Complex valued analytic torsion and dynamical zeta function on locally symmetric spaces
CCOMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETAFUNCTION ON LOCALLY SYMMETRIC SPACES
SHU SHENA
BSTRACT . We show that the Ruelle dynamical zeta function on a closed odd di-mensional locally symmetric space twisted by an arbitrary flat vector bundle hasa meromorphic extension to the whole complex plane and that its leading termin the Laurent series at the zero point is related to the regularised determinantof the flat Laplacian of Cappell-Miller. When the flat vector bundle is close to anacyclic and unitary one, we show that the dynamical zeta function is regular atthe zero point and that its value is equal to the complex valued analytic torsion ofCappell-Miller. This generalises author’s previous results for unitarily flat vectorbundles as well as Müller and Spilioti’s results on hyperbolic manifolds. C ONTENTS
Introduction 21. The generalised Laplacian 111.1. The spectral theory of a generalised Laplacian 111.2. The semigroup of a generalised Laplacian 121.3. The regularised determinant 151.4. Flat vector bundles 151.5. A generalised Laplacian and flat vector bundles 171.6. Flat Laplacian and the Cappell-Miller analytic torsion 202. Müller’s Selberg trace formula 222.1. Real reductive groups 222.2. The Casimir operator 242.3. The symmetric space 242.4. Semisimple elements 252.5. Semisimple orbital integrals 262.6. A discrete subgroup of G g h , g ) 303.4. A lifting property 314. The Ruelle dynamical zeta functions for arbitrary twist 32 Date : September 9, 2020.2020
Mathematics Subject Classification.
Key words and phrases.
Index theory and related fixed point theorems, analytic torsion, Sel-berg trace formula, Ruelle dynamical zeta function.I am indebted to Xiaolong Han, Xiaonan Ma, Jianqing Yu, and Weiping Zhang for reading thepreliminary version of this article and for useful discussions. a r X i v : . [ m a t h . DG ] S e p OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 2
5. The Selberg zeta function for arbitrary twist 345.1. The structure of the reductive group G with δ ( G ) = σ (cid:192) r j C ∞ ( Γ \ G , (cid:98) p ∗ F ) 416.3. Formulas for r η , ρ and r j NTRODUCTION
The purpose of this article is to study the relation between the complex valuedanalytic torsion of Cappell-Miller and the value at the zero point of the Ruelledynamical zeta function associated to a flat vector bundle, which is not necessarilyunitarily flat, on a closed odd dimensional locally symmetric space of reductivetype.Let Z be a smooth closed manifold. Let F be a complex flat vector bundle on Z .Let H • ( Z , F ) be the cohomology of the sheaf of locally constant sections of F . Weassume H • ( Z , F ) = g TZ , g F be the metrics on T Z and F . The analytic torsion is a real positivenumber introduced by Ray and Singer [RaSi71]. It is a spectral invariant definedby the Hodge Laplacian associated with g TZ , g F . If dim Z is odd, they showedthat the analytic torsion does not depend on the metric data, and conjecturedthat, if F is unitarily flat (i.e., the holonomy representation of F is unitary), thereis an equality between the analytic torsion and its topological counterpart, theReidemeister torsion [Re35, Fra35, dR50].Ray and Singer’s conjecture was proved by Cheeger [Ch79] and Müller [Mü78],and is now known under the name of Cheeger-Müller Theorem. Bismut-Zhang[BZ92] and Müller [Mü93] simultaneously considered generalisations of this re-sult. Müller [Mü93] extended his result to odd dimensional oriented manifold andonly det F is required to be unitarily flat. Using the Witten deformation [W82],Bismut and Zhang [BZ92] generalised the original Cheeger-Müller theorem to ar-bitrary flat vector bundles with arbitrary Hermitian metrics on a manifold witharbitrary dimension.On the other hand, Turaev [T89, FaT00] generalised the concept of the Reide-meister torsion to a complex valued invariant whose absolute value provides theoriginal Reidemeister torsion, with the help of the so-called Euler structure.The analytic counterpart of the Turaev torsion was introduced by Braverman-Kappeler, Burghelea-Haller, and Cappell-Miller separately. In [BrKa08, BrKa07], OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 3 using the signature operator, Braverman and Kappeler defined what they calledthe refined analytic torsion for flat vector bundles on odd dimensional manifolds,and showed that it is equal to the Turaev torsion up to a multiplication by a com-plex number of absolute value one. In [BuH07, BuH08], Burghelea and Hallerdefined a generalised analytic torsion associated to a nondegenerate symmetricbilinear form on a flat vector bundle over an arbitrary dimensional manifold. Fol-lowing the idea of [BZ92], Su and Zhang [SuZ08] showed that the Burghelea-Haller torsion coincides with the square of the Turaev torsion. In [BuH10], usingdifferent methods, Burghelea and Haller proved Su-Zhang’s result up to the sign,in the case of odd dimension manifolds. In [CMil10], with the help of the flatLaplacian, Cappell-Miller introduced another version of the analytic torsion as-sociated to a flat vector bundle on an arbitrary dimensional manifold without as-suming the existence of the nondegenerate symmetric bilinear form. By adoptingthe proof of Su-Zhang [SuZ08], Cappell-Miller showed that their torsion invariantis equal to its topological counterpart when the manifold is oriented and is of odddimension.Milnor [Mi68a] initiated the study of the relation between the torsion invariantand a dynamical system. Fried [F86] showed that the Ray-Singer analytic torsionof an acyclic unitarily flat vector bundle on a closed odd dimensional hyperbolicmanifold is equal to the value at the zero point of the Ruelle dynamical zeta func-tion of the geodesic flow. He conjectured [F87, p. 66, Conjecture] that similarresults should hold true for more general flows. In [Sh18], following previouscontributions by Moscovici-Stanton [MoSt91], using Bismut’s orbital integral for-mula [B11], the author affirmed the Fried conjecture for geodesic flows on closedodd dimensional locally symmetric manifolds equipped with a unitarily flat vec-tor bundle. In [ShY17], the authors made a further generalisation to closed odddimensional locally symmetric orbifolds. We refer the reader to [Ma19] for anintroduction to the technique used in [Sh18].The Fried conjecture for arbitrary flat vector bundle is recently consideredby Spilioti [Sp20a, Sp18, Sp20b] and by Müller [Mü20] for closed odd dimen-sional hyperbolic manifolds. When the underlying manifold is hyperbolic, Spilioti[Sp20a, Sp18] showed that the Ruelle dynamical zeta function has a meromor-phic extension to C . Müller [Mü20] related its leading coefficient in the Laurentseries at the zero point to the complex valued analytic torsion of Cappell-Miller.When the flat vector bundle is close to an acyclic and unitary one, it is shown[Mü20, Sp20b] that the dynamical zeta function is regular at the zero point andits value is equal to the Cappell-Miller analytic torsion.In this article, we extend the above results of Müller and Spilioti to closed odddimensional locally symmetric spaces of arbitrary ranks.We refer the reader to [Sh19] for a survey on the Fried conjecture on differentflows, like Morse-Smale flow [ShY18] and Anosov flow [DGRSh20].Now, we will describe our results in more detail, and explain the techniquesused in their proofs.0.1. The analytic torsions of Ray-Singer and Cappell-Miller.
Let Z be asmooth closed manifold of dimension m , and let ( F , ∇ F ) be a complex flat vector OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 4 bundle of rank r on Z with flat connection ∇ F . Let Γ be the fundamental groupof Z . Let ρ : Γ → GL r ( C ) be the holonomy representation of F . Let (cid:161) Ω • ( Z , F ), d Z (cid:162) be the de Rham complex with values in F . Let H • ( Z , F ) be the correspondingcohomology.If X is the universal cover of Z , if ( Ω • ( X ) ⊗ C r ) Γ is the space of C r -valued Γ -invariant forms on X , then Ω • ( Z , F ) = (cid:161) Ω • ( X ) ⊗ C r (cid:162) Γ .(0.1)Let g TZ be a Riemannian metric on T Z . Let (cid:3) Z be the associated flat Laplacianacting on Ω • ( Z , F ). It can be obtained by the restriction of the Hodge Laplacianon X to the Γ -invariant subspace (0.1).If F has a flat metric g F , (cid:3) Z is just the Hodge Laplacian associated with g TZ , g F . By Hodge theory, we haveker (cid:3) Z = H • ( Z , F ).(0.2)The Ray-Singer analytic torsion is a positive real number defined by the followingweighted product of the zeta regularised determinants T RS ( F ) = m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i /2 .(0.3)Formally, det ∗ means the product of non zero eigenvalues counted with multiplic-ities. By a fundamental result of Ray-Singer [RaSi71] (see also [BZ92, Theorem0.1]), if dim Z is odd and if H • ( Z , F ) = T RS ( F ) is independent of g TZ and g F .For general F , (cid:3) Z is a second order elliptic differential operator, which is notnecessarily self-adjoint. Still (cid:3) Z has a Hodge-like theory. Namely, if Ω • ( Z , F ) isthe characteristic space of (cid:3) Z associated with the eigenvalue 0, then ( Ω • ( Z , F ), d Z )is a finite dimensional complex such that H • (cid:179) Ω • ( Z , F ), d Z (cid:180) (cid:39) H • ( Z , F ).(0.4)By (0.4), if (cid:3) Z is invertible, then H • ( Z , F ) =
0. In this case, the Cappell-Millertorsion T CM ( F ) ∈ C is a non vanishing complex number given by the followingweighted product of the zeta regularised determinants (see Section 1.3 for a defi-nition) T CM ( F ) = m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i .(0.5)Note that with our convention, Cappell-Miller’s original definition [CMil10] cor-responds to the inverse of T CM ( F ).For a general F , the Cappell-Miller torsion T CM ( F ) is an element in the dual ofthe determinant line of the graded space H • ( Z , F ⊕ F ∗ ) (see [CMil10, Section 8]).It has been shown that [CMil10, Theorem 8.3] if dim Z is odd, T CM ( F ) does notdepend on g TZ and is a topological invariant.0.2. The dynamical zeta function of Ruelle.
Let us recall the definition ofthe Ruelle dynamical zeta function for geodesic flows introduced by Fried [F87,Section 5] (see also [Sh19, Section 2]).Let ( Z , g TZ ) be a connected manifold with nonpositive sectional curvature. Let[ Γ ] be the set of the conjugacy classes of the fundamental group Γ . For [ γ ] ∈ [ Γ ] , OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 5 let B [ γ ] be the set of closed geodesics in the free homotopy class associated to [ γ ].It is easy to see that all the elements in B [ γ ] have the same length (cid:96) [ γ ] .For simplicity, assume that all the B [ γ ] are smooth finite dimensional subman-ifolds of the loop space of Z . This is the case if ( Z , g TZ ) has a negative sectionalcurvature or if Z is locally symmetric. If γ = B [1] = Z is the space of trivialgeodesics. If γ (cid:54)=
1, the group (cid:83) acts locally freely on B [ γ ] by rotation, so that B [ γ ] / (cid:83) is an orbifold. Let χ orb ( B [ γ ] / (cid:83) ) ∈ Q be the orbifold Euler characteristic[Sa57]. Denote by m [ γ ] = (cid:175)(cid:175) ker (cid:161) (cid:83) → Diff (cid:161) B [ γ ] (cid:162)(cid:162)(cid:175)(cid:175) ∈ N ∗ (0.6)the multiplicity of a generic element in B [ γ ] . Let (cid:178) [ γ ] = ± B [ γ ] induced by the geodesicflow (see [Sh19, (2.17)] for a precise definition). If Z is locally symmetric, then (cid:178) [ γ ] = ρ : Γ → GL r ( C ) be a representation of Γ . The formal dynamical zeta functionof Ruelle is defined for σ ∈ C by R ρ ( σ ) = exp (cid:195) (cid:88) [ γ ] ∈ [ Γ + ] (cid:178) [ γ ] Tr[ ρ ( γ )] χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) m [ γ ] e − σ(cid:96) [ γ ] (cid:33) ,(0.7)where [ Γ + ] = [ Γ ] − { } is the set of the non trivial conjugacy classes of Γ . We will saythat the formal dynamical zeta function is well defined if R ρ ( σ ) is holomorphic forRe ( σ ) (cid:192) σ ∈ C .Observe that if ( Z , g TZ ) is of negative sectional curvature, then B [ γ ] (cid:39) (cid:83) and χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) = ρ is a trivial representation, R ρ ( σ ) was recently shown to be welldefined by Giulietti-Liverani-Pollicott [GiLPo13] and Dyatlov-Zworski [DyZw16].Moreover, Dyatlov and Zworski [DyZw17] and Borns-Weil and Shen [BWSh20]showed that, if ( Z , g TZ ) is a negatively curved surface, the order of the zero of R ρ ( σ ) at σ = Z . For general ρ , the proof of the mero-morphic extension of R ρ is not particularly difficult. In particular, when 0 is nota resonance of the weight dynamical system associated to ρ , then R ρ (0) is welldefined. In this case, Dang, Guillarmou, Rivière, and Shen [DGRSh20] showedthat R ρ (0) remains unchanged under a small perturbation of the geodesic flow.0.3. Results of Fried, Müller, and Spilioti on hyperbolic manifolds.
As-sume that Z is a connected closed odd dimensional orientable hyperbolic mani-fold. Recall that ( F , ∇ F ) is a flat vector bundle on Z with holonomy ρ : Γ → GL r ( C ).If F has a flat metric or equivalently ρ is unitary, using the Selberg trace for-mula, Fried [F86, Theorem 3] showed that there exist explicit constants C ρ ∈ R ∗ and r ρ ∈ Z such that as σ → R ρ ( σ ) = C ρ T RS ( F ) σ r ρ + O ( σ r ρ + ).(0.9) More precisely, the authors [GiLPo13, DyZw16] showed the meromorphic extension when theflow is Anosov.
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 6
Moreover, if H · ( Z , F ) =
0, then C ρ = r ρ = R ρ (0) = T RS ( F ) .(0.11)When ρ is not unitary, in [Mü20, Theorem 1.1], Müller has shown a similarresult. Namely, there exist explicit constants C ρ ∈ R ∗ and r ρ ∈ Z such that as σ → R ρ ( σ ) = C ρ m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i + O ( σ r ρ + ).(0.12)Moreover, the constants C ρ and r ρ depend only on the dimension of Ω • ( Z , F ).If ρ is close to a unitary and acyclic representation of Γ in the sense of C ([BrKa08, Section 6.7], see also Section 1.6), by [BrKa08, Proposition 6.8] (see also[Mü20, Lemma 6.2]), we have Ω • ( Z , F ) =
0. In this case, in [Mü20, Proposition1.3], Müller obtained C ρ = r ρ =
0, and R ρ (0) = T CM ( F ).(0.13)Similar results were also obtained by Spilioti [Sp20b, Theorem 2].0.4. The main result of the article.
Let G be a linear connected real reductivegroup [K86, p. 3], and let θ be the Cartan involution. Let K be the maximalcompact subgroup of the points of G that are fixed by θ . Let k and g be the Liealgebras of K and G , and let g = p ⊕ k be the Cartan decomposition. Let B be anondegenerate bilinear symmetric form on g which is invariant under the adjointaction of G and under θ . Assume that B is positive on p and negative on k . Set X = G / K . Then B | p induces a Riemannian metric g T X on X , such that ( X , g T X )has a parallel nonpositive sectional curvature.Let Γ ⊂ G be a discrete cocompact subgroup of G . Assume for the moment that Γ is torsion free. Set Z = Γ \ X . Then Z is a closed locally symmetric manifold,equipped with the induced Riemannian metric g TZ . Let ρ : Γ → GL r ( C ) be a rep-resentation of Γ . Let F = Γ \ ( X × C r ) be the associated flat vector bundle on Z .When ρ is unitary, in [F87, p. 66, Conjecture], Fried conjectured similar re-sults (0.9)-(0.11) still hold. This conjecture was recently proved by the author[Sh18] and generalised by Shen-Yu [ShY17] on locally symmetric orbifolds where Γ is no longer torsion free, following previous contributions by Moscovici-Stanton[MoSt91], using Bismut’s orbital integral formula [B11].The main result of this article concernes the case where ρ is not always unitary.It generalises [F86, Theorem 3], [Mü20, Theorem 1.1, Proposition 1.3], [Sp20b,Theorem 2] to locally symmetric space, as well as [Sh18, Theorem 1.1] to nonunitary twist. Recall that (cid:3) Z is the flat Laplacian on Ω • ( Z , F ). Theorem 0.1.
Assume that dim
Z is odd. The following statements hold:i) The dynamical zeta function R ρ ( σ ) is holomorphic for Re ( σ ) (cid:192) and extendsmeromorphically to σ ∈ C . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 7 ii) There exist explicit constants C ρ ∈ R ∗ and r ρ ∈ Z (see (5.33) and (5.35) ) suchthat, when σ → , we haveR ρ ( σ ) = C ρ (cid:40) m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i (cid:41) σ r ρ + O ( σ r ρ + ).(0.14) iii) If ρ is close enough (see (1.78) ) to an acyclic and unitary representation of Γ ,then (cid:3) Z is invertible andC ρ = r ρ = so that R ρ (0) = T CM ( F ).(0.16)We will also extend the above result to the case where Γ is not torsion free.Then Z is an orbifold and F is a flat orbifold vector bundle. In this case theanalytic torsion of Ray-Singer is still well-defined [Ma05, DaY17, ShY17], andwe can define the Cappell-Miller torsion in a similar way. Also, the Ruelle zetafunction R ρ ( σ ) was introduced in [ShY17]. Theorem 0.2.
The statement of Theorem 0.1 holds when Z = Γ \ G / K is an orbifold.
If the representation ρ : Γ → GL r ( C ) admits an invariant bilinear form, it in-duces a flat bilinear form b F on F . Since b F is flat, by (0.1), the Laplacian ofBurghelea-Haller [BuH07, (26)] [SuZ08, (2.20)] coincides with the flat Laplacian (cid:3) Z of Cappell-Miller. In particular, we get the following corollary. Corollary 0.3.
If the representation ρ : Γ → GL r ( C ) admits an invariant bilin-ear form, the statements of Theorems 0.1 and 0.2 hold for the complexed valuedanalytic torsion of Burghelea-Haller. Moscovici-Stanton’s vanishing theorem and Kazhdan’s property (T).
Let δ ( G ) ∈ N be the fundamental rank of G , i.e., the difference between the com-plex ranks of G and K . Note that δ ( G ) and dim Z have the same parity.If δ ( G ) (cid:202)
3, our theorem follows easily from an observation originally due toMoscovici-Stanton [MoSt91, Corollary 2.2, Remark 3.7] (see also Bismut [B11,Theorem 7.93]). More precisely, R ρ ( σ ) ≡ m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i = δ ( G ) =
1, Theorem 0.1 i) ii) is more significant (see [BMaZ11, BMaZ17]and also [MüPf13] when ρ is the restriction of a representation of G ). By theclassification theory of real simple Lie algebras ([B11, Remark 7.9.2] and [Sh18,Theorem 6.15]), δ ( G ) = g = g ⊕ R sl ( R ) so ( p , q ) with p (cid:202) q , pq > g is a real semisimple Lie algebra such that δ ( g ) =
0. Here, δ ( g ) isdefined in an obvious way as δ ( G ). Note that so (1, 1) (cid:39) R . However, we single outthis case since R is not considered to be simple. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 8
Theorem 0.1 iii) makes sense only if Γ has an acyclic and unitary represen-tation and if such a representation is not isolated. A construction of acyclic andunitary representations can be founded in [FrN15]. Assume now that Γ admits anacyclic and unitary representation. If Γ has Kazhdan’s property (T), by [Rap99],all unitary representations of Γ are isolated. Therefore, Theorem 0.1 iii) givesinteresting results only if Γ does not have Kazhdan’s property (T). By [BHdlVa08,Theorems 1.7.1 and 3.5.4], this is the case when g contains a factor so ( p , 1) or su ( p , 1). Combining it with (0.18), we get g = g ⊕ (cid:40) R so ( p , 1) with p (cid:202) g = g ⊕ (cid:40) su ( s , 1) with s (cid:202) so ( s , 1) with s (cid:202) ⊕ (cid:40) sl ( R ) so ( p , q ) with p (cid:202) q > pq odd,(0.20)where g is a real semisimple Lie algebra with δ ( g ) = The proof of Theorem 0.1.
Our proof of Theorem 0.1 is inspired by [Sh18].As we have seen that we can reduce the proof to the case where δ ( G ) =
1. In thiscase, our proof relies on the Selberg zeta functions.Assume now δ ( G ) =
1. Let t ⊂ k be a Cartan subalgebra of k . Let h = z ( t ) ⊂ g be the stabiliser of t in g . By [K86, p. 129], h ⊂ g is a θ -invariant fundamentalCartan subalgebra of g . Let h = b ⊕ t be the Cartan decomposition of h . Note thatdim b = δ ( G ) =
1. Let H ⊂ G be the associated Cartan subgroup of G .Let Z ( b ) ⊂ G be the stabiliser of b in G with Lie algebra z ( b ). Let Z ( b ) be theconnected component of the identity in Z ( b ). Then z ( b ), Z ( b ) split z ( b ) = b ⊕ m , Z ( b ) = exp( b ) × M ,(0.21)where M is a connected reductive subgroup of G with Lie algebra m . Let m = p m ⊕ k m be the Cartan decomposition of m . Let z ⊥ ( b ) ⊂ g be the orthogonal space of z ⊥ ( b ) with respect to B .We fix an orientation on b , whose choice is irrelevant. Let n ⊂ g be the directsum of the eigenspaces associated with the positive eigenvalues of a positive ele-ment in b . Then, n is a Lie subalgebra of g so that z ⊥ ( b ) = n ⊕ θ n .(0.22)Let η = η + − η − be a virtual representation of M acting on the finite dimensionalcomplex vector spaces E η = E + η − E − η such that • the Casimir of M acts on η ± by the same scalar; • the restriction of η to K M = K ∩ M lifts uniquely to a virtual representationof K . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 9
The Selberg zeta function for the pair ( η , ρ ) is defined formally for σ ∈ C by Z η , ρ ( σ ) = exp (cid:181) − (cid:88) [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H χ orb ( B [ γ ] / (cid:83) ) m [ γ ] Tr (cid:163) ρ ( γ ) (cid:164) Tr E η s [ k − ] (cid:175)(cid:175) det(1 − Ad( e a k − )) | z ⊥ ( b ) (cid:175)(cid:175) e − σ(cid:96) [ γ ] (cid:182) ,(0.23)where the sum is taken over the non elliptic conjugacy classes [ γ ] of Γ such that γ can be conjugate by elements of G into the Cartan subgroup H .In [Sh18, Section 6], we have shown that the adjoint action of K M on p m liftsuniquely to a virtual representation of K . Let (cid:98) η = (cid:98) η + − (cid:98) η − be the unique virtualrepresentation of K such that (cid:98) η | K M = Λ • ( p ∗ m ) (cid:98) ⊗ R η .(0.24)The Casimir operator of g acts as a generalised Laplacian C g , Z , (cid:98) η ± , ρ on the smoothsections over Z of the locally homogenous vector bundle, induced by (cid:98) η ± , twisted bythe flat vector bundle F (see (2.26)). By the general theory on elliptic differentialoperators, the regularised determinant det (cid:161) C g , Z , (cid:98) η ± , ρ + σ (cid:162) is holomorphic on σ ∈ C .Following [Sh18, Section 7], using Müller’s Selberg trace formula for arbitrarytwists [Mü11], in Section 5, we show that for Re ( σ ) (cid:192)
1, up to a multiplication bya non zero entire function, Z η , ρ ( σ ) is just the graded regularised determinantdet (cid:161) C g , Z , (cid:98) η + , ρ + σ η + σ (cid:162) det (cid:161) C g , Z , (cid:98) η − , ρ + σ η + σ (cid:162) ,(0.25)where σ η ∈ R is some constant. In this way, we deduce the meromorphic extensionof Z η , ρ and we get precise information about the poles and zeros as well as theirmultiplicities.In [Sh18, Section 6], we have shown that the adjoint representations η j of M on Λ j ( n ∗ ) ⊗ R C satisfy our two previous assumptions. Using the fact that R ρ isan alternating product of Z η j , ρ , we deduce the meromorphic extension of R ρ . Inparticular, up to a multiplication by a non zero entire function, R ρ is a productof the graded regularised determinants (0.25). Comparing this with the weightedregularised determinant of the flat Laplacian (0.5), we get (0.14).As in [Sh18, Section 8], the proof of (0.15) requires a detailed analysis on theright regular representation of G on the left Γ -invariant space C ∞ ( G , C r ) Γ . Ingeneral, C ∞ ( G , C r ) Γ is not unitarisable. Let Z ( g C ) be the centre of the envelop-ing algebra of the complexified Lie algebra g C . Given a character χ : Z ( g C ) → C ,let V χ ( Γ , ρ ) ⊂ C ∞ ( G , C r ) Γ be the subspace of C ∞ ( G , C r ) Γ consisting of the K -finiteelements such that for all z ∈ Z ( g C ), z − χ ( z ) acts nilpotently. In Section 6.2, wewill show that V χ ( Γ , ρ ) is a Harish-Chandra ( g C , K )-module with a generalisedinfinitesimal character χ (see Definitions 6.1 and 6.2). Proceeding similarly as in[Sh18, Section 8], we can deduce that C ρ and r ρ are determined by the Euler char-acteristics of certain cohomologies of V χ ( Γ , ρ ), where χ is the trivial character.The proof of (0.15) reduces to showing V χ ( Γ , ρ ) = ρ is unitary, using fundamental results of Vogan-Zuckerman [VoZu84],Vogan [Vo84], and Salamanca-Riba [SR99], in [Sh18, Proposition 8.12], we haveshown that V χ ( Γ , ρ ) = ρ is acyclic. When ρ is not unitary, even if ρ is acyclic, V χ ( Γ , ρ ) does not always vanish (see [Mü20, (1.19)]). OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 10
In Section 6, using a perturbation argument, we will show that V χ ( Γ , ρ ) van-ishes if ρ is close enough to an acyclic and unitary representation ρ . Indeed,if { z i } dim h i = is a system of generators Z ( g C ), we need to show that the operators z i − χ ( z i ) act invertiblely on sections of certain locally homogenous vector bundle(see Proposition 6.7) twisted by the flat vector bundle associated to ρ . Note that z i − χ ( z i ) are differential operators. If ρ is close enough to ρ in the sense of C k with k big enough, all the above operators are lower order small perturbationsof the corresponding operators associated to ρ which are invertible. From theabove considerations, we can deduce the desired invertibility.Remark that in the case of hyperbolic manifolds, only the flat Laplacian is in-volved. Since the flat Laplacian is the square of the flat Hodge-Dirac operator, andsince the flat Hodge-Dirac operator is a perturbation of an invertible self-adjointHodge-Dirac operator associated to ρ by a small 0-th order operator. Therefore,we can take k = • as in the case where ρ is unitary, Bismut’s geometric orbital integral for-mula plays an essential role in our proof of Theorem 0.1. Here, we usesuch formulas inexplicitly, since all the involved orbital integrals havebeen evaluated in [Sh18] and [ShY17]. • it is crucial that the adjoint representations of K M on p m and n lift to vir-tual representations of K . In [Sh18], we prove this fact by the classificationtheory of simple real Lie algebras (0.18). One of the contributions of thecurrent paper is to give a simple new conceptual proof of a complexifiedversion of [Sh18, Theorem 6.11], which is enough for our applications.0.7. Organisation of the article.
This article is organised as follows. In Section1, we introduce the generalised Laplacian, its heat kernel, its regularised deter-minant, as well as the flat Laplacian and the Cappell-Miller analytic torsion.In Section 2, we show the Selberg trace formula with arbitrary twists for theheat operator associated to the Casmir, which is originally due to Müller [Mü11].In Section 3, we give a simple new conceptual proof for the lifting property[Sh18, Theorem 6.11].In Section 4, we introduce the Ruelle zeta function R ρ , and we state Theorem0.1 as Theorem 4.3.In Section 5, we introduce the Selberg zeta function Z η , ρ . We show the mero-morphic extensions of Z η , ρ and R ρ , and we establish Theorem 0.1 i) ii).In Section 6, we show certain cohomological formulas for C ρ and r ρ . We proveTheorem 0.1 iii).In Section 7, we extend Theorem 0.1 to orbifolds.0.8. Notation.
Throughout the paper, we use the superconnection formalism of[Q85] and [BeGeV04, Section 1.3]. If A is a Z -graded algebra and if a , b ∈ A , thesupercommutator [ a , b ] is given by ab − ( − deg a deg b ba .(0.26) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 11 If B is another Z -graded algebra, we denote by A (cid:98) ⊗ B the super tensor productalgebra of A and B . If E = E + ⊕ E − is a Z -graded vector space, the algebra End( E )is Z -graded. If τ = ± E ± and if a ∈ End( E ), the supertrace Tr s [ a ] is definedby Tr[ τ a ].If V is a real vector space, we will use the notation V C = V ⊗ R C for its com-plexification. If M is a topological group, we will denote by M the connectedcomponent of the identity in M . We make the convention that N = {
0, 1, 2, . . . } , N ∗ = {
1, 2, . . . } , R ∗+ = (0, ∞ ).1. T HE GENERALISED L APLACIAN
The purpose of this section is to recall some properties of a generalised Lapla-cian defined on a closed manifold. The generalised Laplacian P is a second order(not necessarily self-adjoint) elliptic differential operator, whose principal symbolis a positive scalar. We show that the regularised determinant det( σ + P ) is a holo-morphic function on σ ∈ C . Our proof of the meromorphic extension of the Ruellezeta function associated to arbitrary twist given in Section 5 relies on this fact.This section is organised as follows. In Sections 1.1-1.3, we introduce the gen-eralised Laplacian, the associated semigroup, and the regularised determinant.In Section 1.4, we introduce a flat vector bundle and its holonomy representa-tion. We establish an estimate on the growth of the holonomy representation.In Section 1.5, we construct a generalised Laplacian acting on the sections ofa flat vector bundle. We relates its heat kernel with the lifted heat kernel on theuniversal cover.Finally, in Section 1.6, given a flat vector bundle, we recall the construction ofthe flat Laplacian of Cappell-Miller, which is a generalised Laplacian. We intro-duce the complex valued analytic torsion of Cappell-Miller under the assumptionthat the flat Laplacian is invertible.1.1. The spectral theory of a generalised Laplacian.
Let Z be a closed man-ifold of dimension m . Let E be a complex vector bundle on Z . Let C ∞ ( Z , E ) be thespace of C ∞ -sections of E .Let g TZ be a Riemannian metric on Z . Denote by dv Z ∈ Ω m ( Z , o ( T Z )) the asso-ciated Riemannian volume form, where o ( T Z ) is the orientation line bundle of Z .Let g E be a Hermitian metric on E . Let 〈 , 〉 g E be the induced Hermitian producton E . We equip C ∞ ( Z , E ) with an L -product defined for u , u ∈ C ∞ ( Z , E ) by 〈 u , u 〉 L = (cid:90) z ∈ Z 〈 u ( z ), u ( z ) 〉 g E dv Z .(1.1)Let L ( Z , E ) be the space of L -sections of E . For s ∈ R , let H s ( Z , E ) be the s -Sobolev space of sections of E .Let P be a generalised Laplacian acting on C ∞ ( Z , E ) in the sense of [BeGeV04,Definition 2.2]. Namely, if | · | g TZ , ∗ is the dual metric on T ∗ Z induced by g TZ , then P is a second order elliptic differential operator with the principal symbol σ ( P )( x , ξ ) = | ξ | g TZ , ∗ x · id E x , for ( x , ξ ) ∈ T ∗ Z .(1.2) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 12
Equivalently, there is a metric connection ∇ E on E and a first order differentialoperator A such that P = ( ∇ E ) ∗ ∇ E + A ,(1.3)where ( ∇ E ) ∗ : C ∞ ( Z , T ∗ Z ⊗ R E ) → C ∞ ( Z , E ) is the formal adjoint of ∇ E with respectto the L -product induced by g E and g TZ .By (1.3), there are c > C > u ∈ C ∞ ( Z , E ),Re 〈 P u , u 〉 L (cid:202) c (cid:107) u (cid:107) H − C (cid:107) u (cid:107) L .(1.4)For s , s ∈ R , if L is a bounded operator from H s ( Z , E ) to H s ( Z , E ), denotedby (cid:107) L (cid:107) H s , H s the corresponding operator norm. Consider P as an operator withdomain H ( Z , E ). Since P is elliptic, the unbounded operator ( P , H ( Z , E )) isclosed. By [S01, Theorem 9.3], ( P , H ( Z , E )) has a compact resolvent, so that itsspectrum Sp( P ) is discrete. Moreover, for any (cid:178) >
0, there is r > P ) ⊂ (cid:110) λ ∈ C : | λ | < r or | Im( λ ) | < (cid:178) Re ( λ ) (cid:111) .(1.5)Also, there are C (cid:178) , r > C (cid:178) , r , n > n ∈ Z such that for all λ ∈ C with | λ | (cid:202) r and | Im( λ ) | (cid:202) (cid:178) Re ( λ ), we have (cid:176)(cid:176) ( λ − P ) − (cid:176)(cid:176) L , L (cid:201) C (cid:178) , r | λ | , (cid:176)(cid:176) ( λ − P ) − (cid:176)(cid:176) H n , H n + (cid:201) C (cid:178) , r , n .(1.6)Take λ ∈ Sp( P ). Let V P ( λ ) ⊂ L ( Z , E ) be the characteristic space of P associatedto the eigenvalue λ . For N (cid:192) V P ( λ ) = ker( λ − P ) N .(1.7)Set m P ( λ ) = dim V P ( λ ) ∈ N .(1.8)Since P is elliptic, V P ( λ ) ⊂ C ∞ ( Z , E ). Since P is a relatively compact pertur-bation of the self-adjoint operator ( ∇ E ) ∗ ∇ E , by the Keldysh theorem [GoKr69,Theorem 10.1], we have L ( Z , E ) = ⊕ λ ∈ Sp( P ) V P ( λ ).(1.9)In particular, Sp( P ) is not empty and is an infinite set.1.2. The semigroup of a generalised Laplacian.
By (1.4), (1.6), and by theLumer-Phillips theorem [Yo95, p. 250], − P is the generator of a C -semigroupexp( − tP ) on L ( Z , E ).Fix r > (cid:178) > Γ = Γ ∪ Γ ∪ Γ be thecontour in the Figure 1.1 with the indicated orientation. Proposition 1.1.
For t > , the following identity of bounded operators in L ( X , E ) holds exp( − tP ) = i π (cid:90) Γ e − t λ ( λ − P ) − d λ .(1.10) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 13 r y = (cid:15)xy = − (cid:15)x Γ Γ Γ Γ (cid:48) y x F IGURE
The heat operator exp( − tP ) has a smooth integral kernel. In particular, it is intrace class such that Tr [exp( − tP )] = (cid:88) λ ∈ Sp( P ) m P ( λ ) e − t λ ,(1.11) where the right hand side of (1.11) converges absolutely.Proof. By the first estimate of (1.6), the integral on the right-hand side of (1.10)converges absolutely in the space of bounded operators. By the general theory of C -semigroup [Pa83, Theorem I.7.7], we get (1.10).Using integration by parts, we find that for any k ∈ N , we haveexp( − tP ) = ( − k k !2 i π t k (cid:90) Γ e − t λ ( λ − P ) − ( k + d λ .(1.12)Take k > m /2. The imbedding H k ( Z , E ) → L ( Z , E ) is in trace class. Denote by (cid:107) · (cid:107) tr the trace norm. By the second estimate of (1.6), there is C > λ ∈ Γ , we have (cid:176)(cid:176)(cid:176) ( λ − P ) − ( k + (cid:176)(cid:176)(cid:176) tr (cid:201) C .(1.13)By (1.13), the right-hand side of (1.12) converges in the space of trace class op-erators. In particular, exp( − tP ) is in trace class. Similar methods show that forany s , s (cid:48) ∈ R , exp( − tP ) maps H s ( Z , E ) to H s (cid:48) ( Z , E ) continuously. In particular,exp( − tP ) has a smooth integral kernel.We claim that Sp(exp( − tP )) = { } ∪ (cid:110) e − t λ : λ ∈ Sp( P ) (cid:111) ,(1.14)and that each e − t λ has the multiplicity m P ( λ ). Indeed, it is clear that e − t λ with λ ∈ Sp( P ) is the spectrum of exp( − tP ). Since Sp( P ) is infinite and since Sp(exp( − tP ))is a closed set, we see that 0 ∈ Sp(exp( − tP )). Assume that µ ∈ C \{ } does not equalto any e − t λ with λ ∈ Sp( P ). Note that exp( − tP ) is compact. For δ > − tP ) to the characteristic space of µ is given by Π µ = i π (cid:90) | λ − µ |= δ d λλ − exp( − tP ) .(1.15) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 14
By our choice of µ , Π µ varnishes on ⊕ λ ∈ Sp( P ) V P ( λ ). By (1.9), Π µ =
0, from whichwe get (1.14). Similarly, Π e − t λ | ⊕ λ ∈ Sp( P ) V P ( λ ) is just the obvious projection onto V P ( λ ).By (1.9), Π e − t λ coincides with the spectral projection of P onto V P ( λ ). Thus, thealgebraic multiplicity of µ = e − t λ is equal to m P ( λ ).By (1.14) and by Lidskii’s Theorem [GoKr69, Theorem 8.4], we get (1.11). (cid:3) Let us study the long time exponential decay and the short time asymptotic ofthe heat trace Tr [exp( − tP )]. We choose r > (cid:178) > P ) ∩ { z ∈ C : | z | = r } = ∅ .(1.16)Let π < be the spectral projection of P onto the space ⊕ λ ∈ Sp( P ): | λ |< r V P ( λ ). Then π < = i π (cid:90) | λ |= r d λλ − P .(1.17)Set π > = − π < .(1.18) Proposition 1.2.
There are c > and C > such that for t (cid:202) , we have (cid:107) exp( − tP ) π > (cid:107) tr (cid:201) C e − ct .(1.19) Proof.
Let Γ (cid:48) = Γ ∪ Γ (cid:48) ∪ Γ be the oriented contour indication in the Figure 1.1.Clearly, there is c > λ ∈ Γ (cid:48) and t > (cid:175)(cid:175)(cid:175) e − t λ (cid:175)(cid:175)(cid:175) (cid:201) e − ct .(1.20)By (1.9), (1.10), (1.12), (1.16)-(1.18), for k ∈ N , we can deduce thatexp( − tP ) π > = ( − k k !2 i π t k (cid:90) Γ (cid:48) e − t λ ( λ − P ) − ( k + d λ .(1.21)Note that the integral converges for the operator norm (cid:107) · (cid:107) L , L .If k > m /2, by the resolvent identity, the maps λ ∈ (Sp( P )) c → (cid:176)(cid:176) ( λ − P ) − ( k + (cid:176)(cid:176) tr iscontinuous. Since Γ (cid:48) is compact, by (1.16), there is C > λ ∈ Γ (cid:48) ,we have (cid:176)(cid:176)(cid:176) ( λ − P ) − ( k + (cid:176)(cid:176)(cid:176) tr (cid:201) C .(1.22)By (1.13), (1.20), and (1.22), if k > m /2, the integral on the right-hand side of (1.21)converges for the trace norm and we get (1.19). (cid:3) By [BeGeV04, Defintion 2.15, Lemma 2.34], the integral kernel of exp( − tP )is exactly the one constructed in [BeGeV04, Section 2.4]. As a consequence of[BeGeV04, Theorem 2.30], we have the following proposition. Proposition 1.3.
There are families of complex numbers { a k ∈ C } k ∈ N and of posi-tive real numbers { C k > } k ∈ N such that for k ∈ N and t ∈ (0, 1] , we have (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) Tr [exp( − tP )] − t m /2 k (cid:88) i = a i t i (cid:175)(cid:175)(cid:175)(cid:175)(cid:175) (cid:201) C k t k + − m /2 .(1.23) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 15
The regularised determinant.
Fix r > (cid:178) > < ( P ) = (cid:89) λ ∈ Sp( P ): | λ |< r λ m ( λ ) ∈ C , det ∗< ( P ) = (cid:89) λ ∈ Sp( P ):0 <| λ |< r λ m ( λ ) ∈ C ∗ .(1.24)For s > m /2, set θ > ( s ) = − Γ ( s ) (cid:90) ∞ Tr [exp( − tP ) π > ] t s − ds .(1.25)By Propositions 1.2 and 1.3, proceeding as [BeGeV04, Section 9.6], we see that θ > ( s ) has a meromorphic extension to s ∈ C , which is holomorphic at s =
0. Setdet > ( P ) = exp (cid:181) ∂∂ s θ > (0) (cid:182) ∈ C ∗ .(1.26) Definition 1.4.
Definedet( P ) = det < ( P ) · det > ( P ) ∈ C , det ∗ ( P ) = det ∗< ( P ) · det > ( P ) ∈ C ∗ .(1.27)By (1.11), it is easy to see that this definition does not depend on the choice of r > (cid:178) > Theorem 1.5.
The function det( σ + P ) is holomorphic on σ ∈ C . Its zeros arelocated at σ = − λ with multiplicity m P ( λ ) , where λ ∈ Sp( P ) .Proof. Since σ + P is a generalised Laplacian, it is enough to show our theoremnear σ =
0. Recall that r > (cid:178) > U ⊂ C of 0 such that U ∩ { − Sp( P ) } ⊂ { } , andthat (1.5), (1.6), and (1.16) still hold for all σ + P with σ ∈ U .We have the identity of functions on U ,det( σ + P ) = det < ( σ + P )det > ( σ + P ).(1.28)By our construction, det > ( σ + P ) is a holomorphic function without zero on U ,and that det < ( P + σ ) is a polynomial function on U where the only possibility ofzeros is situated at σ = m P (0). The proof of our theorem iscompleted. (cid:3) Remark . The statements of Propositions 1.1-1.3, and Theorem 1.5 hold truewhen Z is a Riemannian closed orbifold and when E is a Hermitian orbifold vectorbundle on Z . Indeed, the proofs of Propositions 1.1 and 1.2 are based on (1.5) and(1.6), which are obtained using the peusdodifferential calculus. Using the peus-dodifferential calculus on orbifolds [Ma05, p. 2213], the generalisations of (1.5)and (1.6) to orbifolds are straightforward. The proof of Proposition 1.3 in the orb-ifold setting is originally due to Ma [Ma05, Proposition 2.1]. The generalisationof Theorem 1.5 is a consequence of the orbifold version of Propositions 1.1-1.3.1.4. Flat vector bundles.
Let Z be a connected closed manifold. Let Γ = π ( Z )be the fundamental group of Z . Let (cid:98) π : X → Z be the universal cover of Z . Thegroup Γ acts on the left on X as the deck transformation, so that Z = Γ \ X .(1.29) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 16
Let g TZ be a Riemannian metric on Z . It lifts to a Riemannian metric g T X on X , so that Γ acts on X isometrically. Let d X ( · , · ) be the Riemannian distance on X . If γ ∈ Γ , let d γ : X → R be the displacement function on X, i.e., d γ ( x ) = d X ( x , γ x ).(1.30)Set (cid:96) γ = inf x ∈ X d γ ( x ).(1.31)If i Z > Z , then for γ ∈ Γ and γ (cid:54)= (cid:96) γ (cid:202) i Z .(1.32)Let F be a flat vector bundle of rank r on Z with flat connection ∇ F . Let ρ : Γ → GL r ( C ) be the holonomy representation of F , so that F = Γ \ ( X × C r ).(1.33)Take x ∈ X such that (cid:98) π ( x ) = z . Then ρ ( γ ) can be considered as an element inEnd( F z ) defined by the parallel transport with respect to the flat connection ∇ F along the loop on Z based at z , which is obtained by the projection of a path on X from γ x to x .Let g F be a Hermitian metric on F . Proposition 1.7.
There is C > such that for any γ ∈ Γ , x ∈ X with z = (cid:98) π ( x ) , (cid:175)(cid:175) ρ ( γ ) (cid:175)(cid:175) g Fz ≤ C e Cd γ ( x ) , (cid:175)(cid:175) Tr[ ρ ( γ )] (cid:175)(cid:175) ≤ C e C (cid:96) γ .(1.34) Proof.
Following [BZ92, Definitions 4.1, 4.2], set ω F = (cid:179) g F (cid:180) − (cid:179) ∇ F g F (cid:180) ∈ Ω ( Z , End( F )),(1.35)and ∇ F , u = ∇ F + ω F .(1.36)Then, ∇ F , u is a connection on F which preserves the metric g F .Let ( x t ) (cid:201) t (cid:201) t be a smooth path on X of speed 1 from γ x to x . We trivialise F over (cid:98) π ( x · ) by the parallel transport with respect to the metric connection ∇ F , u . For s , t ∈ [0, t ], let τ st ∈ Hom( F (cid:98) π x s , F (cid:98) π x t ) be the corresponding parallel transport. Let U t ∈ C ∞ ([0, t ] , End( F z )) be the unique solution of the differential equation˙ U t − τ t ω F (cid:98) π x t ( ˙ x t ) τ t U t = U = Id.(1.37)As elements in End( F z ), we have ρ ( γ ) = τ t U t .(1.38)Since Z is compact, the smooth section ω F is uniformly bounded. Since | ˙ x s | = τ st preserves g F , by Gronwall’s inequality, there is C > x , of γ , and of the path x · , such that for all t > | U t | g Fz (cid:201) C e Ct .(1.39)By (1.38) and (1.39), using again that τ t is unitary with respect to g Fz , we get (cid:175)(cid:175) ρ ( γ ) (cid:175)(cid:175) g Fz ≤ C e Ct .(1.40) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 17
Since t is the length of the path ( x t ) t ∈ [0, t ] , by taking the infimum of all suchpaths, we get the first inequality of (1.34).Using the trivial inequality on matrices, by the first inequality of (1.34), wehave (cid:175)(cid:175) Tr (cid:163) ρ ( γ ) (cid:164)(cid:175)(cid:175) (cid:201) r | ρ ( γ ) | g Fz (cid:201) rC e Cd γ ( x ) .(1.41)Since the constants C and r do not depend on x , by taking the infimum on x ∈ X in (1.41), using (1.31), we get the second inequality of (1.34). (cid:3) A generalised Laplacian and flat vector bundles.
We use the notationof Section 1.4. Let (cid:101) E be a Hermitian vector bundle on X with a metric connection ∇ (cid:101) E . Assume that the Γ -action on X lifts to (cid:101) E and preserves the Hermitian metricand the connection ∇ (cid:101) E . Then ( (cid:101) E , ∇ (cid:101) E ) descends to a Hermitian vector bundle E on Z with a metric connection ∇ E .Let P X be a Γ -invariant self-adjoint generalised Laplacian acting on C ∞ ( X , (cid:101) E ).Assume that P X = (cid:179) ∇ (cid:101) E (cid:180) ∗ ∇ (cid:101) E + (cid:101) A ,(1.42)where (cid:101) A ∈ C ∞ ( X , End( (cid:101) E )) Γ . Since the Riemannian manifold ( X , g T X ) is complete,for t >
0, it is classical that the heat operator exp( − tP X ) exists and has a smoothintegral kernel. Let p Xt ( x , y ) be the smooth integral kernel with respect to theRiemannian volume dv X .Recall that F is a flat vector bundle on Z with holonomy ρ : Γ → GL r ( C ). Wehave the identification C ∞ ( Z , E ⊗ F ) = (cid:161) C ∞ ( X , (cid:101) E ) ⊗ C r (cid:162) Γ .(1.43)The operator P X ⊗ id preserves the above Γ -invariant space. It descends to ageneralised Laplacian P Z , ρ acting on C ∞ ( Z , E ⊗ F ). Since ρ is not necessarilyunitary, there is no obvious metric on F such that P Z , ρ is self-adjoint. Still, byProposition 1.1, the heat operator exp( − tP Z , ρ ) exists and has a smooth integralkernel. For t >
0, the smooth integral kernel with respect to the Riemannianvolume dv Z can be identified with a Γ × Γ -invariant section (cid:101) q Z , ρ t ( x , y ) in C ∞ ( X × X , ( (cid:101) E ⊗ C r ) (cid:2) ( (cid:101) E ⊗ C r ) ∗ ).Note that p Xt ( γ − x , y ) ∈ Hom( (cid:101) E y , (cid:101) E γ − x ). Let γ ∗ ∈ Hom( (cid:101) E γ − x , (cid:101) E x ) be the obviouselement. Then, γ ∗ p Xt ( γ − x , y ) ∈ Hom( (cid:101) E y , (cid:101) E x ). Recall that we have fixed a metric g F on F . Theorem 1.8.
Given t > , T > with t < T and given a bounded open setX ⊂ X , for any s ∈ N , the sum on the right-hand side of (1.44) converges absolutelyin the space of C s ([ t , T ] × X × X , C (cid:2) ( (cid:101) E ⊗ C r ) (cid:2) ( (cid:101) E ⊗ C r ) ∗ ) , so that (cid:101) q Zt ( x , y ) = (cid:88) γ ∈ Γ ρ ( γ ) ⊗ γ ∗ p Xt ( γ − x , y ).(1.44) Proof.
We will first show that sum on the right hand side of (1.44) converges toa Γ × Γ -invariant smooth section q t ( x , y ) and then show that q t ( x , y ) satisfies theheat equation. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 18
We fix x ∈ X . We claim that for any δ >
0, we have (cid:88) γ ∈ Γ exp (cid:179) − δ d γ ( x ) (cid:180) < +∞ .(1.45)Indeed, by [Mi68b, Remark p. 1, Lemma 2] or [MaMar15, (3.19)], there is C > r (cid:202) (cid:175)(cid:175)(cid:169) γ ∈ Γ : d γ ( x ) (cid:201) r (cid:170)(cid:175)(cid:175) (cid:201) C e Cr .(1.46)For δ >
0, by (1.46), we have(1.47) (cid:88) γ ∈ Γ exp (cid:179) − δ d γ ( x ) (cid:180) = (cid:88) γ ∈ Γ (cid:90) ∞ δ d γ ( x ) exp( − t ) dt = (cid:90) ∞ exp( − t ) (cid:175)(cid:175)(cid:175)(cid:110) γ ∈ Γ : d γ ( x ) (cid:201) (cid:112) t / δ (cid:111)(cid:175)(cid:175)(cid:175) dt (cid:201) C (cid:90) ∞ e − t + C (cid:112) t / δ dt < +∞ .Since X is bounded, by (1.34) and by triangle inequality, there is C > x ∈ X , we have (cid:175)(cid:175) ρ ( γ ) (cid:175)(cid:175) g F (cid:98) π x (cid:201) C e Cd γ ( x ) .(1.48)By finite propagation speed of solutions of hyperbolic equations [MaMar07,Theorem D.2.1], as in [MaMar15, Theorem 4], there are c > C > t ∈ [ t , T ] and x , y ∈ X , (cid:175)(cid:175)(cid:175) p Xt ( x , y ) (cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:161) − cd X ( x , y ) (cid:162) .(1.49)Moreover, similar estimates hold uniformly on [ t , T ] × X × X , for all derivationson t , x , y . By (1.49) and by triangle inequality, there are c > C > t ∈ [ t , T ], x , y ∈ X , γ ∈ Γ , we have (cid:175)(cid:175)(cid:175) p Xt ( γ − x , y ) (cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:179) − cd γ ( x ) (cid:180) ,(1.50)and similar estimates for all derivations on t , x , y .By (1.45), (1.48), and (1.50), for all s ∈ N , the sum on the right-hand side of(1.44) converges absolutely in C s ([ t , T ] × X × X , C (cid:2) ( (cid:101) E ⊗ C r ) (cid:2) ( (cid:101) E ⊗ C r ) ∗ ) to a Γ × Γ -invariant smooth section q t ( x , y ), so that ∂∂ t q t ( x , y ) = (cid:88) γ ∈ Γ ρ ( γ ) ⊗ ∂∂ t γ ∗ p Xt ( γ − x , y ), P Xx q t ( x , y ) = (cid:88) γ ∈ Γ ρ ( γ ) ⊗ P Xx γ ∗ p Xt ( γ − x , y ),(1.51)where P Xx denotes the differential operator P X acting on the variable x . By (1.51),using ∂∂ t p Xt ( x , y ) = − P Xx p Xt ( x , y ), and using the fact that p Xt is Γ × Γ -invariant, weget ∂∂ t q t ( x , y ) = − P Xx q t ( x , y ).(1.52)Take u ∈ C ∞ ( Z , E ⊗ F ). We identify u with its Γ -invariant lifting (cid:101) u ∈ (cid:161) C ∞ ( X , (cid:101) E ) ⊗ C r (cid:162) Γ .Let F Z ⊂ X be a fundamental domain of Z in X . By (1.34), there is C >
0, for For example, we can take F Z = { x ∈ X : for all γ ∈ Γ , d X ( x , x ) < d X ( x , γ x ) } . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 19 y ∈ F Z and γ ∈ Γ , we have (cid:175)(cid:175)(cid:101) u ( γ y ) (cid:175)(cid:175) (cid:201) C e Cd γ ( y ) .(1.53)By (1.45), (1.50), and (1.53), using Fubini’s theorem, we have (cid:90) y ∈ F Z q t ( x , y ) (cid:101) u ( y ) dv X = (cid:90) y ∈ X p Xt ( x , y ) (cid:101) u ( y ) dv X .(1.54)By [BeGeV04, Definition 2.15, Lemma 2.34], it remains to show that for any u ∈ C ∞ ( Z , E ⊗ F ), as t →
0, we havesup x ∈ F Z (cid:175)(cid:175)(cid:175)(cid:175)(cid:90) y ∈ X p Xt ( x , y ) (cid:101) u ( y ) dv X − (cid:101) u ( x ) (cid:175)(cid:175)(cid:175)(cid:175) → F Z , F Z be two bounded open neighbourhoods of F Z such that F Z ⊂ F Z .Write (cid:101) u = (cid:101) u + (cid:101) u ,(1.56)where (cid:101) u and (cid:101) u are C ∞ -sections on X such that Supp( (cid:101) u ) ⊂ F Z and Supp( (cid:101) u ) ⊂ X \ F Z . Since (cid:101) u has a compact support in X , by the Sobolev imbedding theorem,for k > m /4, there is C k > (cid:176)(cid:176)(cid:176) exp (cid:179) − tP X (cid:180) (cid:101) u − (cid:101) u (cid:176)(cid:176)(cid:176) C ( F Z ) (cid:201) C k (cid:176)(cid:176)(cid:176)(cid:176)(cid:179) + P X (cid:180) k (cid:179) exp (cid:179) − tP X (cid:180) (cid:101) u − (cid:101) u (cid:180)(cid:176)(cid:176)(cid:176)(cid:176) L ( X ) = C k (cid:176)(cid:176)(cid:176)(cid:176) exp (cid:179) − tP X (cid:180) (cid:179) + P X (cid:180) k (cid:101) u − (cid:179) + P X (cid:180) k (cid:101) u (cid:176)(cid:176)(cid:176)(cid:176) L ( X ) .Since (cid:161) + P X (cid:162) k (cid:101) u is L on X , by a property of the semigroup exp( − tP X ) and by(1.57), as t →
0, we havesup x ∈ F Z (cid:175)(cid:175)(cid:175)(cid:175)(cid:90) y ∈ X p Xt ( x , y ) (cid:101) u ( y ) dv X − (cid:101) u ( x ) (cid:175)(cid:175)(cid:175)(cid:175) → c > C >
0, for t ∈ (0, 1], x , y ∈ X , (cid:175)(cid:175)(cid:175) p Xt ( x , y ) (cid:175)(cid:175)(cid:175) (cid:201) Ct m /2 exp (cid:195) − c d X ( x , y ) t (cid:33) .(1.59)Since Supp( (cid:101) u ) ⊂ X \ F Z and since d X ( X \ F Z , F Z ) > c (cid:48) >
0, using (1.45), (1.53), and(1.59), we can deduce that there are c > C >
0, for t ∈ (0, 1],sup x ∈ F Z (cid:175)(cid:175)(cid:175)(cid:175)(cid:90) y ∈ X p Xt ( x , y ) (cid:101) u ( y ) dv X (cid:175)(cid:175)(cid:175)(cid:175) (cid:201) C e − c / t .(1.60)By (1.58) and (1.60), we get (1.55). The proof of our theorem is completed. (cid:3) Corollary 1.9.
For t > , the right-hand side of the following identity convergencesabsolutely, so that Tr (cid:104) exp (cid:179) − tP Z , ρ (cid:180)(cid:105) = (cid:88) γ ∈ Γ Tr[ ρ ( γ )] (cid:90) x ∈ F Z Tr (cid:101) E (cid:104) γ ∗ p Xt (cid:161) γ − x , x (cid:162)(cid:105) dv X .(1.61) Proof.
This is a consequence of [MaMar07, Theorem D.1.5], Proposition 1.7, The-orem 1.8, and the estimates (1.34), (1.45), (1.50). (cid:3)
Set Γ + = Γ − { id } . We have a generalisation of [Sh18, Proposition 4.8]. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 20
Proposition 1.10.
There are c > C > such that for any x ∈ X and t > , (cid:88) γ ∈ Γ + (cid:175)(cid:175) Tr[ ρ ( γ )] (cid:175)(cid:175) (cid:175)(cid:175)(cid:175) p Xt (cid:161) γ − x , x (cid:162)(cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:179) − ct + Ct (cid:180) .(1.62) Proof.
Since X is a cover of a compact manifold, by [Sh18, (4-27)] or [MaMar15,Theorem 4]), there exist c > C > t > x , x (cid:48) ∈ X , we have (cid:175)(cid:175)(cid:175) p Xt (cid:161) x , x (cid:48) (cid:162)(cid:175)(cid:175)(cid:175) (cid:201) Ct m /2 exp (cid:195) − c d γ ( x ) t + Ct (cid:33) .(1.63)By (1.32) and (1.63), there are c > c >
0, and C > t > x ∈ X ,and γ ∈ Γ + , we have (cid:175)(cid:175)(cid:175) p Xt (cid:161) γ − x , x (cid:162)(cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:195) − c t − c d γ ( x ) t + Ct (cid:33) .(1.64)By (1.34) and (1.64), using Cd γ ( x ) (cid:201) c d γ ( x )2 t + C c t , we get (cid:175)(cid:175) Tr[ ρ ( γ )] (cid:175)(cid:175) (cid:175)(cid:175)(cid:175) p X , τ t (cid:161) γ − x , x (cid:162)(cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:195) − c t − c d γ ( x )2 t + C (cid:48) t (cid:33) .(1.65)Now proceeding as in [Sh18, (4-31)], we get (1.62). (cid:3) Flat Laplacian and the Cappell-Miller analytic torsion.
Let Z be acompact manifold of dimension m . Let ( F , ∇ F ) be a complex flat vector bundleon Z with holonomy ρ : Γ → GL r ( C ). Let Ω • ( Z , F ) be the space of smooth differen-tial forms with coefficients in F . Let ( Ω • ( Z , F ), d Z ) be the de Rham complex, andlet H • ( Z , F ) be the corresponding de Rham cohomology.Let g TZ be a Riemannian metric on Z . It induces a Riemannian metric g T X on X . Let 〈 , 〉 L be the L -product on Ω • ( X ) induced by g T X . Let d X , ∗ be the formaladjoint of d X . Set (cid:3) X = (cid:104) d X , d X , ∗ (cid:105) .(1.66)Then, (cid:3) X is a self-adjoint generalised Laplacian acting on Ω • ( X ).We use the convention in Section 1.5 with E = Λ • ( T ∗ Z ) and (cid:101) E = Λ • ( T ∗ X ). By(1.43), d X ⊗ id descends to the de Rham operator d Z . Similarly, d X , ∗ ⊗ id, (cid:3) X ⊗ iddescend to operators d Z , ∗ , (cid:3) Z , so that (cid:3) Z = (cid:104) d Z , d Z , ∗ (cid:105) .(1.67)The operator (cid:3) Z is a generalised Laplacian which will be called the flat Laplacian.Let Ω • ( Z , F ) be the characteristic space of (cid:3) Z associated with the eigenvalue 0.Since d Z commutes with (cid:3) Z , we see that (cid:161) Ω • ( Z , F ), d Z (cid:162) is a complex. By [CMil10,p. 181], H • (cid:179) Ω • ( Z , F ), d Z (cid:180) (cid:39) H • ( Z , F ).(1.68)Set χ (cid:48) CM ( F ) = m (cid:88) i = ( − i i dim Ω i ( Z , F ).(1.69) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 21
Let T ρ ( σ ) be a meromorphic function on C defined by T ρ ( σ ) = m (cid:89) i = det (cid:179) σ + (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i .(1.70)As σ →
0, we have T ρ ( σ ) = (cid:40) m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i (cid:41) σ χ (cid:48) CM ( Z , F ) + O (cid:179) σ χ (cid:48) CM ( Z , F ) + (cid:180) .(1.71) Definition 1.11. If (cid:3) Z is invertible, the complex valued analytic torsion of Cappell-Miller is defined by T CM ( F ) = m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i ∈ C ∗ .(1.72)We refer the reader to [CMil10] for the definition of the Cappell-Miller torsion inthe case where (cid:3) Z is not invertible.Assume that (cid:3) Z is invertible. By [CMil10, p. 179], if Z is orientable and haseven dimension, then T CM ( F ) =
1. By [CMil10, Theorem 8.3], if dim Z is odd, T CM ( F ) does not depend on the metric g TZ . It becomes a topological invariant.Let Rep( Γ , C r ) be the set of all r -dimensional complex representations of Γ . It iswell known that Rep( Γ , C r ) has a natural structure of a complex algebraic variety(see [BrKa08, Section 13.6]). Let us follow [GolM88, Proposition 4.5] and [Mü20,Section 6.2]. Let U ⊂ Rep( Γ , C r ) be a contractible neighbourhood of ρ . Considerthe Γ -action on U × X × C r defined by γ · ( ρ , x , v ) = ( ρ , γ x , ρ ( γ ) v ).(1.73)The projection on the quotient space Γ \ ( U × X × C r ) → U × Z (1.74)define a vector bundle on U × Z , whose restriction to { ρ } × Z is just the flat vectorbundle with holonomy ρ . We write F ρ to emphasise the dependence on ρ .Since U is contractible, we have an identification of vector bundles over U × Z , Γ \ ( U × X × C r ) (cid:39) U × F ρ .(1.75)Note that the identification is non canonical and is only continuous in the vari-ables in U .For ρ ∈ U , by (1.75), we have a bundle isomorphism F ρ (cid:39) F ρ .(1.76)Under this identification, the flat connection on F ρ can be written as ∇ F ρ = ∇ F ρ + A ρ ,(1.77)with A ρ ∈ Ω ( Z , End( F ρ )). For k ∈ N and (cid:178) >
0, we call ρ is C k (cid:178) -close to ρ , if (cid:176)(cid:176) A ρ (cid:176)(cid:176) C k (cid:201) (cid:178) .(1.78)Recall that we have fixed a Riemannian metric on T Z . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 22
Proposition 1.12. If ρ ∈ Rep( Γ , C r ) is unitary and acyclic, then there is (cid:178) > suchthat if ρ is C (cid:178) -close to ρ , then the flat Laplacian on F ρ is invertible, so thatT CM ( F ρ ) = m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ρ ) (cid:180) ( − i i ∈ C ∗ .(1.79) Proof.
This is [Mü20, Lemma 6.2] ([BrKa08, Proposition 6.8]), whose proof isbased on the invertibility of the first order elliptic differential operator d Z + d Z , ∗ (c.f. Proposition 6.7). (cid:3) Remark . Assume that Z is a closed orbifold and that F is a flat orbifold vectorbundle on Z . Given a Riemannian metric on Z , we can define the flat Laplacian,the regularised determinant, Cappell-Miller analytic torsion in the same way, sothat (1.71) and Proposition 1.12 still hold true.2. M ÜLLER ’ S S ELBERG TRACE FORMULA
The purpose of this section is to establish the Selberg trace formula for the heatoperator of the Casimir on the locally symmetric space twisted by an arbitrary flatvector bundle. Such a formula is obtained by Müller [Mü11] for the functional ofthe Casimir with respect to an even Paley-Wiener function. The extension to theheat operator does not contain particular difficulties. We include some detail forcompleteness.This section is organised as follows. In Sections 2.1-2.3, we introduce the realreductive group G , its maximal compact subgroup K , and the Casimir C g , the as-sociated symmetric space X = G / K , and the K -principal bundle p : G → X . Givena finite dimensional unitary representation τ : K → U ( E τ ) of K , we construct theassociated Hermitian vector bundle E τ on X . The Casimir operator on C ∞ ( X , E τ )is a self-adjoint generalised Laplacian C g , X , τ .In Sections 2.4 and 2.5, we introduce the semisimple elements in G and theassociated semisimple orbital integrals with respect to the heat operator of C g , X , τ .Finally, in Section 2.6, we introduce a discrete cocompact subgroup Γ ⊂ G of G and the corresponding locally symmetric space Γ \ X . Given a finite dimensionalrepresentation of Γ , the Casimir operator descends to a generalised Laplacian on Z . We establish the Selberg trace formula for the associated heat operator.2.1. Real reductive groups.
Let G be a linear connected real reductive group[K86, p. 3], and let θ ∈ Aut( G ) be the Cartan involution. That means G is a closedconnected group of real matrices that is stable under transpose, and θ is the com-position of transpose and inverse of matrices. Let K ⊂ G be the subgroup of G fixed by θ , so that K is a maximal compact subgroup of G .Let g , k be the Lie algebras of G , K . The Cartan involution acts by differentialas Lie algebra automorphism on g , which will still be denoted by θ . Then k is theeigenspace of θ associated with the eigenvalue 1. Let p be the eigenspace of θ associated with the eigenvalue −
1, so that g = p ⊕ k .(2.1)Set m = dim p , n = dim k .(2.2) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 23
By [K86, Proposition 1.2], we have the diffeomorphism( Y , k ) ∈ p × K → e Y k ∈ G .(2.3)Let B be a real-valued nondegenerate bilinear symmetric form on g which isinvariant under the adjoint action Ad of G , and also under θ . Then (2.1) is anorthogonal splitting of g with respect to B . We assume B to be positive-definite on p , and negative-definite on k . The form 〈· , ·〉 = − B ( · , θ · ) defines an Ad( K )-invariantscalar product on g such that the splitting (2.1) is still orthogonal. We denote by | · | the corresponding norm.Let Z G ⊂ G be the centre of G with Lie algebra z g ⊂ g . By [K86, Corollary 1.3], Z G is a (possibly non connected) reductive group with maximal compact subgroup Z G ∩ K with the Cartan decomposition z g = z p ⊕ z k .(2.4)Since z p commutes with Z G ∩ K , by (2.3), we have an identification of the groups Z G = exp( z p ) × ( Z G ∩ K ).(2.5)Let G ss ⊂ G be the connected subgroup of G associated with the Lie algebra[ g , g ]. By [K02, Corollary 7.11], G ss is a closed subgroup of G . Moreover, G ss issemisimple and G = G ss · Z G .(2.6) Proposition 2.1.
If G has a compact center, any one dimensional real representa-tion of G is trivial.Proof.
This is a consequence of (2.6) and of the fact that any morphism of groupsfrom a connected semisimple Lie group or a connected compact Lie group to R ∗ istrivial. (cid:3) Let g C = g ⊗ R C be the complexification of g and let u = (cid:112)− p ⊕ k be the compactform of g . By C -linearity, the bilinear form B extends to a complex symmetricbilinear form on g C . Its restriction B | u to u is a real negative-definite symmetricbilinear form.Let G C be the connected group of complex matrices associated with the Liealgebra g C . Let U , U ss ⊂ G C be the connected subgroup of G C associated with theLie algebras u , [ u , u ]. If G has a compact centre, by [K86, Propositions 5.3, 5.6], G C is a reductive group with maximal compact subgroup U . By [K86, Theorem4.32], U ss ⊂ U is a semisimple compact subgroup, so that U = Z G U ss .(2.7)By Weyl’s theorem [K86, Theorem 4.26], the universal cover (cid:101) U ss of U ss is compact.Set (cid:101) U = Z G × (cid:101) U ss .(2.8)By (2.7) and (2.8), the obvious projection (cid:101) U → U is a finite cover of U . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 24
Remark . By Weyl’s unitary trick [K86, Proposition 5.7], if G is semisimpleand if U is simply connected, it is equivalent to consider the finite dimensionalcomplex representations of the Lie groups G , U or of the Lie algebras g , u .For general G , any finite dimensional complex representation of G induces arepresentation of g . It extends uniquely to a representation of u . In generally, the u -representation does not alway lift to U . However, if G has a compact centre, by(2.8), the u -representation lifts to (cid:101) U .Denote by rk C ( G ) (resp. rk C ( K )) the complex rank of G (resp. K ), i.e., thedimension of a Cartan subalgebra of g C (resp. k C ). Definition 2.3.
The fundamental rank of G is defined by δ ( G ) = rk C ( G ) − rk C ( K ) ∈ N .(2.9)Note that m and δ ( G ) have the same parity.In the sequel, if γ ∈ G , we denote by Z ( γ ) ⊂ G the centraliser of γ in G , and by z ( γ ) ⊂ g its Lie algebra. If a ∈ g , let Z ( a ) ⊂ G be the stabiliser of a in G , and let z ( a ) ⊂ g be its Lie algebra. If a ⊂ g is a subset, we define Z ( a ) and z ( a ) similarly.2.2. The Casimir operator.
Let U ( g ) be the enveloping algebra of g , and let Z ( g ) ⊂ U ( g ) be the centre of U ( g ).Let C g ∈ Z ( g ) be the Casimir element associated to B . If e , · · · , e m is anorthonormal basis of ( p , B | p ), and if e m + , · · · , e m + n is an orthonormal basis of( k , − B | k ). Then, C g = − m (cid:88) i = e i + n + m (cid:88) i = m + e i .(2.10)If V is a finite dimensional complex vector space, and if ρ : g → End( V ) is amorphism of Lie algebras, the map ρ extends to a morphism U ( g ) → End( V ) ofalgebras. We denote by C g , V or C g , ρ ∈ End( V ) the corresponding Casimir operatoracting on V , i.e., C g , V = C g , ρ = ρ ( C g ).(2.11)Similarly, the Casimir of u (with respect to B ) acts on V , so that C u , V = C g , V .(2.12)2.3. The symmetric space.
We use the notation in Section 2.1. Let ω g be thecanonical left-invariant 1-form on G with values in g . By (2.1), ω g splits as ω g = ω p + ω k .(2.13)Let X = G / K be the associated symmetric space. Let p : G → X be the naturalprojection. Then p : G → X is a K -principal bundle with connection form ω k . Thegroup K acts isometrically on p . The tangent bundle is given by T X = G × K p .(2.14)By (2.14), the scalar product B | p on p induces a Riemannian metric g T X on X .The connection ∇ T X on T X which is induced by ω k is the Levi-Civita connectionof T X . Its curvature is parallel and nonpositive.
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 25
Let e ( T X , ∇ T X ) ∈ Ω m ( X , o ( T X )) be the Euler characteristic form on X . Let dv X ∈ Ω m ( X , o ( T X )) be the Riemannian volume form on the Riemannian man-ifold ( X , g T X ). Define (cid:163) e ( T X , ∇ T X ) (cid:164) max ∈ R by e (cid:179) T X , ∇ T X (cid:180) = (cid:104) e (cid:179) T X , ∇ T X (cid:180)(cid:105) max dv X .(2.15)An explicit formula for (cid:163) e ( T X , ∇ T X ) (cid:164) max can be found in [Sh18, (4-5)].More generally, if τ is an orthogonal (resp. unitary) representation of K on afinite dimensional Euclidean (resp. Hermitian) space E τ , set E τ = G × K E τ .(2.16)Then E τ is a Euclidean (resp. Hermitian) vector bundle on X , which is equippedwith a connection induced by ω k .We identify the space C ∞ ( X , E τ ) of smooth sections of E τ to the space C ∞ ( G , E τ ) K of smooth E τ -valued K -invariant functions on G . The group G acts on the lefton C ∞ ( X , E τ ). Denote by C g , X , τ the Casimir element of G on C ∞ ( X , E τ ). By(2.10), C g , X , τ is a self-adjoint generalised Laplacian on X satisfying (1.42). When E τ = Λ · ( p ∗ ), we use the notation C g , X . It is classical (see [B11, Proposition 7.8.1])that C g , X is just the Hodge Laplacian on X associated to the trivial line bundle.2.4. Semisimple elements.
The group G acts isometrically on X . If γ ∈ G , let d γ be the corresponding displacement function on X defined in (1.30). Also, (cid:96) γ depends only on the conjugacy class of γ in G , and will be denoted by (cid:96) [ γ ] . Let X ( γ ) ⊂ X be the closed subset where d γ reaches its minimum. Clearly, the group Z ( γ ) acts on X ( γ ).An element γ ∈ G is called semisimple [E96, Definition 2.19.21], if X ( γ ) isnonempty. If γ is semisimple, by [B11, Theorem 3.1.2], there is g γ ∈ G such that γ = g γ e a k − g − γ and a ∈ p , k ∈ K , Ad( k ) a = a .(2.17)Moreover, the norm | a | depends only on the conjugacy class of γ in G , and | a | = (cid:96) [ γ ] .(2.18)A semisimple element γ is called elliptic, if (cid:96) [ γ ] = γ is semisimple, by [K02, Proposition 7.25], Z ( γ ) is a reductive group (withCartan involution g γ θ g − γ ) with maximal compact subgroup K ( γ ) with Cartandecomposition z ( γ ) = p ( γ ) ⊕ k ( γ ).(2.19)By [B11, Theorem 3.1.1], the map g ∈ Z ( γ ) → p g ∈ X induces the identification of Z ( γ )-manifolds, Z ( γ )/ K ( γ ) (cid:39) X ( γ ).(2.20) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 26
Semisimple orbital integrals.
For t >
0, let exp( − tC g , X , τ ) be the heat op-erator of C g , X , τ . Let p X , τ t ( x , x (cid:48) ) be the smooth integral kernel of exp( − tC g , X , τ ) withrespect to the Riemannian volume dv X .Let γ ∈ G be a semisimple element. Since C g , X , τ commutes with the G -actionon C ∞ ( X , E τ ), the function g ∈ G → Tr E τ (cid:104) p X , τ t ( p g , p γ g ) (cid:105) ∈ C (2.21)descends to Z ( γ ) \ G .The form − B ( · , θ · ) induces a volume form dv G on G . We define dv Z ( γ ) similarly.Let dv Z ( γ ) \ G be the induced volume form on Z ( γ ) \ G , so that dv G = dv Z ( γ ) \ G dv Z ( γ ) .In the same way, we can also define dv K ( γ ) \ K and its volume vol( K ( γ ) \ K ). Definition 2.4.
Let γ ∈ G be semisimple. The orbital integral of exp( − tC g , X , τ ) isdefined byTr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105) = K ( γ ) \ K ) (cid:90) g ∈ Z ( γ ) \ G Tr E τ (cid:104) p X , τ t ( p g , p γ g ) (cid:105) dv Z ( γ ) \ G .(2.22)Clearly, Tr [ γ ] (cid:163) exp (cid:161) − tC g , X , τ (cid:162)(cid:164) depends only on the conjugacy class of γ in G . Remark . If E τ is a Z -graded or virtual representation of K , we use the no-tation Tr [ γ ]s [ · ] when the trace on the right-hand side of (2.22) is replaced by thesupertrace on E τ . Remark . An explicit geometric formula for Tr [ γ ] (cid:163) exp (cid:161) − tC g , X , τ (cid:162)(cid:164) is obtainedby Bismut [B11, Theorem 6.1.1]. This formula involves an explicit function J γ defined on the Lie algebra k ( γ ). It can be written down with the help of a rootsystem [BSh19a], [BSh19b, Theorem 4.7]. Remark . Most of the results obtained in [Sh18, ShY17] and as well as in thispaper rely on Bismut’s formula [B11, Theorem 6.1.1]. In this paper, the explicitformula for J γ is not needed, since all the involved orbital integrals have alreadybeen calculated in [Sh18, ShY17] except for a trivial one (5.23).2.6. A discrete subgroup of G . Let Γ ⊂ G be a discrete cocompact subgroupof G . By [Se60, Lemma 1] (see also [Ma19, Proposition 3.9]), Γ contains onlysemisimple elements. Let Γ e ⊂ Γ be the subset of elliptic elements. Then, Γ + = Γ − Γ e consists of nonelliptic elements.The group Γ acts isometrically on the left on X . Take Z = Γ \ X = Γ \ G / K .(2.23)Then Z is a compact orbifold. We denote by (cid:98) p : Γ \ G → Z and (cid:98) π : X → Z the naturalprojections, so that the diagram G p (cid:15) (cid:15) (cid:47) (cid:47) Γ \ G (cid:98) p (cid:15) (cid:15) X (cid:98) π (cid:47) (cid:47) Z (2.24)commutes. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 27
From now on until Section 6, we assume that Γ is torsion free, i.e., Γ e = { id } .Then Z is a connected closed orientable Riemannian locally symmetric manifoldwith nonpositive sectional curvature. Since X is contractible, π ( Z ) = Γ and X isthe universal cover of Z . In Section 7, we will treat the case where Γ is no longertorsion free.The Γ -action on X lifts to all the homogeneous Euclidean or Hermitian vectorbundles E τ on X constructed in (2.16), and preserves the metric connections. Then E τ descends to a Euclidean or Hermitian vector bundle F τ = Γ \ E τ = Γ \ G × K E τ (2.25)on Z , which is equipped with a canonical metric connection.Let F be a flat vector bundle on Z with holonomy representation ρ : Γ → GL r ( C ),so that (1.33) holds. As in (1.43), we have the identification C ∞ ( Z , F τ ⊗ F ) = (cid:161) C ∞ ( X , E τ ) ⊗ C r (cid:162) Γ .(2.26)We use the notation in Section 1.5. In particular, the self-adjoint generalisedLaplacian C g , X , τ ⊗ id descends to a generalised Laplacian operator C g , Z , τ , ρ actingon C ∞ ( Z , F τ ⊗ F ). As before, if E τ = Λ · ( p ∗ ), we denote C g , Z , ρ for simplification.Clearly, C g , Z , ρ is just the flat Laplacian introduced in Section 1.6.For γ ∈ Γ , set Γ ( γ ) = Z ( γ ) ∩ Γ .(2.27)By [Se60, Lemma2] (see also [Sh18, Proposition 4.9], [Ma19, Proposition 3.9]), Γ ( γ ) is cocompact in Z ( γ ).Let [ Γ + ] and [ Γ ] be the sets of conjugacy classes in Γ + and Γ . If γ ∈ Γ , theassociated conjugacy class in Γ is denoted by [ γ ] ∈ [ Γ ]. If [ γ ] ∈ [ Γ ], for all γ (cid:48) ∈ [ γ ],the locally symmetric spaces Γ ( γ (cid:48) ) \ X ( γ (cid:48) )(2.28)are canonically diffeomorphic, which will be denoted by B [ γ ] . Let vol( B [ γ ] ) be theRiemannian volume of B [ γ ] induced by the bilinear form B .We have a generalisation of [Sh18, Theorem 4.10]. Theorem 2.8.
There exist c > , C > such that for t > , we have (cid:88) [ γ ] ∈ [ Γ + ] vol (cid:161) B [ γ ] (cid:162)(cid:175)(cid:175) Tr[ ρ ( γ )] (cid:175)(cid:175) (cid:175)(cid:175)(cid:175) Tr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105)(cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:179) − ct + Ct (cid:180) .(2.29) For t > , the following identity holds, Tr (cid:104) exp (cid:179) − tC g , Z , τ , ρ (cid:180)(cid:105) = (cid:88) [ γ ] ∈ [ Γ ] vol (cid:161) B [ γ ] (cid:162) Tr[ ρ ( γ )] Tr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105) .(2.30) Proof.
Proceeding as the proof of the Selberg trace formula [Se56], by [B11, (4.8.8),(4.8.11), (4.8.16)], for [ γ ] ∈ [ Γ ], we have (cid:88) γ (cid:48) ∈ [ γ ] (cid:90) x ∈ F Z Tr E τ (cid:104) γ (cid:48)∗ p X , τ t (cid:161) ( γ (cid:48) ) − x , x (cid:162)(cid:105) dv X = vol (cid:161) B [ γ ] (cid:162) Tr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105) .(2.31)By Corollary 1.9, Proposition 1.10, and (2.31), we get our proposition. (cid:3) The quantities (cid:96) [ γ ] and Tr [ γ ] [ · ] depend only on the conjugacy class of γ in G . So they are welldefined on the conjugacy classes of Γ . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 28
Remark . In [Mü11, Theorem 1.1], instead of heat operators, Müller obtain asimilar formula for ϕ ( C g , Z , τ , ρ ) where ϕ is an even Paley-Wiener function on R .Let us give a direct application of the Selberg trace formula. Recall the follow-ing theorem due to [MoSt91, p. 194] and [B11, Theorem 7.9.1]. Let N Λ • ( T ∗ X ) bethe number operator on Λ • ( T ∗ X ), which is multiplication by p on Λ p ( T ∗ X ). Theorem 2.10. If δ ( G ) (cid:202) , for any semisimple element γ ∈ G, Tr [ γ ]s (cid:104) N Λ • ( T ∗ X ) exp (cid:179) − tC g , X (cid:180)(cid:105) = Corollary 2.11.
Let F be a flat vector bundle on Z with holonomy ρ . Assume that dim Z is odd and that δ ( G ) (cid:54)= . Then, for t > , we have Tr s (cid:104) N Λ • ( T ∗ Z ) exp (cid:179) − t (cid:3) Z (cid:180)(cid:105) = In particular, m (cid:89) i = det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180) ( − i i = For any acyclic and unitary representation ρ of Γ , there is (cid:178) > such that if ρ isC (cid:178) -close to ρ , then T CM ( F ) = Proof.
Since m is odd, δ ( G ) is odd. Since δ ( G ) (cid:54)=
1, we have δ ( G ) (cid:202)
3. By Theorems2.8 and 2.10, we get (2.33) and (2.34). By Proposition 1.12, we get (2.35). (cid:3)
3. F
UNDAMENTAL C ARTAN SUBALGEBRA AND RELATED CONSTRUCTIONS
The purpose of this section is to give a simple new conceptual proof for a com-plexified version of [Sh18, Theorem 6.11]. There, the corresponding proof is basedon the classification theory of real simple Lie algebras.This section is organised as follows. In Section 3.1, we introduce a θ -invariantfundamental Cartan subalgebra h = b ⊕ t .In Section 3.2, we introduce a splitting of g according to the action of b .In Section 3.3, we introduce a root system of g with respect to the fundamentalCartan subalgebra h . We reinterpret some objets constructed in Section 3.2.Finally, in Section 3.4, we give a simple new conceptual proof for a complexifiedversion of [Sh18, Theorem 6.11].3.1. A fundamental Cartan subalgebra.
Let T ⊂ K be a maximal torus of K .Let t ⊂ k be the Lie algebra of T . Set b = { a ∈ p : [ a , t ] = } .(3.1)Put h = b ⊕ t .(3.2) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 29
By [K86, p. 129], h is a Cartan subalgebra of g . Let H ⊂ G be the associatedCartan subgroup, that is the centraliser of h in G . By [K86, Theorem 5.22], H isa connected abelian reductive subgroup of G , so that H = exp( b ) × T .(3.3)We will call h and H the fundamental Cartan subalgebra of g and the fundamentalCartan subgroup of G .Since h C = h ⊗ R C is a Cartan subalgebra of g C , by (2.9) and (3.2), we have δ ( G ) = dim b .(3.4)3.2. A splitting of g . Recall that Z ( b ) ⊂ G is the stabiliser of b in G with Lie alge-bra z ( b ) ⊂ g . By [K02, Proposition 7.25], Z ( b ) is a possibly non connected reductivesubgroup of G . Also, θ acts on z ( b ) so that we have the Cartan decomposition z ( b ) = p ( b ) ⊕ k ( b ).(3.5)Let m ⊂ z ( b ) be the orthogonal space (with respect to B) of b in z ( b ). Then m is aLie subalgebra of g , and θ acts on m so that m = p m ⊕ k m .(3.6)Let M ⊂ G be the connected Lie group associated to the Lie algebra m . By [B11,(3.3.11) and Theorem 3.3.1], M is closed in G and is a connected reductive sub-group of G with maximal compact subgroup K M = M ∩ K .(3.7)Moreover, we have Z ( b ) = exp( b ) × M , z ( b ) = b ⊕ m , p ( b ) = b ⊕ p m , k ( b ) = k m .(3.8)Since h ⊂ z ( b ) is also a fundamental Cartan subalgebra of z ( b ), we have δ ( M ) = X M = M / K M (3.10)be the associated symmetric space. By (3.9), we see that dim X M is even.Let p ⊥ ( b ), k ⊥ ( b ), z ⊥ ( b ) be respectively the orthogonal spaces (with respect to B )of p ( b ), k ( b ), z ( b ) in p , k , g . Clearly, z ⊥ ( b ) = p ⊥ ( b ) ⊕ k ⊥ ( b ).(3.11)And also p = b ⊕ p m ⊕ p ⊥ ( b ), k = k m ⊕ k ⊥ ( b ), g = b ⊕ m ⊕ z ⊥ ( b ).(3.12)The group K M acts trivially on b . It also acts on p m , p ⊥ ( b ), k m and k ⊥ ( b ), andpreserves the splittings (3.12). Similarly, the groups M and Z ( b ) act trivially on b , act on m , z ⊥ ( b ), and preserves the third splitting in (3.12). OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 30
A root system of ( h , g ) . Let R ⊂ h ∗ C be a root system of ( h , g ) [K02, SectionII.4]. If α ∈ R , let g α ⊂ g C be the weight space associated with α , which is ofdimension 1. Then we have the splitting g C = h C (cid:77) ⊕ α ∈ R g α .(3.13)If α ∈ R , then α ∈ R . A root is called real if α = α , imaginary if α = − α , andcomplex otherwise. By [K02, Proposition 11.16] (see also [BSh19b, Proposition3.7]), since h is fundamental, there are no real roots. Let R im ⊂ R and R c ⊂ R bethe subsets of imaginary and complex roots, so that R = R im (cid:116) R c .(3.14)Let h ⊥ be the orthogonal to h in g with respect to B . Set i = z ( b ) ∩ h ⊥ .(3.15)Let c be the orthogonal to i in h ⊥ . Then g = h ⊕ i ⊕ c .(3.16)By (3.13), we have i C = ⊕ α ∈ R im g α , c C = ⊕ α ∈ R c g α .(3.17)Also, θ acts on i , c , so that we have the splittings i = i p ⊕ i k , c = c p ⊕ c k .(3.18)By (3.5), (3.6), (3.12), (3.15), and (3.16), we have m = t ⊕ i , p m = i p , k m = t ⊕ i k ,(3.19) z ⊥ ( b ) = c , p ⊥ ( b ) = c p , k ⊥ ( b ) = c k .In particular, we can rewrite the third identity of (3.12), g = b ⊕ m ⊕ c .(3.20) Proposition 3.1.
The vector spaces i p , i k have even dimensions, and c p , c k have thesame even dimension. The K M -action preserves the second splitting in (3.18) , sothat the actions on c p , c k are equivalent.Proof. This is [BSh19b, Proposition 3.8], except in the last statement the group K M is replaced by T . By the consideration after (3.12), by the last two identities in(3.19), K M acts on c p , c k . Since K M and b commutes, if b ∈ b is such that 〈 α , b 〉 (cid:54)= α ∈ R c , then ad( b ) : c p → c k defines a K M -equivalent. (cid:3) Let R + ⊂ R be a positive root system. Set R im + = R + ∩ R im , R c + = R + ∩ R c .(3.21)As explained in [BSh19b, Section 3.5], we can choose R + so that R c + is preservedby the complex conjugation.Set c + , C = ⊕ α ∈ R c + g α , c − , C = ⊕ α ∈ R c − g α .(3.22) Proposition 3.2.
The following statements hold.
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 31 i) The vector spaces c + , C , c − , C are the complexifications of real Lie subalgebras c + , c − of g , which have the same even dimension, and are such that c = c + ⊕ c − , c + = θ c − .(3.23) ii) The bilinear form B vanishes on c + , c − and induces the identification, c ∗− (cid:39) c + .(3.24) iii) The group Z ( b ) acts on c ± , so that (3.24) is Z ( b ) -equivalent.iv) The actions of M on c ± are equivalent.v) The projections on p , k map c ± into c p , c k isomorphically.vi) Finally, the actions of K M on c + , c − , c p , c k are equivalent.Proof. The statements i), ii), v) are just [BSh19b, Proposition 3.10], and vi) hasbeen established for the T -action instead of the K M -action .By our choice of the positive root system, we have[ i , c ± , C ] ⊂ c ± , C .(3.25)Therefore, z ( b ) = h ⊕ i preserves c ± , C , and Z ( b ) acts on c ± , C . Since the Z ( b )-actionon c ± , C commutes with the complex conjugation, Z ( b ) acts on c ± . Since B is Z ( b )-invariant, we see that (3.24) is Z ( b )-equivalent, from which we get iii).Since δ ( M ) =
0, by [K86, Probleme XII.14], there is k ∈ K M such that Ad( k ) | m = θ | m . By the second identity of (3.23), Ad( k ) θ : c + → c − is an equivalence of M -representations, from which we get iv).Since the projection c ± → c p is (1 + θ )/2, since K M is fixed by θ , we get vi). (cid:3) Corollary 3.3.
For (cid:201) j (cid:201) dim c ± , we have isomorphisms of representations of M, Λ j ( c ∗± ) (cid:39) Λ dim c ± − j ( c ∗± ).(3.26) Proof.
Since δ ( M ) = M has a compact center. By Proposition 2.1, M acts triv-ially on Λ dim c ± ( c ∗± ). We have an isomorphism of real representations of M , Λ j ( c ∗± ) (cid:39) Λ dim c ± − j ( c ± ).(3.27)By Proposition 3.2 iii) iv) and (3.27), we get (3.26). (cid:3) A lifting property.
Let R ( K ) be the representation ring of K . We can iden-tify R ( K ) with the subring of the Ad( K )-invariant smooth functions on K which isgenerated by the characters of finite dimensional complex representations of K .The restriction induces an injective morphism of rings i : R ( K ) → R ( T ).(3.28)Let W ( T : K ) = N K ( T )/ T be the Weyl group of K , where N K ( T ) is the normaliserof T in K . Then W ( T : K ) acts on R ( T ). By [BrDi85, Proposition VI.2.1], i inducesan isomorphism of rings i : R ( K ) (cid:39) R ( T ) W ( T : K ) .(3.29) Proposition 3.4.
The adjoint action of N K ( T ) preserves the decomposition g = t ⊕ b ⊕ i k ⊕ i p ⊕ c k ⊕ c p .(3.30) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 32
Proof.
Since N K ( T ) preserves t , by (3.1), N K ( T ) preserves b . Also, N K ( T ) pre-serves z ( b ). Since N K ( T ) ⊂ K preserves B , by (3.15), N K ( T ) preserves i . Usingagain N K ( T ) preserves B , by (3.16), N K ( T ) preserves c . By (3.18), N K ( T ) pre-serves i p , i k , c p , c k . (cid:3) Theorem 3.5.
The adjoint action of T on i p , C , i k , C , c p , C (cid:39) c k , C (cid:39) c + , C (cid:39) c − , C liftuniquely to virtual representations in R ( K ) .Proof. By Proposition 3.4, the characters of T on i p , C , i k , C , and c p , C are W ( T : K )-invariant. By (3.28) and (3.29), the T -actions on i p , C , i k , C , and c p , C lift uniquely to R ( K ). (cid:3) Corollary 3.6.
For i , j ∈ N , the adjoint representations of K M on Λ i ( p ∗ m , C ) and Λ j ( c ∗± , C ) have unique lifts in R ( K ) .Proof. This is a consequence of (3.19) and Theorem 3.5, (cid:3)
Remark . Let RO ( K ) be the real representation ring of K . By [BrDi85, Proposi-tion II.7.8], the complexification V ∈ RO ( K ) → V ⊗ R C ∈ R ( K ) induces an injectivemorphism of rings, RO ( K ) → R ( K ).(3.31)In [Sh18, Theorem 6.11], using the classification theory on the real simple Liealgebras, we have shown that when δ ( G ) =
1, the real representation c ± of K M has a unique lift in RO ( K ). However, the above complexified version is enough forapplications both in [Sh18] and the current paper.4. T HE R UELLE DYNAMICAL ZETA FUNCTIONS FOR ARBITRARY TWIST
The purpose of this section is to introduce the Ruelle dynamical zeta functionfor arbitrary twist. We restate the main result of this article as Theorem 4.3. Itsproof will be given in Sections 5 and 6. Theorem 4.3 generalises author’s previousresult [Sh18, Theorem 1.1] where ρ is unitary, as well as the results on hyperbolicmanifolds due to Müller [Mü20, Theorem 1.1, Proposition 1.3] and Spilioti [Sp18,Theorem 1.2], [Sp20b, Theorem 2].We use the notation in Section 2. Recall that Z = Γ \ G / K is a locally symmetricmanifold and ρ : Γ → GL r ( C ) is a representation of Γ .Let us recall the definition of the Ruelle dynamical zeta function introducedby Fried [F87, Section 5]. For [ γ ] ∈ [ Γ ], recall that B [ γ ] (see (2.28)) is the locallysymmetric space Γ ( γ ) \ X ( γ ). By [DuKVa79, Proposition 5.15], the set of nontrivialclosed geodesics on Z consists of a disjoint union (cid:97) [ γ ] ∈ [ Γ + ] B [ γ ] .(4.1)Moreover, if [ γ ] ∈ [ Γ + ], all the elements of B [ γ ] have the same length (cid:96) [ γ ] > γ ] ∈ [ Γ + ], the geodesic flow induces a locally free action of (cid:83) on B [ γ ] , so that B [ γ ] / (cid:83) is a closed orbifold. Let χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) ∈ Q be the orbifold Euler character-istic number [Sa57]. We refer the reader to [Sh18, Proposition 5.1] for an explicit OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 33 formula for χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) . In particular, if δ ( G ) (cid:202)
2, or if δ ( G ) = γ can not beconjugate by an element of G into the fundamental Cartan subgroup H , then χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) = (cid:83) -action on B [ γ ] is not necessarily effective. Let m [ γ ] = (cid:175)(cid:175) ker (cid:161) (cid:83) → Diff( B [ γ ] ) (cid:162)(cid:175)(cid:175) ∈ N ∗ (4.3)be the generic multiplicity.By [Sh18, Theorem 5.6] and by (1.34), there is σ > (cid:88) [ γ ] ∈ [ Γ + ] (cid:175)(cid:175) χ orb (cid:161) B [ γ ] / (cid:83) (cid:162)(cid:175)(cid:175) m [ γ ] (cid:175)(cid:175) Tr (cid:163) ρ ( γ ) (cid:164)(cid:175)(cid:175) e − σ (cid:96) [ γ ] < ∞ .(4.4) Definition 4.1.
For Re ( σ ) (cid:202) σ , set R ρ ( σ ) = exp (cid:195) (cid:88) [ γ ] ∈ [ Γ + ] χ orb (cid:161) B [ γ ] / (cid:83) (cid:162) m [ γ ] Tr (cid:163) ρ ( γ ) (cid:164) e − σ(cid:96) [ γ ] (cid:33) .(4.5) Remark . By (4.2), if δ ( G ) (cid:202)
2, the dynamical zeta function R ρ ( σ ) is the constantfunction 1. Moreover, if δ ( G ) =
1, then the sum on the right-hand side of (4.5) canbe reduced to a sum over [ γ ] ∈ [ Γ + ] such that γ can be conjugate into H .Recall that (cid:3) Z is the flat Laplacian on Z . We restate Theorem 0.1 as follows. Theorem 4.3.
Assume that dim
Z is odd. The following statements hold.i) The dynamical zeta function R ρ ( σ ) has a meromorphic extension to σ ∈ C .ii) There exist explicit constants C ρ ∈ R ∗ and r ρ ∈ Z (see (5.33) and (5.35) ) suchthat when σ → , we haveR ρ ( σ ) = C ρ (cid:40) m (cid:89) i = (cid:179) det ∗ (cid:179) (cid:3) Z | Ω i ( Z , F ) (cid:180)(cid:180) ( − i i (cid:41) σ r ρ + O ( σ r ρ + ).(4.6) iii) There is k ∈ N such that for any acyclic and unitary representation ρ of Γ ,there exists (cid:178) > such that if ρ is C k (cid:178) -close to ρ , then (cid:3) Z is invertible andC ρ = r ρ = so that R ρ (0) = T CM ( F ).(4.8) Proof.
Since m and δ ( G ) have the same parity, we have δ ( G ) (cid:202) δ ( G ) (cid:202)
3, our theorem with k = δ ( G ) =
1, the proofs of i), ii) as well as the proof of iii) when Z G is noncompact arebased on the introducing of the Selberg zeta functions with arbitrary twist andwill be given in Section 5.4. The proof of iii) when Z G is compact will be given inSection 6.3. (cid:3) Let R ρ be the meromorphic function defined for σ ∈ C by R ρ ( σ ) = R ρ ( σ ).(4.9) Proposition 4.4.
The following identities of meromorphic functions on C hold,R ρ ∗ ( σ ) = R ρ ( σ ), R ρ ( σ ) = R ρ ( σ ).(4.10) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 34
Proof.
The second identity in (4.10) is trivial. Let us show the first one. We haveTr (cid:163) ρ ∗ ( γ ) (cid:164) = Tr (cid:163) ρ (cid:161) γ − (cid:162)(cid:164) .(4.11)Note that γ → γ − induces a bijection on [ Γ + ]. Since d γ − ( x ) = d γ ( x ), we have (cid:96) [ γ − ] = (cid:96) [ γ ] , X (cid:161) γ − (cid:162) = X ( γ ),(4.12)By (4.12), using Γ (cid:161) γ − (cid:162) = Γ ( γ ), we get B [ γ − ] = B [ γ ] , m [ γ − ] = m [ γ ] .(4.13)By (4.5) and (4.11)-(4.13), we get the first identity in (4.10). (cid:3)
5. T HE S ELBERG ZETA FUNCTION FOR ARBITRARY TWIST
The purpose of this section is to extend the results of [Sh18, Theorems 7.6 and7.7] on the zeta functions of Ruelle and Selberg to arbitrary twist with the help ofthe Müller’s Selberg trace formula (Theorem 2.8), inexplicitly of Bismut’s orbitalintegral formula [B11, Theorem 6.1.1], and of the lifting properties (Theorem 3.5,Corollary 3.6).More precisely, in [Sh18, Section 7], if δ ( G ) =
1, we associate a Selberg zetafunction Z η , ρ to a representation η of M satisfying [Sh18, Assumption 7.1] and toa unitary representation ρ of Γ . Moreover, we show that Z η , ρ has a meromorphicextension to C and satisfies a functional equation. Also, we prove that the Ruellezeta function with a unitary twist is an alternating product of certain Selberg zetafunctions. In this way, we obtained the meromorphic extension of the Ruelle zetafunction. In this section, we extend all the above results to arbitrary ρ and to aslightly larger class of η .This section is organised as follows. In Section 5.1, we recall some results onthe structure of real reductive groups with δ ( G ) = δ ( G ) =
1, we introduce a class of virtual represen-tations η of the group M . If ρ : Γ → GL r ( C ) is a representation of Γ , we introducethe Selberg zeta function Z η , ρ associated to ( η , ρ ). We show the meromorphic ex-tension and a functional equation for Z η , ρ .Finally, in Section 5.4, we show that R ρ is an alternating product of certainSelberg zeta functions. In particular, we show Theorem 4.3 i) ii), as well as iii)when Z G is non compact.In the whole section, we assume δ ( G ) = The structure of the reductive group G with δ ( G ) = . We use the no-tation in Sections 2 and 3. Assume that δ ( G ) =
1. To make this section readable,instead of writing c ± , we use notation as [Sh18], i.e., n = c + , n = c − .(5.1)When G has a non compact centre, thanks to the following proposition, G has avery simple structure. Proposition 5.1.
If G has a non compact centre, thenG = exp( b ) × M , z ⊥ ( b ) = OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 35
Proof.
Since G has a non compact centre, we have dim z p (cid:202)
1. Since z p ⊂ b andsince dim b =
1, we have z p = b . So g = z ( b ). By (3.8), we get (5.2). (cid:3) Assume in the rest of this subsection that G has a compact centre. By Proposi-tions 3.1 and 3.2 i), dim n is even. Set (cid:96) =
12 dim n .(5.3)Since G has a compact centre, we have b (cid:54)⊂ z g . Therefore, z ⊥ ( b ) (cid:54)= (cid:96) > R + . Proposition 5.2.
There is α ∈ b ∗ such that for all α ∈ R c + , α | b = α .(5.4) Equivalently, b acts on n and n by ± α ∈ b ∗ , i.e., for a ∈ b , f ∈ n , f ∈ n , we have [ a , f ] = 〈 α , a 〉 f , [ a , f ] = −〈 α , a 〉 f .(5.5) In particular, [ n , n ] ⊂ z ( b ), [ n , n ] = (cid:163) n , n (cid:164) = and [ z ( b ), z ( b )] ⊂ z ( b ), (cid:163) z ( b ), z ⊥ ( b ) (cid:164) ⊂ z ⊥ ( b ), (cid:163) z ⊥ ( b ), z ⊥ ( b ) (cid:164) ⊂ z ( b ).(5.7) Proof.
This is [Sh18, Propositions 6.2 and 6.3]. (cid:3)
Let u ( b ) ⊂ u , u m ⊂ u be the compact forms of z ( b ) and m . Then u ( b ) = (cid:112)− b ⊕ u m , u m = (cid:112)− p m ⊕ k m .(5.8)Let u ⊥ ( b ) be the orthogonal space of u ( b ) in u with respect to − B | u . Then, u ⊥ ( b ) = (cid:112)− p ⊥ ( b ) ⊕ k ⊥ ( b ).(5.9)By (5.7), we have[ u ( b ), u ( b )] ⊂ u ( b ), (cid:163) u ( b ), u ⊥ ( b ) (cid:164) ⊂ u ⊥ ( b ), (cid:163) u ⊥ ( b ), u ⊥ ( b ) (cid:164) ⊂ u ( b ).(5.10)Thus, ( u , u ( b )) is a compact symmetric pair.Let U ( b ) ⊂ U , U M ⊂ U , A ⊂ U be the connected subgroups of U associated tothe Lie subalgebra u ( b ), u m , (cid:112)− b of u , so that U ( b ) = A U M .(5.11) Remark . All of the above three groups are compact. Indeed, U ( b ) is the sta-biliser of (cid:112)− b in U , which is compact in U . Since δ ( M ) = M has compactcentre. Therefore, U M is the compact form of M . The compactness of A is estab-lished in [Sh18, Proposition 6.6] OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 36
The Selberg zeta function.
Recall that K M and K have the maximal torus T . By (3.29), the restriction to K M induces an injective morphism of rings R ( K ) → R ( K M ). Assumption 5.4.
Assume that η = η + − η − is a virtual M-representation on thefinite dimensional complex vector space E η = E + η − E + η such that(1) η | K M = η + | K M − η − | K M ∈ R ( K M ) has a unique lift in R ( K ) .(2) the Casimir C u m of u m acts on η ± by the same scalar C u m , η ∈ R .Remark . If Z G is non compact, then K M = K . Assumption (1) is empty. Remark . If Z G is compact, Assumption 5.4 is slightly different from [Sh18, As-sumption 7.1]. Indeed, Assumption 5.4 is the complexified virtual version [Sh18,Assumption 7.1 (1) (3)]. Moreover, the statement (2) in [Sh18, Assumption 7.1],which requires the u m -action on η lifts to U M , is a technique condition. Since δ ( M ) = M has a compact centre. By Remark 2.2, the u m -action on η lifts to a fi-nite cover of U M . Thanks to this observation, most of arguments (see also Remark5.9 and Section 5.3.2) go though up to evident modifications.Recall that ρ : Γ → GL r ( C ) is a representation of Γ . Following [Sh18, Definition7.4], let us define the Selberg zeta function associated to the pair ( η , ρ ). Recallthat H = exp( b ) × T is the fundamental Cartan subgroup of G . For e a k − ∈ H , wewrite γ ∼ e a k − if there is g γ ∈ G such that γ = g γ e a k − g − γ . By [Sh18, (7-62)] andby (1.34), there is σ > (cid:88) [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H (cid:175)(cid:175) χ orb (cid:161) B [ γ ] / (cid:83) (cid:162)(cid:175)(cid:175) m [ γ ] (cid:175)(cid:175) Tr (cid:163) ρ ( γ ) (cid:164) (cid:175)(cid:175) e − σ (cid:96) [ γ ] (cid:175)(cid:175) det(1 − Ad( e a k − )) | z ⊥ ( b ) (cid:175)(cid:175) < ∞ .(5.12) Definition 5.7.
For Re ( σ ) (cid:202) σ , set Z η , ρ ( σ ) = exp (cid:181) − (cid:88) [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H χ orb ( B [ γ ] / (cid:83) ) m [ γ ] Tr (cid:163) ρ ( γ ) (cid:164) Tr E η s [ k − ] (cid:175)(cid:175) det(1 − Ad( e a k − )) | z ⊥ ( b ) (cid:175)(cid:175) e − σ(cid:96) [ γ ] (cid:182) .(5.13)Recall that by Corollary 3.6, Λ • ( p ∗ m , C ) has a unique lift in R ( K ). Definition 5.8.
Let (cid:98) η ∈ R ( K ) be the unique virtual representation of K on E (cid:98) η = E + (cid:98) η − E − (cid:98) η such that the following identity in R ( K M ) holds, E (cid:98) η | K M = Λ • ( p ∗ m , C ) (cid:98) ⊗ E η | K M ∈ R ( K M ).(5.14)Let C g , Z , (cid:98) η , ρ be the generalised Laplacian acting on C ∞ ( Z , F (cid:98) η ⊗ F ) introducedafter (2.26). For λ ∈ C , set m η , ρ ( λ ) = dim ker (cid:179) C g , Z , (cid:98) η + , ρ − λ (cid:180) N − dim ker (cid:179) C g , Z , (cid:98) η − , ρ − λ (cid:180) N ,(5.15)where N (cid:192)
1. When λ =
0, set r η , ρ = m η , ρ (0).(5.16)Let det gr (cid:179) C g , Z , (cid:98) η , ρ + σ (cid:180) = det (cid:161) C g , Z , (cid:98) η + , ρ + σ (cid:162) det (cid:161) C g , Z , (cid:98) η − , ρ + σ (cid:162) (5.17) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 37 be a graded determinant of C g , Z , (cid:98) η , ρ + σ . By Theorem 1.5, (5.17) is a meromorphicfunction on σ ∈ C . Its zeros and poles belong to the set { − λ : λ ∈ Sp( C g , Z , (cid:98) η , ρ ) } . If λ ∈ Sp( C g , Z , (cid:98) η , ρ ), the order of the zero at σ = − λ is m η , ρ ( λ ).Following in [Sh18, (7-60)], set σ η =
18 Tr u ⊥ ( b ) (cid:104) C u ( b ), u ⊥ ( b ) (cid:105) − C u m , η .(5.18)Let P η ( σ ) be the odd polynomial defined in [Sh18, (7-61)]. When G has non com-pact centre, we have u ⊥ ( b ) =
0, so σ η = − C u m , η ,(5.19)and the polynomial P η is given by P η ( σ ) = − (cid:179) dim E + η − dim E − η (cid:180) (cid:104) e (cid:179) T X M , ∇ T X M (cid:180)(cid:105) max σ .(5.20) Remark . In [Sh18, (7-61)], we assume the group U M acts on η . This actionextends to U ( b ) = A U M by requiring A acts trivially on η . To define P η ( σ ), weuse characteristic forms of the homogenous vector bundle F b , η = U × U ( b ) E η on U / U ( b ). In current situation, the u m -action on E η does not necessarily lift to U M .So the vector bundle F b , η is not well defined globally on U / U ( b ). However, it iswell defined in a neighborhood of [ e ] ∈ U / U ( b ), so the right-hand sides of [Sh18,(7-8) (7-61)] are still well defined. In particular, P η ( σ ) is still well defined.We have a generalisation of [Sh18, Theorem 7.6]. Theorem 5.10.
The Selberg zeta function Z η , ρ ( σ ) has a meromorphic extension to σ ∈ C such that the following identity of meromorphic functions on C holds,Z η , ρ ( σ ) = det gr (cid:179) C g , Z , (cid:98) η , ρ + σ η + σ (cid:180) exp (cid:161) r vol( Z ) P η ( σ ) (cid:162) .(5.21) The zeros and poles of Z η , ρ ( σ ) belong to the set (cid:169) ± i (cid:112) λ + σ η : λ ∈ Sp (cid:161) C g , Z , (cid:98) η , ρ (cid:162)(cid:170) . If λ ∈ Sp (cid:161) C g , Z , (cid:98) η , ρ (cid:162) and λ (cid:54)= − σ η , the order of the zero at σ = ± i (cid:112) λ + σ η is m η , ρ ( λ ) . Theorder of the zero at σ = is m η , ρ ( − σ η ) . Also,Z η , ρ ( σ ) = Z η , ρ ( − σ ) exp (cid:161) r vol( Z ) P η ( σ ) (cid:162) .(5.22) Proof.
For the first part of the theorem, it is enough to show (5.21) for σ ∈ R largeenough, which will be given in Section 5.3. The rest parts of our theorem is aconsequence of Theorem 1.5 and (5.21). (cid:3) Proof of (5.21) for σ (cid:192) . The case where δ ( G ) = and Z G is non compact. For γ (cid:48) ∈ M , let X M ( γ (cid:48) ) bethe symmetric space defined in (2.20) when G and γ are replaced by M and γ (cid:48) . Wehave a generalisation of [Sh18, Proposition 4.14] and [ShY17, Proposition 5.7]. Proposition 5.11. If γ = e a k − ∈ H with a ∈ b and k ∈ T, then (5.23) Tr [ γ ]s (cid:183) exp (cid:181) − t C g , X , (cid:98) η (cid:182)(cid:184) = (cid:112) π t exp (cid:181) − | a | t − t C u m , η (cid:182) (cid:104) e (cid:179) T X M ( k ), ∇ T X M ( k ) (cid:180)(cid:105) max Tr E η s (cid:163) k − (cid:164) . OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 38 If γ can not be conjugate into H, then Tr [ γ ]s (cid:183) exp (cid:181) − t C g , X , (cid:98) η (cid:182)(cid:184) = Proof.
Since K = K M ⊂ M and p = b ⊕ p m , by (5.14), we have C ∞ (cid:161) G , E (cid:98) η (cid:162) K = C ∞ ( R ) ⊗ C ∞ (cid:179) M , Λ • ( p ∗ m , C ) (cid:98) ⊗ E η | K M (cid:180) K M .(5.25)By the identification (5.25), if ∆ R is the usual Laplacian operator on R , we have C g , X , (cid:98) η = − ∆ R + C m , X M , Λ • ( p ∗ m ) (cid:98) ⊗ η | KM .(5.26)Let γ be a semisimple element in G . Since G = exp( b ) × M , write γ = e a γ (cid:48) where a ∈ b and γ (cid:48) is semisimple in M . By (5.26), we haveTr [ γ ]s (cid:183) exp (cid:181) − t C g , X , (cid:98) η (cid:182)(cid:184) = Tr [ e a ] (cid:183) exp (cid:181) t ∆ R (cid:182)(cid:184) Tr [ γ (cid:48) ]s (cid:183) exp (cid:181) − t C m , X M , Λ • ( p ∗ m ) (cid:98) ⊗ η | KM (cid:182)(cid:184) .(5.27)Clearly, we have Tr [ e a ] (cid:183) exp (cid:181) t ∆ R (cid:182)(cid:184) = (cid:112) π t e − | a | t .(5.28)If γ can not be conjugate in H , then γ (cid:48) is not elliptic in M , by [Sh18, (8-82)], wehave Tr [ γ (cid:48) ]s (cid:183) exp (cid:181) − t C m , X M , Λ • ( p ∗ m ) (cid:98) ⊗ η | KM (cid:182)(cid:184) = γ ∈ H , then γ (cid:48) = k − ∈ T . By [Sh18, (8-83)], we have(5.30) Tr [ k − ]s (cid:183) exp (cid:181) − t C m , X M , Λ • ( p ∗ m ) (cid:98) ⊗ η | KM (cid:182)(cid:184) = (cid:104) e (cid:179) T X M ( k ), ∇ T X M ( k ) (cid:180)(cid:105) max Tr E η s (cid:163) k − (cid:164) exp (cid:181) − t C u m , η (cid:182) .By (5.27)-(5.30), we get (5.23) and (5.24). (cid:3) Proceeding as in the proof of [Sh18, Theorem 5.6], using Theorem 2.8 andProposition 5.11 instead of [Sh18, Theorem 4.10 and Proposition 4.14], we get(5.21) for σ (cid:192) (cid:3) The case where δ ( G ) = and Z G is compact. Although, Assumption 5.4 on η is different from [Sh18, Assumption 7.1], by Remarks 5.6 and 5.9, the statement of[Sh18, Theorem 7.3] still holds. Therefore, the proof of (5.21) for σ (cid:192) (cid:3) Proof of Theorem 4.3.
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 39
The case where δ ( G ) = and Z G is non compact. Since G has a non compactcentre, by the second equation of (5.2), the denominator (cid:175)(cid:175) det(1 − Ad( e a k − )) (cid:175)(cid:175) z ⊥ ( b ) | in (5.13) disappears. When η = is the trivial representation, by (4.5), (5.13),(5.14), and (5.19), we have R ρ ( σ ) = (cid:161) Z , ρ ( σ ) (cid:162) − , E (cid:98) = m (cid:88) i = ( − i − i Λ i ( p ∗ C ) ∈ R ( K ), σ = R ρ ( σ ) has a meromorphicextension and R ρ ( σ ) = exp (cid:179) r vol( Z ) (cid:104) e (cid:179) T X M , ∇ T X M (cid:180)(cid:105) max σ (cid:180) T ρ ( σ ).(5.32)Set C ρ = r ρ = χ (cid:48) CM ( F ).(5.33)By (1.71), (5.32), and (5.33), we get (4.6). By Proposition 1.12, (1.69), and (5.33),we see that (4.7) holds with k =
0. The proof of Theorem 4.3 is complete when δ ( G ) = Z G is not compact. (cid:3) The case where δ ( G ) = and Z G is compact. For 0 (cid:201) j (cid:201) (cid:96) , let η j be therepresentation of M on Λ j ( n ∗ C ). By Corollary 3.6 (or [Sh18, Corollary 6.12]) and[Sh18, Proposition 6.13], η j satisfies Assumption 5.4. By Theorem 5.10, we seethat Z η j , ρ has a meromorphic extension to C . Using the relation ([Sh18, (7-65)]) R ρ ( σ ) = (cid:96) (cid:89) j = Z η j , ρ ( σ + ( j − (cid:96) ) | α | ) ( − j − ,(5.34)we see that R ρ ( σ ) has a meromorphic extension.Write r j = r η j , ρ . As in [Sh18, (7-74),(7-75)], set C ρ = (cid:96) − (cid:89) j = (cid:161) − (cid:96) − j ) | α | (cid:162) ( − j − r j , r ρ = (cid:96) (cid:88) j = ( − j − r j .(5.35)Proceeding exactly as in [Sh18, (7-71)-(7-78)], we get (4.6). The proof of (4.7) willbe given in Section 6.3. (cid:3) Remark . Note that r ρ is even. By (5.35) and by [Sh18, (7-74), (7-80)], we have C ρ = ( − χ (cid:48) CM ( X , F ) + r ρ (cid:96) − (cid:89) j = (cid:161) (cid:96) − j ) | α | (cid:162) ( − j − r j , r ρ = χ (cid:48) CM ( Z , F ) + ( − (cid:96) − r (cid:96) = χ (cid:48) CM ( Z , F ) + (cid:96) − (cid:88) j = ( − j r j .(5.36) 6. A COHOMOLOGICAL FORMULA FOR r j The purpose of this section is to establish (4.7) when δ ( G ) = Z G is com-pact. Its proof relies on some results from the representation theory of real reduc-tive groups.This section is organised as follows. In Section 6.1, we recall the definition ofinfinitesimal characters of g C -modules and some basic properties of the Harish-Chandra ( g C , K )-modules. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 40
In Section 6.2, we study the right regular representation of G on C ∞ ( Γ \ G , (cid:98) p ∗ F ).Given a character χ : Z ( g C ) → C , we introduce a Harish-Chandra ( g C , K )-modules V χ ( Γ , ρ ) ⊂ C ∞ ( Γ \ G , (cid:98) p ∗ F ) with generalised infinitesimal character χ .Finally, in Section 6.3, assuming that δ ( G ) = Z G is compact, we write r j as a sum of Euler characteristic numbers of certain cohomologies of V χ ( Γ , ρ ),where χ is the trivial character. We establish (4.7).In Sections 6.1 and 6.2, we assume neither δ ( G ) = Z G is compact.6.1. Preliminary on the representation theory.
Recall that G is a linear con-nected real reductive group. We use the notation in Sections 2.1 and 2.2. Recallthat Z ( g C ) is the centre of the enveloping algebra of g C .A morphism of algebras χ : Z ( g C ) → C will be called a character of Z ( g C ).Clearly, for a ∈ C , we have χ ( a ) = a .(6.1) Definition 6.1.
A complex representation of g C is said to have an infinitesimalcharacter χ if z ∈ Z ( g C ) acts as a scalar χ ( z ) ∈ C .A complex representation of g C is said to have a generalised infinitesimal char-acter χ if there is i (cid:192) z ∈ Z ( g C ), ( z − χ ( z )) i acts like 0.The infinitesimal character of a trivial representation will be called the trivialcharacter χ . Note that in [Sh18, Section 8], such a character is denoted by 0. Wewill not use this notation here. Definition 6.2.
A complex U ( g C )-module V , equipped with an action of K , iscalled a ( g C , K )-module, if(1) every v ∈ V is K -finite, i.e., { k · v } k ∈ K spans a finite dimensional vectorspace;(2) the actions of g C and K are compatible.A ( g C , K )-module V is called a Harish-Chandra ( g C , K )-module, if(1) the space V is finitely generated as a U ( g C )-module;(2) each irreducible K -module occurs only for a finite number of times in V .Let V be a Harish-Chandra ( g C , K )-module. By [K86, Proposition 10.43], sub-modules or quotient modules of V is still a Harish-Chandra ( g C , K )-module. When V is irreducible, by [K86, Corollary 8.13], V has an infinitesimal character. Forgeneral V , by [K86, Proposition 10.41], there are finitely many Harish-Chandra( g C , K )-submodules V χ ⊂ V of V with generalised infinitesimal character χ , so that V = (cid:77) χ V χ .(6.2)By [K86, Corollary 10.42], V has a finite composition series, i.e., there existfinitely many Harish-Chandra ( g C , K )-submodules V = V N ⊃ V N − ⊃ · · · ⊃ V ⊃ V − = V s / V s − with 0 (cid:201) s (cid:201) N is irreducible. Moreover, thelength N ∈ N , the set of all irreducible quotients and their multiplicities are the OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 41 same for all the composition series. Also, if E is a finite dimensional representa-tion of K , then dim ( V ⊗ E ) K = N (cid:88) s = dim (cid:161) ( V s / V s − ) ⊗ E (cid:162) K .(6.4) Proposition 6.3.
Given a character χ : Z ( g C ) → C , there is a finite dimensionalrepresentation E of K such that if V is a Harish-Chandra ( g C , K ) -module withgeneralised infinitesimal character χ , thenV = ⇐⇒ ( V ⊗ E ) K = Proof.
It is enough to show the inverse direction. By a fundamental theorem ofHarish-Chandra [K86, Corollary 10.37], there are only finitely many irreducibleHarish-Chandra ( g C , K )-modules V , . . . , V k with infinitesimal character χ . Let E i be a K -type of V i . Set E = k (cid:77) i = E ∗ i .(6.6)By (6.6), for all 1 (cid:201) i (cid:201) k , ( V i ⊗ E ) K (cid:54)= (cid:3) The right regular representation on C ∞ ( Γ \ G , (cid:98) p ∗ F ) . Recall that Γ ⊂ G isa discrete cocompact and torsion free subgroup of G and that ρ : Γ → GL r ( C ) isa representation of Γ . Recall also that F is the flat vector bundle on Z = Γ \ G / K defined in (1.33). We fix a Hermitian metric g F on F . Note that the choice of g F is irrelevant.Let g (cid:98) p ∗ F be the pull back metric on (cid:98) p ∗ F . Let L ( Γ \ G , (cid:98) p ∗ F ) be the L -spaceassociated to g (cid:98) p ∗ F and to an invariant measure on Γ \ G . The group G acts onthe right on L ( Γ \ G , (cid:98) p ∗ F ). Since ρ is not necessarily unitary, this G -action isnot always unitary. Since g (cid:98) p ∗ F is K -invariant, the above G -action restricts to aunitary representation of K . Let (cid:98) K be the set of equivalent classes of irreduciblerepresentations of K . If τ ∈ (cid:98) K , let L τ ( Γ \ G , (cid:98) p ∗ F ) be the τ -isotropic subspace of L ( Γ \ G , (cid:98) p ∗ F ). By (2.16) and (2.25), we have L τ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) (cid:39) L (cid:161) Z , F ∗ τ ⊗ F (cid:162) ⊗ E τ .(6.8)By Peter-Weyl’s theorem, we have a direct sum of Hilbert spaces L (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) = Hil (cid:77) τ ∈ (cid:98) K L τ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.9)By (6.8), the Casimir operator C g , Z , τ ∗ , ρ defines an unbounded operator on theHilbert space L τ ( Γ \ G , (cid:98) p ∗ F ). If λ ∈ Sp( C g , Z , τ ∗ , ρ ), as in (1.7), let C ∞ τ , λ ( Γ \ G , (cid:98) p ∗ F ) bethe corresponding finite dimensional characteristic space. As the notation indi-cates, C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ C ∞ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.10) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 42
Let V ( Γ , ρ ) be the subspace of C ∞ ( Γ \ G , (cid:98) p ∗ F ) defined by the algebraic sum V ( Γ , ρ ) = (cid:77) τ ∈ (cid:98) K (cid:77) λ ∈ Sp( C g , Z , τ ∗ , ρ ) C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.11)Clearly, the group K acts on V ( Γ , ρ ) and the elements of V ( Γ , ρ ) are K -finite.By (1.9), (6.9), and (6.11), we have L (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) = V ( Γ , ρ ).(6.12) Proposition 6.4.
The algebra U ( g C ) acts on V ( Γ , ρ ) , so that V ( Γ , ρ ) is a ( g C , K ) -module.Proof. We have seen in the arguments below (6.11) that elements of V ( Γ , ρ ) are K -finite. We need to show that g C acts on V ( Γ , ρ ), or equivalently, for any pair( τ , λ ), g · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ V ( Γ , ρ ).(6.13)Let us follow the proof of [K86, Proposition 8.5]. Since the space C ∞ τ , λ ( Γ \ G , (cid:98) p ∗ F ) ⊂ C ∞ ( Γ \ G , (cid:98) p ∗ F ) is K -invariant, for k ∈ K and a ∈ g , we have(6.14) k · a · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ (Ad( k ) a ) · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ g · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .By (6.14), the finite dimensional space g · C ∞ τ , λ ( Γ \ G , (cid:98) p ∗ F ) is K -invariant. In par-ticular, the elements of g · C ∞ τ , λ ( Γ \ G , (cid:98) p ∗ F ) are K -finite. Therefore, g · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ (cid:77) τ ∈ (cid:98) K C ∞ τ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.15)Since the Casimir commutes with the g -actions, by (6.15), we have g · C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) ⊂ (cid:77) τ ∈ (cid:98) K C ∞ τ , λ (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.16)By (6.11) and (6.16), we get (6.13). (cid:3) Note that Z ( g C ) acts on the finite dimensional space C ∞ τ , λ ( Γ \ G , (cid:98) p ∗ F ). For acharacter χ of Z ( g C ), set V χ ( Γ , ρ ) = (cid:110) v ∈ V ( Γ , ρ ) : ∃ i ∈ N , ∀ z ∈ Z ( g C ), (cid:161) z − χ ( z ) (cid:162) i v = (cid:111) .(6.17)By (6.11) and (6.17), we have an infinite algebraic sum V ( Γ , ρ ) = (cid:77) χ V χ ( Γ , ρ ).(6.18)By Proposition 6.4, V χ ( Γ , ρ ) is a ( g C , K )-module. Proposition 6.5.
Each V χ ( Γ , ρ ) is a Harish-Chandra ( g C , K ) -module.Proof. By (6.17), we have V χ ( Γ , ρ ) ⊂ (cid:77) τ ∈ (cid:98) K C ∞ τ , χ ( C g ) (cid:161) Γ \ G , (cid:98) p ∗ F (cid:162) .(6.19)In particular, the multiplicity of each K -type in V χ ( Γ , ρ ) is finite.It remains to show that V χ ( Γ , ρ ) is a finitely generated U ( g C )-module. Other-wise, we can find a sequence ( v n ) n ∈ N in V χ ( Γ , ρ ) such that0 (cid:218) U ( g C ) v (cid:218) . . . (cid:218) n − (cid:88) i = U ( g C ) v i (cid:218) n (cid:88) i = U ( g C ) v i (cid:218) · · · (6.20) OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 43
For n (cid:202) (cid:80) ni = U ( g C ) v i is a Harish-Chandra ( g C , K )-module. Since there areonly finitely many irreducible Harish-Chandra ( g C , K )-modules with infinitesimalcharacter χ ([K86, Corollary 10.37]), there is at least one, denoted by V , occurs aninfinite number of times in (cid:80) ni = U ( g C ) v i / (cid:80) n − i = U ( g C ) v i with n (cid:202)
1. If τ ∈ (cid:98) K is a K -type in V , then τ occurs infinitely many times in V χ ( Γ , ρ ), which is in contradictionwith (6.19). (cid:3) The following proposition is an immediate consequence of (6.11) and (6.17).
Proposition 6.6. If ( τ , E τ ) is a representation of K , then we have a finite sumC ∞ λ ( Z , F τ ⊗ F ) = (cid:77) χ : χ ( C g ) = λ (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K .(6.21)We use the notation in Section 1.6. Recall that Rep( Γ , C r ) is the set of all r -dimensional representations of Γ . Proposition 6.7.
There is k ∈ N such that if χ is a character of Z ( g C ) and if ρ ∈ Rep( Γ , C r ) is such that V χ ( Γ , ρ ) = , then there is (cid:178) > such that if ρ ∈ Rep( Γ , C r ) is C k (cid:178) -close to ρ , then V χ ( Γ , ρ ) = Proof.
We fix a character χ of Z ( g C ). Let E τ be the K -representation constructedin Proposition 6.3, so that (6.22) is equivalent to (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K = Z ( g C ) is a polynomial algebra of dim h variables. Let z = C g , z , . . . , z dim h be a set of generators. If z ∈ Z ( g C ), denote by z Z , τ , ρ theaction of z on C ∞ ( Z , F τ ⊗ F ρ ). Clearly, z Z , τ , ρ is a differential operator. Recallthat we have choosen a Hermitian metric on F ρ and it induces an L -product on C ∞ ( Z , F τ ⊗ F ρ ). Take d ∈ N such that 2 d is strictly larger than the order of thedifferential operators z Z , τ , ρ i with 2 (cid:201) i (cid:201) dim h . Set(6.24) L ρ = (cid:110)(cid:179) C g , Z , τ , ρ − χ ( C g ) (cid:180) ∗ (cid:179) C g , τ , E , ρ − χ ( C g ) (cid:180)(cid:111) d + dim h (cid:88) i = (cid:179) z Z , τ , ρ i − χ ( z i ) (cid:180) ∗ (cid:179) z Z , τ , ρ i − χ ( z i ) (cid:180) ,where ∗ is the adjoint with respect to the L -product. Then L ρ is a self-adjointelliptic differential operator of order 4 d .We claim that (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K = ⇐⇒ ker L ρ = L ρ = dim h (cid:92) i = ker (cid:179) z Z , τ , ρ i − χ ( z i ) (cid:180) ⊂ (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K ,(6.26)which implies the direction =⇒ . For the other direction, if (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K (cid:54)= z Z , τ , ρ i commute and act on the finite dimensional vector space (cid:161) V χ ( Γ , ρ ) ⊗ E τ (cid:162) K , they have a common eigenvector. This means ker L ρ (cid:54)= OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 44
Fix now ρ ∈ Rep( Γ , C r ) such that V χ ( Γ , ρ ) =
0. By (6.25), L ρ is invertible. Asin (1.76), if ρ is in a small C -neighborhood of ρ , we can identity ( F ρ , ∇ F ρ ) with( F ρ , ∇ F ρ + A ρ ). We take g F ρ = g F ρ . Then, L ρ and L ρ act on the same space C ∞ ( Z , F τ ⊗ F ρ ) with the same principal symbol. Set B ρ = L ρ − L ρ .(6.27)Then, B ρ is a differential operator of order 4 d −
1. By (1.77), there is C > (cid:107) B ρ (cid:107) H d − , L (cid:201) C (cid:107) A ρ (cid:107) C d − .(6.28)By (6.27), since L ρ is invertible, we have L ρ = (cid:179) + B ρ L − ρ (cid:180) L ρ .(6.29)Take k = d −
1. By (6.28), if (cid:107) A ρ (cid:107) C k is small enough, then (cid:107) B ρ L − ρ (cid:107) L , L < L ρ is invertible. By (6.25), we get (6.23). (cid:3) We have an analogue [Sh18, (8-68)].
Theorem 6.8.
Let ρ ∈ Rep( Γ , C r ) be an acyclic and unitary representation of Γ .There is (cid:178) > such that if ρ is C k (cid:178) -close to ρ , thenV χ ( Γ , ρ ) = Proof.
In [Sh18, Proposition 8.12], using fundamental results of Vogan-Zuckerman[VoZu84], Vogan [Vo84], and Salamanca-Riba [SR99], we have shown that ρ isunitary and acyclic, if and only if V χ ( Γ , ρ ) = ρ is close enough to ρ in the sense of C k , we get (6.30). (cid:3) Formulas for r η , ρ and r j . Assume now δ ( G ) = Z G is compact. Let ρ : Γ → GL r ( C ) be a representation of Γ , and let η be a virtual representation of M satisfying Assumption 5.4. Recall that (cid:98) η is defined in (5.14).If V is a Harish-Chandra ( g C , K )-module, denote by H • ( g , K ; V ) the associated( g C , K )-cohomology and by H • ( n , V ) the n -homology. By [HeSc83, Proposition2.24], H • ( n , V ) is a Harish-Chandra ( m C , K M )-module. Let χ ( K / K M ) be the Eu-ler characteristic number of K / K M . Proposition 6.9.
The following identity holds, (6.32) r η , ρ = χ ( K / K M ) (cid:88) χ : χ ( C g ) = (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k (cid:110) dim H i (cid:179) m , K M ; H k (cid:161) n , V χ ( Γ , ρ ) (cid:162) ⊗ E + η (cid:180) − dim H i (cid:179) m , K M ; H k (cid:161) n , V χ ( Γ , ρ ) (cid:162) ⊗ E − η (cid:180) (cid:111) , where the above sum is finite. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 45
Proof.
We claim that it is enough to show that(6.33) dim (cid:179) V χ ( Γ , ρ ) ⊗ E + (cid:98) η (cid:180) K − dim (cid:179) V χ ( Γ , ρ ) ⊗ E − (cid:98) η (cid:180) K = χ ( K / K M ) (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k (cid:110) dim (cid:179) Λ i ( p ∗ m ) ⊗ H k (cid:161) n , V χ ( Γ , ρ ) (cid:162) ⊗ E + η (cid:180) K M − dim (cid:179) Λ i ( p ∗ m ) ⊗ H k (cid:161) n , V χ ( Γ , ρ ) (cid:162) ⊗ E − η (cid:180) K M (cid:111) .Indeed, by (5.16) and (6.21), we have a finite sum r η , ρ = (cid:88) χ : χ ( C g ) = (cid:189) dim (cid:179) V χ ( Γ , ρ ) ⊗ E + (cid:98) η (cid:180) K − dim (cid:179) V χ ( Γ , ρ ) ⊗ E − (cid:98) η (cid:180) K (cid:190) .(6.34)Equation (6.32) follows from (6.33), (6.34), and the Euler formula [Sh18, (8-85)].If V χ ( Γ , ρ ) is unitary and irreducible, then (6.33) is just [Sh18, Theorem 8.14].However, the proof extends to any irreducible ( g C , K )-module without any change.In particular, if V χ ( Γ , ρ ) has a composition series (6.3), for 0 (cid:201) s (cid:201) N , we have(6.35) dim (cid:179) ( V s / V s − ) ⊗ E + (cid:98) η (cid:180) K − dim (cid:179) ( V s / V s − ) ⊗ E − (cid:98) η (cid:180) K = χ ( K / K M ) (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k (cid:110) dim (cid:179) Λ i ( p ∗ m ) ⊗ H k ( n , V s / V s − ) ⊗ E + η (cid:180) K M − dim (cid:179) Λ i ( p ∗ m ) ⊗ H k ( n , V s / V s − ) ⊗ E − η (cid:180) K M (cid:111) .Using the exact sequence of the Harish-Chandra ( g C , K )-modules,0 → V s − → V s → V s / V s − → (cid:179) ( V s / V s − ) ⊗ E ± (cid:98) η (cid:180) K = dim (cid:179) V s ⊗ E ± (cid:98) η (cid:180) K − dim (cid:179) V s − ⊗ E ± (cid:98) η (cid:180) K .(6.37)On the other hand, by (6.36), we have the long exact sequence of Harish-Chandra ( m C , K M )-modules, · · · → H k ( n , V s − ) → H k ( n , V s ) → H k ( n , V s / V s − ) → H k − ( n , V s − ) → · · · (6.38)By (6.38), for 0 (cid:201) i (cid:201) dim p m and 0 (cid:201) s (cid:201) N , we have(6.39) (cid:96) (cid:88) k = ( − k dim (cid:179) Λ i ( p ∗ m ) ⊗ H k ( n , V s / V s − ) ⊗ E ± η (cid:180) K M = (cid:96) (cid:88) k = ( − k (cid:110) dim (cid:179) Λ i ( p ∗ m ) ⊗ H k ( n , V s ) ⊗ E ± η (cid:180) K M − dim (cid:179) Λ i ( p ∗ m ) ⊗ H k ( n , V s − ) ⊗ E ± η (cid:180) K M (cid:111) .By (6.35), (6.37), and (6.39), we get (6.33). (cid:3) Recall that for 0 (cid:201) j (cid:201) (cid:96) , η j is the adjoint representation of M on Λ j ( n ∗ C ). InSection 5.4.2, we have seen that η j satisfies Assumption 5.4. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 46
Proposition 6.10.
If V is an irreducible Harish-Chandra ( g C , K ) -module withinfinitesimal character χ such that χ ( C g ) = and (cid:77) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) H i (cid:179) m , K M ; H k ( n , V ) ⊗ Λ j ( n ∗ C ) (cid:180) (cid:54)= then χ = χ .(6.41) Proof.
The proof given in [Sh18, Proposition 8.17] where V is unitary extends toany irreducible Harish-Chandra ( g C , K )-module without any change. (cid:3) Corollary 6.11.
If V is Harish-Chandra ( g C , K ) -module with generalised infini-tesimal character χ such that χ ( C g ) = and (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k dim H i (cid:179) m , K M ; H k ( n , V ) ⊗ Λ j ( n ∗ C ) (cid:180) (cid:54)= then χ = χ .(6.43) Proof. If V has a composition series (6.3), by (6.39) and the Euler formula [Sh18,(8-85)], we have(6.44) (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k dim H i (cid:179) m , K M ; H k ( n , V ) ⊗ Λ j ( n ∗ C ) (cid:180) = N (cid:88) s = (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k dim H i (cid:179) m , K M ; H k ( n , V s / V s − ) ⊗ Λ j ( n ∗ C ) (cid:180) .By (6.42), (6.44), there is 0 (cid:201) s (cid:201) N such that (cid:77) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) H i (cid:179) m , K M ; H k ( n , V s / V s − ) ⊗ Λ j ( n ∗ C ) (cid:180) (cid:54)= (cid:3) We have a generalisation of [Sh18, Corollary 8.18].
Theorem 6.12.
For (cid:201) j (cid:201) (cid:96) , we haver j = χ ( K / K M ) (cid:88) (cid:201) i (cid:201) dim p m (cid:201) k (cid:201) (cid:96) ( − i + k dim H i (cid:179) m , K M ; H k (cid:161) n , V χ ( Γ , ρ ) (cid:162) ⊗ Λ j ( n ∗ C ) (cid:180) .(6.46) In particular, if V χ ( Γ , ρ ) = , for (cid:201) j (cid:201) (cid:96) , we haver j = Proof.
This is a consequence of Proposition 6.9 and Corollary 6.11. (cid:3)
Remark . By (5.35), Theorems 6.8 and 6.12, we get (4.7) in the case δ ( G ) = Z G is compact. We complete the proof of Theorem 4.3 in full generality. OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 47
7. A
N EXTENSION TO ORBIFOLDS
In this section, we no longer assume that Γ ⊂ G is torsion free. Then Z = Γ \ G / K is a closed orbifold. The purpose of this section is to extend Theorem 4.3 to orb-ifolds.This section is organised as follows. In Section 7.1, we establish Müller’s Sel-berg trace formula for orbifolds.In Section 7.2, we indicate the essential steps in generalising Theorem 4.3 toorbifolds.7.1. Müller’s Selberg trace formula on locally symmetric orbifolds.
Weuse the notation in Section 2.6. Recall that Γ ⊂ G is a discrete cocompact subgroupof G . Then, Z = Γ \ G / K is a closed orbifold. Recall also that ρ : Γ → GL r ( C ) is arepresentation of Γ , and that ( τ , E τ ) is a representation of K . Define F and F τ asin (1.33) and (2.25). Then F is a flat orbifold vector bundle and F τ is a Hermitianorbifold vector bundle. As in the case where Γ is torsion free, the Casimir operator C g induces a generalised Laplacian C g , Z , τ , ρ acting on C ∞ ( Z , F τ ⊗ F ).For γ ∈ Γ , we have seen in Section 2.6 that γ is semisimple. The group K ( γ ) actson the right on Γ ( γ ) \ Z ( γ ). For h ∈ Γ ( γ ) \ Z ( γ ), let K ( γ ) h be the stabiliser of h in K ( γ ). Since Γ ( γ ) \ X ( γ ) is connected, the cardinality of a generic stabiliser K ( γ ) h iswell defined (see [BLa99, Section 3.1]) and depends only on the conjugacy class of γ in Γ . We denote it by n [ γ ] . By [BLa99, (3.10)], we havevol( Γ ( γ ) \ Z ( γ ))vol( K ( γ )) = vol( Γ ( γ ) \ X ( γ )) n [ γ ] .(7.1)By [ShY17, Proposition 5.3] and by (2.19),(2.20), we have n [ γ ] = (cid:175)(cid:175) K ∩ Γ ( γ ) ∩ Z ( p ( γ )) (cid:175)(cid:175) = (cid:175)(cid:175) ker (cid:161) Γ ( γ ) → Diffeo (cid:161) X ( γ ) (cid:162)(cid:162)(cid:175)(cid:175) .(7.2)Recall that [ Γ + ] ⊂ [ Γ ] is the set of non elliptic conjugacy classes of Γ . For [ γ ] ∈ [ Γ ], B [ γ ] is defined in (2.28). We have a generalisation of [ShY17, Theorem 5.4] andTheorem 2.8. Theorem 7.1.
There exist c > , C > such that for t > , we have (cid:88) [ γ ] ∈ [ Γ + ] vol (cid:161) B [ γ ] (cid:162) n [ γ ] (cid:175)(cid:175) Tr[ ρ ( γ )] (cid:175)(cid:175) (cid:175)(cid:175)(cid:175) Tr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105)(cid:175)(cid:175)(cid:175) (cid:201) C exp (cid:179) − ct + Ct (cid:180) .(7.3) For t > , the following identity holds, Tr (cid:104) exp (cid:179) − tC g , Z , τ , ρ (cid:180)(cid:105) = (cid:88) [ γ ] ∈ [ Γ ] vol (cid:161) B [ γ ] (cid:162) n [ γ ] Tr[ ρ ( γ )] Tr [ γ ] (cid:104) exp (cid:179) − tC g , X , τ (cid:180)(cid:105) .(7.4) Proof.
The proof is similar to the one given in [ShY17, Theorem 5.4] and Theorem2.8. (cid:3)
By Theorem 7.1, proceeding as in Corollary 2.11, we get the following corollary.
Corollary 7.2.
The statement of Corollary 2.11 holds for orbifolds.
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 48
The Ruelle zeta functions on locally symmetric orbifolds.
Let us fol-low [ShY17, Section 5.6]. By [ShY17, Remark 5.6, (5.59)], as in (4.1), the closedgeodesics (see [GuHa06] or [ShY17, Remark 2.26]) on the orbifold Z with positivelength are given by (cid:97) [ γ ] ∈ [ Γ + ] B [ γ ] .(7.5)Moreover, if [ γ ] ∈ [ Γ + ], all the elements of B [ γ ] have the same length (cid:96) [ γ ] >
0. Also,the group (cid:83) acts locally freely on the orbifold B [ γ ] by rotation. So, B [ γ ] / (cid:83) is stilla closed orbifold. Set m [ γ ] = n [ γ ] (cid:175)(cid:175) ker (cid:161) (cid:83) → Diffeo (cid:161) B [ γ ] (cid:162)(cid:162)(cid:175)(cid:175) ∈ N ∗ .(7.6)Following [ShY17, Definition 5.10], for Re ( σ ) (cid:192) R ρ ( σ ) = exp (cid:195) (cid:88) [ γ ] ∈ [ Γ + ] χ orb ( B [ γ ] / (cid:83) ) m [ γ ] Tr (cid:163) ρ ( γ ) (cid:164) e − σ(cid:96) [ γ ] (cid:33) ,(7.7)as in (4.5). By [ShY17, Proposition 5.9], the statement of Remark 4.2 still holdsfor orbifolds. Theorem 7.3.
The statements of Theorem 4.3 holds for orbifolds.Proof.
As previous, by Corollary 7.2, we need only consider the case δ ( G ) =
1. If η satisfies Assumption 5.4, we can define the Selberg zeta function by the sameformula (5.13) with m [ γ ] defined in (7.6). Using Theorem 7.1 instead of Theorem2.8, by Remark 1.6, we can deduce that the statements of Theorem 5.10, (5.31),(5.34) still hold when Γ is not torsion free. Moreover, all the results in Section 6hold when Γ is not torsion free. In this way, we get our theorem. (cid:3) R EFERENCES [BHdlVa08] B. Bekka, P. de la Harpe, and A. Valette,
Kazhdan’s property (T) , New MathematicalMonographs, vol. 11, Cambridge University Press, Cambridge, 2008. MR 2415834[BeGeV04] N. Berline, E. Getzler, and M. Vergne,
Heat kernels and Dirac operators , GrundlehrenText Editions, Springer-Verlag, Berlin, 2004, Corrected reprint of the 1992 original.MR 2273508 (2007m:58033)[B11] J.-M. Bismut,
Hypoelliptic Laplacian and orbital integrals , Annals of MathematicsStudies, vol. 177, Princeton University Press, Princeton, NJ, 2011. MR 2828080[BGS88] J.-M. Bismut, H. Gillet, and C. Soulé,
Analytic torsion and holomorphic determinantbundles. I. Bott-Chern forms and analytic torsion , Comm. Math. Phys. (1988), no. 1,49–78. MR 929146 (89g:58192a)[BMaZ11] J.-M. Bismut, X. Ma, and W. Zhang,
Opérateurs de Toeplitz et torsion analytique asymp-totique , C. R. Math. Acad. Sci. Paris (2011), no. 17-18, 977–981. MR 2838248(2012k:58055)[BMaZ17] J.-M. Bismut, X. Ma, and W. Zhang,
Asymptotic torsion and Toeplitz operators , J. Inst.Math. Jussieu (2017), no. 2, 223–349. MR 3615411[BLa99] J.-M. Bismut and F. Labourie, Symplectic geometry and the Verlinde formulas , Surveysin differential geometry: differential geometry inspired by string theory, Surv. Differ.Geom., vol. 5, Int. Press, Boston, MA, 1999, pp. 97–311. MR 1772272 (2001i:53145)[BSh19a] J.-M. Bismut and S. Shen,
Intégrales orbitales semi-simples et centre de l’algèbre en-veloppante , C. R. Math. Acad. Sci. Paris (2019), no. 11-12, 897–906. MR 4038265
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 49 [BSh19b] J.-M. Bismut and S. Shen,
Geometric orbital integrals and the center of the envelopingalgebra , arXiv: 1910.11731 (2019).[BZ92] J.-M. Bismut and W. Zhang,
An extension of a theorem by Cheeger and Müller ,Astérisque (1992), no. 205, 235, With an appendix by François Laudenbach.MR 1185803 (93j:58138)[BWSh20] Y. Borns-Weil and S. Shen,
Dynamical zeta functions in the nonorientable case ,arXiv:2007.08043 (2020).[BrKa07] M. Braverman and T. Kappeler,
Refined analytic torsion as an element of the determi-nant line , Geom. Topol. (2007), 139–213. MR 2302591[BrKa08] M. Braverman and T. Kappeler, Refined analytic torsion , J. Differential Geom. (2008), no. 2, 193–267. MR 2394022[BrDi85] T. Bröcker and T. tom Dieck, Representations of compact Lie groups , Graduate Texts inMathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344 (86i:22023)[BuH07] D. Burghelea and S. Haller,
Complex-valued Ray-Singer torsion , J. Funct. Anal. (2007), no. 1, 27–78. MR 2329682[BuH08] D. Burghelea and S. Haller,
Torsion, as a function on the space of representations , C ∗ -algebras and elliptic theory II, Trends Math., Birkhäuser, Basel, 2008, pp. 41–66.MR 2408135[BuH10] D. Burghelea and S. Haller, Complex valued Ray-Singer torsion II , Math. Nachr. (2010), no. 10, 1372–1402. MR 2744135[CMil10] S.E. Cappell and E.Y. Miller,
Complex-valued analytic torsion for flat bundles and forholomorphic bundles with (1,1) connections , Comm. Pure Appl. Math. (2010), no. 2,133–202. MR 2588459[Ch79] J. Cheeger, Analytic torsion and the heat equation , Ann. of Math. (2) (1979), no. 2,259–322. MR 528965 (80j:58065a)[DGRSh20] N. V. Dang, C. Guillarmou, G. Rivière, and S. Shen,
The Fried conjecture in smalldimensions , Invent. Math. (2020), no. 2, 525–579. MR 4081137[DaY17] X. Dai and J. Yu,
Comparison between two analytic torsions on orbifolds , Math. Z. (2017), no. 3-4, 1269–1282. MR 3623749[dR50] G. de Rham,
Complexes à automorphismes et homéomorphie différentiable , Ann. Inst.Fourier Grenoble (1950), 51–67 (1951). MR 0043468 (13,268c)[DuKVa79] J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan, Spectra of compact locallysymmetric manifolds of negative curvature , Invent. Math. (1979), no. 1, 27–93.MR 532745 (82a:58050a)[DyZw16] S. Dyatlov and M. Zworski, Dynamical zeta functions for Anosov flows via microlocalanalysis , Ann. Sci. Éc. Norm. Supér. (4) (2016), no. 3, 543–577. MR 3503826[DyZw17] S. Dyatlov and M. Zworski, Ruelle zeta function at zero for surfaces , Invent. Math. (2017), no. 1, 211–229. MR 3698342[E96] P. B. Eberlein,
Geometry of nonpositively curved manifolds , Chicago Lectures in Math-ematics, University of Chicago Press, Chicago, IL, 1996. MR 1441541[FaT00] M. Farber and V. Turaev,
Poincaré-Reidemeister metric, Euler structures, and torsion ,J. Reine Angew. Math. (2000), 195–225. MR 1748274[Fra35] W. Franz,
Über die Torsion einer Überdeckung. , J. Reine Angew. Math. (1935),245–254 (German).[F86] D. Fried,
Analytic torsion and closed geodesics on hyperbolic manifolds , Invent. Math. (1986), no. 3, 523–540. MR 837526 (87g:58118)[F87] D. Fried, Lefschetz formulas for flows , The Lefschetz centennial conference, Part III(Mexico City, 1984), Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987,pp. 19–69. MR 893856 (88k:58138)[FrN15] S. Friedl and M. Nagel, , Int. Math. Res. Not.IMRN (2015), no. 24, 13360–13378. MR 3436149[GiLPo13] P. Giulietti, C. Liverani, and M. Pollicott,
Anosov flows and dynamical zeta functions ,Ann. of Math. (2) (2013), no. 2, 687–773. MR 3071508
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 50 [GolM88] W.M. Goldman and J.J. Millson,
The deformation theory of representations of funda-mental groups of compact Kähler manifolds , Inst. Hautes Études Sci. Publ. Math.(1988), no. 67, 43–96. MR 972343[GoKr69] I. C. Gohberg and M. G. Kre˘ın,
Introduction to the theory of linear nonselfadjoint opera-tors , Translated from the Russian by A. Feinstein. Translations of Mathematical Mono-graphs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142[GuHa06] K. Guruprasad and A. Haefliger,
Closed geodesics on orbifolds , Topology (2006),no. 3, 611–641. MR 2218759[HeSc83] H. Hecht and W. Schmid, Characters, asymptotics and n-homology of Harish-Chandramodules , Acta Math. (1983), no. 1-2, 49–151. MR 716371 (84k:22026)[Hi74] N. Hitchin,
Harmonic spinors , Advances in Math. (1974), 1–55. MR 0358873 (50 Representation theory of semisimple groups , Princeton Mathematical Se-ries, vol. 36, Princeton University Press, Princeton, NJ, 1986, An overview based onexamples. MR 855239 (87j:22022)[K02] A. W. Knapp,
Lie groups beyond an introduction , second ed., Progress in Mathematics,vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)[Ma05] X. Ma,
Orbifolds and analytic torsions , Trans. Amer. Math. Soc. (2005), no. 6,2205–2233 (electronic). MR 2140438[Ma19] X. Ma,
Geometric hypoelliptic Laplacian and orbital integrals [after Bismut, Lebeauand Shen] , no. 407, 2019, Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120–1135,pp. Exp. No. 1130, 333–389. MR 3939281[MaMar07] X. Ma and G. Marinescu,
Holomorphic Morse inequalities and Bergman kernels ,Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007. MR 2339952(2008g:32030)[MaMar15] X. Ma and G. Marinescu,
Exponential estimate for the asymptotics of Bergman kernels ,Math. Ann. (2015), no. 3-4, 1327–1347. MR 3368102[Mi68a] J. Milnor,
Infinite cyclic coverings , Conference on the Topology of Manifolds (MichiganState Univ., E. Lansing, Mich., 1967), Prindle, Weber & Schmidt, Boston, Mass., 1968,pp. 115–133. MR 0242163 (39
A note on curvature and fundamental group , J. Differential Geometry (1968), 1–7. MR 0232311[MoSt91] H. Moscovici and R. J. Stanton, R-torsion and zeta functions for locally symmetric man-ifolds , Invent. Math. (1991), no. 1, 185–216. MR 1109626 (92i:58199)[Mü78] W. Müller,
Analytic torsion and R-torsion of Riemannian manifolds , Adv. in Math. (1978), no. 3, 233–305. MR 498252 (80j:58065b)[Mü93] W. Müller, Analytic torsion and R-torsion for unimodular representations , J. Amer.Math. Soc. (1993), no. 3, 721–753. MR 1189689 (93m:58119)[Mü11] W. Müller, A Selberg trace formula for non-unitary twists , Int. Math. Res. Not. IMRN(2011), no. 9, 2068–2109. MR 2806558[Mü20] W. Müller,
On Fried’s conjecture for compact hyperbolic manifolds , arXiv:2005.01450(2020).[MüPf13] W. Müller and J. Pfaff,
Analytic torsion and L -torsion of compact locally symmetricmanifolds , J. Differential Geom. (2013), no. 1, 71–119. MR 3128980[Pa83] A. Pazy, Semigroups of linear operators and applications to partial differential equa-tions , Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.MR 710486[Q85] D. Quillen,
Superconnections and the Chern character , Topology (1985), no. 1, 89–95. MR 790678 (86m:58010)[RaSi71] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds ,Advances in Math. (1971), 145–210. MR 0295381 (45 On the finite-dimensional unitary representations of Kazhdan groups ,Proc. Amer. Math. Soc. (1999), no. 5, 1557–1562. MR 1476387
OMPLEX VALUED ANALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 51 [Re35] K. Reidemeister,
Homotopieringe und Linsenräume , Abh. Math. Sem. Univ. Hamburg (1935), no. 1, 102–109. MR 3069647[SR99] S. A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and theA q ( λ ) modules: the strongly regular case , Duke Math. J. (1999), no. 3, 521–546.MR 1671213 (2000a:22023)[Sa57] I. Satake, The Gauss-Bonnet theorem for V -manifolds , J. Math. Soc. Japan (1957),464–492. MR 0095520 (20 Harmonic analysis and discontinuous groups in weakly symmetric Rie-mannian spaces with applications to Dirichlet series , J. Indian Math. Soc. (N.S.) (1956), 47–87. MR 0088511[Se60] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces , Contri-butions to function theory (internat. Colloq. Function Theory, Bombay, 1960), TataInstitute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324[Sh18] S. Shen,
Analytic torsion, dynamical zeta functions, and the Fried conjecture , Anal.PDE (2018), no. 1, 1–74. MR 3707290[Sh19] S. Shen, Analytic torsion and dynamical flow: a survey on the Fried conjecture .[ShY17] S. Shen and J. Yu,
Flat vector bundles and analytic torsion on orbifolds , arXiv:1704.08369 (2017). To appear in Comm. Anal. Geom.[ShY18] S. Shen and J. Yu,
Morse-Smale flow, Milnor metric, and dynamical zeta function ,arXiv: 1806.00662 (2018).[S01] M. A. Shubin,
Pseudodifferential operators and spectral theory , second ed., Springer-Verlag, Berlin, 2001, Translated from the 1978 Russian original by Stig I. Andersson.MR 1852334[Sp18] P. Spilioti,
Selberg and Ruelle zeta functions for non-unitary twists , Ann. Global Anal.Geom. (2018), no. 2, 151–203. MR 3766581[Sp20a] P. Spilioti, The functional equations of the Selberg and Ruelle zeta functions for non-unitary twists , arXiv:1507.05947. To appear Ann. Global Anal. Geom. (2020)[Sp20b] P. Spilioti,
Twisted Ruelle zeta function and complex-valued analytic torsion ,arXiv:2004.13474 (2020).[SuZ08] G. Su and W. Zhang,
A Cheeger-Müller theorem for symmetric bilinear torsions , Chin.Ann. Math. Ser. B (2008), no. 4, 385–424. MR 2429629[T89] V. G. Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions ,Izv. Akad. Nauk SSSR Ser. Mat. (1989), no. 3, 607–643, 672. MR 1013714[Vo84] D. A. Vogan, Jr., Unitarizability of certain series of representations , Ann. of Math. (2) (1984), no. 1, 141–187. MR 750719 (86h:22028)[VoZu84] D. A. Vogan, Jr. and G. J. Zuckerman,
Unitary representations with nonzero cohomol-ogy , Compositio Math. (1984), no. 1, 51–90. MR 762307 (86k:22040)[W82] E. Witten, Supersymmetry and Morse theory , J. Differential Geom. (1982), no. 4,661–692 (1983). MR 683171 (84b:58111)[Yo95] K. Yosida, Functional analysis , Classics in Mathematics, Springer-Verlag, Berlin,1995, Reprint of the sixth (1980) edition. MR 1336382 (96a:46001)I
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