Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion

Abstract

This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $s=0$ and its value at $s=0$ equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.

1 Introduction

Let X be a d-dimensional closed, oriented hyperbolic manifold. Then there exists a discrete torsion free subgroup $\Gamma \subset {\text {SO}}_0(d,1)$ such that $X=\Gamma \backslash {\mathbb H}^d$, where ${\mathbb H}^d={\text {SO}}_0(d,1)/{\text {SO}}(d)$ is the d-dimensional hyperbolic space. Every $\gamma \in \Gamma \setminus \{e\}$ is loxodromic and the $\Gamma$-conjugacy class $[\gamma ]$ corresponds to a unique closed geodesic $\tau _\gamma$. Let $\ell (\gamma )$ denote the length of $\tau _\gamma$. A conjugacy class is called prime if $\gamma$ is not a non-trivial power of some other element of $\Gamma$. Let $\chi :\Gamma \rightarrow {\text {GL}}(V_\chi )$ be a finite dimensional complex representation of $\Gamma$ and let $s\in {\mathbb C}$. Then the Ruelle zeta function $R(s,\chi )$ is defined by the following Euler product

$$\begin{aligned} {R(s,\chi ):=\prod _{\begin{array}{c} [\gamma ]\ne e\\ {[}\gamma ]\,\text{ prime } \end{array}}{\text {det}}\left( {\text {Id}}-\chi (\gamma ) e^{-s\ell (\gamma )}\right) .} \end{aligned}$$

(1.1)

The infinite product is absolutely convergent in a certain half plane ${\text {Re}}(s)>C$ and admits a meromorphic extension to the entire complex plane [11, 31]. The Ruelle zeta function is a dynamical zeta function associated to the geodesic flow on the unit sphere bundle S(X) of X. There are formal analogies with the zeta functions in number theory such as the Artin L-function associated to a Galois representation. Analogues to the role of zeta functions in number theory, one expects that special values of the Ruelle zeta function provide a connection between the length spectrum of closed geodesics and geometric and topological invariants of the manifold.

In [10], Fried has established such a connection. To explain his result we need to introduce some notation. Recall that a representation $\chi$ is called acyclic, if the cohomology $H^*(X,F_\chi )$ of X with coefficients in the flat bundle $F_\chi \rightarrow X$ associated to $\chi$ vanishes. Let $\chi$ be an orthogonal acyclic representation. Then $F_\chi$ is equipped with a canonical fiber metric which is compatible with the flat connection. Let $\Delta _{k,\chi }$ be the Laplacian acting in the space $\Lambda ^k(X,F_\chi )$ of $F_\chi$-valued k-forms. Regarded as operator in the space of $L^2$-forms, it is essentially self-adjoint with a discrete spectrum ${\text {Spec}}(\Delta _{k,\chi })$ consisting of eigenvalues $\lambda$ of finite multiplicity $m(\lambda )$. Let $\zeta _k(s;\chi )= \sum _{\lambda \in {\text {Spec}}(\Delta _{k,\chi })}m(\lambda )\lambda ^{-s}$ be the spectral zeta function of $\Delta _{k,\chi }$ [30]. The series converges absolutely in the half plane ${\text {Re}}(s)>d/2$ and admits a meromorphic extension to the complex plane, which is holomorphic at $s=0$. Then the Ray–Singer analytic torsion $T^{RS}(X,\chi )\in {\mathbb R}^+$ is defined by

$$\begin{aligned} \log T^{RS}(X,\chi ):=\frac{1}{2}\sum _{k=1}^d (-1)^kk\frac{d}{ds}\zeta _k(s;\chi ) \big |_{s=0}, \end{aligned}$$

(1.2)

[28]. Now we can state the result of Fried [10, Theorem 1]. He proved that for an acyclic unitary representation $\chi$ the Ruelle zeta function $R(s,\chi )$ is holomorphic at $s=0$ and

$$\begin{aligned} |R(0,\chi )^\varepsilon |=T^{RS}(X,\chi )^2, \end{aligned}$$

(1.3)

where $\varepsilon =(-1)^{d-1}$ and the absolute value can be removed if $d>2$. If $\chi$ is not acyclic, but still orthogonal, $R(s,\chi )$ may have a pole or zero at $s=0$. Fried [10] has determined the order of $R(s,\chi )$ at $s=0$ and the leading coefficient of the Laurent expansion around $s=0$. Let $b_k(\chi ):=\dim H^k(X,F_\chi )$. Assume that $d=2n+1$. Put

$$\begin{aligned} h=2\sum _{k=0}^n(n+1-k)(-1)^kb_k(\chi ). \end{aligned}$$

Then by [10, Theorem 3], the order of $R(s,\chi )$ at $s=0$ is h and the leading term of the Laurent expansion of $R(s,\chi )$ at $s=0$ is

$$\begin{aligned} C(\chi )\cdot T^{RS}(X,\chi )^2s^h, \end{aligned}$$

(1.4)

where $C(\chi )$ is a constant that depends on the Betti numbers $b_k(\chi )$. In [13, p. 66] Fried conjectured that (1.3) holds for all compact locally symmetric manifolds X and acyclic orthogonal bundles over S(X). This conjecture was recently proved by Shen [29].

Let $\chi$ be a unitary acyclic representation of $\Gamma$. Let $\tau (X,\chi )$ be the Reidemeister torsion [25, 28]. It is defined in terms of a smooth triangulation of X. However, it is independent of the particular $C^\infty$-triangulation. Since $\chi$ is acyclic, $\tau (X,\chi )$ is a topological invariant, i.e., it does not depend on the metrics on X and in $F_\rho$. By [8, 24] we have $T^{RS}(X,\chi )=\tau (X,\chi )$. Assume that d is odd. Then (1.3) can be restated as

$$\begin{aligned} R(0,\chi )=\tau (X,\chi )^2. \end{aligned}$$

(1.5)

This provides an interesting relation between the length spectrum of closed geodesics and a secondary topological invariant.

Another class of interesting representations arises in the following way. Let $G:={\text {SO}}_0(d,1)$. Let $\rho$ be a finite dimensional complex or real representation of G. Then $\rho |_\Gamma$ is a finite dimensional representation of $\Gamma$. In general, $\rho |_\Gamma$ is not an orthogonal representation. However, the flat vector bundle $F_\rho$ associated with $\rho |_\Gamma$ can be equipped with a canonical fiber metric which allows the use of methods of harmonic analysis to study the Laplace operators $\Delta _{k,\rho }$. Put

$$\begin{aligned} R(s,\rho ):=R(s,\rho |_\Gamma ). \end{aligned}$$

The behavior of $R(s,\rho )$ at $s=0$ has been studied by Wotzke [35]. Let $\theta :G\rightarrow G$ be the Cartan involution of G with respect to $K={\text {SO}}(d)$. Let $\rho _\theta :=\rho \circ \theta$. Also denote by $T^{RS}(X,\rho )$ the analytic torsion of X with respect to $\rho |_\Gamma$ and an admissible metric in $F_\rho$. Assume that $\rho$ is irreducible. If $\rho \not \cong \rho _\theta$, then the cohomology $H^*(X,F_\rho )$ vanishes [2, Chapt. VII, Theorem 6.7]. Moreover, in this case Wotzke [35] has proved that $R(s,\rho )$ is holomorphic at $s=0$ and

$$\begin{aligned} |R(0,\rho )|=T^{RS}(X,\rho )^2. \end{aligned}$$

(1.6)

If $\rho \cong \rho _\theta$, then $R(s,\rho )$ may have a zero or a pole at $s=0$. Wotzke [35] has also determined the order of $R(s,\rho )$ at $s=0$ and the coefficient of the leading term of the Laurent expansion of $R(s,\rho )$ at $s=0$. As in (1.4) the main contribution to the coefficient is the analytic torsion.

Let $\tau (X,\rho )$ be the Reidemeister torsion [25] of X with respect to $\rho |_\Gamma$. Assume that $\rho$ is irreducible and $\rho \not \cong \rho _\theta$. As mentioned above, the cohomology $H^*(X,F_\rho )$ vanishes in this case and therefore, $\tau (X,\rho )$ is independent of the metrics on X and in $F_\rho$. By [25, Theorem 1] we have $T^{RS}(X,\rho )=\tau (X,\rho )$. Thus (1.6) can be restated as

$$\begin{aligned} |R(0,\rho )|=\tau (X,\rho )^2. \end{aligned}$$

(1.7)

This equality has interesting consequences for arithmetic subgroups $\Gamma$. Assume that there exists a $\Gamma$-invariant lattice $M_\rho \subset V_\rho$. Let ${\mathcal M}_\rho \rightarrow X$ be the associated local system of free ${\mathbb Z}$-modules of finite rank. The cohomology $H^*(X,{\mathcal M}_\rho )$ is a finitely generated abelian group. If $\rho _\Gamma$ is acyclic, $H^*(X,{\mathcal M}_\rho )$ is a finite abelian group. Denote by $|H^k(X,{\mathcal M}_\rho )|$ the order of $H^k(X,{\mathcal M}_\rho )$. By [8, (1.4)], [1, Sect. 2.2], $\tau (X,\rho )$ can be expressed in terms of $|H^k(X,{\mathcal M}_\rho )|$, $k=0,\ldots ,d$. Combined with (1.7) we get

$$\begin{aligned} |R(0,\rho )|= \prod _{k=0}^d |H^k(X,{\mathcal M}_\rho )|^{(-1)^{k+1}}. \end{aligned}$$

(1.8)

This is another interesting relation between the length spectrum of X and topological invariants of X.

For arithmetic subgroups $\Gamma \subset G$, representations of G with $\Gamma$-invariant lattices in the corresponding representation space exist. See [1, 20].

The main purpose of this paper is to extend the above results about the behavior of the Ruelle zeta function at $s=0$ to every finite dimensional representation $\chi$ of $\Gamma$. To this end we use a complex version $T^{\mathbb C}(X,\chi )$ of the analytic torsion, which was introduced by Cappell and Miller [7] and which is closely related to the refined analytic torsion of Braverman and Kappeler [3]. It is defined in terms of the flat Laplacians $\Delta _{k,\chi }^\sharp$, $k=0,\ldots ,d$, which are obtained by coupling the Laplacian $\Delta _k$ on k-forms to the flat bundle $F_\chi$ (see Sect. 2 for its definition). In general, the flat Laplacian $\Delta _{k,\chi }^\sharp$ is not self-adjoint. However, its principal symbol equals the principal symbol of a Laplace type operator. Therefore, it has good spectral properties which allows to carry over most of the results from the self-adjoint case. The Cappell–Miller torsion $T^{\mathbb C}(X,\chi )$ is defined as an element of the determinant line

$$\begin{aligned} T^{\mathbb C}(X,\chi )\in {\text {det}}H^*(X,F_\chi )\otimes ({\text {det}}H_*(X,F_\chi ))^*. \end{aligned}$$

For an acyclic representation $T^{\mathbb C}(X,\chi )$ is a complex number and

$$\begin{aligned} |T^{\mathbb C}(X,\chi )|=T^{RS}(X,\chi )^2, \end{aligned}$$

(1.9)

where $T^{\mathbb C}(X,\chi )$ is the Ray–Singer analytic torsion with respect to any choice of a fiber metric in $F_\chi$. Since $\chi$ is acyclic, $T^{RS}(X,\chi )$ is independent of the choice of the metric in $F_\chi$.

Let $V_0^k$ be the generalized eigenspace of $\Delta _{k,\chi }^\sharp$, $k=0,\ldots ,d$, with generalized eigenvalue 0. Let $d^{*,\sharp }_\chi$ be the coupling of the co-differential $d^*_\chi :\Lambda ^*(X)\rightarrow \Lambda ^*(X)$ to the flat bundle $F_\chi$. Then $(V_0^*,d_\chi ,d^{*,\sharp }_\chi )$ is a double complex in the sense of [7, §6]. Let

$$\begin{aligned} T_0(X,\chi )\in {\text {det}}H^*(X,F_\chi )\otimes ({\text {det}}H_*(X,F_\chi ))^*. \end{aligned}$$

(1.10)

be its torsion [7, §6]. We note that $T^{\mathbb C}(X,\chi )$ and $T_0(X,\chi )$ are both non-zero elements of the determinant line ${\text {det}}H^*(X,F_\chi )\otimes ({\text {det}}H_*(X,F_\chi ))^*$. Hence there exists $\lambda \in {\mathbb C}$ with $T^{\mathbb C}(X,\chi )=\lambda T_0(X,\chi )$. Set

$$\begin{aligned} \frac{T^{\mathbb C}(X,\chi )}{T_0(X,\chi )}:=\lambda . \end{aligned}$$

Put

$$\begin{aligned} h_k:=\dim V_0^k,\quad k=0,\ldots ,d. \end{aligned}$$

(1.11)

Furthermore, let $d=2n+1$ and put

$$\begin{aligned} h:=2\sum _{k=0}^n(n+1-k)(-1)^kh_k \end{aligned}$$

(1.12)

and

$$\begin{aligned} C(d,\chi ):=\prod _{k=0}^{d-1}\prod _{\begin{array}{c} p=k\\ p\ne n \end{array}}^{d-1} \left( 2(n-p)\right) ^{(-1)^kh_k}. \end{aligned}$$

(1.13)

Then our main result is the following theorem.

Theorem 1.1

Let $\chi$ be a finite dimensional complex representation of $\Gamma$. Let h be defined by (1.12). Then the order of the singularity of $R(s,\chi )$ at $s=0$ is h and

$$\begin{aligned} \lim _{s\rightarrow 0} s^{-h}R(s,\chi )=C(d,\chi )\cdot \frac{T^{\mathbb C}(X,\chi )}{T_0(X,\chi )}. \end{aligned}$$

(1.14)

As above there is a combinatorial formula. Let $\tau _{\text{ comb }}(X,\chi )\in {\text {det}}H^*(X,F_\chi )\otimes$${\text {det}}H^*(X,F_\chi ^*)$ be the combinatorial torsion defined by Cappell and Miller [7, Sect. 9], which is defined in terms of a triangulation of X, but is independent of the choice of the triangulation. An equivalent definition is as follows. For a unimodular complex representation $\rho$ of $\Gamma$, one can define a complex valued Reidemeister torsion $\tau ^{\mathbb C}(X,\rho )\in {\text {det}}H^*(X,F_\rho )$ [9, §3]. See also Sect. 6.1. Now let $\chi ^*$ be the contragredient representation to $\chi$. The representation $\chi \oplus \chi ^*$ is unimodular and therefore, the complex Reidemeister torsion $\tau ^{\mathbb C}(X,\chi \oplus \chi ^*)\in {\text {det}}H^*(X,F_\chi \oplus F_\chi ^*)$ is well defined. Then by [5] it follows that with respect to the canonical isomorphism

$$\begin{aligned} {\text {det}}H^*(X,F_\chi )\otimes {\text {det}}H^*(X,F_\chi ^*)\cong {\text {det}}H^*(X,F_\chi \oplus F_\chi ^*) \end{aligned}$$

we have ${\tau _{\text{ comb }}(X,\chi )=\pm \tau ^{\mathbb C}(X,\chi \oplus \chi ^*)}$. By [7, Theorem 10.1] we have ${T^{\mathbb C}(X,\chi )=\tau _{\text{ comb }}(X,\chi )}$. Thus we can restate Theorem 1.1 as

$$\begin{aligned} {\lim _{s\rightarrow 0} s^{-h}R(s,\chi )=C(d,\chi )\cdot \frac{\tau _{\text{ comb }}(X,\chi )}{T_0(X,\chi )}.} \end{aligned}$$

(1.15)

If $\chi$ is acyclic, then $T^{\mathbb C}(X,\chi )$, $T_0(X,\chi )$ and ${\tau _{\text{ comb }}(X,\chi )}$ are complex numbers and on the right hand side of (1.14) and (1.15) appear quotients of complex numbers.

Now we apply Theorem 1.1 to representations of $\Gamma$ which are restrictions of representations of G. Denote by ${\text {Rep}}(G)$ the space of finite dimensional complex representations of G. Then we have

Corollary 1.2

Let $\rho \in {\text {Rep}}(G)$ be irreducible and assume that $\rho \not \cong \rho _\theta$. Then $R(s,\rho )$ is holomorphic at $s=0$ and

$$\begin{aligned} R(0,\rho )=C(d,\rho )\cdot \frac{T^{\mathbb C}(X,\rho )}{T_0(X,\rho )}. \end{aligned}$$

(1.16)

Using (1.6) and (1.9), it follows that

$$\begin{aligned} |T_0(X,\rho )|=C(d,\rho ). \end{aligned}$$

(1.17)

Let $d=3$. Then ${\mathbb H}^3\cong {\text {SL}}(2,{\mathbb C})/{\text {SU}}(2)$. For $m\in {\mathbb N}$ let $\rho _m:{\text {SL}}(2,{\mathbb C})\rightarrow {\text {SL}}(S^m({\mathbb C}^2))$ be the m-th symmetric power of the standard representation of ${\text {SL}}(2,{\mathbb C})$ on ${\mathbb C}^2$. For a compact, oriented hyperbolic 3-manifold $X=\Gamma \backslash {\mathbb H}^3$ and the representations $\rho _m$, $m\in {\mathbb N}$, Corollary 1.2 was proved by Park [27, (5.5)]. He also determined the constant $C(3,\rho _m)$ and $|T_0(X,\rho _m)|$. By [27, Prop. 5.1] we have

$$\begin{aligned} \begin{aligned}&h_0=1\quad \text {and}\quad |T_0(X,\rho _m)|=2,\quad \text {if}\;m\;\text {is}\; \text {even}\\&h_0=0\quad \text {and}\;\;\quad T_0(X,\rho _m)=1,\quad \text {if}\;m\;\text {is}\; \text {odd}. \end{aligned} \end{aligned}$$

(1.18)

Moreover $C(3,\rho _m)=(-4)^{h_0}$. The order h of $R(s,\rho _m)$ at $s=0$ is zero. Thus by (1.12) we have $h_1=2h_0$. Note that $\rho _m$ is acyclic. Let $\Delta _{k,\rho _m}$ be the usual Laplacian in $\Lambda ^k(X,F_{\rho _m})$ with respect to the admissible metric in $F_{\rho _m}$. Then for $m\in {\mathbb N}$ even we have

$$\begin{aligned} \ker \Delta _{k,\rho _m}=0,\quad \ker \Delta _{k,\rho _m}^\sharp \ne 0,\quad k=0,\ldots , \end{aligned}$$

(1.19)

which shows that for acyclic representations $\chi$, in general, the flat Laplacian $\Delta _{k,\chi }^\sharp$ need not be invertible.

Again we can replace $T^{\mathbb C}(X,\rho )$ in (1.16) by the combinatorial torsion ${\tau _{\text{ comb }}(X,\rho )}$. Using (1.16) we get

$$\begin{aligned} {R(0,\rho )=C(d,\rho )\cdot \frac{\tau _{\text{ comb }}(X,\rho )}{T_0(X,\rho )}.} \end{aligned}$$

(1.20)

Since G is a connected semi-simple Lie group and $\rho$ a representation of G, it follows from [25, Lemma 4.3] that $\rho$ is actually a representation in ${\text {SL}}(n,{\mathbb C})$. Thus $\rho |_\Gamma$ and $\rho ^*|_\Gamma$ are unimodular. Moreover, if $\rho \not \cong \rho _\theta$, we have $H^*(X,F_\chi \oplus F^*_\chi )=0$. Thus ${\tau _{\text{ comb }}(X,\rho )=\tau ^{\mathbb C}(X,\rho \oplus \rho ^*) \in {\mathbb C}^*}$ and

$$\begin{aligned} \tau ^{\mathbb C}(X,\rho \oplus \rho ^*)=\pm \tau ^{\mathbb C}(X,\rho )\cdot \tau ^{\mathbb C}(X,\rho ^*). \end{aligned}$$

(1.21)

Another case to which Theorem 1.1 can be applied are deformations of unitary acyclic representations. Let ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ be the set of all m-dimensional complex representations of $\Gamma$ equipped with the usual topology. Let ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m) \subset {\text {Rep}}(\Gamma ,{\mathbb C}^m)$ be the subset of all unitary acyclic representations (see Sect. 6.2). If $d=3$, then by [14, Theorem 1.1] we have ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)\ne \emptyset$. Hence there exists a neighborhood V of ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)$ in ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ such that $\Delta _{\chi }^\sharp$ is invertible for all $\chi \in V$. In dimensions $>3$ this is not known.

Using Theorem 1.1, we get

Proposition 1.3

Assume that ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)\ne \emptyset$. There exists an open neighborhood V of ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)$ in ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ such that for every $\chi \in V$, $R(s,\chi )$ is regular at $s=0$ and

$$\begin{aligned} R(0,\chi )=T^{\mathbb C}(X,\chi ). \end{aligned}$$

This proposition was first proved by Spilioti [33] using the odd signature operator [3]. She also discusses the relation with the refined analytic torsion. As above we can also express $R(0,\chi )$ in terms of the combinatorial torsion and by (1.21) we have

$$\begin{aligned} R(0,\chi )=\pm \tau ^{\mathbb C}(X,\rho )\cdot \tau ^{\mathbb C}(X,\rho ^*). \end{aligned}$$

(1.22)

This agrees with (1.7).

2 Coupling Differential Operators to a Flat Bundle

We recall a construction of the flat extension of a differential operator introduced in [7]. Let X be a smooth manifold and $E_1$ and $E_2$ complex vector bundles over X. Let

$$\begin{aligned} D:C^\infty (X,E_1)\rightarrow C^\infty (X,E_2) \end{aligned}$$

be a differential operator. Let $F\rightarrow X$ be a flat vector bundle. Then there is a canonically operator

$$\begin{aligned} D_F^\sharp :C^\infty (X,E_1\otimes F)\rightarrow C^\infty (X,E_2\otimes F) \end{aligned}$$

associated to D, which is defined as follows. Let $U\subset X$ be an open subset such that $F|_U$ is trivial. Let $s_1,\dots ,s_k\in C^\infty (U,F|_U)$ be a local frame field of flat sections. Every section $\varphi$ of $(E_1\otimes F)|_U$ can be written as

$$\begin{aligned} \varphi =\sum _{i=1}^k \psi _i\otimes s_i \end{aligned}$$

for some sections $\psi _1,\ldots ,\psi _k\in C^\infty (U,E_1|_U)$. Then define

$$\begin{aligned} D_F^\sharp |_U:C^\infty (U,(E_1\otimes F)|_U)\rightarrow C^\infty (U,(E_2\otimes F)|_U) \end{aligned}$$

by

$$\begin{aligned} (D_F^\sharp |_U)(\varphi ):=\sum _{i=1}^k D(\psi _i)\otimes s_i. \end{aligned}$$

Let $s_1^\prime ,\ldots ,s_k^\prime$ be another local frame field of flat sections of $F|_U$. Then $s_i=\sum _{j=1}^k f_{ij}s_j^\prime$, $i=1,\ldots ,k$, with $f_{ij}\in C^\infty (U)$, and it follows that the transition functions $f_{ij}$ are constant. Since D is linear, $(D_F^\sharp |_U)(\varphi )$ is independent of the choice of the local frame field of flat sections and therefore, $D_F^\sharp$ is globally well defined. Let $\sigma (D)$ be the principal symbol of D. Then the principal symbol $\sigma (D_F^\sharp )$ of $D_F^\sharp$ is given by $\sigma (D_F^\sharp )=\sigma (D)\otimes {\text {Id}}_F$. Thus if D is elliptic, then $D_F^\sharp$ is also an elliptic differential operator.

As an example consider a Riemannian manifold X and the Laplace operator $\Delta _p$ on p-forms. Let F be a flat bundle over X. Denote by $\Lambda ^p(X,F)$ the space of smooth F-valued p-forms, i.e., $\Lambda ^p(X,F)=C^\infty (X,\Lambda ^pT^*(X)\otimes F)$. By the construction above we obtain the flat Laplacian $\Delta _{p,F}^\sharp :\Lambda ^p(X,F)\rightarrow \Lambda ^p(X,F)$. If the flat bundle is fixed, we will denote the flat Laplacian simply by $\Delta _p^\sharp$. The flat Laplacian can be also described as the usual Laplacian. Let $d_F:\Lambda ^{p-1}(X,F)\rightarrow \Lambda ^p(X,F)$ be the exterior derivative defined as above. Let $\star :\Lambda ^p(X)\rightarrow \Lambda ^{n-p}(X)$ denote the Hodge $\star$-operator. Then the flat extension

$$\begin{aligned} d_F^{*,\sharp }:\Lambda ^p(X,F)\rightarrow \Lambda ^{p-1}(X,F) \end{aligned}$$

of the co-differential $d^*:\Lambda ^*(X)\rightarrow \Lambda ^{p-1}(X)$ is given by

$$\begin{aligned} d_F^{*,\sharp }=(-1)^{np+n+1}(\star \otimes {\text {Id}}_F)\circ d_F\circ (\star \otimes {\text {Id}}_F). \end{aligned}$$

Then $d_F^{*,\sharp }$ satisfies $d_F^{*,\sharp }\circ d_F^{*,\sharp }=0$ and we have

$$\begin{aligned} \Delta _{F}^\sharp =(d_F+d_F^{*,\sharp })^2. \end{aligned}$$

If we choose a Hermitian fiber metric on F, we can define the usual Laplace operator $\Delta _F$ in $\Lambda ^p(X,F)$, which is defined by

$$\begin{aligned} \Delta _F=(d_F+d_F^*)^2=d_Fd_F^*+d_F^*d_F, \end{aligned}$$

which is formally self-adjoint. Now note that $d_F^{*,\sharp }=d_F^*+B$, where B is a smooth homomorphism of vector bundles. Thus it follows that $\Delta _F^\sharp =\Delta _F+(Bd_F+d_FB)$. Hence the principal symbol $\sigma (\Delta _F^\sharp )(x,\xi )$ of $\Delta _F^\sharp$ is given by

$$\begin{aligned} \sigma (\Delta _F^\sharp )(x,\xi )=\Vert \xi \Vert ^2_x{\text {Id}}_{\Lambda ^p T^*_x(X)\otimes F_x}, \quad x\in X,\;\xi \in F_x. \end{aligned}$$

(2.1)

More generally, let $E\rightarrow X$ be a Hermitian vector bundle over X. Let $\nabla$ be a covariant derivative in E which is compatible with the Hermitian metric. We denote by $C^\infty (X,E)$ the space of smooth sections of E. Let

$$\begin{aligned} \Delta _E=\nabla ^*\nabla \end{aligned}$$

be the Bochner–Laplace operator associated to the connection $\nabla$ and the Hermitian fiber metric. Then $\Delta _E$ is a second order elliptic differential operator. Its leading symbol $\sigma (\Delta _E):\pi ^*E\rightarrow \pi ^*E$, where $\pi$ is the projection of $T^*X$, is given by

$$\begin{aligned} \sigma (\Delta _E)(x,\xi )=\Vert \xi \Vert ^2_x\cdot {\text {Id}}_{E_x},\quad x\in X,\; \xi \in T_x^*X. \end{aligned}$$

(2.2)

Let $F\rightarrow X$ be a flat vector bundle and

$$\begin{aligned} \Delta _{E\otimes F}^\sharp :C^\infty (X,E\otimes F)\rightarrow C^\infty (X,E\otimes F) \end{aligned}$$

the coupling of $\Delta _E$ to F. Then the principal symbol of $\Delta _{E\otimes F}^\sharp$ is given by

$$\begin{aligned} \sigma (\Delta _{E\otimes F}^\sharp )(x,\xi )=\Vert \xi \Vert ^2_x\cdot {\text {Id}}_{E_x\otimes F_x}. \end{aligned}$$

(2.3)

3 Regularized Determinants and Analytic Torsion

Let $\Delta _E$ be as above. Let

$$\begin{aligned} P:C^\infty (X,E)\rightarrow C^\infty (X,E) \end{aligned}$$

be an elliptic second order differential operator which is a perturbation of $\Delta _E$ by a first order differential operator, i.e.,

$$\begin{aligned} P=\Delta _E+D, \end{aligned}$$

(3.1)

where $D:C^\infty (X,E)\rightarrow C^\infty (X,E)$ is a first order differential operator. This implies that P is an elliptic second order differential operator with leading symbol $\sigma (P)(x,\xi )$ given by

$$\begin{aligned} \sigma (P)(x,\xi ):=\Vert \xi \Vert ^2_x\cdot {\text {Id}}_{E_x}. \end{aligned}$$

(3.2)

Though P is not self-adjoint in general, it still has nice spectral properties [30, Chapt. I, §8]. We recall the basic facts. For $I\subset [0,2\pi ]$ let

$$\begin{aligned} \Lambda _I=\{re^{i\theta }:0\le r<\infty ,\;\theta \in I\}. \end{aligned}$$

(3.3)

The following lemma describes the structure of the spectrum of P.

Lemma 3.1

For every $0<\varepsilon <\pi /2$ there exists $R>0$ such that the spectrum of P is contained in the set $D_R(0)\cup \Lambda _{[-\varepsilon ,\varepsilon ]}$. Moreover the spectrum of P is discrete.

Proof

The first statement follows from [30, Theorem 9.3]. The discreteness of the spectrum follows from [30, Theorem 8.4]. $\square$

For $\lambda \in {\mathbb C}\setminus {\text {spec}}(P)$ let $R_\lambda (P):=(P-\lambda {\text {Id}})^{-1}$ be the resolvent. Given $\lambda _0\in {\text {spec}}(P)$, let $\Gamma _{\lambda _0}$ be a small circle around $\lambda _0$ which contains no other points of ${\text {spec}}(P)$. Put

$$\begin{aligned} \Pi _{\lambda _0}=\frac{i}{2\pi }\int _{\Gamma _{\lambda _0}}R_\lambda (P)\;d\lambda . \end{aligned}$$

(3.4)

Then $\Pi _{\lambda _0}$ is the projection onto the root subspace $V_{\lambda _0}$. This is a finite-dimensional subspace of $C^\infty (X,E)$ which is invariant under P and there exists $N\in {\mathbb N}$ such that $(P-\lambda _0{\text {I}})^NV_{\lambda _0}=0$. Furthermore, there is a closed complementary subspace $V_{\lambda _0}^\prime$ to $V_{\lambda _0}$ in $L^2(X,E)$ which is invariant under the closure ${\bar{P}}$ of P in $L^2$ and the restriction of $(\bar{P}-\lambda _0{\text {I}})$ to $V_{\lambda _0}^\prime$ has a bounded inverse. The algebraic multiplicity $m(\lambda _0)$ of $\lambda _0$ is defined as

$$\begin{aligned} m(\lambda _0):=\dim V_{\lambda _0}. \end{aligned}$$

Moreover $L^2(X,E)$ is the closure of the algebraic direct sum of finite-dimensional P-invariant subspaces $V_k$

$$\begin{aligned} L^2(X,E)=\overline{\bigoplus _{k\ge 1} V_k} \end{aligned}$$

(3.5)

such that the restriction of P to $V_k$ has a unique eigenvalue $\lambda _k$, for each k there exists $N_k\in {\mathbb N}$ such that $(P-\lambda _k{\text {I}})^{N_k}V_k=0$, and $|\lambda _k|\rightarrow \infty$. In general, the sum (3.5) is not a sum of mutually orthogonal subspaces. See [23, Sect. 2] for details.

Assume that P is invertible. Recall that an angle $\theta \in [0,2\pi )$ is called an Agmon angle for P, if there exists $\varepsilon >0$ such that

$$\begin{aligned} {\text {spec}}(P)\cap \Lambda _{[\theta -\varepsilon ,\theta +\varepsilon ]}=\emptyset . \end{aligned}$$

(3.6)

By Lemma 3.1 it is clear that an Agmon angle always exists for P. Choose an Agmon angle $\theta$ for P. Define the complex power $P_\theta ^{-s}$, $s\in {\mathbb C}$, as in [30, §10]. Let $d=\dim X$. For ${\text {Re}}(s)>d/2$, the complex power $P_\theta ^{-s}$ is a trace class operator and the zeta function $\zeta _\theta (s,P)$ of P is defined by

$$\begin{aligned} \zeta _\theta (s,P):={\text {Tr}}(P_\theta ^{-s}),\quad {\text {Re}}(s)>\frac{d}{2}. \end{aligned}$$

(3.7)

The zeta function admits a meromorphic extension to the entire complex plane which is holomorphic at $s=0$ [30, Theorem 13.1]. Let $R_\theta := \{\rho e^{i\theta }:\rho \in {\mathbb R}^+\}$. Denote by $\log _\theta (\lambda )$ the branch of the logarithm in ${\mathbb C}\setminus R_\theta$ with $\theta<{\text {Im}}\log _\theta <\theta +2\pi$. We enumerate the eigenvalues of P such that

$$\begin{aligned} {\text {Re}}(\lambda _1)\le {\text {Re}}(\lambda _2)\le \cdots \le {\text {Re}}(\lambda _k)\le \cdots . \end{aligned}$$

By Lidskii’s theorem [16, Theorem 8.4] if follows that for ${\text {Re}}(s)>d/2$ we have

$$\begin{aligned} \zeta _\theta (s,P)={\text {Tr}}(P_\theta ^{-s})=\sum _{k=1}^\infty m(\lambda _k) (\lambda _k)_\theta ^{-s}, \end{aligned}$$

(3.8)

where $(\lambda _k)_\theta ^{-s}=e^{-s\log _\theta (\lambda _k)}$. We will need a different description of the zeta function in terms of the heat operator $e^{-tP}$, which can be defined using the functional calculus developed in [23, Sect. 2] by

$$\begin{aligned} e^{-tP}:=\frac{i}{2\pi }\int _\Gamma e^{-t\lambda ^2}(P^{1/2}-\lambda )^{-1}d\lambda , \end{aligned}$$

(3.9)

where $\Gamma \subset {\mathbb C}$ is the same contour as in [23, (2.18)]. As in [23, Lemma 2.4] one can show that $e^{-tP}$ is an integral operator with a smooth kernel. By [23, Prop. 2.5] it follows that $e^{-tP}$ is a trace class operator. Using Lidskii’s theorem as above we get

$$\begin{aligned} {\text {Tr}}(e^{-tP})=\sum _{k=1}^\infty m(\lambda _k) e^{-t\lambda _k}. \end{aligned}$$

(3.10)

The absolute convergence of the right hand side follows from Weyl’s law [23, Lemma 2.2]. Assume that there exists $\delta >0$ such that ${\text {Re}}(\lambda _k)\ge \delta$ for all $k\in {\mathbb N}$. Then by (3.10) and Weyl’s law it follows that there exist $C,c>0$ such that

$$\begin{aligned} |{\text {Tr}}(e^{-tP})|\le C e^{-ct} \end{aligned}$$

(3.11)

for $t\ge 1$. Since ${\text {spec}}(P)$ is contained in the half plane ${\text {Re}}(s)>0$, we can choose the Agmon angle as $\theta =\pi$. Using the asymptotic expansion of ${\text {Tr}}(e^{-tP})$ as $t\rightarrow 0$, it follows from (3.8) and (3.11) that

$$\begin{aligned} \zeta (s,P)=\frac{1}{\Gamma (s)}\int _0^\infty {\text {Tr}}(e^{-tP})t^{s-1}dt \end{aligned}$$

for ${\text {Re}}(s)>d/2$.

Then the regularized determinant of P is defined by

$$\begin{aligned} {\text {det}}_\theta (P):=\exp \left( -\frac{d}{ds}\zeta _\theta (s,P)\Big |_{s=0}\right) . \end{aligned}$$

(3.12)

As shown in [3, 3.10], ${\text {det}}_\theta (P)$ is independent of $\theta$. Therefore we will denote the regularized determinant simply by ${\text {det}}(P)$.

Assume that the vector bundle E is ${\mathbb Z}/2{\mathbb Z}$-graded, i.e., $E=E^+\oplus E^-$ and P preserves the grading, i.e., assume that with respect to the decomposition

$$\begin{aligned} C^\infty (Y,E)=C^\infty (Y,E^+)\oplus C^\infty (Y,E^-) \end{aligned}$$

P takes the form

$$\begin{aligned} P=\begin{pmatrix}P^+&{}0\\ 0&{}P^-\end{pmatrix}. \end{aligned}$$

Then we define the graded determinant ${\text {det}}_\mathrm {gr}(P)$ of P by

$$\begin{aligned} {\text {det}}_{\mathrm {gr}}(P)=\frac{{\text {det}}(P^+)}{{\text {det}}(P^-)}. \end{aligned}$$

(3.13)

Next we introduce the analytic torsion defined in terms of the non-self-adjoint operators $\Delta _{p,\chi }^\sharp$. We use the definition given in [7, Sect. 8]. Recall that the principal symbol of $\Delta _{p,\chi }^\sharp$ is given by (2.3). Therefore, $\Delta _{p,\chi }^\sharp$ satisfies the assumptions of Sect. 3.

Let $r>0$ be such that ${\text {Re}}(\lambda )\ne r$ for all generalized eigenvalues $\lambda$ of $\Delta _{p,\chi }^\sharp$. Let $\Pi _{p,r}$ be the spectral projection on the span of the generalized eigenvectors with eigenvalues with real part less than r. Let $\Delta _{p,\chi ,r}^\sharp :=(1-\Pi _{p,r})\Delta _{p,\chi }^\sharp$. Let $S(p,\chi ,r)$ be the set of all nonzero generalized eigenvalues with real part less than r. Furthermore, let $V^p_0$ be the generalized eigenspace of $\Delta ^\sharp _{p,\chi }$ with generalized eigenvalues 0. Then $(V_0^*,d,d^{*,\sharp })$ is double complex in the sense of [7]. Let

$$\begin{aligned} T_0(X,\chi )\in ({\text {det}}H^*(X,F_\chi ))\otimes ({\text {det}}H_*(X,F_\chi ))^*\end{aligned}$$

(3.14)

be the torsion of the double complex. Then the Cappell–Miller torsion is defined by

$$\begin{aligned} T^{\mathbb C}(X,\chi ):=\prod _{p=1}^d{\text {det}}(\Delta _{p,\chi ,r}^\sharp )^{(-1)^{p+1}p}\cdot \prod _{p=1}^d\Bigl (\prod _{\lambda \in S(p,\chi ,r)}\lambda ^{m(\lambda )}\Bigr )^{(-1)^{p+1}p}\cdot T_0(X,\chi ),\nonumber \\ \end{aligned}$$

(3.15)

where $m(\lambda )$ denotes the algebraic multiplicity of $\lambda$. Let $\Pi _{k,0}$ be the spectral projection on the generalized eigenspace of $\Delta _{k,\chi }^\sharp$ with generalized eigenvalue 0. Let

$$\begin{aligned} (\Delta _{k,\chi }^\sharp )^\prime :=({\text {Id}}-\Pi _{k,0})\Delta _{k,\chi }^\sharp . \end{aligned}$$

(3.16)

If we choose an Agmon angle we can also write

$$\begin{aligned} T^{\mathbb C}(X,\chi )=\prod _{k=1}^d\left[ {\text {det}}(\Delta _{k,\chi }^\sharp )^\prime \right] ^{(-1)^{k+1}k} \cdot T_0(X,\chi ). \end{aligned}$$

(3.17)

If $\chi$ is acyclic, i.e., $H^*(X,E_\chi )=0$, then $T_0(X,\chi )$ and $T^{\mathbb C}(X,\chi )$ are complex numbers.

4 Twisted Ruelle Zeta Functions

In this section we consider compact oriented hyperbolic manifolds of odd dimension $d=2n+1$ and we recall some basic properties of Ruelle type zeta functions.

To begin with we fix some notation. Let $G={\text {SO}}_0(d,1)$ and $K={\text {SO}}(d)$. Then G/K equipped with the normalized invariant metric is isometric to the d-dimensional hyperbolic space ${\mathbb H}^d$. Let $G=KAN$ be the standard Iwasawa decomposition. Let M be the centralizer of A in K. Then $M\cong {\text {SO}}(d-1)$. Let ${\widehat{K}}$ and ${\widehat{M}}$ be the set of equivalence classes of irreducible unitary representations of K and M, respectively.

Denote by ${\mathfrak g}$, ${\mathfrak k}$, ${\mathfrak m}$, ${\mathfrak n}$, and ${\mathfrak a}$ the Lie algebras of G, K, M, N, and A, respectively. Let $W(A)\cong {\mathbb Z}/2{\mathbb Z}$ be the Weyl group of $({\mathfrak g},{\mathfrak a})$. Let $\alpha \in {\mathfrak a}^*$ be the unique positive root of $({\mathfrak g},{\mathfrak a})$. Let ${\mathfrak a}^+:=\{H\in {\mathfrak a}:\alpha (H)>0\}$. Put $A^+:=\exp ({\mathfrak a}^+)$. Let ${\mathfrak g}_{\mathbb C}:={\mathfrak g}\otimes {\mathbb C}$ and denote by ${\mathcal Z}({\mathfrak g}_{\mathbb C})$ the center of the universal enveloping algebra of ${\mathfrak g}_{\mathbb C}$.

Let $\Gamma \subset G$ be a discrete, torsion free, co-compact subgroup. Then $\Gamma$ acts fixed point free on ${\mathbb H}^d$. The quotient $X=\Gamma \backslash {\mathbb H}^d$ is a closed, oriented hyperbolic manifold and each such manifold is of this form. Given $\gamma \in \Gamma$, we denote by $[\gamma ]$ the $\Gamma$-conjugacy class of $\gamma$. The set of all conjugacy classes of $\Gamma$ will be denoted by $C(\Gamma )$. Let $\gamma \ne 1$. Then there exist $g\in G$, $m_\gamma \in M$, and $a_\gamma \in A^+$ such that

$$\begin{aligned} g\gamma g^{-1}=m_\gamma a_\gamma . \end{aligned}$$

(4.1)

By [34, Lemma 6.6], $a_\gamma$ depends only on $\gamma$ and $m_\gamma$ is determined up to conjugacy in M. Let $H\in {\mathfrak a}$ such that $\alpha (H)=1$. By definition there exists $\ell (\gamma )>0$ such that

$$\begin{aligned} a_\gamma =\exp \left( \ell (\gamma )H\right) . \end{aligned}$$

(4.2)

Then $\ell (\gamma )$ is the length of the unique closed geodesic in X that corresponds to the conjugacy class $[\gamma ]$. An element $\gamma \in \Gamma -\{e\}$ is called primitive, if it can not be written as $\gamma =\gamma _0^k$ for some $\gamma _0\in \Gamma$ and $k>1$. For every $\gamma \in \Gamma -\{e\}$ there exist a unique primitive element $\gamma _0\in \Gamma$ and $n_\Gamma (\gamma )\in {\mathbb N}$ such that $\gamma =\gamma _0^{n_\Gamma (\gamma )}$. Let $\chi :\Gamma \rightarrow {\text {GL}}(V_\chi )$ be a finite dimensional complex representation of $\Gamma$ and let $s\in {\mathbb C}$. Then the Ruelle zeta function $R(s,\chi )$ is defined by the following Euler product

$$\begin{aligned} {R(s,\chi ):=\prod _{\begin{array}{c} [\gamma ]\ne e\\ {[}\gamma ]\,\text{ prime } \end{array}}{\text {det}}\left( {\text {Id}}-\chi (\gamma ) e^{-s\ell (\gamma )}\right) .} \end{aligned}$$

(4.3)

In order to verify that the product converges in some half plane, we first recall that there exists $C>0$ such that for all $R>0$ we have

$$\begin{aligned} \#\left\{ [\gamma ]\in C(\Gamma ):\ell (\gamma )\le R\right\} \le C e^{(n-1)R} \end{aligned}$$

(4.4)

[6, (1.31)]. We also need the following auxiliary lemma.

Lemma 4.1

Let $\chi :\Gamma \rightarrow {\text {GL}}(V)$ be a finite dimensional representation of $\Gamma$. There exist $C,c>0$ such that

$$\begin{aligned} |{\text {tr}}(\chi (\gamma ))|\le C e^{c\ell (\gamma )},\quad \forall \gamma \in \Gamma -\{e\}. \end{aligned}$$

(4.5)

For the proof see [31, Lemma 3.3]. It follows from (4.4) and Lemma 4.1 that the product on the right hand side of (4.3) converges absolutely and uniformly in some half-plane ${\text {Re}}(s)>C$ (see [31, Prop. 3.5]). Furthermore, $R(s,\chi )$ admits a meromorphic extensions to the entire complex plane [31] and satisfies the following functional equation [32]:

$$\begin{aligned} R(s,\chi )=R(-s,\chi )\cdot \exp \left( (-1)^{n+1}\frac{2\pi (d+1){\text {vol}}(V_\chi ){\text {vol}}(X)}{{\text {vol}}(S^d)}s\right) . \end{aligned}$$

(4.6)

For unitary representations $\chi$, these results were proved by Bunke and Olbrich [6]. The main technical tool is the Selberg trace formula. For the extension to the non-unitary case the Selberg trace formula is replaced by a Selberg trace formula for non-unitary twists, developed in [23]. The proofs are similar except that on has to deal with non-self-adjoint operators.

There are also expressions of the zeta functions in terms of determinants of certain elliptic operators. To explain the formulas we need to recall the definition of the relevant differential operators. Given $\tau \in {\widehat{K}}$, let $\widetilde{E}_\tau \rightarrow \widetilde{X}$ be the homogeneous vector bundle associated to $\tau$ and let $E_\tau :=\Gamma \backslash \widetilde{E}_\tau$ be the corresponding locally homogeneous vector bundle over X. Denote by $C^\infty (X,E_\tau )$ the space of smooth sections of $E_\tau$. There is a canonical isomorphism

$$\begin{aligned} C^\infty (X,E_\tau )\cong (C^\infty (\Gamma \backslash G)\otimes V_\tau )^K \end{aligned}$$

(4.7)

[21, §1]. Let $\Omega \in {\mathcal Z}({\mathfrak g}_{{\mathbb C}})$ be the Casimir element and denote by $R_\Gamma$ the right regular representation of G in $C^\infty (\Gamma \backslash G)$. Then $R_\Gamma (\Omega )$ acts on the right hand side of (4.7) and via this isomorphism, defines an operator in $C^\infty (X,E_\tau )$. We denote the operator induced by $-R_\Gamma (\Omega )$ by $A_\tau$.

Denote by $\widetilde{\nabla }^\tau$ the canonical connection in $\widetilde{E}_\tau$ and let $\nabla ^\tau$ be the induced connection in $E_\tau$. Let $\Delta _\tau :=(\nabla ^\tau )^*\nabla ^\tau$ be the associated Bochner–Laplace operator acting in $C^\infty (X,E_\tau )$ . Let $\Omega _K\in {\mathcal Z}({\mathfrak k}_{\mathbb C})$ be the Casimir element of K. Assume that $\tau$ is irreducible. Let $\lambda _\tau :=\tau (\Omega _K)$ denote the Casimir eigenvalue of $\tau$. Then we have

$$\begin{aligned} A_\tau :=\Delta _\tau -\lambda _\tau {\text {Id}}. \end{aligned}$$

(4.8)

[21, §1]. Thus $A_\tau$ is a formally self-adjoint second order elliptic differential operator. Let $F_\chi \rightarrow X$ be the flat vector bundle defined by $\chi$. Let

$$\begin{aligned} A^\sharp _{\tau ,\chi }:C^\infty (X,E_\tau \otimes F_\chi )\rightarrow C^\infty (X,E_\tau \otimes F_\chi ) \end{aligned}$$

be the coupling of $A_\tau$ to $F_\chi$.

Denote by R(K) and R(M) the representation rings of K and M, respectively. Let $i:M\rightarrow K$ be the inclusion and $i^*:R(K)\rightarrow R(M)$ the induced map of the representation rings. The Weyl group W(A) acts on R(M) in the canonical way. Let $R^\pm (M)$ denote the $\pm 1$-eigenspaces of the non-trivial element $w\in W(A)$. Let $\sigma \in R(M)$. It follows from the proof of Proposition 1.1 in [6] that there exist $m_\tau (\sigma )\in \{-1,0,1\}$, depending on $\tau \in {\widehat{K}}$, which are equal to zero except for finitely many $\tau \in \widehat{K}$, such that

$$\begin{aligned} \sigma =\sum _{\tau \in {\widehat{K}}}m_\tau (\sigma ) i^*(\tau ), \end{aligned}$$

(4.9)

if $\sigma \in R^+(M)$, and

$$\begin{aligned} \sigma +w\sigma =\sum _{\tau \in {\widehat{K}}}m_\tau (\sigma ) i^*(\tau ), \end{aligned}$$

(4.10)

if $\sigma \ne w\sigma$. Let

$$\begin{aligned} E(\sigma ):=\bigoplus _{\begin{array}{c} \tau \in {\widehat{K}}\\ m_\tau (\sigma )\ne 0 \end{array}} E_\tau . \end{aligned}$$

(4.11)

Then $E(\sigma )$ has a grading

$$\begin{aligned} E(\sigma )=E^+(\sigma )\oplus E^-(\sigma ) \end{aligned}$$

defined by the sign of $m_\tau (\sigma )$. Let $\sigma \in {\widehat{M}}$. Denote by $\nu _\sigma$ the highest weight of $\sigma$. Let ${\mathfrak b}$ be the standard Cartan subalgebra of ${\mathfrak m}$ [26, Sect. 2]. Let $\varrho _{{\mathfrak m}}$ be the half-sum of positive roots of $({\mathfrak m}_{\mathbb C},{\mathfrak b}_{\mathbb C})$ and $\varrho _{\mathfrak a}$ the half-sum of positive roots of $({\mathfrak g}_{\mathbb C},{\mathfrak a}_{\mathbb C})$. Put

$$\begin{aligned} c(\sigma ):=-\Vert \varrho _{\mathfrak a}\Vert ^2-\Vert \varrho _{{\mathfrak m}}\Vert ^2+\Vert \nu _\sigma +\varrho _{{\mathfrak m}}\Vert ^2. \end{aligned}$$

(4.12)

We define the operator $A^\sharp _{\chi }(\sigma )$ acting in $C^\infty (X,E(\sigma )\otimes F_\chi )$ by

$$\begin{aligned} A^\sharp _{\chi }(\sigma ):=\bigoplus _{\begin{array}{c} \tau \in {\widehat{K}}\\ m_\tau (\sigma )\ne 0 \end{array}} A^\sharp _{\tau ,\chi }+c(\sigma ). \end{aligned}$$

(4.13)

For $p\in \{0,\ldots ,d-1\}$ let $\sigma _p$ be the standard representation of $M={\text {SO}}(d-1)$ on $\Lambda ^p{\mathbb R}^{d-1}\otimes {\mathbb C}$. We note that $\sigma _p$ is irreducible except when $p=n$, in which case $\sigma _n$ is the direct sum of the two spin representations $\sigma _n^+,\sigma _n^-$. Moreover, $\sigma _n^\mp = w\sigma _n^\pm$. Thus $\sigma _n=\sigma _n^++w\sigma _n^+$ and

$$\begin{aligned} A^\sharp _{\chi }(\sigma _n)=A^\sharp _{\chi }(\sigma _n^+)\oplus A^\sharp _{\chi }(\sigma _n^-). \end{aligned}$$

(4.14)

Recall that $A_\chi ^\sharp (\sigma _p)$ acts in the space of sections of a graded vector bundle. Then by [32, Prop. 1.7] we have the following determinant formula.

Proposition 4.2

For every $p=0,\dots ,d-1$ we have

$$\begin{aligned} \begin{aligned} R(s,\chi )=&\prod _{p=0}^{d-1} {\text {det}}_{\mathrm {gr}}\biggl (A_\chi ^\sharp (\sigma _p)+ (s+n-p)^2\biggr )^{(-1)^p}\\&\times \exp \left( -\frac{2\pi (n+1)\dim (V_\chi ){\text {vol}}(X)}{{\text {vol}}(S^d)}s \right) , \end{aligned} \end{aligned}$$

(4.15)

where ${\text {vol}}(S^d)$ denotes the volume of the d-dimensional unit sphere.

For unitary $\chi$ this was proved in [6, Prop. 4.6].

5 Proof of the Main Theorem

To prove Theorem 1.1 we apply Proposition 4.2. Let ${\mathfrak h}={\mathfrak a}\oplus {\mathfrak b}$ be the standard Cartan subalgebra of ${\mathfrak g}$ and let $e_1,\ldots ,e_{n+1}\in {\mathfrak h}^*_{\mathbb C}$ be the standard basis [26, Sect. 2]. Thus $e_2,\ldots ,e_{n+1}$ is a basis of ${\mathfrak b}$. Then $\rho _{{\mathfrak m}}=\sum _{j=2}^{n+1}(n+1-j)e_j$ and

$$\begin{aligned} \nu _{\sigma _p}={\left\{ \begin{array}{ll} e_2+\cdots +e_{p+1},&{} \text {if}\;\;p\le n,\\ e_2+\cdots +e_{2n+1-p},&{}\text {if}\;\; p>n, \end{array}\right. } \end{aligned}$$

[17, Chap. IV, §7]. Moreover, $|\rho |=n$. An explicit computation shows that $c(\sigma _p)=-(n-p)^2$. Let $\lambda _p$ be the p-th exterior power of the standard representation of ${\text {SO}}(d)$. Then for $p=0,\ldots ,d-1$ we have $i^*(\lambda _p)=\sigma _p+ \sigma _{p-1}$. Put $\tau _p:=\sum _{k=0}^p (-1)^k\lambda _{p-k}$. Then it follows that $i^*(\tau _p)=\sigma _p$, $p=0,\ldots ,d-1$. Using (4.13) and (4.14), we obtain

$$\begin{aligned} A^\sharp _\chi (\sigma _p)+(n-p)^2=\bigoplus _{k=0}^p A^\sharp _{\lambda _{p-k},\chi }= \bigoplus _{k=0}^p A^\sharp _{\lambda _k,\chi }. \end{aligned}$$

(5.1)

Now recall that $A^\sharp _{\lambda _k,\chi }$ is the coupling of $A_{\lambda _k}$ to $F_\chi$. Furthermore, with respect to the isomorphism (4.7), $A_{\lambda _k}$ corresponds to the action of $-R_\Gamma (\Omega )$ on $(C^\infty (\Gamma \backslash G)\otimes \Lambda ^k{\mathbb C}^d)$. By the Lemma of Kuga, this operator corresponds to the Laplacian $\Delta _k$ on $\Lambda ^k(X)$. Let $\Delta ^\sharp _{k,\chi }$ be the coupling of $\Delta _k$ to $F_\chi$. Then by (5.1) we get

$$\begin{aligned} A^\sharp _\chi (\sigma _p)+(n-p)^2=\bigoplus _{k=0}^p \Delta ^\sharp _{k,\chi }. \end{aligned}$$

(5.2)

Inserting this equality into the alternating product of graded determinants on the right hand side of (4.15), we obtain

$$\begin{aligned} \begin{aligned}&\prod _{p=0}^{d-1}{\text {det}}_{\mathrm {gr}}(A^\sharp _\chi (\sigma _p)+(s+n-p)^2)^{(-1)^p}\\&\quad =\prod _{p=0}^{d-1}\prod _{k=0}^p{\text {det}}(\Delta ^\sharp _{p-k,\chi }+s(s+2(n-p)))^{(-1)^{p+k}}\\&\quad =\prod _{k=0}^{d-1}\prod _{p=k}^{d-1}{\text {det}}(\Delta ^\sharp _{k,\chi }+s(s+2(n-p)))^{(-1)^{k}}. \end{aligned} \end{aligned}$$

(5.3)

Applying the determinant formula (4.15) together with (5.3), we finally get

$$\begin{aligned} \begin{aligned} R(s;\chi ) =&\prod _{k=0}^{d-1}\prod _{p=k}^{d-1}{\text {det}}(\Delta ^\sharp _{k,\chi }+s(s+2(n-p)))^{(-1)^{k}}\\&\times \exp \left( -\frac{2\pi (n+1)\dim (V_\chi ){\text {vol}}(X)}{{\text {vol}}(S^d)}s \right) . \end{aligned} \end{aligned}$$

(5.4)

Let $h_k$ be the dimension of the generalized eigenspace of $\Delta ^\sharp _{k,\chi }$ with eigenvalue zero.

Lemma 5.1

We have $h_p=h_{d-p}$ for $p=0,\ldots ,d$.

Proof

Let $\star :\Lambda ^p(X,F_\chi )\rightarrow \Lambda ^{d-p}(X,F_\chi )$ be the extension of the Hodge $\star$-star operator, which acts locally as $\star (\omega \otimes f) =(\star \omega )\otimes f$, where $\omega$ is a usual p-form and f a local section of $F_\chi$. Since $\star \Delta _p=\Delta _{d-p}\star$, it follows from the definition of the Laplacians coupled to $F_\chi$ that $\star \Delta ^\sharp _{p,\chi } =\Delta ^\sharp _{d-p,\chi }\star$. It follows that for every $k\in {\mathbb N}$ we have $\star (\Delta ^\sharp _{p,\chi })^k=(\Delta ^\sharp _{d-p,\chi })^k\star$.This proves the lemma. $\square$

Denote by h the order of the singularity of $R(s,\chi )$ at $s=0$. Using (5.4) and Lemma 5.1 it follows that

$$\begin{aligned} h=\sum _{k=0}^n(d+1-k)(-1)^kh_k+\sum _{k=n+1}^{d-1}(d-k)(-1)^kh_k=\sum _{k=0}^n(d+1-2k)(-1)^kh_k.\nonumber \\ \end{aligned}$$

(5.5)

Let $(\Delta _{k,\chi }^\sharp )^\prime$ be the operator defined by (3.16). We note that for $s\in {\mathbb C}$, $|s|\ll 1$, there is a common Agmon angle for the operator $(\Delta _{k,\chi }^\sharp )^\prime +s(s+2(n-p))$. Therefore, in order to study the limit of ${\text {det}}((\Delta _{k,\chi }^\sharp )^\prime +s(s+2(n-p)))$ as $s\rightarrow 0$, we can use one and the same Agmon angle.

If $p\ne n$, we get

$$\begin{aligned} \begin{aligned} \lim _{s\rightarrow 0} s^{-h_k}{\text {det}}(\Delta ^\sharp _{k,\chi }+s(s+2(n-p)))&=\lim _{s\rightarrow 0} \Bigl [{\text {det}}((\Delta ^\sharp _{k,\chi })^\prime +s(s+2(n-p))\\&\quad \times \frac{(s(s+2(n-p)))^{h_k}}{s^{h_k}}\Bigr ]\\&=(2(n-p))^{h_k}\cdot {\text {det}}((\Delta ^\sharp _{k,\chi })^\prime ). \end{aligned} \end{aligned}$$

(5.6)

For $p=n$ we get a similar formula

$$\begin{aligned} \lim _{s\rightarrow 0} s^{-2h_k}{\text {det}}(\Delta ^\sharp _{k,\chi }+s^2)=\lim _{s\rightarrow 0} {\text {det}}((\Delta ^\sharp _{k,\chi })^\prime +s^2) ={\text {det}}((\Delta ^\sharp _{k,\chi })^\prime ). \end{aligned}$$

(5.7)

Let

$$\begin{aligned} C(d,\chi ):=\prod _{k=0}^{d-1}\prod _{\begin{array}{c} p=k\\ p\ne n \end{array}}^{d-1} \left( 2(n-p)\right) ^{(-1)^kh_k}. \end{aligned}$$

(5.8)

Using (5.5), (5.6) and (5.7) we get

$$\begin{aligned} \begin{aligned} \lim _{s\rightarrow 0} s^{-h}R(s;\chi )&=\prod _{k=0}^{d-1} \prod _{\begin{array}{c} p=k\\ p\ne n \end{array}}^{d-1} \lim _{s\rightarrow 0}\left[ s^{-h_k}{\text {det}}(\Delta ^\sharp _{k,\chi }+s(s+2(n-p)))\right] ^{(-1)^{k}}\\&\quad \times \prod _{k=0}^n\lim _{s\rightarrow 0} \left[ s^{-2h_k}{\text {det}}(\Delta ^\sharp _{k,\chi }+s^2)\right] ^{(-1)^k}\\&=C(d,\chi )\cdot \prod _{k=0}^{d-1}{\text {det}}((\Delta _{k,\chi }^\sharp )^\prime )^{(d-k)(-1)^k} =C(d,\chi )\cdot \prod _{k=1}^d{\text {det}}((\Delta _{k,\chi }^\sharp )^\prime )^{k(-1)^{k+1}}. \end{aligned}\nonumber \\ \end{aligned}$$

(5.9)

For the last equality we used that $\Delta _{k,\chi }^\sharp \cong \Delta _{d-k,\chi }^\sharp$. Let $T_0(X,\chi )$ be the torsion (3.14) of the double complex $(V_0^*,d,d^{*,\sharp })$ and $T^{\mathbb C}(X,\chi )$ the Cappell–Miller torsion defined by (3.15). We note that $T^{\mathbb C}(X,\chi )$ and $T_0(X,\chi )$ are both non-zero elements of the determinant line ${\text {det}}H^*(X,F_\chi )\otimes ({\text {det}}H_*(X,F_\chi ))^*$. Hence there exists $\lambda \in {\mathbb C}$ with $T^{\mathbb C}(X,\chi )=\lambda T_0(X,\chi )$. Set

$$\begin{aligned} \frac{T^{\mathbb C}(X,\chi )}{T_0(X,\chi )}:=\lambda . \end{aligned}$$

If we combine this convention with the definition of the Cappell–Miller torsion (3.17), then (5.9) implies Theorem 1.1.

6 Acyclic Representations

In this section we assume that $\chi$ is acyclic. Then $T^{\mathbb C}(X,\chi )$, $T_0(X,\chi )$ and ${\tau _{\text{ comb }}(X,\chi )}$ are complex numbers and the right hand side of (1.14) is the quotient of the two complex numbers. Besides the Cappell–Miller torsion we need another version of a complex analytic torsion for arbitrary flat vector bundles $F_\chi$. This is the refined analytic torsion $T^{\text{ ran }}(X,\chi ) \in {\text {det}}(H^*(X,F_\chi ))$ introduced by Braverman and Kappeler [4]. The definition is based on the consideration of the odd signature operator $B_\chi$ [3, 2.1]. It is defined as follows. Let

$$\begin{aligned} \alpha :\Lambda ^*(X,F_\chi )\rightarrow \Lambda ^*(X,F_\chi ) \end{aligned}$$

be the chirality operator defined by

$$\begin{aligned} \alpha (\omega ):=i^{n+1}(-1)^{k(k+1)/2}\star \omega ,\quad \omega \in \Lambda ^k(M,F_\chi ). \end{aligned}$$

Let $\nabla _\chi$ be the flat connection in $F_\chi$. Then the odd signature operator is defined as

$$\begin{aligned} B_\chi :=\alpha \nabla _\chi +\nabla _\chi \alpha . \end{aligned}$$

(6.1)

It leaves the even subspace ${\Lambda ^{\text{ ev }}(X,F_\chi )}$ invariant. Let ${B_{\text{ ev },\chi }}$ be the restriction of $B_\chi$ to ${\Lambda ^{\text{ ev }}(X,F_\chi )}$. Then ${T^{\text{ ran }}(X,\chi ) \in {\text {det}}(H^*(X,F_\chi ))}$ is defined in terms of ${B_{\text{ ev },\chi }}$. If $\chi$ is acyclic, then ${T^{\text{ ran }}(X,\chi )}$ is a complex number. In [5], Braverman and Kappeler determined the relation between the Cappell–Miller torsion and the refined analytic torsion. Let $\eta (B)$ be the eta-invariant of ${B_{\text{ ev },\chi }}$. In general, $B_\chi$ is not self-adjoint and therefore, $\eta (B)$ is in general not real. Furthermore, let $\eta _0$ be the eta-invariant of the trivial line bundle. Then by Proposition 4.2 and Theorem 5.1 of [5] it follows that

$$\begin{aligned} {T^{\mathbb C}(X,\chi )=\pm T^{\text{ ran }}(X,\chi )^2\cdot e^{2\pi i(\eta (B)-\dim (\chi )\eta _0)}.} \end{aligned}$$

(6.2)

On the other hand, it follows from [4, Theorem 1.9] that

$$\begin{aligned} {|T^{\text{ ran }}(X,\chi )|=T^{RS}(X,\chi )\cdot e^{\pi {\text {Im}}(\eta (B))}.} \end{aligned}$$

(6.3)

Combining (6.2) and (6.3), we obtain (1.9).

6.1 Restriction of Representations of the Underlying Lie Group

The first case that we consider are representations which are restrictions to $\Gamma$ of representations of G.

Let $\rho :G\rightarrow {\text {GL}}(V_\rho )$ be a finite dimensional real ( resp. complex) representation of G. Denote by $F_\rho \rightarrow X$ the flat vector bundle associated to $\rho |_\Gamma$. Let $\widetilde{E}_\rho \rightarrow G/K$ be the homogeneous vector bundle associated to $\rho |_K$. By [18, Part I, Prop. 3.3] there is a canonical isomorphism

$$\begin{aligned} F_\rho \cong \Gamma \backslash {\widetilde{E}}_\rho . \end{aligned}$$

(6.4)

Let ${\mathfrak g}={\mathfrak k}\oplus {\mathfrak p}$ be the Cartan decomposition of ${\mathfrak g}$. By [18, Part I, Lemma 3.1], there exists an inner product $\langle \cdot , \cdot \rangle$ in $V_\rho$ such that

• (1) $\left<\tau (Y)u,v\right>=-\left<u,\tau (Y)v\right>$ for all $Y\in \mathfrak {k}$, $u,v\in V_{\tau }$

• (2) $\left<\tau (Y)u,v\right>=\left<u,\tau (Y)v\right>$ for all $Y\in \mathfrak {p}$, $u,v\in V_{\tau }$.

Such an inner product is called admissible. It is unique up to scaling. Fix an admissible inner product. Since $\rho |_{K}$ is unitary with respect to this inner product, it induces a metric in $\Gamma \backslash {\widetilde{E}}_{\tau }$ and by (6.4) also in $F_\rho$. Denote by $T^{RS}(X,\rho )$ the Ray–Singer analytic torsion of $(X,F_\rho )$ with respect to the metric on X and the metric in $F_\rho$. Denote by $\theta :G\rightarrow G$ the Cartan involution. Let $\rho _\theta :=\rho \circ \theta$. Assume that $\rho$ is irreducible and $\rho \not \cong \rho _\theta$. Then $H^*(X,F_\rho )=0$ [2, Chap. VII, Theorem 6.7]. Thus $\rho |_\Gamma$ is acyclic. In this case $T^{RS}(X,\rho )$ is independent of the metrics on X and in $F_\rho$ [25, Corollary 2.7]. Let $R^\rho (s):=R(s,\rho )R(s,\rho _\theta )$. Then by [35, Theorem 8.13] $R^\rho (s)$ is holomorphic at $s=0$ and

$$\begin{aligned} R^\rho (0)=T^{RS}(X,\rho )^4. \end{aligned}$$

(6.5)

Furthermore, from the discussions in [35, Sect. 9.1] follows that both $R(s,\rho )$ and $R(s,\rho _\theta )$ are holomorphic at $s=0$ and $R(0,\rho _\theta )=\overline{R(0,\rho )}$. Thus it follows that

$$\begin{aligned} |R(0,\rho )|=T^{RS}(X,\rho )^2. \end{aligned}$$

(6.6)

Hence $R(s,\rho )$ is regular at $s=0$ and $R(0,\rho )\ne 0$. Applying Theorem 1.1 we obtain Corollary 1.2.

Next we briefly recall the definition of the complex Reidemeister torsion [9]. Let V be ${\mathbb C}$-vector space of dimension m. Let $v=(v_1,\ldots ,v_m)$ and $w=(w_1,\ldots ,w_m)$ be two basis of V. Let $T=(t_{ij})$ be the matrix of the change of basis from v to w, i.e., $w_i=\sum _j t_{ij}v_j$. Put $[w/v]:={\text {det}}(T)$. Let

$$\begin{aligned} C^*:0\rightarrow C^0\xrightarrow {\delta _0} C^1\xrightarrow {\delta _1} \cdots \xrightarrow {\delta _{n-2}} C^{n-1}\xrightarrow {\delta _{n-1}}C^n \rightarrow 0 \end{aligned}$$

be a co-chain complex of finite dimensional complex vector spaces. We assume that $C^*$ is acyclic. Let $Z_q=\ker (\delta _q)$ and $B_q:={\text {Im}}(\delta _{q-1})\subset C^q$. Let $c_q$ be a preferred base of $C_q$. Choose a basis $b_q$ for $B_q$, $q=0,\ldots ,n$, and let ${\tilde{b}}_{q+1}$ be an independent set in $C_q$ such that $\delta _q({\tilde{b}}_{q+1})=b_{q+1}$. Then $(b_q,{\tilde{b}}_{q+1})$ is a basis of $C^q$ and $[b_q,\tilde{b}_{q+1}/c_q]$ depends only on $b_q$ and $b_{q+1}$. Therefore, we denote it by $[b_q,b_{q+1}/c_q]$. Then the complex Reidemeister torsion $\tau ^{\mathbb C}(C^*)\in {\mathbb C}$ of the co-chain complex $C^*$ is defined by

$$\begin{aligned} \tau ^{\mathbb C}(C^*):=\prod _{q=0}^n[b_q,b_{q+1}/c_q]^{(-1)^q}. \end{aligned}$$

(6.7)

Let K be a $C^\infty$-triangulation of X and ${\widetilde{K}}$ the lift of K to a triangulation of the universal covering ${\mathbb H}^d$ of X. Then $C^q({\widetilde{K}},{\mathbb C})$ is a module over the complex group algebra ${\mathbb C}[\Gamma ]$. Now let $\rho$ be an acyclic representation of $\Gamma$ in ${\text {SL}}(N,{\mathbb C})$. Let

$$\begin{aligned} C^q(K,\rho ):=C^q({\widetilde{K}},{\mathbb C})\otimes _{C[\Gamma ]}{\mathbb C}^N \end{aligned}$$

be the twisted co-chain group and

$$\begin{aligned} C^*(K,\rho ):0\rightarrow C^0(K,\rho )\xrightarrow {\partial _\rho }C^1(K,\rho )\xrightarrow {\partial _\rho } \cdots \xrightarrow {\partial _\rho }C^d(K,\rho )\rightarrow 0 \end{aligned}$$

the corresponding co-chain complex. Then $C^*(K,\rho )$ is acyclic. Let $e_1,\ldots ,e_{r_q}$ be a preferred basis of $C^q(\tilde{K},{\mathbb C})$ as a ${\mathbb C}[\Gamma ]$-module consisting of the duals of lifts of q-simplexes and let $v_1,\ldots ,v_N$ be a basis of ${\mathbb C}^N$. Then $\{e_i\otimes v_j:i=1,\ldots ,r_q,\quad j=1,\ldots ,N\}$ is a preferred basis of $C^q(K,\rho )$. Now consider the complex-valued Reidemeister torsion $\tau ^{\mathbb C}(C^*(K,\rho ))$. Since $\rho$ is a representation in ${\text {SL}}(N,{\mathbb C})$, a different choice of the preferred basis $\{e_i\}$ leads at most to a sign change of $\tau ^{\mathbb C}(C^*(K,\rho ))$. If $v^\prime$ is a different basis of ${\mathbb C}^N$, then $\tau ^{\mathbb C}(C^*(K,\rho ))$ changes by $[v^\prime /v]^{\chi (X)}$. Hence, if $\chi (X)=0$, $\tau ^{\mathbb C}(C^*(K,\rho ))$ is well defined as an element of ${\mathbb C}^*/\{\pm 1\}$. Since every two smooth triangulations of X admit a common subdivision, it follows from [22] that $\tau ^{\mathbb C}(C^*(K,\rho ))$ is independent of the smooth triangulation K. Put

$$\begin{aligned} \tau ^{\mathbb C}(X,\rho ):=\tau ^{\mathbb C}(C^*(K,\rho )). \end{aligned}$$

(6.8)

This is the complex valued Reidemeister torsion of X and $\rho$. If $\rho$ is not acyclic, then the same construction gives an element $\tau ^{\mathbb C}(X,\rho )\in {\text {det}}H^*(X,F_\rho )$ [9].

6.2 Deformations of Acyclic Unitary Representations

Let ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ be the set of all m-dimensional complex representations of $\Gamma$. It is well known that ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ has a natural structure of a complex algebraic variety [3, 13.6]. Recall that $\chi \in {\text {Rep}}(\Gamma ,{\mathbb C}^m)$ is called acyclic, if $H^*(X,F_\chi )=0$, where $F_\chi \rightarrow X$ is the flat vector bundle associated to $\chi$. Denote by ${\text {Rep}}_0(\Gamma ,{\mathbb C}^m)\subset {\text {Rep}}(\Gamma ,{\mathbb C}^m)$ the subset of all acyclic representations. A representation $\chi \in {\text {Rep}}(\Gamma ,{\mathbb C}^m)$ is called unitary, if there exists a Hermitian scalar product $\langle \cdot ,\cdot \rangle$ on ${\mathbb C}^m$ which is preserved by all maps $\chi (\gamma )$, $\gamma \in \Gamma$. Let ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)\subset {\text {Rep}}_0(\Gamma ,{\mathbb C}^m)$ be the subset of all unitary acyclic representations. By [14, Theorem 1.1] we get

Proposition 6.1

For every compact hyperbolic 3-manifold $\Gamma \backslash {\mathbb H}^3$, we have ${\text {Rep}}_0^u(\Gamma ,{\mathbb C}^m)\ne \emptyset$.

For $d>3$ it is not known if a d-dimensional compact oriented hyperbolic manifold $\Gamma \backslash {\mathbb H}^d$ admits a unitary flat bundle. Assume that ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)\ne \emptyset$. Let $\chi \in {\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)$. For such a representation the flat Laplacian $\Delta _{k,\chi }^\sharp$ equals the usual Laplace operator $\Delta _{k,\chi }$ and $T^{\mathbb C}(X,\chi )=T^{RS}(X,\chi )^2$. Moreover, $h_k=0$, $k=0,\ldots ,d$, which implies $h=0$ and $T_0(X,\chi )=1$. Thus $R(s,\chi )$ is regular at $s=0$ and from Theorem 1.1 we recover Fried’s result [10]

$$\begin{aligned} R(0,\chi )=T^{RS}(X,\chi )^2. \end{aligned}$$

(6.9)

We equip ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ with the topology obtained from its structure as complex algebraic variety. The complement of the singular set is a complex manifold. Let $W\subset {\text {Rep}}(\Gamma ,{\mathbb C}^m)$ be the connected component of ${\text {Rep}}(\Gamma ,{\mathbb C}^m)$ which contains ${\text {Rep}}^u_0(\Gamma ,{\mathbb C}^m)$. Let $\chi _0\in W$ be a unitary acyclic representation and let $E_0$ be the associated flat vector bundle. By [15, Prop. 4.5] every vector bundles $E_\chi$, $\chi \in W$, is isomorphic to $E_0$. Thus the flat connection on $E_\chi$, which is induced by the trivial connection on ${\widetilde{X}}\times {\mathbb C}^m$, corresponds to a flat connection $\nabla _\chi$ on $E_0$. Now recall that

$$\begin{aligned} \Delta _\chi ^\sharp =(d_\chi +d_\chi ^{*,\sharp })^2, \end{aligned}$$

where

$$\begin{aligned} d_\chi ^{*,\sharp }\big |_{\Lambda ^p(X,E_\chi )}=(-1)^p(\star \otimes {\text {Id}})d_\chi (\star \otimes {\text {Id}}). \end{aligned}$$

Via the isomorphism $E_\chi \cong E_0$, the operator $d_\chi +d_\chi ^{*,\sharp }$ corresponds to the operator

$$\begin{aligned} D_\chi ^\sharp :=\nabla _\chi +\nabla _\chi ^{*,\sharp }:\Lambda ^*(X,E_0)\rightarrow \Lambda ^*(X,E_0), \end{aligned}$$

where

$$\begin{aligned} \nabla _\chi ^{*,\sharp }=(-1)^p(\star \otimes {\text {Id}})\nabla _\chi (\star \otimes {\text {Id}}). \end{aligned}$$

Let $\nabla _0$ be the unitary flat connection on $E_0$. Let ${\mathcal C}(E_0)$ denote the space of connections on $E_0$. Recall that ${\mathcal C}(E_0)$ can be identified with $\Lambda ^1(X,{\text {End}}(E_0))$ by associating to a connection $\nabla \in {\mathcal C}(E_0)$ the 1-form $\nabla -\nabla _0\in \Lambda ^1(X,{\text {End}}(E_0))$. We equip ${\mathcal C}(E_0)$ with the $C^0$-topology defined by the sup-norm $\Vert \omega \Vert _{\sup }:=\max _{x\in X} |\omega (x)|$, $\omega \in \Lambda ^1(X,{\text {End}}(E_0))$, where $|\cdot |$ denotes the natural norm on $\Lambda ^1(T^*X)\otimes E_0$. Since $E_0$ is acyclic, $D_0:=\nabla _0+\nabla _0^*$ is invertible. If $\Vert \nabla _\chi -\nabla _0\Vert \ll 1$ it follows as in [3, Prop 6.8] that $D_\chi ^\sharp$ is invertible and hence $\Delta _{\chi }^\sharp =(D_\chi ^\sharp )^2$ is invertible too. Thus we get

Lemma 6.2

There exists an open neighborhood $V\subset W$ of ${\text {Rep}}_0^u(\Gamma ,{\mathbb C}^m)$ such that $\Delta _{\chi }^\sharp$ is invertible for all $\chi \in V$.

Let $\chi \in V$. Then we have $h_k=\dim (\Delta _{k,\chi }^\sharp )=0$, $k=0,\ldots ,d$, and therefore the order h of $R(s,\chi )$ at $s=0$ vanishes. Also $C(d,\chi )=1$ and $T_0(X,\chi )=1$. Thus by Theorem 1.1 we obtain Proposition 1.3.