The twisted Selberg trace formula and the twisted Selberg zeta function for compact orbifolds

We propose a version of the Selberg trace formula for compact hyperbolic orbifolds Γ\H2n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma \backslash {\mathbb {H}}^{2n+1}$$\end{document} for non-unitary representations of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} and establish that the associated Selberg zeta function admits a meromorphic continuation to C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document}.


Introduction
The Selberg zeta function is an important tool in the study of the spectral theory of locally symmetric Riemannian spaces. This zeta function is defined by an infinite product over the closed geodesics that only converges in a complex half-space. For its investigation it is useful to understand if it admits a meromorphic continuation. The purpose of this paper is to prove the existence of a meromorphic continuation of the Selberg zeta functions on compact odd-dimensional hyperbolic orbifolds by establishing a suitable Selberg trace formula.

Theorem 1 Suppose O = \H 2n+1 is a compact odd-dimensional hyperbolic orbifold, χ is a (possibly non-unitary) finite-dimensional representation of , and τ is a unitary finite-dimensional representation of S O(2n). Then the Selberg zeta function Z (s, τ, χ) (see Definition 33 below) admits a meromorphic continuation to C.
The key point of the proof of Theorem 1 is to show that the residues of Z (s, τ, χ)/ Z (s, τ, χ) are integers. If O is a compact hyperbolic manifold and χ is unitary, this was proven by [1]. Later on their result was extended to non-compact finite volume hyperbolic manifolds with cusps when χ is unitary [7] and when χ is a restriction of a representation of SO 0 (1, 2n + 1) [17]. Using a slightly different approach, the theorem was proved in [24] for compact orbifolds when χ and τ are trivial representations. The approach of [1,7,17] is due to Selberg and consists in applying the Selberg trace formula to a certain test function which makes Z (s, τ, χ)/Z (s, τ, χ) appear as one of the terms in the geometric side of the formula. In order to adapt their approach, we need to prove a more general version of the Selberg trace formula.
The Selberg trace formula has a rich history starting from the classical work [19], but has mostly been constrained to unitary representations χ of . The non-unitary case was first studied in [16] under the assumption that contains no non-trivial elements of finite order, also called elliptic elements, and no non-trivial unipotents elements, also called parabolic elements, which means O is a compact manifold. We drop the former restriction on and prove: To complete the proof of Theorem 1 we apply Theorem 2 to the case G/K = H 2n+1 . The major remaining problem is to study the orbital integrals I ϕ (γ ) for the elliptic elements γ ∈ . We show that the Fourier transform of orbital integrals is given by polynomials:

Lemma 3
In the above setup G/K = H 2n+1 the orbital integral I ϕ (γ ) for elliptic γ ∈ equals where SO(2n) is the unitary dual of S O(2n), σ ,λ (ϕ) is the character of the unitarily induced representation π σ ,λ of G, and P γ (λ) is a certain even polynomial.
Orbital integrals have so far been computed for G/K = H 2 , H 3 and H 2n in [4,9] and [23] respectively. The computation of orbital integrals is not only useful for the proof of Theorem 1, but also for other applications of the Selberg trace formula. For example, we will use Lemma 3 in the upcoming papers to study the behavior of the analytic torsion of odd-dimensional compact and non-compact finite-volume orbifolds.

Locally homogeneous vector bundles
Let G = SO 0 (1, 2n + 1), K = SO(2n + 1). Let G = N AK be an Iwasawa decomposition of G with respect to K . For each g ∈ G there are uniquely determined elements n(g) ∈ N , a(g) ∈ A, κ(g) ∈ K such that g = n(g)a(g)κ(g). Let M be the centralizer of A in K , thus M = SO(2n). Denote the Lie algebras of G, K , A, M and N by g, k, a, m and n, respectively. Define the Cartan involution θ : g → g by θ(Y ) = −Y t , Y ∈ g, and let g = k ⊕ p be the Cartan decomposition of g with respect to θ .
Definition 9 [13, p. 4] Denote by the associated homogeneous vector bundle, where K acts on G × V ν by Similarly, we denote by C ∞ c (G; ν) the subspace of compactly supported functions in C ∞ (G; ν) and by L 2 (G; ν) the completion of C ∞ c (G; ν) with respect to the inner product Proposition 4 [13, p. 4] There is a canonical isomorphism Similarly, there are isomorphisms of topological vector spaces where f ∈ C ∞ (G; ν) and Y ∈ p.
Let ∈ Z (g C ) and K ∈ Z (k C ) be the Casimir elements of G and K , respectively. Assume that ν is irreducible. Let R denote the right regular representation of G on C ∞ (G; ν).

Lie groups
Let H : G → a be defined by There is a G-invariant metric on G/K which is unique up to scaling. Suitably normalized, it is the hyperbolic metric, and G/K is isometric to H 2n+1 .
Denote by E i, j the matrix in g whose (i, j)'th entry is 1 and the other entries are 0. Let Then a = RH 1 , where a is from Sect. 2.3.

Definition 13 Define
Let b = √ −1 · RH 2 + · · · + √ −1 · RH n+1 be the standard Cartan subalgebra of m. Moreover, h = a ⊕ b is a Cartan subalgebra of g. Denote by h C , g C , m C , b C , k C the complexification of h, g, m, b, k, respectively. Define e i ∈ h * C with i = 1, . . . , n + 1, by The sets of roots of (g C , h C ) and (m C , b C ) are given by We fix a positive systems of roots by The half-sum of the positive roots Let M be the normalizer of A in K , and let be the restricted Weyl group. It has order 2 and acts on finite-dimensional representations of M [17, p. 18]. Denote by w 0 the non-identity element of W (A).

Representations
Let σ : M → End(V σ ) be a finite-dimensional irreducible representation of M.

Definition 14
We define H σ to be the space of measurable functions f : K → V σ such that Recall H : G → a, κ : G → K as in Sect. 2.3, and e 1 ∈ h * C is as in Sect. 2.3. For λ ∈ R, define the representation π σ,λ of G on H σ by the following formula:

Compact hyperbolic orbifolds
The main subject of study of the article are compact hyperbolic orbifolds \H 2n+1 . Compactness implies that all non-identity elements of are either hyperbolic or elliptic.
where d(x, y) denotes the hyperbolic distance between x and y.
Remark 2 Some authors use the term "loxodromic" instead of "hyperbolic". An alternative definition is the following: an element γ is elliptic if and only if it is conjugate to a non-identity element in K , so without loss of generality we may assume γ is of the form: where n − k + 1 = 0 and R φ = cos φ sin φ − sin φ cos φ , φ ∈ (0, 2π). There is an even number of eigenvalue 1 in (11), because an element γ should belong to SO 0 (1, 2n + 1).

Definition 17
An elliptic element γ is regular if the centralizer G γ of γ in G is isomorphic to SO 0 (1, 1) × SO(2) n−1 .

Sobolev spaces
To define Sobolev norms on an orbifold O, we first define Sobolev norms locally. Let U and G U be as in Definition 2. Note that if G U is finite, then where is equipped with usual Sobolev norm || · || s , and this norm restricts to G U -invariant sections. We equip ) with the following norm: and the corresponding element f ∈ C ∞ 0 ( U , U × R k ). Next we use an orbifold atlas and a partition of unity to define the Sobolev norm on the space of smooth sections of an orbibundle E → O. Sobolev norms defined using equivalent atlases will be themselves equivalent.

Remark 3
The isomorphism (12) does not necessarily hold if G U is infinite. For example, let γ act on R by x · γ = x + 1 and put G U = {γ n : n ∈ Z}.

Pseudodifferential operators
We recall some basic facts about pseudodifferential operators on orbibundles. For more details see [2, p. 28 is given by the restriction to G U -invariant functions of a pseudodifferential operator A of order m on C ∞ ( U , V ρ ) that commutes with the induced G U -action on C ∞ ( U , V ρ ).

Proof of Propositions 7 and 8
Instead of the original proofs [21, p. 60], one chooses a partition of unity and reduces the theorems to their local versions in a single chart. As sections over orbifold charts are G U -invariant sections over the corresponding smooth charts, the desired proofs are obtained by repeating the local arguments from [21] verbatim for the subspaces of G U -invariant sections.

Remark 4
For another proof of the Sobolev embedding and the Kondrachov-Rellich theorem on orbifolds, see [3].

Remark 5
Let O be compact. Note that any pseudodifferential operator of order 0 extends to a bounded operator in L 2 (O, E); compare [21,Theorem 6.5]. Moreover, the Proposition 8 implies that any pseudodifferential operator of negative order is compact; compare [21, Corollary 6.2].

Theorem 9
Let H be a second order elliptic pseudodifferential operator acting on sections of an orbibundle E over a compact good orbifold O with the leading symbol For a subset I ⊂ [−π, π], let Then for every 0 < ε < π/2, there exists R > 0 such that the spectrum of H is contained in the set B R (0) ∪ [−ε,ε] . Moreover, the spectrum of H is discrete, and there exists R ∈ R such that for |λ| > R and λ / ∈ [−ε,ε] , Proof The proof of theorem is similar to the smooth case for which we refer to [21, Theorems 9.3 and 8.4], except for the following: in the case of manifolds a partition of unity reduces the proof to R n , whereas in our case it is G U \R n , where G U is a finite group.

Functional analysis
In this section we refine the necessary facts from functional analysis from [16,Sect. 2] for the case of compact orbifolds. The main difference from the case of compact manifolds is that we replace all theorems involving Sobolev spaces to their orbifold analogues from the previous section. Note that though we assume our orbifold is good this assumption is not used until the end of this section. The requirement on O to be compact is crucial, because we will need Remark 5. Let E → O be a Hermitian orbibundle, pick a Hermitian metric in E and let ∇ be a covariant derivative in E which is compatible with the Hermitian metric.

Definition 20
The operator is the Bochner-Laplacian associated to the connection ∇ and the Hermitian fiber metric.
By [2, Theorem 3.5], the Bochner-Laplace operator E is essentially selfadjoint. We denote its selfadjoint extension by the same symbol. Consider the class of elliptic operators which are perturbations of the Laplace operator E by a first order differential operator, i.e. where is a first order differential operator. For every 0 < < π/2 there exists R > 0 such that the spectrum of H is contained in B R (0) ∪ [− ,+ ] by Theorem 9. Though H is not self-adjoint in general, it has nice spectral properties. The reason is the following: D 1 ( E − λ) −1 is a pseudodifferential operator of order −1, and hence by Remark 5 is compact. This implies [12] such that the restriction of H to V k has a unique eigenvalue λ k , and for each k there exists Denote by spec(H ) the spectrum of H . Suppose that 0 / ∈ spec(H ). It follows from Theorem 9 that there exists an Agmon angle θ for H , and we can define the square root H By the spectral theorem we can define where B 0 is a pseudodifferential operator of order zero.

Lemma 10
The resolvent of H 1/2 is compact, and the spectrum of H 1/2 is discrete. There exists b > 0 and d ∈ R such that the spectrum of Proof The proof is similar to [16,Lemma 2.3]. First note that H 1/2 is an elliptic pseudodifferential operator of order 1, hence by Remark 5 its resolvent is compact, that implies the spectrum of H 1/2 is discrete. Second, the operator B 0 extends to a bounded operator in Recall that [8, Chapter V, (3.16)] for λ / ∈ spec( The equations (21) and (22) imply Moreover, that together with (22) implies It follows from the spectral decomposition (18) that H 1/2 has the same spectral decomposition as H with eigenvalues λ 1/2 , λ ∈ spec(H ) and multiplicities m(λ 1/2 ) = m(λ). We need to introduce some class of function for further use.

Definition 21
Denote by PW (C) be the space of Paley-Wiener functions on C, that is be the Fourier-Laplace transform of h. Then ϕ satisfies (23) Recall that we are assuming 0 / ∈ spec(H ).

Lemma 12 ϕ(H 1/2 ) is an integral operator with a smoothing kernel.
Proof The proof follows in the same way as in [16,Lemma 2.4]. For k, l ∈ N we have is rapidly decreasing on L ±b,d . One can easily observe that as in the case of manifolds,

The heat kernel
Let E → O be an orbibundle from Definition 6, equip it with a Hermitian fibre metric and let be the Bochner-Laplace operator acting on sections of E; assume that E is a complex orbibundle of rank k. Our goal is to construct and study the heat kernel for .

Existence and uniqueness of the heat kernel
is a heat kernel, if it satisfies: 1. K is C 0 in all three variables, C 1 in the first, and C 2 in the second, If O = \H 2n+1 is a compact hyperbolic orbifold (hence, is finitely generated), the existence and the uniqueness of the heat kernel follow from the following two lemmas: wherex andỹ are elements of π −1 (x) and π −1 (y), respectively.

Computation of the heat asymptotics
where Above a k , a γ k are some coefficients in C, and N γ is the fixed point set of γ in M.
where dvol M and dvol O denotes the Riemannian measure on M and O, respectively. We study the asymptotic behavior of (28) following [5]. The asymptotic expansion of the first summand in the right hand side of (28) follows from the following theorem:

Remark 6 The leading coefficient is given by
The asymptotic expansion of the second summand of the right hand side of (28) follows with minor modification from [5, Lemma 1.8.2]: Putting together (29), (30) and (31) implies Theorem 17.
Applying Theorems 16 and 17 finishes the proof of Lemma 15.
As usual, we establish the Weyl law.
Proof Follows from the Tauberian theorem, Lemma 15 and Remark 6.

Functional analysis, part II
Proof The Weyl law for E from Theorem 18 and the compactness of We need to establish an auxiliary result about smoothing operators. The proof of the following lemma literally repeats [16, Proposition 2.5]:

Lemma 20 Let
be an integral operator with a smooth kernel K ; denote by dμ(x) the Riemanian measure on O. Then A is a trace class operator and Proof The proof generalizes [11, Chapter VII, §1]. Let {ϕ j } j∈N be an orthonormal basis of L 2 (O, E) consisting of eigensections of E with eigenvalues 0 ≤ λ 1 ≤ λ 2 ≤ . . . → ∞. We can expand K in the orthonormal basis as where a i, j = Aφ i , φ j . Note that for any i, j ∈ N. Together with Theorem 18 it implies that the right hand side of (33) converges in the C ∞ -topology.

The wave equation
In this subsection we give a description of the kernel K ϕ of the smoothing operator ϕ(H 1/2 ) in terms of the solution of the wave equation. Let O = \H n be a hyperbolic orbifold; further denote G = SO 0 (1, n) and K = SO(n). Let 0 be as in Lemma 13, then 0 \G/K is a manifold. Let ρ : → GL(C n ) be a finite-dimensional representation of , and let be the associated vector orbibundle. Let ρ 0 be the restriction of ρ to 0 , and denote by  (O, E), respectively. Then there exist C, c > 0 such that for any f and f 0 as above, the following inequality holds: Proof Follows from the fact that 0 \G/K is a finite covering of O.
Consider the wave equation: .  For σ > 0, define the operator cos(t H 1/2 )e −σ (H +c) by: Note that for f ∈ C ∞ (O, E), and that the right hand side of (39) converges to ϕ(H 1/2 ) f as σ → 0, whereas the left hand side converges to (2π) −1 Rφ (t)u(t, ·; f ) dt. Proof of convergence is analogous to the manifold case. Now we would like to lift the wave equation once again, but now to G/K . Let be the lift of H to G/K . Let u(u, x; f ) and f be the pull-back to G/K of u(t, x; f ) and f , respectively. Then the following holds: As in [16, (3.15)], with the help of the finite propagation speed argument one can show that it does not matter if: 1. either we solve the wave equation (36) on O and then pull the solution back to G/K , 2. or we first pull back the initial condition to G/K and then solve the wave equation (40).

Lemma 25 There exists δ > 0 and H
Proof The proof follows [16,Proposition 3.3] and is based on the finite propagation speed argument, that is valid for orbifolds as well.
Using Lemmas 24 and 25 we obtain for all f ∈ C ∞ (X , E). Let F ⊂ M be a fundamental domain for the action of on G/K , and let be the induced bundle map for γ ∈ . Note that Arguing as in [16] by rewriting G/K as γ ∈ γ F in (41) and using (43), one can show that the kernel K ϕ of ϕ(H 1/2 ) is given by where x, y are any lifts of x and y to F. Together with Lemma 21 we obtain an analogue of [16, Proposition 3.4]: Lemma 26 Let ϕ ∈ P W be even (or ϕ(·) = e −t(·) 2 ). Then we have
Let χ : → G L(V χ ) be a finite-dimensional representation of , and let F → O be the associated orbibundle over O; let ∇ F be a canonical flat connection on F. Let E be a Hermitian vector orbibundle over O with a Hermitian connection ∇ E .

Definition 26
We equip E ⊗ F with a product connection ∇ E⊗F , defined by

Definition 27
The twisted connection Laplacian # E,χ associated to ∇ E⊗F is given by where (∇ E⊗F ) 2 is the invariant second covariant derivative.

Remark 7
The principal symbol of # E,χ is given by Let E and F be the pullback to G/K of E and F, respectively. Note that Let # E,χ and E be the lift of # E,χ and E to G/K , respectively. Note that the operator # E,χ splits as follows: where E ⊗ Id acts on C ∞ (G/K , E) ⊗ V χ . Then for any ψ ∈ C ∞ c (G/K , E), the unique solution of the equation (∂ 2 /∂t 2 + # E,χ )u(t, ·; ψ) = 0, u(0, ·; ψ) = ψ, u t (0, ·; ψ) = 0 splits as well and is given by where cos(t( E ) 1/2 ) is defined by the spectral theorem. Let ϕ ∈ PW (C) be even and let k ϕ (x,ỹ) be the kernel of Then H ϕ from Lemma 25 is given by H ϕ (x,ỹ) = k ϕ (x,ỹ) ⊗ Id. Then it follows from (44) that the integral kernel of the operator ϕ ( # E,χ ) 1/2 is given by Lemma 26 implies the following Lemma:

Locally symmetric spaces and the pre-trace formula
In this subsection we apply Lemma 27 to the case when E is a locally homogeneous orbibundle.
Let χ : → G L(V χ ) be a finite-dimensional (possibly non-unitary) complex representation and let F → O be the associated flat vector bundle over O as in previous subsection. Let ν : K → GL(V ν ) be a unitary representation of K and let E ν → O be the locally homogeneous orbibundle as in Definition 12.
Denote by # E ν ,χ be the twisted connection Laplacian acting on C ∞ (O, E ν ⊗ F) as in Definition 27. To simplify notations, denote: Let ν be as in Definition 5. We are now interested in rewriting k ϕ in a different way with respect to the information that E ν is a locally homogeneous orbibundle. Note that ϕ( With respect to the isometry (2) it can be identified with a compactly supported C ∞ function Then ϕ( 1/2 ν ) acts by convolution: and the kernel K ϕ of ϕ(( # ν,χ ) 1/2 ) is given by By Lemma 27, we get Definition 29 For γ ∈ , denote by {γ } its -conjugacy class.

Definition 30
For γ ∈ , denote by γ and G γ the centralizers of γ in and G, respectively.
Collect the terms in the right hand side of (49) according to their conjugacy classes. Separating {e} , we obtain a pre-trace formula.

Orbital integrals for hyperbolic elements
Definition 31 For a hyperbolic γ ∈ , define its primitive element as an element γ 0 ∈ such that γ = γ k 0 , and for any γ 0 ∈ such that γ = (γ 0 ) n , it follows that n ≤ k.
A primitive element γ 0 is not necessarily unique. It is defined up to Let π σ,λ be as in (10), and denote by σ,λ the character of π σ,λ . For hyperbolic γ , we slightly modify [25,Theorem 6.7] in order to get the following lemma: Lemma 29 Let γ ∈ be a hyperbolic element. Then the following holds: and l(γ 0 ) is from Definition 15, m γ and a γ is from Lemma 6.
Without loss of generality assume that all φ i from (11) are different, then the stabilizer G γ of γ is equal to SO(2) k × SO 0 (1, 2k − 1). The root system for G γ can be written as We can choose an ordering such that Lemma 32 [23, (5.2)] There exists M γ ∈ R\{0} such that We are ready to prove the main theorem in this subsection: Theorem 33 There exists an even polynomial P γ σ ( Proof Theorem 33 holds with by Lemmas 31 and 32. We need to show that P γ σ ( √ −1λ) is an even polynomial. Note that every α ∈ + γ is a root with α, α = 2, hence Let s( (σ ) + ρ M ) = 2≤i≤n+1 k i e i with ρ M as in (8). For simplicity assume k 1 = 0 and denote by δ i, j the Kronecker delta. Then Note that above δ j ,1 is always equal to 0. Now we would like to study the dependance of (61) on λ, for this we split the product above as: and first notice that does not depend on λ. Second, Putting together (58-60) gives us where (−1) | + γ |+1 · C(k) does not depend on λ. Note that (61) is an even polynomial in λ and by Remark 9, ξ −s( (σ )+ρ M )− √ −1e 1 λ (γ ) does not depend on λ. Hence, (57) and (56) are even polynomials in λ as well.
We would like to mention the resemblance of Theorem 33 to the following: Proposition 34 [9, Theorem 13.2] There exists an even polynomial P σ ( For further use we need to show one property of the polynomial P γ σ ( √ −1λ). Let σ be a finite-dimensional representation of M with the highest weight then the highest weight of a representation w 0 σ , where w 0 is the non-identity element of W (A) from (9), equals Lemma 35 The polynomial P γ σ is invariant under the action of W (A): Proof Recall that s ∈ W acts on the roots by even sign changes and the permutations. Then it follows from (62)

Definition 33
The Selberg zeta function is defined as: Proposition 36 There exists a constant c = C( , χ) > 0 such that Z (s, σ, χ) converges absolutely and locally uniformly for Re(s) > c.
It follows by definition that | 1 γ | ≥ 1, n ≥ 1 and tr(σ (m γ )) ≤ dim(σ ). We need the following lemma to estimate the number of closed geodesics: Lemma 37 There exists a constant C 3 > 0 such that for all x ∈ H 2n+1 , the following estimate holds: where ρ(x, y) denotes the hyperbolic distance between x and y.
Proof Let x ∈ H 2n+1 , denote by B R (x) the hyperbolic ball around x of radius R; note that vol(B R (x)) ≤ C 2 · e 2n R for some C 2 > 0. Note that because is cocompact, there exists ε > 0 such that that implies (65). Moreover, hence there exists a constant C 4 such that for every γ hyperbolic, Collecting all together proves Proposition 36.
Definition 34 For Re(s) > c with the constant c as in Proposition 36, we define the symmetric Selberg zeta function by where w 0 is the non-identity element of W (A) from (9).
In this subsection we prove the existence of the meromorphic continuation of the symmetric Selberg zeta function. We follow the approach of [17] which associates a vector bundle E(σ ) to every representation σ ∈ SO(2n). This vector bundle is graded and there exists a canonical graded differential operator A(σ, χ) which acts on smooth sections of E(σ ). The next step is to apply the Selberg trace formula to A(σ, χ) with a certain test function.
We are almost ready to apply the Selberg trace formula.
Definition 38 Let B ν , ν ∈K , be trace class operators acting on sections of E ν . Let Then define Proposition 28, Lemma 29, Theorem 33 and Proposition 34 imply:

Lemma 41
The operator N j=1 R(s 2 j ) is of trace class.
Proof In [1] Lemma 40 was proven for manifolds by the following argument: each of the factors is a pseudodifferential operator of order −2/(2n + 1), hence their product is a pseudodifferential operator of order −2N /(2n + 1) that is of trace class for sufficiently large N by the Weyl law. Now, let R be the value of the resolvent of any self-adjoint Laplacian on E(σ ) at a point x ∈ R, that is not in its spectrum and such that R ≥ 0.

Lemma 42
The operator R N is of trace class for N > (2n + 1)/2.
Proof Let λ k be the k-th eigenvalue of R. Then by Theorem 19,λ Lemma 43 The operator R −N · N j=1 R(s 2 j ) is bounded. Proof It is a pseudodifferential operator of order 0 and hence bounded by Remark 5.
By the above two lemmas, From now on let all s j , j = 1, . . . , N satisfy spec(s j + A(σ, χ) 2 ) ∈ {z ∈ C, Re(z) > 0}. We can choose such s j , because the real parts of eigenvalues of A(σ, χ) are bounded from below. Put s := s 1 and c j := c j /c 1 for 1 ≤ j ≤ N , then by Lemmas 40 and 41, c j e −ts 2 Then (76) can be rewritten as where (s, σ ) denotes the differentiation with respect to the first variable. It follows from (76) and (77), that S(s, σ ) extends meromorphically to C if and only if (s, σ ) does, moreover, its singularities coincide. Let λ i , i = 1, 2, . . . be the eigenvalues of A(σ, χ). For each λ i let E(λ i ) be the eigenspace of A(σ ) with eigenvalue λ i . Put where E (λ i ) is the eigenspace of A ν,χ with eigenvalue λ i . Put where √ λ i is chosen to have the non-negative imaginary part. Note that 1 λ i +s 2 and c j λ i +s 2 j are the eigenvalues of R(s 2 ) and c j R(s 2 j ), hence by (78) and Lidskii's theorem,  (λ i , σ ). The order of the singularity at s = 0 is 2m s (0, σ ).

Antisymmetric Selberg zeta function
Suppose that σ = w 0 σ , otherwise the symmetric Selberg zeta function equals the Selberg zeta function and this section can be skipped. For Re(s) > c with the constant c as in Proposition 36 we define the antisymmetric Selberg zeta function as S a (s, σ, χ) := Z (s, σ, χ)/Z (s, w 0 σ, χ).
In this subsection we prove the meromorphic continuation of the antisymmetric Selberg zeta function S a (s, σ, χ).

Dirac bundles and twisted Dirac operators
Let Cl(p) be the Clifford algebra of p with respect to the scalar product on p. Let κ be the spin-representation of K and put 2n := C 2 n ; denote byS = G × κ 2n be the spinor bundle on H 2n+1 and equip it with a connection ∇ S .
Consider an open subset U of O such that E χ | U is trivial. Then E σ,χ | U is isomorphic to the direct sum of rank(E χ ) copies of E σ | U . Let v j be the basis of flat sections of E χ | U , then each ϕ ∈ C ∞ (U , E σ,χ | U ) can be written as: where φ j ∈ C ∞ (U , E σ | U ). The Dirac operator D(σ, χ) acting on sections of E σ,χ is defined as follows: for each ϕ as above, The Dirac operator D(σ ) acting on sections of E ν k (σ ) is defined as follow: Note that D(σ, χ) 2 a second order elliptic differential operator and by Theorem 9, its spectrum is discrete and there exist R ∈ R and ε > 0 such that spec(D(σ, χ) 2 ) ∈ L := [−ε,ε] ∪ B(R). (81)

Proposition 45
The right hand side of (82) converges.
Proof Follows from Theorem 9.
Applying Proposition 47 to (84), we get Moreover, the first and the second summand in the right hand side of (85) vanish by the following two remarks.