residue Π 0 is infinite rank. One way to prove it is to consider the asymptotic of

Π 0 φ if φ ∈ C ∞ 0 ( X; 0 Σ). First, both R -(0) and R + (0) have infinite rank since DR ± (0) = Id on C ∞ 0 ( X; 0 Σ), but moreover if φ is smooth compactly supported, R ± (0)φ has an asymptotic of the form x d 2 ψ ± at the boundary where ψ ± ∈ C ∞ (∂ X; 0 Σ ∓ ) according to Proposition 3.2. If ψ ± = 0, then R ± (0)φ = O(x d 2 +1
), and by the indicial equation (3.6) it must vanish to infinite order at ∂ X, which by Mazzeo's unique continuation theorem implies that R ± (0)φ = 0, a contradiction. This then shows that the range of R + (0) on C ∞ 0 ( X; 0 Σ) does not intersect the range of R -(0) acting on the same space, concluding the proof.

Remark. By self-adjointness of D 2 , one deduces easily that R(λ) * = R( λ), or in terms of kernels

R(λ; m, m ′ ) * = R( λ; m ′ , m) ∀m, m ′ ∈ X, m = m ′ , here A * ∈ Σ m ′ ⊠ Σ m means the adjoint of A ∈ Σ m ⊠ Σ m ′ if we identify the dual Σ *
m with Σ m via the Hermitian product induced by the metric g.

3.6.

Another parametrix construction when the curvature is constant near ∞. When (X, g) is asymptotically hyperbolic with constant curvature outside a compact set (which is the case of a convex co-compact quotient X Γ = Γ\H d+1 ), one may use a simplified construction similar to that of Guillopé-Zworski [START_REF] Guillopé | Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity[END_REF] for the Laplacian on functions.

Indeed, there exists a covering of a neighbourhood of ∂ X by open sets (U j ) with isometries

ι j : (U j , g) → (B, g H d+1 ),
where

B := {(x 0 , y 0 ) ∈ (0, ∞) × R d , x 2 0 + |y 0 | 2 < 1} and g H d+1 = dx 2 0 + |dy 0 | 2 x 2 0 .
We denote by B := {(x 0 , y 0 ) ∈ [0, ∞)×R d , x 2 0 +|y 0 | 2 < 1} a half-ball in H d+1 and ∂ B := B ∩{x 0 = 0}. We shall also use the notation ι j for the restriction ι j | Uj ∩∂ X .

Note that the function x 0 in B is not pulled-back to a boundary defining function putting the metric g under the form g = (dx 2 + h x )/x 2 , but we have (x/ι * j x 0 )| ∂ X = ι * j η j for some functions η j ∈ C ∞ (∂ B). Through ι j , the spinor bundle on H d+1 pulls-back to the spinor bundle 0 Σ( X)| Uj but the splitting induced by cl(x 0 ∂ x0 ) does not correspond to the splitting 0 Σ + ⊕ 0 Σ -, except at the boundary x 0 = 0, since the eigenspaces of cl(x 0 ∂ x0 ) are the eigenspaces of cl(ι j * (x∂ x )) when restricted to the boundary.

The Dirac operator D H d+1 pulls back to D in U j , consequently one may choose a partition of unity ι

* j χ 1 j near ∂ X associated to (U j ) j , that is j ι * j χ 1 j = χ where χ ∈ C ∞ ( X) is equal to 1 near ∂ X. Take now χ 2 j ∈ C ∞ 0 ( B)
some functions which are equal to 1 on the support of χ 1 j . Let us define φ i j on ∂ B by χ i j (0, y 0 ) = φ i j (y 0 ) so that j ι * j φ 1 j = 1 on ∂ X and φ 2 j φ 1 j = φ 1 j . The first parametrix we can use for (

D 2 + λ 2 ) -1 is (3.9) R 0 (λ) = j ι * j χ 2 j R H d+1 (λ)χ 1 j ι j * + Q 0 (λ)
where

Q 0 (λ) ∈ Ψ -2;∞,∞ 0 ( X; 0 Σ) is holomorphic, compactly supported and solves (D 2 +λ 2 )Q 0 (λ) = (1 -χ)Id + K 0 (λ) for some K 0 (λ) ∈ C ∞ 0 (X × X; E).
Here ι * j denotes the pull back on sections of the spinor bundle and ι j * := (ι -1 j ) * . We obtain

(D 2 + λ 2 )R 0 (λ) = Id + j ι * j [D 2 H d+1 , χ 2 j ]R H d+1 (λ)χ 1 j ι j * + K 0 (λ).
The last term K 0 (λ) is clearly compact on all weighted spaces x N L 2 (X; 0 Σ) while the first one is not. Since on H d+1 one has

D 2 H d+1 = x d 2 0 -(x 0 ∂ x0 ) 2 Id + x 2 0 D R d + ix 0 AD R d x -d 2 0 , A := 1 0 0 -1 = -icl(x 0 ∂ x0 ),
in the splitting induced by cl(x 0 ∂ x0 ). The operator [D 2 H d+1 , χ 2 j ] can be written as follows:

[D 2 H d+1 , χ 2 j ] = dx 0 (∂ x0 χ 2 j (x 0 , y 0 ))Id -x 2 0 (∂ 2 x0 χ 2 j (x 0 , y 0 ))Id + x 2 0 [D R d , χ 2 j (x 0 , y 0 )] + ix 0 [AD R d , χ 2 j (x 0 , y 0 )].
Using the fact that (∇χ 2 j )χ 1 j = 0 and the expression of R H d+1 (λ), we deduce that

[D 2 H d+1 , χ 2 j ]R H d+1 (λ)χ 1 j ∈ x λ+ d 2 +1 0 x ′ 0 λ+ d 2 C ∞ ( B × B; E)
where (x 0 , y 0 , x ′ 0 , y ′ 0 ) are the natural coordinates on B × B. This error term can be solved away using the indicial equation explained above for the general AH case and one can thus construct, for all N ∈ N, an operator

R N (λ) ∈ x λ+ d 2 +1 x ′ λ C ∞ ( X × X; E) such that (D 2 + λ 2 )(R 0 (λ) + R N (λ)) = Id + K N (λ), K N (λ) ∈ x λ+ d 2 +N x ′ λ+ d 2 C ∞ ( X × X; E) and K N (λ) is compact on x N ′ L 2 (X; 0 Σ) if 0 < N ′ < N and ℜ(λ) > -N + N ′ and ℜ(λ) > -N ′ .
All these terms are holomorphic in λ except possibly at -N/2 where first order poles come from the indicial equation and at λ = 0 where R H d+1 (λ) has an infinite rank pole. As above for the general case, we can take an asymptotic series using Borel Lemma, which gives an operator

R ∞ (λ) ∈ x λ+ d 2 +1 x ′ λ+ d 2 C ∞ ( X × X; E), holomorphic in λ / ∈ -N 0 /2 so that (D 2 + λ 2 )(R 0 (λ) + R ∞ (λ)) = Id + K ∞ (λ) for some K ∞ (λ) ∈ x ∞ x ′ λ+ d 2 C ∞ ( X × X; E).
And again, as in the proof of Proposition 3.2, up to the addition of a residual finite rank term for R ∞ (λ), we can assume that there is λ 0 with ℜ(λ 0 ) > 0 such that Id + K ∞ (λ 0 ) is invertible on x N L 2 (X) for all N > 0. The extended resolvent of D 2 + λ 2 is thus given by

R(λ) = (R 0 (λ) + R ∞ (λ))(Id + K ∞ (λ)) -1 , it is finite meromorphic in C\(-N 0 /2). Moreover standard arguments show that (Id+K ∞ (λ)) -1 = Id + S ∞ (λ) for some S ∞ (λ) ∈ x ∞ x ′ λ+ d 2 C ∞ ( X × X; E) and so R(λ) -R 0 (λ)(Id + S ∞ (λ)) ∈ x λ+ d 2 +1 x ′ λ+ d 2 C ∞ ( X × X; E).
Using the composition results of Mazzeo [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF]Th. 3.15], we get

R 0 (λ)S ∞ (λ) ∈ (xx ′ ) λ+ d 2 C ∞ ( X × X; E) so (3.10) R(λ) -R 0 (λ) ∈ x λ+ d 2 x ′ λ+ d 2 C ∞ ( X × X; E).
Similarly, using the remark following Corollary 3.5, we deduce that the kernel

(3.11) R(λ; m, m ′ ) -R 0 ( λ; m ′ , m) * ∈ x λ+ d 2 x ′ λ+ d 2 C ∞ ( X × X; E) where R 0 ( λ; m ′ , m) * is given, by symmetry of R H d+1 (λ), by (3.12) R 0 ( λ; m ′ , m) * = j (ι * j χ 1 j R H d+1 (λ)χ 2 j ι j * )(m, m ′ ) + Q 0 ( λ) * .
The expressions (3.10), (3.11) and (3.12) will be very useful in what follows for obtaining an explicit formula of the scattering operator modulo a smoothing term.

Scattering and Eisenstein series

4.1. Definitions and properties. Similarly to the Laplacian on functions, we can define Eisenstein series and scattering operator for Dirac operator. The Eisenstein series E(λ) is an operator mapping C ∞ (∂ X; 0 Σ) → C ∞ (X; 0 Σ) and for all ψ, E(λ)ψ is a non L 2 -solution of (D 2 + λ 2 )σ = 0; more precisely it is defined using the following Lemma 4.1. Let ψ ∈ C ∞ (∂ X; 0 Σ), and λ ∈ C \ (-N/2) not a pole of R(λ), then there exists σ ∈ C ∞ (X; 0 Σ) solution of (D 2 + λ 2 )σ = 0, unique when ℜ(λ) ≥ 0, and such that there exist

σ ± ∈ C ∞ ( X; 0 Σ) with σ -| ∂ X = ψ and σ = x d 2 -λ σ -+ x d 2 +λ σ + . Moreover σ ± are meromorphic in λ ∈ C \ (-N/2).
Proof. This is essentially the same construction as for the Laplacian on functions in [START_REF] Graham | Scattering matrix in conformal geometry[END_REF]: using the indicial equation (3.6) and Borel lemma, it is possible to construct a spinor

σ ∞ ∈ x -λ+ d 2 C ∞ ( X; 0 Σ), holomorphic in C\(Z/2) such that (D 2 +λ 2 )σ ∞ = O(x ∞ ) and (x λ-d 2 σ ∞ )| ∂ X = ψ.
Note that this spinor is meromorphic in λ ∈ C with only simple poles at N/2 coming from the roots of the indicial equation. Then we can set

σ := σ ∞ -R(λ)(D 2 + λ 2 )σ ∞ .
If λ is not a pole of R(λ), this solves the problem and defines σ ± by using the mapping property of R(λ) stated in Corollary 3.5. The meromorphy of σ ± is also a consequence of the construction and of the meromorphy of R(λ). The uniqueness of the solution is due to the fact that for two solutions σ 1 , σ 2 of the problem, the indicial equation implies that σ 1 -σ 2 ∈ x λ+ d 2 C ∞ ( X; 0 Σ), and then for ℜ(λ) > 0 this would be L 2 and λ would be a pole of the resolvent in the physical plane. For ℜ(λ) = 0, this can be proved using an application of Green formula like in [START_REF] Graham | Scattering matrix in conformal geometry[END_REF]: if σ 1 , σ 2 are two solutions of the problem, then the difference σ 1 -σ 2 is in x d 2 +λ C ∞ ( X; 0 Σ) by the indicial equation and it is also in the kernel of D 2 + λ 2 , so we may apply the Lemma 3.3 with σ 1 = σ 1 -σ 2 and σ 2 := R(-λ)ϕ where ϕ ∈ Ċ∞ ( X; 0 Σ) is chosen arbitrarily. This clearly implies that X σ 1 -σ 2 , ϕ dv g = 0 and thus σ 1 = σ 2 .

Remark. By uniqueness of the solution, σ and σ ± | ∂ X depend linearly on ψ. It is rather easy to prove that the scattering operator is off-diagonal with respect to the splitting 0 Σ(∂ X) = 0 Σ + (∂ X)⊕ 0 Σ -(∂ X). To that end, we give an alternative construction of the Eisenstein series E(λ)ψ when ψ ∈ 0 Σ + or ψ ∈ 0 Σ -. Let us first define a useful meromorphic function on

C (4.1) C(λ) := 2 -2λ Γ( 1 2 -λ) Γ( 1 2 + λ)
.

which satisfies C(λ)C(-λ) = 1. Lemma 4.4. Let ψ ∈ C ∞ (∂ X; 0 Σ ± )
, and λ ∈ C\ (-N/2) be not a pole of R ± (λ), then there exists a unique σ ∈ C ∞ (X; 0 Σ) solution of (D ± iλ)σ = 0 and such that there exists

σ ± ∈ C ∞ ( X; 0 Σ) with σ -| ∂ X = ψ and σ = x d 2 -λ σ -+ x d 2 +λ σ + . Moreover one has σ + | ∂ X ∈ C ∞ (∂ X; 0 Σ ∓ ) and σ ± are meromorphic in λ ∈ C \ (-N/2).
If in addition the metric g is even, then σ ± /C(λ) are holomorphic in {ℜ(λ) ≥ 0} where C(λ) is the function in (4.1).

Proof. Recall the indicial equation for (D ± iλ): let j ∈ N and ψ ± ∈ C ∞ (∂ X; 0 Σ ± ) then there exist some smooth spinor F λ,j near ∂ X such that

(4.2) x λ-d 2 (D ± iλ)x -λ+ d 2 +j (ψ + + ψ -) = ix j (j -λ ± λ)ψ + + (λ -j ± λ)ψ -+ x j+1 F λ,j .
Using this indicial equation inductively, we can construct for all ψ ∈ C ∞ (∂ X; 0 Σ ± ) a formal Taylor series, and thus a true spinor

σ ∞,± ∈ x -λ+ d 2 C ∞ ( X; 0 Σ) by Borel lemma, such that (D±iλ)σ ∞,± = O(x ∞ ) and (x λ-d 2 σ ∞,± )| ∂ X = ψ.
This can be done holomorphically in λ as long as λ is not a root of the indicial equation (4.2). The λ such that the indicial numbers j -λ ± λ and λ -j ± λ in (4.2) vanish are N/2 and they vanish only on the Σ ∓ part of the bundle, therefore since C(λ) has first order poles at 1/2 + N 0 , we see that σ ∞ /C(λ) can be chosen holomorphically in λ ∈ C \ N and that it has at most poles of order 1 at each λ = k with k ∈ N; since it does not involve new arguments we refer the reader who needs more details to the paper of Graham-Zworski [START_REF] Graham | Scattering matrix in conformal geometry[END_REF] where it was studied in the case of the Laplacian on functions. Now consider the case of a metric g even. Since A + , A -defined in (3.4)

are preserved by x -d 2 Dx d 2 if g is even, then clearly x λ-d 2 (D ± iλ)x -λ+ d 2 = λ(-cl(x∂ x ) ± i) + x -d 2 Dx d 2 also preserves both A + , A -.
In particular, if ψ -= 0 in (4.2), then x 2j+1 F λ,2j ∈ A + and x 2j+2 F λ,2j+1 ∈ A -, while the converse is true if ψ + = 0. This implies that the spinor σ ∞,± can be taken so that x λ-d 2 σ ∞,± ∈ A ± and the λ which are actually solution of the indicial equation (4.2) for D ± iλ are only at 1/2 + N 0 . The spinor σ ∞ /C(λ) can be taken holomorphic also at λ ∈ N. It remains to set

(4.3) σ := σ ∞,± -R ± (λ)(D ± iλ)σ ∞,±
which solves our problem, using the mapping property of R ± (λ) stated in Proposition 3.2.

By uniqueness, the solution in Lemma 4.4 is clearly the same as the one of Lemma 4.1 when the initial data ψ is either in

σ + | ∂ X or σ -| ∂ X , which implies Corollary 4.5. The scattering operator S(λ) maps C ∞ (∂ X; 0 Σ ± ) to C ∞ (∂ X; 0 Σ ∓ ).
Let us define the natural projection and inclusion

P ± : C ∞ (∂ X; 0 Σ) → C ∞ (∂ X; 0 Σ ± ); I ± : C ∞ (∂ X; 0 Σ ± ) → C ∞ (∂ X; 0 Σ)
and also the maps corresponding to the two off-diagonal components of S(λ)

S ± (λ) := P ∓ S(λ)I ± : C ∞ (∂ X; 0 Σ ± ) → C ∞ (∂ X; 0 Σ ∓ ) E ± (λ) := E(λ)I ± P ± : C ∞ (∂ X; 0 Σ) → C ∞ (X; 0 Σ).
4.2. Some relations between resolvent, scattering operator and Eisenstein series. Like for the Laplacian on functions, the Schwartz kernels of R(λ), E(λ) and S(λ) are related by the following Proposition 4.6. Let λ ∈ C be such that λ / ∈ -N/2 and λ not a pole of R(λ), then the Schwartz kernel E(λ; m, y ′ ) and E ± (λ; m, y ′ ) in C -∞ ( X × ∂ X; E) of respectively E(λ) and E ± (λ) can be expressed by

(4.4) E(λ; m, y ′ ) = 2λ[x ′ -d 2 -λ R(λ; m, x ′ , y ′ )]| x ′ =0 , E ± (λ; m, y ′ ) = [x ′ -d 2 -λ R ± (λ; m, x ′ , y ′ )]| x ′ =0 cl(ν)
where we use the product decomposition (x ′ , y ′ ) ∈ [0, ǫ)×∂ X near the boundary in the right variable of X × X. If in addition ℜ(λ) < -d 2 , the Schwartz kernel S(λ; y, y ′ ) of S(λ) is in C 0 (∂ X × ∂ X; E) and can be expressed by

(4.5) S(λ; y, y ′ ) = [x -d 2 -λ E(λ; x, y, y ′ )] x=0 , S ± (λ; y, y ′ ) = [x -d 2 -λ E ± (λ; x, y, y ′ )]| x=0 . Proof. Let σ ∞ as in Lemma 4.1, then E(λ)ψ = σ ∞ -R(λ)(D 2 + λ 2 )σ ∞ .
The first statement of Proposition is simply obtained by integration by part in (x ′ , y ′ ) of

x ′ ≥ǫ R(λ; m, x ′ , y ′ ), (D 2 + λ 2 )σ ∞ (x ′ , y ′ ) dv g (x ′ , y ′ )
and letting ǫ → 0, this gives the term σ ∞ (m) plus a term

∂ X x ′ -d R(λ; m, x ′ , y ′ ), ∇ x ′ ∂ x ′ σ ∞ (x ′ , y ′ ) -∇ x ′ ∂ ′ x R(λ; m, x ′ , y ′ ), σ ∞ (x ′ , y ′ ) x ′ =0 dv h0 .
But from the analysis of the resolvent R(λ), we have for m ∈ X and as

x ′ → 0 R(λ; m, x ′ , y ′ ) = x ′ d 2 +λ (L(λ; m, y ′ ) + O(x ′ )), ∇ x ′ ∂ x ′ R(λ; m, x ′ , y ′ ) = ( d 2 + λ)x ′ d 2 +λ (L(λ; m, y ′ ) + O(x ′ )) σ ∞ (x ′ , y ′ ) = x ′ d 2 -λ (ψ(y ′ ) + O(x ′ )), ∇ x ′ ∂ x ′ σ ∞ (x ′ , y ′ ) = ( d 2 -λ)x ′ d 2 -λ (ψ(y ′ ) + O(x ′ ))
for some L ∈ C ∞ (X ×∂ X; E) and where ψ ∈ C ∞ (∂ X, Σ) is arbitrarily chosen. We can then deduce that E(λ; m, y ′ ) = 2λL(λ; m, y ′ ) as distributions in C ∞ (X × ∂ X; E). Using the structure of R(λ) in Proposition 3.2, we observe that the kernel of

E(λ) is also a distribution in C -∞ ( X × ∂ X; E) since its lift to X × 0 ∂ X is a conormal distribution on X × 0 ∂ X, more precisely it is an element in ρ λ+ d 2 lb ρ -λ-d 2 ff C ∞ ( X × 0 ∂ X; E)
. This is exactly the same argument for the E ± (λ) formula in (4.4) by using the representation (4.3) and integration by part. Now for the scattering operator, we take ℜ(λ) < -d 2 and use the definition of S(λ)ψ to deduce that

S(λ)ψ = (x -λ-d 2 E(λ)ψ)| x=0 . From the fact that the lift of the kernel x -λ-d 2 E(λ) to X × 0 ∂ X is in ρ -2λ-d ff C ∞ ( X × 0 ∂ X; E), thus in C 0 ( X × ∂ X), we see that ∂ X x -λ-d 2 E(λ; x, y, y ′ )ψ(y ′ )dv h0 (y ′ )
is C 0 ( X; 0 Σ) and its restriction to ∂ X is given by the pairing of [x -λ-d 2 E(λ)]| x=0 with ψ, which ends the proof for S(λ). The argument is the same for S ± (λ).

In the same way as E(λ), we define the operator E ♯ (λ), E ♯ ± (λ) so that the Schwartz kernel of

E ♯ (λ), E ♯ ± (λ) are given for λ ∈ C \ (-N/2) not a pole of R(λ) by E ♯ (λ; y, m ′ ) :=2λ[x -λ-d 2 R(λ; x, y, m ′ )]| x=0 , E ♯ ± (λ; y, m ′ ) := -cl(ν)[x -λ-d 2 R ± (λ; x, y, m ′ )]| x=0 (4.6)
using the product decomposition [0, ǫ) x × ∂ X near ∂ X. Like for the analysis of the kernel of E(λ) above, the structure of the kernel E ♯ (λ) on the blow-up ∂ X × 0 X is clear from the analysis of R(λ). Note also that E ♯ ± (λ) maps Ċ∞ ( X; 0 Σ) to C ∞ (∂ X; 0 Σ ± ) by using Corollary 3.5 and

(E ♯ ± (λ)f )(y) = -cl(ν) lim x→0 X x -λ-d 2 R ± (λ; x, y, m ′ )σ(m ′ )dv g (m ′ ).
We see also from the remark following Corollary 3.5 that

E ♯ ( λ; y, m ′ ) = E(λ; m ′ , y) * , E ♯ ± ( λ; y, m ′ ) = E ∓ (λ; m ′ , y) * (4.7)
when these are considered as linear maps from 0 Σ m ′ to 0 Σ y . Lemma 4.7. Let m, m ′ ∈ X, then for λ / ∈ Z/2 neither a pole of R(λ) nor of R(-λ), we have

(4.8) R(λ; m, m ′ ) -R(-λ; m, m ′ ) = (2λ) -1 ∂ X E(λ; m, y)E ♯ (-λ; y, m ′ )dv h0 (y), R ± (λ; m, m ′ ) -R ∓ (-λ; m, m ′ ) = - ∂ X E ± (λ; m, y)cl(ν)E ♯ ∓ (-λ; y, m ′ )dv h0 (y)
or in terms of operators

R(λ) -R(-λ) = (2λ) -1 E(λ)E ♯ (-λ), R ± (λ) -R ∓ (-λ) = -E ± (λ)cl(ν)E ♯ ∓ (-λ).
Proof. This is a straightforward application of Green formula and does not involve anything more than in the proof given by Guillopé [START_REF] Guillopé | Fonctions Zêta de Selberg et surfaces de géométrie finie[END_REF] for the Laplacian on functions on a surface. It is based on the fact that (

D 2 + λ 2 )R(λ; m, m ′ ) = (D 2 + λ 2 )R(-λ; m, m ′ ) = δ(m -m ′ ) and (D ± iλ)R ± (λ; m, m ′ ) = (D ± iλ)R ∓ (-λ; m, m ′ ) = δ(m -m ′ )
, where δ(m -m ′ ) denotes the Dirac mass on the diagonal.

A corollary of this is some functional equations relating E(λ), E ♯ (λ) and S(λ).

Corollary 4.8. The following meromorphic identities hold

E(λ) =E(-λ)S(λ), E ♯ (λ) =S(λ)E ♯ (-λ), E ∓ (λ) =E ± (-λ)S ∓ (λ), E ♯ ± (λ) =S ± (λ)E ♯ ∓ (-λ).
Proof. Let us consider the second identity: assume ℜ(λ) < -d 2 , it suffices to multiply the first line of (4.8) by x(m) -λ-d 2 and take the limit as x(m) → 0 when m ′ ∈ X is fixed, the limit makes sense in view of our analysis of the Schwartz kernels of R(λ), E(λ), E ♯ (λ) and S(λ). Then we use Proposition 4.6 and the definition of E ♯ (λ) and this gives the proof of the second identity of the Corollary, at least for ℜ(λ) < -d 2 , but this extends meromorphically to λ ∈ C. The proofs of other identities are similar.

Properties of S(λ).

Proposition 4.9. For λ such that ℜ(λ) < -d 2 , λ / ∈ -N/2 and λ not a pole of R(λ), the operator S(λ) is a classical pseudo-differential operator on ∂ X of order 2λ, with principal symbol

σ pr (S(λ))(ξ) = C(λ)cl(ν)|ξ| 2λ-1 h0 icl(ξ), with C(λ) := 2 -2λ Γ(-λ + 1/2) Γ(λ + 1/2) . (4.9)
Moreover S(λ) can be meromorphically extended to C \ (-N/2) as a family of pseudo-differential operators in Ψ 2λ (∂ X; 0 Σ).

Proof. Let β : X × 0 X → X × X be the blow-down map, ∂ X × 0 ∂ X := [∂ X, ∂ X, ∆ ∂ ] be the blow-up of ∂ X × ∂ X around the diagonal ∆ ∂ and β ∂ : ∂ X × 0 ∂ X the associated blow-down map.
Then the expression (4.5) can also be written for ℜ(λ) < -d 2 (S(λ) and R(λ) denote also the Schwartz kernel) (4.10)

S(λ) = 2λβ ∂ * β * ((xx ′ ) -λ-d 2 R(λ))| lb∩rb
where lb ∩ rb is naturally identified with ∂ X × 0 ∂ X. For more details, we refer to the article of Joshi-Sa Barreto [START_REF] Joshi | Inverse scattering on asymptotically hyperbolic manifolds[END_REF] which deals with the Laplacian on functions. Now using the fact that

R(λ) ∈ Ψ -2,λ+ d 2 ,λ+ d 2 0
( X; 0 Σ), we deduce that

((xx ′ ) -λ-d 2 R(λ))| lb∩rb ∈ ρ -2λ-d ff,∂ C ∞ (∂ X × 0 ∂ X; E)
where ρ ff,∂ := ρ ff | lb∩rb is a boundary defining function of the boundary (i.e., the face obtained by blowing-up) of ∂ X × 0 ∂ X. This shows that the kernel S(λ) is classically (or polyhomogeneous) conormal to the diagonal and the leading singularity at y = y ′ = p given in polar coordinates in the conormal bundle is given by

S(λ; y, y ′ ) ∼ c(λ)|y -y ′ | -2λ-d U p (p ′ ), p ′ = y -y ′ |y -y ′ | ∈ S d-1
for some c(λ) ∈ C and where U p (p ′ ) ∈ End( 0 Σ p ( X)) denote the limit of the parallel transport U (e p , m) in the fiber ff p when m → p ′ ∈ lb∩rb∩ff p ≃ S d-1 (here e p is the center of ff p defined by the intersection of the interior diagonal with ff p and identified with the center of hyperbolic space).

Thus we obtain S(λ) ∈ Ψ 2λ (∂ X; 0 Σ), moreover the expression (4.10) can be meromorphically extended to C \ (-N/2) as a distribution classically conormal to the diagonal, thus as a family S(λ) ∈ Ψ 2λ (∂ X; 0 Σ). As for the principal symbol, we use the expression of U p (p ′ ) for H d+1 given in Corollary A.4 and Fourier transform to obtain (4.9). Notice that there might be first order poles of infinite multiplicity at N/2 coming from the meromorphic extension of the distribution |y -y ′ | 2λ+j to λ ∈ C. This phenomenon is described in [START_REF] Graham | Scattering matrix in conformal geometry[END_REF] for the case of the Laplacian on functions.

Like for functions, the scattering operator is a unitary operator on the continuous spectrum and satisfies a functional equation. Proof. The analyticity in the right half-plane is a consequence of the last statement in Lemma 4.4 and the fact that S(λ)ψ = σ + | ∂ X with the notation of this Lemma. We already know the meromorphic extension outside -N/2 so we can write, using Proposition 4.9,

S(λ)/C(λ) = cl(ν)(D h0 + i)(|D h0 | + 1) 2λ-1 (Id + K(λ))
for some K(λ) compact on L 2 (∂ X; 0 Σ) and analytic in {ℜ(λ) ≥ 0}. We know from Lemma 4.10 that Id + K(λ) is invertible for almost all λ ∈ C, so we may use Fredholm analytic theorem to show that (S(λ)/C(λ)) -1 is a meromorphic family of operators with poles of finite multiplicity at most in ℜ(λ) > 0, so by the functional equation in Lemma 4.10, we deduce that S(λ)/C(λ) is meromorphic in ℜ(λ) < 0 with poles of finite multiplicity.

We give another corollary of the properties of S(λ).

Corollary 4.12. For an AH manifold with a metric g even in the sense of (3.2), the resolvent R ± (λ) is finite meromorphic.

Proof. According to Proposition 3.2, the only problem for the meromorphy of R ± (λ) can be at -N/2, so consider the half plane {ℜ(λ) < 0}. Since R(-λ), E(-λ), E ♯ (-λ) are holomorphic in {ℜ(λ) < 0}, the result is a straightforward consequence of Corollary 4.11 together with the formula

R(λ) = R(-λ) + (2λ) -1 C(λ)E(-λ) S(λ) C(λ) E ♯ (-λ),
itself a consequence of Lemma 4.7 and Corollary 4.8.

4.4.

Representation of E(λ) and S(λ) in the case X Γ . Using Proposition 4.6 and (3.10), we obtain directly an explicit representation modulo a smoothing term. We use the functions η j , φ i j , χ i j of Section 3.6. We denote by S H d+1 (λ) and E H d+1 (λ) the scattering operator and Eisenstein series on H d+1 , defined using a defining function of ∂H d+1 which is equal to x 0 on the half ball B in the model H d+1 = {(x 0 , y 0 ) ∈ (0, ∞) × R d-1 }, i.e., in terms of distribution kernel on B × ∂ B and on

∂ B × ∂ B E H d+1 (λ; x 0 , y 0 , y ′ 0 ) :=2λ[x ′ 0 -λ-d 2 R H d+1 (λ; x 0 , y 0 , x ′ 0 , y ′ 0 )] x ′ 0 =0 , S H d+1 (λ; y 0 , y ′ 0 ) :=[x -λ-d 2 0 E H d+1 (λ; x 0 , y 0 , y ′ 0 )]| x0=0 . Lemma 4.13. If λ ∈ C \ (-N 0 /2) is not a pole of R(λ), then the Eisenstein series E(λ) for D 2 on a convex co-compact quotient X := Γ\H d+1 has the kernel E(λ) = E 0 (λ) + E ∞ (λ) where E 0 (λ) := j ι * j χ 2 j E H d+1 (λ)φ 1 j η -λ-d 2 j ι j * , E ∞ (λ) = 2λ[x ′ -λ-d 2 (R(λ) -R 0 (λ))]| x ′ =0 ∈ x λ+ d 2 C ∞ ( X × ∂ X; E).
Similarly, the scattering operator S(λ) for D 2 on X has the kernel S(λ) = S 0 (λ) + S ∞ (λ) where

S 0 (λ) := j ι * j η -λ-d 2 j φ 2 j S H d+1 (λ)φ 1 j η -λ-d 2 j ι j * , S ∞ (λ) = 2λ[(xx ′ ) -λ-d 2 (R(λ) -R 0 (λ))]| x=x ′ =0 ∈ C ∞ (∂ X × ∂ X; E).

Selberg zeta function of odd type

In this section, we will assume that the dimension d + 1 = 2n + 1 is odd except in Lemma 5.2.

5.1. Odd heat kernel of Dirac operator on H d+1 . By the identification (2.4), the kernel of the odd heat operator D H d+1 e -tD 2 H d+1 on L 2 (H d+1 ; Σ(H d+1 )) can be considered as a τ d -radial function P t over G. Hence there exists a function P t from G to End(V τ d ) satisfying the K-equivariance condition

P t (k 1 gk 2 ) = τ d (k 2 ) -1 P t (g)τ d (k 1 ) -1 for g ∈ G, k 1 , k 2 ∈ K (5.1) such that D H d+1 e -tD 2 H d+1 (gK, hK) = P t (h -1 g) for g, h ∈ G. (5.2)
Let us remark that P t (h -1 g) and P t (k -1 1 h -1 gk 2 ) for k 1 , k 2 ∈ K give the same map by the condition (5.1), so that the right hand side of (5.2) does not depend on the choice of the representatives of the K-cosets.

Recalling the Cartan decomposition G = KA + K with A + := {a r = exp(rH) | r > 0}, any element g ∈ G can be written as g = ha r k where a r = exp(rH) and r is the same as the hyperbolic distance d H d+1 (eK, gK) between two points eK and gK in H d+1 ∼ = G/K. Here e denotes the identity element in G. Now let us recall that the spin representation τ d decomposes into two half spin representations σ + ,σ -when restricting to M = Spin(d),

τ d | M = σ + ⊕ σ -,
hence the representation space V τ d also decomposes into V σ + ⊕ V σ -as M -representation spaces. By Schur's lemma there exists a function p ± t : R → C such that P t (a r )| Vσ ± = p ± t (r) Id Vσ ± where a r ∈ A + . As in the proof of Theorem 8.5 of [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF] using Theorem 8.3 of [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF], one can easily derive Proposition 5.1. The scalar components of D H d+1 e -tD 2 H d+1 are given by

(5.3) p ± t (r) = ± sinh(r/2) i2 3n+3/2 Γ(n + 3/2)t 3/2 - d d(cosh r) n r sinh -1 (r/2) e -r 2 4t .
Let us observe that the equalities (5.1) and (5.3) determine the odd heat kernel D H d+1 e -tD 2 H d+1 by the Cartan decomposition G = KA + K.

5.2.

Odd heat kernels over convex co-compact hyperbolic manifolds. By abuse of notation, g, h will also denote the points in H d+1 corresponding to the cosets gK, hK in G/K. By a usual construction, the kernel of the odd heat operator De -tD 2 over X Γ is given (as an automorphic kernel) by

(5.4) De -tD 2 (g, h) = γ∈Γ D H d+1 e -tD 2 H d+1 (g, γh)
where g, h denote points in X Γ = Γ\H d+1 which we view as a fundamental domain in H d+1 with sides identified through Γ. Using d H d+1 (g, h) = d H d+1 (e, g -1 h) where e denotes the origin in the unit disc model of H d+1 , then by (5.1), (5.3) and some elementary calculations, we have

(5.5) ||D H d+1 e -tD 2 H d+1 (g, γh)|| ≤ C t -3 2 e -d 2 rγ (g,h)- rγ (g,h) 2 4t 0≤j≤n+1 0≤k≤n r γ (g, h) j t -k .
Here C is a positive constant independent on t and r γ (g, h) := d H d+1 (g, γh). In particular the number r γ (g, g) is independent of g in the axis of γ and is called the translation length of γ, denoted by ℓ γ .

Lemma 5.2. Let F be a fundamental domain of Γ and x be a boundary defining function of X Γ which we view as well as a function on F. There are positive constants C 1 , C 2 such that for all γ ∈ Γ with translation length ℓ γ > C 1 and all g, h ∈ F, e -rγ (g,h) ≤ C 2 e -ℓγ x(g) x(h).

Proof. By conjugating by an isometry of H d+1 , we can assume that, in the half space model

H d+1 = R + x × R d y ,
the point at ∞ is not in the limit set Λ(Γ) of the group Γ. Then, since the group is convex co-compact, there exists a fundamental domain F which satisfies the following: there exists C > 0 and ǫ > 0 such that

F ∪ Λ(Γ) ⊂ B := {z ∈ [0, ∞) × R d ; |z| ≤ C, d eucl (z; Λ(Γ)) ≥ ǫ}
where d eucl denotes the Euclidean distance in R d+1 . Notice that the function x is comparable to the function x on F in the sense that 1/A < x/x < A for some constant A > 0. Let now γ ∈ Γ be an isometry, whose fixed points

p 1 γ , p 2 γ must belong to R d y ∩ B. Composing a translation z → z -(p 1 γ + p 2 γ
)/2 with a rotation in the R d y variable, we define an isometry q γ which maps p 1 γ to p γ := (0, |p 1 γ -p 2 γ |/2, 0 . . . , 0) and p 2 γ to -p γ . Notice that, since q γ is also an Euclidean isometry, then

q γ (F) ⊂ B ′ := {z ∈ [0, ∞) × R d ; |z| ≤ 2C, d eucl (z; ±p γ ) ≥ ǫ}.
We identify the (x, y 1 ) half-plane inside H d+1 with H 2 by setting z 0 := y 1 + ix, in particular ±p γ belong to the boundary of H 2 . We consider the isometry s γ of H 2 defined by (5.6)

s γ : z 0 → z 0 + p γ -z 0 + p γ .
This isometry maps p γ to ∞ and -p γ to 0 in H d+1 . We extend s γ recursively to H d+1 as follows: the isometry s γ of H k is extended to an isometry of H k+1 by identifying H k+1 with (0, π) θ × H k via the map ι k : (θ, x, y 1 , . . . , y k-1 ) → (x sin θ, y 1 , . . . , y k-1 , x cos θ) and defining the extension, still denoted s γ , by s γ (ι k (θ, w)) := ι k (θ, s γ (w)). Note that |ι k (θ, w)| = |w| for each k, and thus |s γ (z 0 , y 2 , . . . , y d )| = |s γ (z 0 )|. Using this fact and the explicit formula (5.6) in H 2 , it is easy to see that for all z ∈ B ′ , we have

ǫ 4C ≤ |s γ (z)| ≤ 4C ǫ .
We conclude that t

γ := s γ • q γ maps F into {z ∈ R + × R d , ε ≤ |z| ≤ ε -1 } for some ε > 0 which does not depend on γ. But t γ • γ • t -1
γ is an isometry fixing the line {y = 0} and thus it can be written under the form

t γ • γ • t -1 γ : (x, y) → e ℓγ (x, O γ (y)) for some O γ ∈ SO(d) where ℓ γ is the translation length of γ. Then we have for m = (x, y), m ′ = (x ′ , y ′ ) in the half-space model cosh 2 (d H d+1 (m, m ′ )/2) = |y -y ′ | 2 + |x + x ′ | 2
4xx ′ and by writing t γ g = (x, y) and t γ h = (x ′ , y ′ ) in the half-space model,

(5.7) cosh 2 (r γ (g, h)/2) = e ℓγ |e -ℓγ y -O γ (y ′ )| 2 + |e -ℓγ x + x ′ | 2 4xx ′ . But since t γ g, t γ h ∈ t γ (F), one has ε ≤ (x 2 + |y| 2 ) 1 2 ≤ ε -1
and the same for (x ′ , y ′ ), which from (5.7) implies that (5.8)

ε 2 ≤ cosh 2 (r γ (g, h)/2)e -ℓγ xx ′ ≤ ε -2
for ℓ γ large enough (depending only on ε). Observe now that, using the embeddings H 2 ⊂ • • • ⊂ H d+1 as above using the maps ι k , we can view the point p γ as an element of (the boundary of) each H k for k = 2, . . . , d + 1. Moreover, as Euclidean norms, one clearly has |ι k (θ, ω)p γ | = |ω -p γ | for all ω ∈ H k , and thus |z -p γ | = |z 0 -p γ | where z ∈ H d+1 corresponds to a unique (θ 1 , . . . , θ d-1 , z 0 ) ∈ (0, π) d-1 × H 2 by the maps ι k described above. Let us denote Im(z) := x when z = (x, y 1 , . . . , y d ) ∈ H d+1 , then each z ∈ H d+1 is associated to a unique (θ 1 , . . . , θ d-1 , z 0 ) ∈ (0, π) d-1 × H 2 by the maps ι k (here z 0 is a complex coordinate on H 2 viewed as the half-space Im(z 0 ) > 0) and we have Im(z) = Im(z 0 ) sin(θ 1 ) . . . sin(θ d-1 ) and Im(s γ (z)) = Im(s γ (z 0 )) sin(θ 1 ) . . . sin(θ d-1 ) by construction of s γ acting on H n+1 . But a short computation gives

Im(s γ (z 0 )) = 2p γ Im(z 0 ) |z 0 -p γ | 2 = 2p γ Im(z 0 ) |z -p γ | 2
where p γ ∈ H 2 is viewed as a positive real number in C, which therefore implies Im(s γ (z))/ Im(z) = 2p γ /|z -p γ | 2 for all z ∈ H d+1 . We have also shown that inf z∈qγ (F) |z -p γ | > ǫ and 0 < p γ < 1/ǫ for some ǫ > 0 uniform in γ ∈ Γ by convex co-compactness of Γ, thus we can combine this with (5.8) and the fact that x • q γ = x where x is comparable to x on F to deduce that there exists a constant C > 0 uniform in γ so that (x • t γ )/ x ≤ C on F. This ends the proof.

By (1.2), (5.5) and Lemma 5.2, we get that the right hand side of (5.4) converges uniformly in g, h in a fundamental domain. We denote by tr the local trace over 0 Σ m (X Γ ) ∼ = V τ d for m in a fundamental domain of Γ.

Proposition 5.3. For m in a fundamental domain of Γ, we have where dv(m) denotes the metric over X Γ induced from the hyperbolic metric dv H d+1 . By our assumption on Γ, Γ \ {e} consists of hyperbolic elements and decomposes into Γconjugacy classes of hyperbolic elements. We denote by Γ hyp the set of Γ-conjugacy classes of hyperbolic elements. Each element [γ] in the set Γ hyp corresponds to a closed geodesic C γ in X Γ . We denote by l(C γ ) the length of C γ and by j(γ) the positive integer such that γ = γ j(γ) 0 with a primitive γ 0 . A primitive hyperbolic element γ means that it can not be given by a power of any other elements in Γ, so that Γ-conjugacy class of a primitive γ corresponds to a prime geodesic C γ in X Γ . The trace of the monodromy in Σ(X Γ ) ∼ = Γ\ G × τ d V τ d along a closed geodesic C γ is given by χ σ+ (m γ ) + χ σ-(m γ ) since any hyperbolic element γ can be conjugated to m γ a γ ∈ M A + . A closed geodesic C γ corresponds to a fixed point of the geodesic flow on the unit sphere bundle over X Γ . The Poincaré map P (C γ ) is the differential of the geodesic flow at C γ , which is given by P (C γ ) = Ad(m γ a γ ) if γ = m γ a γ . The unit sphere bundle SX Γ of X Γ is given by Γ\G/M , and its tangent bundle T SX Γ is given by

(5.9) tr(De -tD 2 )(m) = γ∈Γ\{e} tr(D H d+1 e -tD 2 H d+1 )(m, γm). Proof. It is enough to show that tr(D H d+1 e -tD 2 H d+1 )(m, m) = 0, which is a consequence of tr(D H d+1 e -tD 2 H d+1 )(m, m) = tr(P t )(e) = d(σ ± )(p + t (0) + p - t (
T SX Γ = Γ\G × M (n ⊕ a ⊕ n)
where n = θ(n) and M acts on n ⊕ a ⊕ n by the adjoint action Ad. Hence P (C γ ) preserves the decomposition n ⊕ a ⊕ n. We denote by P (C γ ) |n , P (C γ ) |n its restriction to n, n part respectively. Now we put (5.11)

D(γ) := | det(P (C γ ) |n⊕n -Id)| 1/2 = e -nl(Cγ ) | det(P (C γ ) |n -Id)| = e nl(Cγ ) det(Id -P (C γ ) |n ).
Proposition 5.4. The following identity holds

Tr(De -tD 2 ) = 2πi (4πt) 3 2 [γ]∈Γ hyp l(C γ ) 2 j(γ)D(γ) (χ σ+ (m γ ) -χ σ-(m γ ))e -l(Cγ ) 2 4t . (5.12)
Proof. By equalities (5.2), (5.9) and (5.10), (5.13) Tr(De

-tD 2 ) = γ∈Γ\{e} Γ\G tr P t (g -1 γg) d(Γg).
By Theorem 2.2 in [START_REF] Millson | Closed geodesics and the η-invariant[END_REF], the scalar function p t (g) := tr P t (g) is in the Harish-Chandra L 1 -space. (Note that p t (g) should not be confused with p ± t (r) in the subsection 5.1.) Hence we can follow the well known path of Selberg on p. 63-66 of his famous paper [START_REF] Selberg | Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series[END_REF] to obtain (5.14)

γ∈Γ\{e} Γ\G p t (g -1 γg) d(Γg) = [γ]∈Γ hyp vol(Γ γ \G γ ) Gγ \G p t (g -1 γg) d(G γ g)
where Γ γ , G γ denote the centralizer of γ in Γ and G respectively. Now we show the following equality, (5.15)

vol(Γ γ \G γ ) Gγ \G p t (g -1 γg) d(G γ g) = 2πi (4πt) 3 2 l(C γ ) 2 j(γ)D(γ) (χ σ+ (m γ ) -χ σ-(m γ ))e -l(Cγ ) 2 4t
.

We may assume that a hyperbolic element γ ∈ Γ has the form

m γ a γ ∈ M A + . If γ ∈ M A + , (5.16
)

Gγ \G p t (g -1 γg) d(G γ g) = vol(G γ /A) -1 G/A p t (gγg -1 ) d(gA).
We also have (5.17)

G/A p t (gγg -1 ) d(gA) = D(γ) -1 F pt (m γ a γ )
where the Abel transform of p t is given by

F pt (m γ a γ ) = a ρ γ N K p t (k m γ a γ nk -1
) dk dn with a ρ γ = exp(nl(C γ )). By Theorem 6.2 in [START_REF] Wallach | On the Selberg trace formula in the case of compact quotient[END_REF], we have

(5.18) Θ σ,λ (p t ) = M ∞ -∞
F pt (m exp(rH)) tr σ(m) e iλr dr dm.

Here Θ σ,λ (p t ) is defined by

Θ σ,λ (p t ) := Tr π σ,λ (p t ) = Tr G p t (g)π σ,λ (
H σ,λ := { f : G → H σ | f (xman) = a -(iλ+ρ) σ(m) -1 f (x), f | K ∈ L 2 (K) } by the left translation π σ,λ (g)f (x) = f (g -1 x
). Applying the Fourier inversion theorem and the Peter-Weyl theorem to the equality (5.18), we get (5. [START_REF] Joshi | Inverse scattering on asymptotically hyperbolic manifolds[END_REF])

F pt (m γ a γ ) = σ∈ M tr σ(m γ ) 1 2π ∞ -∞ Θ σ,λ (p t )e -il(Cγ )λ dλ. Now let observe that Θ σ,λ (p t ) vanishes unless σ = σ ± since τ d | M = σ + ⊕ σ -. Moreover, we have (5.20) Θ σ±,λ (p t ) = ±λe -tλ 2
as in Proposition 3.1 in [START_REF] Park | Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps[END_REF] by (4.5) in [START_REF] Moscovici | Eta invariants of Dirac operators on locally symmetric manifolds[END_REF]. Note that the analysis for this does not depend on Γ, but is performed over G. Combining (5.16), (5.17), (5.19), (5.20) and observing that vol

(Γ γ \G γ )/vol(G γ /A) = l(C γ )/j(γ), we conclude vol(Γ γ \G γ ) Gγ \G p t (g -1 γg) d(G γ g) = σ∈ M l(C γ )tr σ(m γ ) j(γ)D(γ) 1 2π ∞ -∞ Θ σ,λ (h)e -il(Cγ )λ dλ = l(C γ ) χ σ+ (m γ ) -χ σ-(m γ ) j(γ)D(γ) 1 2π ∞ -∞ λe -tλ 2 e il(Cγ )λ dλ = 2πi (4πt) 3 2 l(C γ ) 2 χ σ+ (m γ ) -χ σ-(m γ ) j(γ)D(γ) e -l(Cγ ) 2 4t
.

Taking the sum over [γ] ∈ Γ hyp of this equality and by (5.13) and (5.14), we obtain (5.12).

From Proposition 5.4, putting c := min

Z Γ (σ ± , λ) := exp - [γ]∈Γ hyp χ σ± (m γ ) j(γ)D(γ) e -λ l(Cγ ) for ℜ(λ) > δ Γ -n. It is easy to see that Z Γ (σ ± , λ) absolutely converges for ℜ(λ) > δ Γ -n by (1.2). Proposition 5.6. For ℜ(λ) > δ Γ -n, (5.22)
Z Γ (σ ± , λ) = [γ]∈PΓ hyp ∞ k=0 det Id -σ ± (m γ ) ⊗ S k (P (C γ ) |n )e -(λ+n)l(Cγ )
where PΓ hyp is the set of Γ-conjugacy classes of primitive hyperbolic elements, and for an endomorphism L : V → V , S k (L) denotes the action of L on the symmetric tensor product V ⊗k sym . Proof. It is easy to see that log of the right hand side (5.22) is the same as

[γ]∈PΓ hyp ∞ k=0 tr log Id -σ ± (m γ ) ⊗ S k (P (C γ ) |n )e -(λ+n)l(Cγ ) = - [γ]∈PΓ hyp ∞ k=0 ∞ j=1 j -1 tr σ ± (m γ ) ⊗ S k (P (C γ ) |n )e -(λ+n)l(Cγ ) j = - [γ]∈Γ hyp ∞ k=0 j(γ) -1 tr(σ ± (m γ )) tr(S k (P (C γ ) |n ))e -(λ+n)l(Cγ ) = - [γ]∈Γ hyp j(γ) -1 det(Id -P (C γ ) |n ) -1 tr(σ ± (m γ )) e -(λ+n)l(Cγ ) .
Now these equalities complete the proof if we use (5.11).

The Selberg zeta function of odd type is defined by

(5.23) Z o Γ,Σ (λ) = Z Γ (σ + , λ) Z Γ (σ -, λ) for ℜ(λ) > δ Γ -n.
Note that the definition in (5.23) is shifted by -n from the one in [START_REF] Millson | Closed geodesics and the η-invariant[END_REF], [START_REF] Park | Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps[END_REF]. From this definition, the following equality follows easily (5.24) Cγ ) .

∂ λ log Z o Γ,Σ (λ) = [γ]∈Γ hyp l(C γ )j(γ) -1 D(γ) -1 (χ σ+ (m γ ) -χ σ-(m γ ))e -λl(
By Proposition 5.4, and from the identity

∞ 0 e -tλ 2 (4πt) -3 2 e -r 2 4t dt = e -λr 4πr for ℜ(λ 2 ) > 0, we have Corollary 5.7. For λ such that ℜ(λ) > δ Γ -n and ℜ(λ 2 ) > 0, ∞ 0 e -tλ 2 Tr(De -tD 2 ) dt = i 2 ∂ λ log Z o Γ,Σ (λ). (5.25)
Let us define the function ω λ on the fundamental domain X Γ by (5.26)

ω λ (m) := tr[DR(λ; m, m ′ ) -DR H d+1 (λ; m, m ′ )] m=m ′ where R H d+1 (λ) is the meromorphic extension of the resolvent of D 2 H d+1 . The kernel of D(R(λ) - R H d+1 (λ)) is smooth in X Γ × X Γ
and, since the function ω λ is automorphic on H d+1 with respect to Γ, it induces a smooth function on X × X. From the analysis of the resolvent above, we see that ω λ is in x 2λ+d C ∞ ( X) and thus integrable when ℜ(λ) > 0; moreover it is meromorphic in C. By reversing the order of integration and trace in (5.25), we can write for ℜ(λ) > max(δ Γ -n, 0)

(5.27) ∂ λ log Z o Γ,Σ (λ) = ∂ λ Z o Γ,Σ (λ) Z o Γ,Σ (λ) = -2i XΓ ω λ (m) dv(m)
and the integral of ω λ (m) can be decomposed under the form x>ǫ0 and x<ǫ0 for some boundary defining function x, so that it can be decomposed under the sum of a meromorphic function of λ and of

lim ǫ→0 ǫ0 ǫ x 2λ-1 ω ′ λ (x, y) dx dv h0 (y)
for some ω ′ λ smooth and meromorphic in λ. As ǫ → 0, this has an expansion of the form A(λ) + ∞ j=0 ǫ 2λ+j C j (λ) for some meromorphic C j (λ), A(λ), and for ℜ(λ) > max(δ Γ -n, 0) we have (5.27) which is equal to A(λ) + x(m)>ǫ0 ω λ (m) dv(m). This shows Lemma 5.8. The function

∂ λ Z o Γ,Σ (λ)/Z o Γ,Σ (λ) has a meromorphic extension to C given by the value ∂ λ Z o Γ,Σ (λ) Z o Γ,Σ (λ) = -2i FP ǫ→0 x(m)>ǫ ω λ (m) dv(m)
where FP ǫ→0 means the finite part as ǫ → 0, that is the constant coefficient in the expansion in powers of ǫ, and ω λ is given in (5.26).

In addition, using that tr

[DR H d+1 (λ; m, m ′ ) -DR H d+1 (-λ; m, m ′ )] m=m ′ = 0 for all λ ∈ C, we obtain (5.28) ∂ λ Z o Γ,Σ (λ) Z o Γ,Σ (λ) - ∂ λ Z o Γ,Σ (-λ) Z o Γ,Σ (-λ) = -2i FP ǫ→0 x(m)>ǫ tr[DΠ(λ; m, m ′ )] m=m ′ dv(m) where Π(λ; m, m ′ ) := R(λ; m, m ′ ) -R(-λ; m, m ′ ).

Spectral side of trace formula, Maass-Selberg relation

With the only exception of Theorem 6.9, the dimension d + 1 in this Section can be either odd or even, and X can be any asymptotically hyperbolic manifold with constant curvature near infinity. By convention, if J(λ) is an operator depending on λ, we shall use the following notation throughout the section

J(λ) := J(λ)/C(λ), with C(λ) = 2 -2λ Γ(-λ + 1/2) Γ(λ + 1/2)
where the function C(λ), already introduced in (4.9), satisfies C(λ)C(-λ) = 1.

6.1. The Maass-Selberg relation. We now describe the Maass-Selberg relation in order to study the singularities of the odd Selberg zeta function in terms of scattering data. A corollary of the Lemma 4.7 is that the kernel of Π(λ) := R(λ) -R(-λ) is smooth on X × X. Actually, in the Mazzeo-Melrose construction described before, one can choose the same term Q 0 (λ) for the parametrix of R(λ) and R(-λ), proving directly that Π(λ) is the sum of a term whose lift under

β is smooth on X × 0 X \ (lb ∪ rb) with a term in (xx ′ ) λ+ d 2 C ∞ ( X × X; E) + (xx ′ ) -λ+ d 2 C ∞ ( X × X; E).
The local trace of Π(λ), i.e., the trace of the endomorphism Π(λ; m, m),

satisfies tr(Π(λ; m, m)) ∈ C ∞ ( X) + x 2λ+d C ∞ ( X) + x -2λ+d C ∞ ( X).
From the composition properties of Ψ * , * , * 0 ( X; 0 Σ) in Mazzeo [START_REF] Mazzeo | Elliptic theory of differential edge operators I[END_REF], the operator 1 DΠ(λ) has a kernel which has the exact same properties as Π(λ) and thus its local trace satisfies (6.1) tr

((DΠ)(λ; m, m)) ∈ C ∞ ( X) + x 2λ+d C ∞ ( X) + x -2λ+d C ∞ ( X). Lemma 6.1. Let ǫ > 0 and λ ∈ C neither a pole of R(λ) nor of R(-λ), then x(m)>ǫ tr(DΠ)(λ; m, m)dv g (m) = - ǫ -d 2 x(m)=ǫ ∂ X tr cl(ν)∂ λ E(λ;
x, y, y ′ ) E ♯ (-λ; y ′ , x, y) dv h0 (y ′ )dv hǫ (y).

Proof. First observe that

D E(λ) = -λ E(λ)cl(ν), E ♯ (-λ)D = -λcl(ν) E ♯ (-λ)
which is a consequence of Lemma 4.4 and the remark that follows. Then we get for small t ∈ C

1 2t D E(λ + t) E ♯ (-λ) -E(λ + t)( E ♯ (-λ)D) = - 1 2 E(λ + t)cl(ν) E ♯ (-λ).
From Lemma 4.7, the limit as t → 0 on the right hand side is -1 2 E(λ)cl(ν) E ♯ (-λ) = DΠ(λ), which by taking the local trace and using the fact that tr(AB) = tr(BA) gives

tr(DΠ(λ; m, m)) = lim t→0 1 2t tr E ♯ (-λ)D E(λ + t)(m, m) -( E ♯ (-λ)D) E(λ + t)(m, m) .
In particular, remark that the local trace on the right hand side has to vanish at t = 0, which will be used in the last equality below. We use this expression and Green's formula on {x(m) > ǫ} in 1 The notation DΠ should not be confused with the usual notation dΠ for the spectral measure! the dv g (m) integral to get

x(m)>ǫ tr(DΠ(λ; m, m))dv g (m) = lim t→0 1 2t ∂ X x(m)>ǫ tr E ♯ (-λ; y ′ , m)D E(λ + t; m, y ′ ) -( E ♯ (-λ)D)(y ′ , m) E(λ + t; m, y ′ ) dv g (m)dv h0 (y ′ ) = -ǫ -d lim t→0 1 2t x(m)=ǫ ∂ X tr E ♯ (-λ; y ′ , m)cl(ν) E(λ + t; m, y ′ ) dv h0 (y ′ ) dv hǫ (y) = - ǫ -d 2 x(m)=ǫ ∂ X tr E ♯ (-λ; y ′ , m)cl(ν)∂ λ E(λ; m, y ′ ) dv h0 (y ′ ) dv hǫ (y).
The Lemma is proved using the cyclicity of the local trace once again.

We now consider the limit of the expression in Lemma 6.1 as ǫ → 0. For that we introduce the representation of E(λ) given in Lemma 4.13 and a similar one for E ♯ (λ):

E ♯ (λ) = E ♯ 0 (λ) + E ♯ ∞ (λ) obtained by restricting (3.12) times x(m) -λ-d
2 at m ∈ ∂ X and using (3.11), that is

E ♯ 0 (λ) := j ι * j φ 1 j η -λ-d 2 j E ♯ H d+1 (λ)χ 2 j ι j * , E ♯ ∞ (λ) := 2λ[x -λ-d 2 ( R(λ) -R ♯ 0 (λ))] x=0 ∈ x ′ λ+ d 2 C ∞ (∂ X × X; E) (6.2)
where again E ♯ H d+1 (λ) is the corresponding operator on H d+1 like in Lemma 4.13 and R ♯ 0 (λ) := R 0 ( λ) * . Notice that, using the same arguments as in (4.7), we obtain

E ♯ 0 (λ; y ′ , m) = E 0 ( λ; m, y ′ ) * . Similarly we have S(λ) = S ♯ 0 (λ) + S ♯ ∞ (λ) with S ♯ 0 (λ) := j ι * j φ 1 j η -λ-d 2 j S H d+1 (λ)η -λ-d 2 j φ 2 j ι j * , S ♯ ∞ (λ) := 2λ[xx ′ -λ-d 2 ( R(λ) -R ♯ 0 (λ))] x=x ′ =0 ∈ C ∞ (∂ X × ∂ X; E) (6.3)
and S H d+1 (λ) is the operator on H d+1 like in Lemma 4.13. Then we can prove Proposition 6.2. The meromorphic identity holds in λ ∈ C,

x(m)>ǫ tr(DΠ(λ; m, m))dv g (m) = - ǫ -d 2 x(m)=ǫ ∂ X tr cl(ν)∂ λ E(λ; x, y, y ′ ) E ♯ ∞ (-λ; y ′ , x, y) + tr cl(ν)∂ λ E ∞ (λ; x, y, y ′ ) E ♯ 0 (-λ; y ′ , x, y) dv h0 (y ′ )dv hǫ (y) (6.4)
where tr means the local trace on End( 0 Σ).

Proof. The point is to prove the vanishing of

tr cl(ν)∂ λ E 0 (λ; x, y, y ′ ) E ♯ 0 (-λ; y ′ , x, y)
so we use the explicit formula for E 0 (λ) and E ♯ 0 (λ) given in Lemma 4.13 and (6.2). We have to deal with terms of the form

(6.5) tr ι * j cl(X j )∂ λ (χ 2 j E H d+1 (λ)φ 1 j η -λ-d 2 j )γ * jk φ 1 k η λ-d 2 k E ♯ H d+1 (-λ)χ 2 k ι k * (m, m)
where γ jk is the unique isometry of H d+1 extending ι k • ι -1 j : ι j (U k ∩ U j ) → ι k (U k ∩ U j ) (and which acts smoothly up to the boundary) and X j is the vector field X j := ι j * (ν). We use the fact that

γ * jk R H d+1 (λ) = R H d+1 (λ)γ * jk since γ jk is an isometry so if α jk := [γ * jk (x 0 )/x 0 ]| x0=0 ∈ C ∞ (R d ), then one deduces that γ * jk E ♯ H d+1 (-λ) = α λ-d 2 jk E ♯ H d+1 (-λ)γ * jk .
Let us consider (6.5), it can be written as A jk (λ; x 0 , y 0 , y ′ 0 ) :=χ 2 j (x 0 , y 0 )χ 2 k (γ jk (x 0 , y 0 ))

tr cl(X j )(∂ λ [ E H d+1 (λ; x 0 , y 0 , y ′ 0 )η j (y ′ 0 ) -λ-d 2 ]β jk (λ; y ′ 0 ) E ♯ H d+1 (-λ; y ′ 0 , x 0 , y 0 )
where ι j (m) = (x 0 , y 0 ) and β jk (λ; y ′ ) := η j (y ′ ) λ-d 2 φ 1 j (y ′ )φ 1 k (γ jk (y ′ )). We shall show that A jk vanishes for algebraic reasons. First we recall from (2.5) that

E H d+1 (λ; x 0 , y 0 , y ′ 0 ) = f (λ)x λ+ d 2 0 (x 2 0 + |y 0 -y ′ 0 | 2 ) -λ-d 2
U ((x 0 , y 0 ), y ′ 0 ) and a similar expression for E ♯ H d+1 , here U is the parallel transport on H d+1 extended to the boundary (see Appendix A) and f (λ) some explicit meromorphic function. Thus using the fact that U (m 0 , y ′ 0 )U (y ′ 0 , m 0 ) = Id, A jk can be written under the form A jk (λ; m 0 , y ′ 0 ) = b jk (λ; m 0 , y ′ 0 )tr(cl(X j )U (m 0 , y ′ 0 )U (y ′ 0 , m 0 )) = b jk (λ; m 0 , y ′ 0 )tr(cl(X j )) for some b jk where m 0 = (x 0 , y 0 ). But since the dimension d > 1, the trace vanishes.

We deduce from this formula Proposition 6.3. For λ ∈ C not a pole of S(λ) and S(-λ), the right-hand side term in Proposition 6.2 has a limit as ǫ → 0, given by

(6.6) lim ǫ→0 x(m)>ǫ tr(DΠ(λ; m, m))dv g (m) = - 1 2 Tr cl(ν)[∂ λ S(λ) S ♯ ∞ (-λ) + ∂ λ S ∞ (λ) S ♯ 0 (-λ)] ,
where Tr denotes the trace for trace-class operators.

Proof. First when d + 1 is odd, we know from the discussion before Lemma 5.8 that the term (6.4) has an expansion as ǫ → 0 of the form A(λ)

+ ∞ j=0 ǫ -2λ+j C - j (λ) + ∞ j=0 ǫ 2λ+j C + j (λ)
for some meromorphic functions A(λ), C ± j (λ). But actually the same result holds for the general AH manifolds where the metric has constant curvature near ∞ and d + 1 odd or even: indeed, using the parametrix (3.10) and the fact that the local trace tr(D(R H d+1 (λ) -R H d+1 (-λ))) vanishes as explained in the Remark following Proposition 2.1, it is clear that tr(DΠ(λ; m, m)) is a function in the class x 2λ C ∞ ( X) + x -2λ C ∞ ( X). Let us then take the limit as ǫ → 0 in (6.4). For instance consider

x -d tr cl(ν)∂ λ E(λ; x, y, y ′ )) E ♯ ∞ (-λ; y ′ , x, y) , we can use the arguments used in the proof of Theorem 3.10 of [START_REF] Guillarmou | Generalized Krein formula, Determinants and Selberg zeta function in even dimension[END_REF] (in the present case this is even simpler since they correspond only to the mixed terms there, which comes from the regularity Thus the sum of (6.7) and (6.8) integrates in y, y ′ to a function of x of the form α(λ) log(x) + β(λ) + O(x log(x)) for some meromorphic function α(λ), β(λ) which we can express in terms of the scattering operators. But from the discussion before, we also know that this trace has no log(x) coefficients and so α(λ) = 0, which ends the proof by letting x → 0 and writing β(λ) in terms of S(λ), S ∞ (λ), S ♯ ∞ (-λ) and S ♯ 0 (-λ) from (6.7), (6.8). Proof. Since ∂ λ S(λ) S ♯ ∞ (-µ) and ∂ λ S ∞ (λ) S ♯ 0 (-µ) have smooth kernels, it is clear that their supertrace extends meromorphically to C and is analytic at µ = λ by assumption on λ. Now if we show (6.10) s-Tr(∂ λ S 0 (λ) S ♯ 0 (-µ)) = 0 then we have proved the corollary in view of Proposition 6.3. We have to study terms of the form (6.11) tr

E ♯ ∞ (-λ) ∈ x -λ+ d 2 C ∞ (∂ X × X; E))
ι * j cl(ν)∂ λ [η -λ-d 2 j φ 2 j S H d+1 (λ)φ 1 j η -λ-d 2 j ]γ * jk η µ-d 2 k φ 1 k S H d+1 (-µ)φ 2 k η µ-d 2 k ι k *
where γ jk is the unique isometry of

H d+1 extending ι k • ι -1 j : ι j (U k ∩ U j ) → ι k (U k ∩ U j
), which acts also as a conformal transformation on ∂ B ⊂ R d . As above we use the fact that γ

* jk R H d+1 (-µ) = R H d+1 (-µ)γ * jk since γ jk is an isometry so if α jk := [γ * jk (x 0 )/x 0 ]| x=0 ∈ C ∞ (R d ), then one deduces that γ * jk S H d+1 (-µ) = α µ-d 2 jk S H d+1 (-µ)α µ-d 2 jk γ * jk .
So the term (6.11) is equal to (6.12)

∂ B×∂ B tr cl(ν)∂ λ η j (y) -λ-d 2 φ 2 j (y) S H d+1 (λ; y, y ′ )φ 1 j (y ′ )η j (y ′ ) -λ-d 2 η j (y ′ ) µ-d 2 φ 1 k (γ jk (y ′ )) S H d+1 (-µ; y ′ , y)φ 2 k (γ jk (y))η j (y) µ-d 2 (η j (y)η j (y ′ )) d dydy ′ .
Using the explicit formula of S H d+1 (λ; y, y ′ ), we see that the local trace of the operator above, i.e., the integrand in (6.12), can be written under the form f jk (λ, µ; y, y ′ )tr(cl(ν)U (y, y ′ )U (y ′ , y))

for some function f jk and where U (y, y ′ ) is the parallel transport map on spinors on H d+1 studied in Appendix A and extended down to the boundary R d . Thus since U (y, y ′ )U (y ′ , y) = Id and tr(cl(ν)) = 0, we obtain that (6.12) vanishes, which finishes the proof. It is clear from Corollary 6.5 that F (λ) is meromorphic in λ ∈ C, but we want to prove that it has only first order poles, the residues of which are integers. Since S(λ) is unitary on {ℜ(λ) = 0} it is analytic at λ = 0, so one can define (6.14)

S ± (λ) := S ± (λ) S ∓ (0) : C ∞ (∂ X; 0 Σ ∓ ) → C ∞ (∂ X; 0 Σ ∓ )
which are the two diagonal components of S(λ) := S(λ) S(0) in the splitting 0 Σ + ⊕ 0 Σ -. These two operators are elliptic pseudo-differential operators of complex order 2λ by Proposition 4.9

S ± (λ) ∈ Ψ 2λ (∂ X; 0 Σ ∓ ⊠ 0 Σ * ∓ )
, and their principal symbol is |ξ| 2λ . Let D be the Dirac operator on (∂ X, h 0 ) and let

D ∓ = P ± DI ∓ : C ∞ (∂ X; 0 Σ ∓ ) → C ∞ (∂ X; 0 Σ ± ) be the off diagonal components of D. If |D| ∓ := (D ± D ∓ ) 1 2
, it is possible to factorize S ± (λ) by (Id + |D| ∓ ) -λ S ± (λ)(Id + |D| ∓ ) -λ and this operator is of the form Id + K(λ) for some meromorphic family of compact operators on L 2 (∂ X; 0 Σ ∓ ), it is thus Fredholm on this space. Then we can use the theory of Gohberg-Sigal [START_REF] Gohberg | An Operator Generalization of the logarithmic residue theorem and the theorem of Rouché[END_REF] like in [START_REF] Guillopé | Scattering asymptotics for Riemann surfaces[END_REF] or Section 2 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] for these operators. In particular, one can define the null multiplicities N λ0 (S ± (λ)) of S ± (λ) at a point λ 0 as follows: by the theory of [START_REF] Gohberg | An Operator Generalization of the logarithmic residue theorem and the theorem of Rouché[END_REF], for a meromorphic family of operators L(λ) = Id + K(λ) acting on a Hilbert space H with K(λ) compact, with L(λ) invertible for some λ, there exist holomorphically invertible operators U 1 (λ), U 2 (λ) near λ 0 , some (k l ) l=0,...,m ∈ Z\{0} (with m ∈ N) and some orthogonal projectors P l on L 2 (∂ X; 0 Σ ± ) such that rank(P l ) = 1 if l > 0, P i P j = δ ij and (6.15)

L(λ) = U 1 (λ) P 0 + m l=1 (λ -λ 0 ) k l P l U 2 (λ),
then we define the null multiplicity at λ 0 by (6.16)

N λ0 (L(λ)) := k l >0 k l .
Note that, by [START_REF] Gohberg | An Operator Generalization of the logarithmic residue theorem and the theorem of Rouché[END_REF], this is an integer depending only on S ± (λ) and not on the factorization (6.15) and that N λ0 (L(λ)) = 0 if and only if L(λ) -1 is holomorphic at λ = λ 0 .

Proposition 6.6. The function F (λ) of (6.13) is meromorphic in λ ∈ C, one has (6.17)

F (λ) = FP µ=λ TR(∂ λ S -(λ)S -1 -(µ)) -FP µ=λ TR(∂ λ S + (λ)S -1 + (µ))
where TR is the Kontsevich-Vishik trace of [START_REF] Kontsevich | Determinants of elliptic pseudo-differential operators[END_REF] and FP µ=λ means the finite part (or regular value) of the meromorphic function of µ at µ = λ. The poles of F (λ) are first order poles, with residue at a pole λ 0 given by

Res λ=λ0 F (λ) = Tr(Res λ=λ0 (∂ λ S -(λ)S -(λ) -1 )) -Tr(Res λ=λ0 (∂ λ S + (λ)S + (λ) -1 )) = (N λ0 (S -(λ)) -N λ0 (S -(λ) -1 )) -(N λ0 (S + (λ)) -N λ0 (S + (λ) -1 ))
where N λ0 is the null multiplicity defined in (6.16).

Proof. The first statement is straightforward since s-Tr(∂ λ S(λ) S(-µ)) = TR(∂ λ S -(λ)S -1 -(µ)) -TR(∂ λ S + (λ)S -1 + (µ)) and we know from the work of Lesch [21] that the Kontsevich-Vishik trace of an analytic family of log-polyhomogeneous operators A(µ) extend meromorphically to µ ∈ C, so it suffices to use the fact that s-Tr(∂ λ S(λ) S(-µ)) analytically continues to µ ∈ C and is analytic at µ = λ to prove (6.17).

As shown in Proposition 6.3, F (λ) can be written as a trace of a meromorphic family of trace class operators, more precisely, using the fact that cl(ν) anti-commutes with S(λ) and S 0 (λ) for all λ where they are defined,

F (λ) = 1 i tr cl(ν)(∂ λ S(λ) S(λ) -1 -∂ λ S 0 (λ) S ♯ 0 (-λ))
. Consequently, the polar part of F (λ) at a pole λ 0 is given by the trace of the polar part (which is finite rank

) of cl(ν)(∂ λ S(λ) S(λ) -1 -∂ λ S 0 (λ) S ♯ 0 (-λ)). But clearly from the explicit formula of S H d+1 (λ), we see that ∂ λ S 0 (λ) S ♯ 0 (-λ) is holomorphic in λ ∈ C. Now use that S(0) 2 = Id to write cl(ν)∂ λ S(λ) S(λ) -1 = + iI + ∂ λ ( S -(λ) S + (0))( S -(λ) S + (0)) -1 P + -iI -∂ λ ( S + (λ) S -(0))( S + (λ) S -(0)) -1 P - (6.18)
and write a factorization of the form (6.15) for S ± (λ), from which it is clear that ∂ λ S ± (λ)S ± (λ) -1 has only first order poles except possibly

U 1 (λ)(P 0 + m l=1 (λ -λ 0 ) k l P l )∂ λ U 2 (λ)U 2 (λ) -1 ( m l=1 (λ -λ 0 ) -k l P l )U 1 (λ) -1 +U 1 (λ)( m l=1 (λ -λ 0 ) k l P l )∂ λ U 2 (λ)U 2 (λ) -1 (P 0 + m l=1 (λ -λ 0 ) -k l P l )U 1 (λ) -1 .
Theorem 6.8. The function F (λ) is meromorphic with first order poles and integer residues given by Res

λ=λ0 F (λ) = m + (λ 0 ) -m -(λ 0 ) + 1l -1/2-N0 (λ 0 )Ind(S -(-λ 0 )) = m + (λ 0 ) -m -(λ 0 ).
for ℜ(λ 0 ) ≤ 0, where m ± (λ 0 ) is defined in (6.19).

Proof. Apply Proposition 6.7 with Proposition 6.6. To see the index of S -(-λ 0 ) appearing, we also use that S + (-λ 0 ) * = S -(-λ 0 ) for λ 0 ∈ R, which comes from the self-adjointness of S(-λ 0 ). The fact that the index of S -(-λ 0 ) vanishes comes from the invariance of the index by continuous deformation and the invertibility of S(λ) except on a discrete set of λ ∈ C.

We deduce directly our main theorem from this Theorem, Corollary 6.5 and the identity (5.28):

Theorem 6.9. The odd Selberg zeta function Z o Γ,Σ (λ) on a spin convex co-compact hyperbolic manifold X Γ of dimension 2n + 1 has a meromorphic extension to C, is analytic in a neighborhood of the right half plane {ℜ(λ) ≥ 0}, and λ 0 is a zero or pole if and only if the meromorphic extension R + (λ) or R -(λ) of (D ± iλ) -1 from {ℜ(λ) > 0} to C have a pole at λ 0 , in which case the order of λ 0 as a zero or pole of Z o Γ,Σ (λ) (with the positive sign convention for zeros) is given by rank Res λ0 R -(λ)rank Res λ0 R + (λ).

Eta invariant of Dirac operator

In this section, we will assume that the dimension of X Γ is odd, that is, d + 1 = 2n + 1. First, we prove Proposition 7.1. Using notation (5.10), the following estimate holds

Tr(De -tD 2 ) = O(t -1 )
as t → ∞.

Proof. Since the claim easily follows from (5.5) when δ Γ < n (and actually in that case one gets directly O(t -3/2 ) instead of just O(t -1 )), we assume that δ Γ ≥ n in the following proof.

Let us write the operator De -tD 2 as a contour integral (7.1)

De -tD 2 = 1 2πi Λ e -tλ 2 D(D 2 -λ 2 ) -1 2λdλ
where Λ = {re i(π+π/8) ; r > 0} ∪ {re -iπ/8 ; r > 0} oriented from +∞e -iπ/8 towards +∞e i(π+π/8) . Let us check the identity (7.1): by Corollary 3.5, we have that D(D 2 -λ 2 ) -1 = DR(iλ) is holomorphic in ℑ(λ) ≤ 0 as an operator bounded from x ǫ L 2 to x -ǫ L 2 for all ǫ > 0, moreover

||DR(iλ)|| L 2 →L 2 = O(1/|ℑ(λ)|)
when |λ| → ∞ thus the integral converges in the operator norm of L(x ǫ L 2 , x -ǫ L 2 ). Moreover, applying the integral (7.1) to a C ∞ 0 (X) function f defines a function u(t) and since -λ 2 (D 2 -λ 2 ) -1 = Id-D 2 (D 2 -λ 2 ) -1 , we see that the integral converges in C ∞ (X) uniformly in t ∈ [0, ∞) and solves ∂ t u = -D 2 u with u(0) = Df . It is easy to prove that the C k norms of the integral kernels in (7.1) also converge by applying powers of D 2 on the right and the left and using Sobolev embeddings. The same is true for the integral kernel of the operator D H d+1 e -tD We thus use that for ℜ(λ) > d, one has

)(m) = 1 2πi Λ e -tλ 2 ω iλ (m)2λdλ with ω λ (m) = tr(DR(λ)-DR H d+1 (λ))(m). Since ω iλ ∈ x 2iλ+d C ∞ ( X) is holomorphic in ℑ(λ) ≤ 0, it is in L 1 (X) if ℑ(λ) < 0,
ω λ (m) = tr [DR(λ; m, m ′ ) -DR H d+1 (λ; m, m ′ )] m=m ′ ) = γ∈Γ\Id tr(DR H d+1 (λ; m, γm)γ * m )
where γ * m : 0 Σ m → 0 Σ γm is the action induced by γ on the spinor bundle. The map γ * is given by γ (2.4). Using (2.5), the Euler integral formula for the hypergeometric function

* [g, v] = [γg, v] for [g, v] ∈ 0 Σ m , [γg, v] ∈ 0 Σ γm under the identification in
F d + 1 2 + λ, λ + 1, 2λ + 1; z = Γ(2λ + 1) Γ(λ + 1)Γ(λ) 1 0 t λ (1 -t) λ-1 (1 -tz)
-tz > 1 -z = tanh 2 (d H d+1 (m, γm)/2) when z = cosh -2 (d H d+1 (m, γm)/2), we obtain |ω λ (m)| ≤ C Γ( d+1 2 +λ)Γ(λ+1) Γ(2λ+1) γ∈Γ\{Id} sinh(d H d+1 (m, γm)/2) -d-2ℜ(λ) .
Using the Legendre duplication formula, the term containing Gamma functions is uniformly bounded for ℜ(λ) > 0. Since the injectivity radius of X = Γ\H d+1 is strictly positive, i.e., Now for those (finitely many) γ ∈ Γ \ {Id} for which Lemma 5.2 possibly does not hold we still have the weaker inequality e -d H d+1 (m,γm) ≤ Cx(m) 2 for some C > 0, where x is a boundary defining function. In particular e -λd H d+1 (m,γm) descends to an L 1 function on X if ℜ(λ) > d/2. Combining finally with Lemma 5.2 and (7.2), the convergence of Poincaré series in {ℜ(λ) > d} implies the bound in the same half-space

||ω λ || L 1 (X) ≤ e Cℜ(λ)
and this ends the proof.

We define the eta invariant of D by

(7.3) η(D) = 1 √ π ∞ 0 t -1 2 Tr(De -tD 2 ) dt
where the trace Tr means the integral of the local trace like in (5.10). Note that the integral on the right hand side of (7.3) is finite by Corollary 5.5 and Proposition 7.1. Theorem 6.9 about meromorphic extension of Z o Γ,Σ (λ) and its analyticity on [0, ∞) implies directly the following result: Theorem 7.2. The eta invariant of the Dirac operator over a convex co-compact hyperbolic manifold X Γ satisfies (7.4) exp(πiη(D)) = Z o Γ,Σ (0).

Proof. We start by writing

t -1/2 = 2 √ π ∞ 0 e -λ 2 t dλ,
then we have by (5.25)

η(D) = 2 π ∞ 0 ∞ 0 e -λ 2 t Tr(De -tD 2 ) dλ dt = i π ∞ 0 ∂ λ Z o Γ,Σ (λ) Z o Γ,Σ (λ)
dλ and this concludes the proof by Theorem 6.9, in particular, the meromorphic extension of Z o Γ,Σ (λ) over C with λ = 0 as a regular value.

8. Eta invariant of odd signature operator and its structure on Schottky space For a (4m -1)-dimensional convex co-compact hyperbolic manifold X Γ = Γ\SO 0 (4m -1, 1)/SO(4m -1), we consider the odd signature operator A on odd forms Λ odd = ⊕ 2m p=0 Λ 2p-1 acting by (-1) m+p (⋆d+ d⋆) over Λ 2p-1 as in Millson's paper [START_REF] Millson | Closed geodesics and the η-invariant[END_REF]. Recall that A 2 = ∆ and A is self adjoint. We want to make a sense of (8.1)

η(A) := 1 √ π ∞ 0 t -1 2 XΓ Tr Λ (Ae -t∆ )(m) dv(m) dt
where Tr Λ is the local trace on the bundle Λ odd . First it is easy to see that Tr Λ (Ae -t∆ ) is the same as Tr Λ 2m-1 (⋆d e -t∆ ) since the other parts are off diagonal if we write Ae -t∆ as a matrix with respect to the natural basis of Λ odd . First we show that the local trace tr Λ 2m-1 (⋆d e -t∆ ) is integrable on X Γ . As in the spinor bundle Σ, the bundle of (2m -1)-forms can be understood as a homogeneous vector bundle given by the representation Λ 2m-1 φ with the standard representation φ of SO(4m -1), which decomposes into

(8.2) Λ 2m-1 φ | SO(4m-2) = Λ 2m-1 + φ ⊕ Λ 2m-1 - φ ⊕ Λ 2m-2 φ
where φ denotes the standard representation of SO(4m -2). As in the subsection 5.1, there is the Λ 2m-1 φ-radial function P t associated to ⋆d e -t∆ , and we have the corresponding scalar functions p ± t (r), p 2m-2 t (r) of P t restricting to the representation spaces on the right hand side of (8.2). Now, as in Proposition 5.1, we have Proposition 8.1. The scalar components p ± t (r), p 2m-2 t (r) are given by

p ± t (r) = ± (4m -1) sinh(r) i2 2m-1/2 π 2m+1/2 t 3/2 - d d(cosh r) 2m-1 r sinh -1 (r) e -r 2 4t , p 2m-2 t (r) ≡ 0.
Proof. The equalities follow from Lemma 7.4 and Theorem 7.6 in [START_REF] Pedon | Harmonic analysis for differential forms on real hyperbolic spaces[END_REF] and Theorem 1.1 in [START_REF] Millson | Closed geodesics and the η-invariant[END_REF].

Using this proposition and repeating the same argument as in Section 5, one can easily show that tr Λ 2m-1 (⋆d e -t∆ ) is integrable over X Γ . By the same argument as in Proposition 5.4 and Corollary 5.5, one can also obtain the corresponding results, which implies that the small time part of the integral (⋆d e -t∆ ) is integrable on X for all t > 0, so that the integral (8.1) converges and defines the eta invariant η(A). Moreover, we also have

(8.
3) e πiη(A) = Z o Γ,Λ (0).

Proof. We already showed the first claim. The equality (8.3) also easily follows from the corresponding results to Proposition 5.4 and the assumption δ Γ < 2m -1, from which we do not need to show the meromorphic extension of Z o Γ,Λ (λ) at λ = 0. It turns out that this eta invariant η(A) of the odd signature operator A has an intimate relationship with the deformation space of the hyperbolic structures when X Γ is 3-dimensional. To explain this, first we review the work of Zograf [START_REF] Zograf | Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces[END_REF].

8.1. Zograf factorization formula. A marked Schottky group is a discrete subgroup Γ of the linear fractional transformations PSL(2, C), with distinguished free generators γ 1 , γ 2 , . . . , γ g satisfying the following condition: there exist 2g smooth Jordan curves C r , r = ±1, . . . , ±g, which form the oriented boundary of a domain Ω 0 in Ĉ = C ∪ {∞} such that γ r C r = -C -r for r = 1, . . . , g.

If Ω is the union of images of Ω 0 under Γ, then Y Γ = Γ\Ω is a compact Riemann surface of genus g. The action of Γ on C naturally extends to the action on H 3 where ∂H 3 = C and the quotient space X Γ = Γ\H 3 is a Schottky hyperbolic 3-manifold whose boundary is the Riemann surface Y Γ . Here let us remark that δ Γ is the Hausdorff dimension of the limit set Λ in ∂H 3 of Γ and δ Γ is also the smallest number such that {γ} (1 -q s γ ) absolutely converges whenever ℜ(s) > δ Γ . The function {γ} (1 -q s γ ) was briefly described in [START_REF] Bowen | Hausdorff dimension of quasicircles[END_REF] where it was asserted without the proof that with the values of q s γ chosen appropriately, the infinite product is defined for ℜ(s) > δ Γ and has an analytic continuation to C.

Each nontrivial element γ ∈ Γ is loxodromic: there exists a unique number q γ ∈ C (the multiplier) such that 0 < |q γ | < 1 and γ is conjugate in PSL(2, C) to z → q γ z, that is, γz -a γ γz -b γ = q γ z -a γ z -b γ for some a γ , b γ ∈ Ĉ (the attracting and repelling fixed points respectively). A marked Schottky group with an ordered set of free generators γ 1 , . . . , γ g is normalized if a γ1 = 0, b γ1 = ∞, a γ2 = 1.

The Schottky space S g is the space of marked normalized Schottky groups with g generators. It is a complex manifold of dimension 3g -3, covering the Riemann moduli space M g and with universal cover the Teichmüller space T g . Like the Teichmüller space T g , the Schottky space S g has a natural Kähler metric, the Weil-Petersson metric. In [START_REF] Takhtajan | On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces[END_REF] where ∂ and ∂ are the (1, 0) and (0, 1) components of the de Rham differential d on S g respectively, and ω W P is the symplectic form of the Weil-Petersson metric. On the other hand, from the local index theorem for families of ∂-operators in Takhtajan-Zograf [START_REF] Takhtajan | A local index theorem for families of ∂-operators on Riemann surfaces[END_REF], the following equality also follows where c g is a constant depending only on g. For points in S g corresponding to Schottky groups Γ with δ Γ < 1, the function F (Γ) is given by the following absolutely convergent product: which are smooth on the stretched product H d+1 × 0 H d+1 defined in subsection 3.2.

Proof. Let A be the translation by (0, y ′ ) in R d+1 , composed to the left by the homothety of factor 1/x ′ . This isometry of H d+1 maps m ′ to (1, 0). In the above trivialization of the tangent bundle, A * acts as the identity. Moreover, since it is an isometry, A transforms the geodesic from m to m ′ into the geodesic from A(m) to (1, 0) and preserves parallelism. Thus (as matrices)

τ (m, m ′ ) = τ (A(m), (1, 0)), where A(m) = x x ′ , y -y ′ x ′ . (A.1)

We now concentrate on τ (m, (1, 0)).

If y = 0 it is clear that τ ((x, 0), (1, 0)) is just the identity matrix. Suppose y = 0. Let ∂ r := 1 r d j=1 y j ∂ yj denote the radial vector field, defined outside the vertical line through the origin. Define R := x∂ r , X := x∂ x . For each j ≥ 1 set e j := x∂ yj , and let T j := e j -e j , R R denote the component of e j which is tangent to the sphere S d-1 . The geodesic from m to (1, 0) lives in the totally geodesic plane Π m passing through (1, 0) and m, which is a copy of the hyperbolic 2-space. Along this plane the vector fields T j extend smoothly at the vertical line through the origin. It is clear that the vector fields T j are parallel along Π m . Lemma A.2. In the plane Π m , parallel transport between m and (1, 0) is given by the complex number -r + i(1 + x) r + i(1 + x)

Proof. We use as (real) basis for T Π m the orthonormal vector fields X and R. The complex structure rotates R to X. The formula is deduced from the similar formula in H 2 .

Equivalently, in the basis {X, R}, parallel transport is given by the 2 × 2 orthogonal matrix ρ -2 ff (x + 1) 2 -r 2 2r(x + 1) -2r(x + 1) (x + 1) 2 -r 2 .

We decompose a vector V = a 0 X + d j=1 a j e j into its tangent, respectively orthogonal components to Π m as follows: Corollary A.4. Let m ′ = (1, 0), m = (0, rω). In the limit r → ∞, the parallel transport U (m, m ′ ) tends to -cl(X)cl(R).

V = a 0 X +  

(2. 3 )

 3 dg = a 2ρ dk da dn = a -2ρ dn da dk where g = kan is the Iwasawa decomposition and a 2ρ = exp(2ρ(log a)). Here dk is the Haar measure over K with K dk = 1, da is the Euclidean Lebesgue measure on A given by the identification A ∼ = R via a r = exp(rH) with H ∈ a, β(H) = 1, and dn is the Euclidean Lebesgue measure on N induced by the normalized Cartan-Killing form •, • given in (2.1).

Lemma 3 . 1 .

 31 The analytically extended resolvent R H d+1 ± (λ) := (D H d+1 ±iλ) -1 of the Dirac operator

Definition 4 . 2 . 1 . 4 . 3 .

 42143 The Eisenstein series is the operator E(λ) : C ∞ (∂ X; 0 Σ) → C ∞ (X; 0 Σ) defined by E(λ)ψ := σ where σ is the smooth spinor in Lemma 4.Definition The scattering operator S(λ) : C ∞ (∂ X; 0 Σ) → C ∞ (∂ X; 0 Σ) is defined by S(λ)ψ := σ + | ∂ X where σ + is the smooth spinor in Lemma 4.1.

Lemma 4 . 10 . 1 . 4 . 11 .

 4101411 The operator S(λ) is unitary on {ℜ(λ) = 0}, it satisfies S(λ)S(-λ) = Id for λ such that S(±λ) is defined, and it is conformally covariant in the sense that for another choice x = e ω x of geodesic boundary defining function, the corresponding scattering operator isŜ(λ) = e -( d 2 +λ)ω0 S(λ)e ( d 2 -λ)ω0, where ω 0 = ω| ∂ X .Proof. The functional equation is a straightforward consequence of the uniqueness in Lemma 4.1 or the first equality of Corollary 4.8. The unitarity follows easily from Lemma 3.3 by taking the solutions σ 1 , σ 2 of Lemma 4.1 for two initial data ψ 1 , ψ 2 ∈ C ∞ (∂ X; 0 Σ). The conformal covariance of S(λ) is straightforward by using the uniqueness of the solution in Lemma 4.Corollary If the metric g is even in the sense of (3.2), the operator S(λ) := S(λ)/C(λ) is finite meromorphic in C, and it is holomorphic in {ℜ(λ) ≥ 0}.

  0)) = 0 where d(σ ± ) denotes the dimension of V σ± and the last equality follows from (5.3). By equations (5.5), (5.7) and Proposition 5.3, we deduce that there is ǫ > 0 such that tr(De -tD 2 )(m) ≤ C ǫ (t) x(m) 2( d 2 +ǫ) γ∈Γ\{e} e -(d+ǫ)rγ where C ǫ (t) is a constant depending only on ǫ, t and x a boundary defining function. Hence the local trace function tr(De -tD 2 )(m, m) is integrable over X Γ . Now we can define (5.10) Tr(De -tD 2 ) := XΓ tr(De -tD 2 )(m) dv(m)

  g) dg and for (σ, H σ ) ∈ M (where M denotes the set of equivalence classes of irreducible unitary representations of M ) and λ ∈ a * C the principal representation π σ,λ := Ind G MAN (σ ⊗ e iλ ⊗ Id) of G acts on the space

  [γ]∈Γ hyp l(C γ ) > 0 we obtain the Corollary 5.5. The following estimate holds Tr(De -tD 2 ) = O(e -c 2 /t ) as t → 0. 5.3. Selberg zeta function of odd type. We define the Selberg zeta functions attached to half spinor representations σ ± by (5.21)

  and we obtainx -d tr cl(ν)∂ λ E(λ; x, y, y ′ ) E ♯ ∞ (-λ; y ′ , x, y) = log(x)tr cl(ν) S(λ; y, y ′ ) S ♯ ∞ (-λ; y ′ , y) + tr cl(ν)∂ λ S(λ; y, y ′ ) S ♯ ∞ (-λ; y ′ , y) + O(x log(x)). (6.7)Similarly we havex -d tr cl(ν)∂ λ E ∞ (λ; x, y, y ′ ) E ♯ 0 (-λ; y ′ , x, y)= log(x)tr cl(ν) S ∞ (λ; y, y ′ ) S ♯ 0 (-λ; y ′ , y) + tr cl(ν)∂ λ S ∞ (λ; y, y ′ ) S ♯ 0 (-λ; y ′ , y) + O(x log(x)).

Remark 6 . 4 .Corollary 6 . 5 .

 6465 By holomorphy on the continuous spectrum, the terms O(x log x) in (6.7) and (6.8) are continuous functions of λ ∈ iR, and thus G(ǫ, λ) := x(m)>ǫ tr(DΠ(λ; m, m))dv g (m) is a continuous function on [0, ǫ 0 ] × iR for some small ǫ 0 . Let us define the super trace of a trace class operator A on L 2 (∂ X, Let λ ∈ C be such that S(z) and S(-z) are analytic at z = λ, then the super trace s-Tr(∂ λ S(λ) S(-µ)) extends meromorphically in µ from ℜ(λ -µ) < -d to µ ∈ C, it is analytic in µ = λ, and the following identity holds lim ǫ→0 x(m)>ǫ tr(DΠ(λ; m, m))dv g (m) = -i 2 s-Tr(∂ λ S(λ) S(-µ))| µ=λ .

6. 2 .

 2 Analysis of residues of s-Tr(∂ λ S(λ) S(-λ)). Let us define F (λ) for the value at µ = λ of the meromorphic extension in µ of s-Tr(∂ λ S(λ) S(-µ)) (6.13) F (λ) := s-Tr(∂ λ S(λ) S(-µ))| µ=λ .

  and for ℑ(λ) ≤ 0 one has the asymptoticK(λ, ǫ) := 2λ x(m)>ǫ ω iλ (m)dv(m) = ǫ 2iλ F (λ) + λG(λ, ǫ)where F (λ) is a holomorphic function of λ and G(λ, ǫ) is continuous in (λ, ǫ) down to ǫ = 0 and holomorphic in λ. In particular for |λ| < 1, we have |K(λ, ǫ)| ≤ C for some constant C > 0 independent of (λ, ǫ) and lim ǫ→0 K(λ, ǫ) = λG(λ, 0) as long as ℑ(λ) < 0. We can thus use Lebesgue's dominated convergence theorem to deduce that lim ǫ→0 x(m)>ǫ Λ∩{|λ|<1} e -tλ 2 ω iλ (m)2λdλ dv(m) = Λ∩{|λ|<1} e -tλ 2 λG(λ, 0)dλ. Now if we can show that , for λ ∈ Λ ∩{|λ| > 1}, the estimate ||ω iλ || L 1 (X) = O(e α|λ| ) holds for some α > 0 we can use Lebesgue's dominated convergence theorem again for the integral corresponding to λ ∈ Λ ∩ {|λ| > 1}, and then changing variable u = √ tλ in the whole Λ integral gives directly lim ǫ→0 x(m)>ǫ tr(De -tD 2 )(m)dv(m) = O(t -1 ) as t → ∞.

  the expression of the Beta function B(λ + 1, λ) in terms of Gamma functions, and the obvious bound 1

  inf m∈H d+1 ,γ∈Γ\{Id} {sinh(d H d+1 (m, γm)/2)} > ǫ for some ǫ > 0, we deduce the estimate (7.2) |ω λ (m)| ≤ e Cℜ(λ) γ∈Γ\{Id} e -(d/2+ℜ(λ))d H d+1 (m,γm) .

∞ 0 • 1 ±Theorem 8 . 2 .

 0182 dt in (8.1) converges. The convergence of the large time part also follows from the corresponding computations to (5.5) and Lemma 5.2 under the conditionδ Γ < 2m -1. Hence the eta invariant η(A) given in (8.1) is well defined if δ Γ < 2m-1. For ℜ(λ) > δ Γ -(2m-1), we also have the Selberg zeta function of odd type Z o Γ,Λ (λ) just putting σ ± = Λ 2m-φ in (5.21) and (5.23), which coincides with the one in (1.1) introduced by Millson [27]. We first have a result similar to the case of spinor bundle dealt with above: If X Γ := Γ\H 4m-1 is a convex co-compact hyperbolic manifold with the Poincaré exponent δ Γ < 2m -1, then the local trace Tr Λ 2m-1

  , Takhtajan-Zograf constructed a Kähler potential S called classical Liouville action of the Weil-Petersson metric on S g , that is, (8.4) ∂∂S = 2i ω W P

where

  Det∆ and τ denote the ζ-regularized determinant of the Laplacian ∆ of hyperbolic metric and the period matrix respectively over the Riemann surface corresponding to an inverse image in T g of a point in S g . Let us remark that Det∆ and det Im τ descend to well-defined functions on S g . Comparing (8.4) with (8.5), one can expect a nontrivial relationship between S and log Det∆ det Im τ . Indeed, in[START_REF] Zograf | Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces[END_REF],[START_REF] Zograf | Determinants of Laplacians, Liouville action, and an analogue of the Dedekind η-function on Teichmüller space[END_REF] Zograf proved Theorem 8.3. [Zograf ] There exists a holomorphic function F (Γ) : S g → C such that (8.6) Det∆ det Im τ = c g exp -1 12π S |F (Γ)| 2

1 .

 1 Let r := |y -y ′ |, ρ ff := (x + x ′ ) 2 + r 2 . The special orthogonal matrix τ (m, m ′ ) has the following coefficients:τ 00 = 1 -2r 2 /ρ 2 ff τ 0j = -2(x + x ′ )(y j -y ′ j )/ρ 2 ff for j = 1, . . . , d τ j0 = 2(x + x ′ )(y j -y ′ j )/ρ 2 ff for j = 1, . . . , d τ jl = δ l j -2(y j -y ′ j )(y l -y ′ l )/ρ 2 fffor j, l ∈ {1, . . . , d}.

R 1 0 T H d+1 ⊠ π * 2 0

 12 j .Since T j are parallel, the orthogonal component is constant during parallel transport. Using (A.1) and Lemma A.2, we writeτ (m, m ′ )(a 0 X + d j=1a j e j ) =a 0 τ (X) + d j=1 a j e j , R τ (R) + from which the proposition follows, since (ρ ff , (y -y ′ )/ρ ff , y ′ , x/ρ ff , x ′ /ρ ff ) are smooth coordinates on the blow-up H d+1 × 0 H d+1 .The above oriented basis {X, e 1 , . . . , e d } of T H d+1 extends smoothly to the boundary {0} × R d of Hd+1 as an orthonormal basis of the zero tangent bundle with respect to the hyperbolic metric.Therefore, by the Proposition above, τ (m, m ′ ) is a smooth section on H d+1 × 0 H d+1 in the pull-back vector bundleβ * (π * T * H d+1 ).The orthonormal frame bundle P SO of the 0-tangent bundle is trivialized over H d+1 by the frame p = {X, e 1 , . . . , e d }, therefore the (unique) spin structure P Spin is identified with H d+1 ×Spin(d+1).Denote by p one of the lifts of p to P Spin . By definition of the lifted connection, parallel transport in P Spin of the section p along the geodesic from m to m ′ is p U (m, m ′ ), where U (m, m ′ ) the unique lift of the SO(d + 1)-valued function τ (m, m ′ ) to the Spin(d + 1) group, starting at the identity for m = m ′ . Thus, parallel transport of a constant section σ (with respect to the trivialization p) in the spinor bundle is simplyτ m ′ m [p, σ] = [p U (m, m ′ ), σ] = [p, U (m, m ′ )σ]where multiplication in the last term is the spinor representation. By abuse of notation we writeU (m, m ′ ) for τ m ′ m . Proposition A.3. Let m = (x, y), m ′ = (x ′ , y ′ ) ∈ H d+1 .In the above trivialization of the spinor bundle, parallel transport takes the formU (m, m ′ ) = x + x ′ ρ -r ρ cl(X)cl(R).Proof. We view the Spin(d + 1) group inside the Clifford algebra as the group generated by even Clifford products of unit vectors. The projection π : Spin(d + 1) → SO(d + 1) is given by the adjoint action in the Clifford algebra on vectors:π(c)(V ) := cV c -1 ,the kernel being precisely {±1}. We must therefore examine the adjoint action of A(m, m ′ ) := x+x ′ ρ ff -r ρ ff cl(X)cl(R) on 0 T H d+1 . Note that any Clifford element of the form α + βcl(X)cl(R) withα 2 + β 2 = 1 belongs to the Spin group. Next, A -1 (m, m ′ ) = x+x ′ ρ ff + r ρ ff cl(X)cl(R) so π(A(m, m ′ ))X = (x + x ′) the action of τ (m, m ′ ) on X from Proposition A.1. Similarly, for the vector fields T j from the proof of Proposition A.1 we have π(A(m, m ′ ))T j = T j = τ (m, m ′ )T j . Thus π(A(m, m ′ )) = τ (m, m ′ ). The proof is finished by noting that A(m, m ′ ) was normalized so that A(m, m) = 1.

  2 H d+1 in terms of the resolvent kernel D H d+1 (D 2 H d+1 -λ 2 ) -1 and by Proposition 5.3 we deduce that tr(De -tD 2

Acknowledgements The authors thank the KIAS Seoul (South Korea) where part of this work was started and completed. We are also grateful to the CIRM in Luminy for funding us through a 'Research in Pairs' in January 2008. C.G. was supported by grants ANR05-JCJC-0107091 and NSF0500788, he thanks the MSRI at Berkeley where part of this work was done. S.M. was supported by grant PN-II-ID-PCE 1188 265/2009. We finally thank Josh Kantor for useful comments.

But this is a finite rank operator and so by the cyclicity of the trace, we deduce that the trace of this term is holomorphic in λ. To finish the proof, it suffices to apply the main result of [START_REF] Gohberg | An Operator Generalization of the logarithmic residue theorem and the theorem of Rouché[END_REF]: tr(Res λ=λ0 (∂ λ S ± (λ)S ± (λ) -1 )) = N λ0 (S ± (λ)) -N λ0 (S ± (λ) -1 ).

Let us define the multiplicity of resonances as follows (6.19) m ± (λ 0 ) := rank(Res λ=λ0 R ± (λ))

We want to identify scattering poles and resonances.

Proposition 6.7. Let λ 0 ∈ C, then the following identity holds (6.20) N λ0 (S ∓ (-λ)) = m ± (λ 0 ) + 1l -1/2-N0 (λ 0 ) dim ker S ∓ (-λ 0 ).

Proof. We just sketch the proof since it is very similar to that of Theorem 1.1. of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF], and we strongly encourage the reader to look at [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF]. The first thing to notice is that

Remark that R ± (λ) and S ± (λ) are analytic in {ℜ(λ) ≥ 0} and so the identity (6.20) is trivial (all terms are 0) for ℜ(λ 0 ) ≥ 0. Now suppose that ℜ(λ 0 ) < 0. First we prove that (6.21)

By (4.4) and (4.5), S ± (λ) can be represented for

and the expression can be extended to λ ∈ C meromorphically as a singular integral kernel using the blow-down maps like in (4.10). Then we can apply mutatis mutandis Lemma 3.2 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF], where S(λ) there is replaced by S ± (λ) here, the function z(λ) there is λ here, and we have to multiply the factorization (3.11) of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] by S ∓ (0) on the right, which is harmless since it does not depend on λ. We want to apply the factorization of S(λ) obtained from this Lemma 3.2 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] to prove (6.21), in a way similar to Corollary 3.3 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF]. First Corollary 3.3 in [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] can also be rewritten (using the notations of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF]) under the form

by using equation (3.19) in [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] if λ 0 ∈ n/2 -N and the fact that c(n -λ) is holomorphic at all λ 0 / ∈ n/2 -N. Then the proof of this Corollary 3.3 in [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] can be copied word by word by replacing S(n -λ) and c(n -λ) there by S ∓ (-λ) and C(-λ) here, and m λ0 (z ′ (λ)R(λ)) by m ± (λ 0 ). This finally proves (6.21).

Then we need to prove the converse inequality of (6.21). From Lemma 4.7, Corollary 4.8 and the fact that S ∓ (0) S ± (0) = Id|0 Σ∓ , we deduce λ) which is the equivalent in our setting to the identity (3.15) of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF]. Since Lemma 3.4 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] is only based on the identity (3.15) in [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF], the structure of the resolvent kernel at the boundary and the unique continuation principle of Mazzeo [START_REF] Mazzeo | Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds[END_REF], the same proof applies and is actually easier in our case since there is no pure point spectrum thus no resonance in the physical sheet {ℜ(λ) ≥ 0}. This implies

). The idea of the proof of Lemma 3.4 of [START_REF] Guillarmou | Resonances and scattering poles on asymptotically hyperbolic manifolds[END_REF] is to use (6.22) to write the residue of R ± (λ) at λ 0 with ℜ(λ 0 ) < 0 in terms of the singular part of the Laurent expansion of S ± (λ), itself obtained from a factorization of the form (6.15), then use the fact that R ∓ (-λ), E ∓ (-λ) and E ♯ ∓ (-λ) are holomorphic in {ℜ(λ) < 0} and finally count the rank of the residue in terms of the k l of the factorization (6.15).

where q γ is the multiplier of γ ∈ Γ, and {γ} runs over all distinct primitive conjugacy classes in Γ excluding the identity.

Combining the equalities (8.6) and (8.7), these are called Zograf factorization formula. This result was extended by McIntyre-Takhtajan to the Schottky groups without the condition for δ Γ in [START_REF] Mcintyre | Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula[END_REF]. Here they used the ζ-regularized determinant of ∆ n acting on the space of n-differentials so that the corresponding holomorphic function is

) which absolutely converges for any Schottky group Γ if n > 1.

8.2. Eta invariant as a functional over the Schottky space. By the construction of X Γ and its boundary Y Γ , the eta invariant η(A) can be understood as a functional over the Schottky space S g . Now a natural question is to describe the eta invariant η(A) as a functional over S g . For this, we have Theorem 8.4. Let S 0 g be a subset of S g consisting of normalized Schottky groups Γ's with the property δ Γ < 1. Then we have

The proof is a simple application of the equality (8.3). For this, as in Proposition 5.6, we rewrite Z o Γ,Λ (λ) with respect to the group PSL(2, C) as follows:

Here γ runs over the set of Γ-conjugacy classes of the loxodromic elements in Γ and a loxodromic element γ can be conjugated to a diagonal matrix with the diagonal elements

. Let us remark that the infinite product on the right hand side of (8.8) absolutely converges for ℜ(λ) > δ Γ -1, in particular, at λ = 0 since δ Γ < 1. Now comparing the definition of q γ and µ γ , one can see that q γ = µ -2 γ , that is, q 1/2 γ = µ -1 γ . Hence the odd Selberg zeta function Z o Γ,Λ (0) has the following expression in terms of q γ ,

1 -(q γ q -1 γ )

Combining this and (8.3) completes the proof.

Remark. In the proof of Theorem 8.4, we assume the condition δ Γ < 1 which simplifies the proof in several steps. But, one can expect that a similar result still holds over the whole Schottky space S g . This extension to S g is also related to the proof of the assertion of Bowen in [START_REF] Bowen | Hausdorff dimension of quasicircles[END_REF] about the meromorphic extension of {γ} (1 -q s γ ) over C. These problems will be discussed elsewhere.

Appendix A. Computation of parallel transport in the spinor bundle Let τ m ′ m denote parallel transport in the tangent bundle of the upper half-space model H d+1 of hyperbolic space, between points m = (x, y), m ′ = (x ′ , y ′ ), along the unique geodesic linking them. We identify T m H d+1 with R d+1 using the orthonormal basis at m given by {x∂ x , x∂ y1 , . . . , x∂ y d }. We denote by τ (m, m ′ ) the matrix of the transformation τ m ′ m written in these bases.