Analytic torsion, dynamical zeta function, and the Fried conjecture for admissible twists
aa r X i v : . [ m a t h . DG ] S e p ANALYTIC TORSION, DYNAMICAL ZETA FUNCTION, AND THEFRIED CONJECTURE FOR ADMISSIBLE TWISTS
SHU SHENA
BSTRACT . We show an equality between the analytic torsion and the absolutevalue at the zero point of the Ruelle dynamical zeta function on a closed odddimensional locally symmetric space twisted by an acyclic flat vector bundle ob-tained by the restriction of a representation of the underlying Lie group. Thisgeneralises author’s previous result for unitarily flat vector bundles, and the re-sults of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds. C ONTENTS
Introduction 21. The analytic torsion 92. Reductive groups and finite dimensional representations 102.1. Real reductive groups 112.2. The Casimir operator 122.3. The Dirac operator 122.4. Semisimple elements 132.5. The fundamental Cartan subalgebra 132.6. A splitting of g according to the b -action 142.7. Admissible metrics 153. The zeta functions of Ruelle and Selberg 163.1. Symmetric spaces 163.2. Locally symmetric spaces 173.3. The Ruelle zeta function 183.4. The Selberg zeta function 194. The Fried conjecture and admissible metrics 214.1. Hermitian metrics on flat vector bundles 214.2. The statement of the main result 224.3. Proof of Corollary 0.2 224.4. Proof of Theorem 0.1 234.5. Proof of Corollary 0.3 234.6. Proof of Theorem 4.4 when δ ( G ) = Z G is noncompact 245. The proof of (4.8) when δ ( G ) = Z G is compact 265.1. The structure of the reductive group G with δ ( G ) = ρ according to the b -action 275.3. The representation η β Date : September 11, 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. Z η β r η β H • ( Z , F ) 336.3. A formula for r η β NTRODUCTION
The purpose of this article is to study the relation between the analytic torsionand the value at the zero point of the Ruelle dynamical zeta function associated toa flat vector bundle, which is not necessarily unitary, on a closed odd dimensionallocally symmetric space of reductive type.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 sheaf of locally constant sections of F . Weassume H • ( Z , F ) = g T Z , g F be metrics on T Z and F . The analytic torsion T ( F ) of Ray-Singer[RaSi71] is a spectral invariant defined by a weighted product of zeta regulariseddeterminants the Hodge Laplacian associated with g T Z , g F . When dim Z is odd,they showed that T ( F ) does not depend on the metric data.Ray and Singer [RaSi71] conjectured, which was proved later by Cheeger [Ch79]and Müller [Mü78], that if F is unitarily flat (i.e., the holonomy representationof F is unitary) the analytic torsion coincides with its topological counterpart,the Reidemeister torsion [Re35, Fr35, dR50]. Bismut-Zhang [BZ92] and Müller[Mü93] simultaneously considered generalisations of this result. Müller [Mü93]extended his result to odd dimensional oriented manifolds where only det F is re-quired to be unitary. Bismut and Zhang [BZ92] generalised the original Cheeger-Müller theorem to arbitrary flat vector bundles with arbitrary Hermitian metricson a manifold with arbitrary dimension orientable or not.Milnor [Mi68] initiated the study of the relation between the torsion invariantand a dynamical system. When Z is an orientable hyperbolic manifold, Fried[F86a, F86b] showed an identity between the analytic torsion of an acyclic uni-tarily flat vector bundle and the value at the zero point of the Ruelle dynamicalzeta function of the geodesic flow of Z . He conjectured [F87, p. 66, Conjecture]that similar results should hold true for more general flows. In [Sh18], followingprevious contributions by Moscovici-Stanton [MoSt91], using Bismut’s orbital in-tegral formula [B11], the author affirmed the Fried conjecture for geodesic flowson closed odd dimensional locally symmetric manifolds equipped with an acyclicunitarily flat vector bundle. In [ShY17], the authors made a further generalisa-tion to closed locally symmetric orbifolds. We refer the reader to [Ma19] for anintroduction to the technique used in [Sh18]. The case of even dimension is trivial [Sh19, Remark 5.12] (c.f. Remark 4.5).
NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 3
When the flat vector bundle is not unitary, Müller [Mü20] and Spilioti [Sp18,Sp20a, Sp20b] related the leading coefficients of the Laurent series of the Ruelledynamical zeta function at the zero point to a weighted product of zeta regulariseddeterminants of the flat Laplacian of Cappell-Miller [CMi10] on orientable odd di-mensional hyperbolic manifolds. When the flat vector bundle is near to an acyclicand unitary one, the authors have shown that the Ruelle dynamical zeta functionis regular at the zero point and its value is equal to the complexed valued analytictorsion of Cappell-Miller [CMi10]. In [Sh20], we generalised the above results toodd dimensional locally symmetric spaces.In this article, we prove the Fried conjecture on odd dimensional locally sym-metric spaces for a class of flat vector bundles, which is not necessarily close toa unitary one, and whose holonomy representations are the restrictions of rep-resentations of the underlying reductive groups. This generalises the previousresults of Bröcker [Brö98], Müller [Mü12], and Wotzke [Wo08] on orientable odddimensional hyperbolic manifolds.We refer the reader to [ShY18, DaGRSh20] for the Fried conjecture for theMorse-Smale flow and the Anosov flow, to [Sh19] for a survey on the Fried conjec-ture.Now, we will describe our results in more detail, and explain the techniquesused in their proofs.0.1. The analytic torsion.
Let Z be a smooth closed manifold, and let F be acomplex flat vector bundle on Z .Let g T Z be a Riemannian metric on
T Z , and let g F be a Hermitian metric on F .To g T Z and g F , we can associate an L -metric on Ω • ( Z , F ), the space of differentialforms with values in F . Let (cid:3) Z be the Hodge Laplacian acting on Ω • ( Z , F ). ByHodge theory, we have a canonical isomorphismker (cid:3) Z ≃ H • ( Z , F ).(0.1)The analytic torsion T ( F ) is a positive real number defined by the followingweighted product of the zeta regularised determinants (see Section 1) T ( F ) = dim Z Y i = det ³ (cid:3) Z | Ω i ( Z , F ) ∩ ( ker (cid:3) Z ) ⊥ ´ ( − i i /2 .(0.2)By [RaSi71] and [BZ92, Theorem 0.1], if dim Z is odd and if H • ( Z , F ) =
0, then T ( F ) is independent of g T Z and g F . Therefore, it is a topological invariant.When Z is a closed orbifold, the analytic torsion is still well defined [Ma05,DaiY17]. In [ShY17, Corollary 4.9], the authors show that if Z as well as all thesingular strata have odd dimension, then the analytic torsion of an acyclic orbifoldflat vector bundle is still a topological invariant.0.2. The Ruelle dynamical zeta function.
Let us recall the definition of theRuelle dynamical zeta function associated to a geodesic flow introduced by Fried[F87, Section 5] (see also [Sh19, Section 2]). By Margulis’ super-rigidity [M91, Section VII.5] (see also [BoW00, Section XIII.4.6]), this isthe most interesting case, when the real rank of the locally symmetric space is ≥ NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 4
Let ( Z , g T Z ) be a connected manifold with nonpositive sectional curvature. Let Γ be the fundamental group of Z , and let [ Γ ] be the set of the conjugacy classes of Γ . For [ γ ] ∈ [ Γ ] , let B [ γ ] be the set of closed geodesics in the free homotopy classassociated to [ γ ]. It is easy to see that all the elements in B [ γ ] have the samelength ℓ [ 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 T Z ) has a negative sectionalcurvature or if Z is locally symmetric. If γ
1, the group S acts locally freely on B [ γ ] by rotation, so that B [ γ ] / S is an orbifold. Let χ orb ( B [ γ ] / S ) ∈ Q be the orbifoldEuler characteristic [Sa57]. Denote by m [ γ ] = ¯¯ ker ¡ S → Diff( B [ γ ] ) ¢¯¯ ∈ N ∗ (0.3)the multiplicity of a generic element in B [ γ ] . Let ǫ [ γ ] = ± Z is locally symmetric, then ǫ [ γ ] = r ∈ N , let ρ : Γ → GL r ( C ) be a representation of Γ . The formal dynamical zetafunction is defined for σ ∈ C by R ρ ( σ ) = exp à X [ γ ] ∈ [ Γ + ] ǫ [ γ ] Tr[ ρ ( γ )] χ orb ( B [ γ ] / S ) m [ γ ] e − σℓ [ γ ] ! ,(0.4)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 ) ≫ σ ∈ C .If ( Z , g T Z ) has negative sectional curvature, the geodesic flow on the spherebundle of ( Z , g T Z ) is Anosov. In this case, if ρ is a trivial representation, R ρ ( σ )has been shown to be well defined by Giulietti-Liverani-Pollicott [GiLPo13] andDyatlov-Zworski [DyZ16]. For general ρ , the proof of the meromorphic extensionof R ρ is not particularly difficult. For behaviour of the Ruelle zeta function near σ =
0, we refer the reader to the work of Dyatlov and Zworski [DyZ17], Dang,Guillarmou, Rivière, and Shen [DaGRSh20], as well as Borns-Weil and Shen[BWSh20].0.3.
Results of Fried, Bröcker, Wotzke, and Müller on hyperbolic mani-folds.
Assume that Z is an odd dimensional connected orientable closed hyper-bolic manifold. Let F be the unitarily flat vector bundle on Z with holonomy ρ : Γ → U( r ).Using the Selberg trace formula, Fried [F86a, Theorem 3] showed that thereexist explicit constants C ρ ∈ R ∗ and r ρ ∈ Z such that as σ → R ρ ( σ ) = C ρ T ( F ) σ r ρ + O ( σ r ρ + ).(0.5)Moreover, if H • ( Z , F ) =
0, then C ρ = r ρ = R ρ (0) = T ( F ) .(0.7) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 5
When ρ is not unitary but a restriction of a representation of the orientationpreserving isometric group of Z , similar results (see Theorem 0.1) have beenshown by Bröcker [Brö98] and Wotzke [Wo08], as well as Müller [Mü12, Theo-rem 1.5].0.4. The main result of the article.
Let G be a linear connected real reductivegroup [Kn86, p. 3], and let θ be the Cartan involution. Let K be the maximalcompact subgroup of G of the points of G that are fixed by θ . Let k and g be theLie algebras of K and G , and let g = p ⊕ k be the Cartan decomposition. Let B bea nondegenerate bilinear symmetric form on g which is invariant under G and θ .Assume that B is positive on p and negative on k . Set X = G / K . Then B | p inducesa Riemannian metric on X , which has nonpositive sectional curvature.Let Γ ⊂ G be a discrete torsion free cocompact subgroup of G . Let Z = Γ \ X bethe associated locally symmetric manifold, which is equipped with the inducedRiemannian metric g T Z . Let ρ : Γ → GL( E ) be a finite dimensional complex repre-sentation of Γ . Let F be the associated flat vector bundle on Z . In [Sh20, Theorem0.1 i)], we have shown that if dim Z is odd, the Ruelle zeta function R ρ has ameromorphic extension to C .Suppose now that ρ extends to a representation of G , which is still denoted by ρ . We assume also that E has an admissible metric 〈 , 〉 E , i.e., p acts symmetricallyand k acts antisymmetrically on ( E , 〈 , 〉 E ). By a construction due to Matsushima-Murakami [MatMu63], 〈 , 〉 E induces canonically a Hermitian metric g F on F (seealso Section 4.1).Let T ( F ) be the analytic torsion of F associated to ( g T Z , g F ). The followingtheorem generalises [Sh18, Theorem 0.1] where ρ is assumed to be unitary, andBröcker [Brö98], Müller [Mü12], and Wotzke [Wo08] where Z is hyperbolic. Theorem 0.1.
Assume that dim
Z is odd and that ρ : G → GL( E ) is a finite dimen-sional complex representation of G with an admissible metric. Let ( F , g F ) be theassociated Hermitian flat vector bundle. Then there exist constants C ρ ∈ C ∗ andr ρ ∈ Z such that when σ → , we haveR ρ ( σ ) = C ρ T ( F ) σ r ρ + O ( σ r ρ + ).(0.8) Moreover, if H • ( Z , F ) = , then ¯¯ C ρ ¯¯ = r ρ = so that ¯¯ R ρ (0) ¯¯ = T ( F ) .(0.10)Set ρ θ = ρ ◦ θ . Then ρ θ is still a representation of G with an admissible metric. If ρ ≃ ρ θ , we will show in Theorem 4.4 that the constant C ρ ∈ R ∗ and we can removethe absolute value in (0.9) and (0.10). For general ρ , by [Sh20, Theorem 0.1 ii)iii)], the argument of C ρ is determined by the argument of a (0.2)-like product ofzeta regularised determinants of the flat Laplacian of Cappell and Miller, whichis related to the complex valued analytic torsion of Cappell and Miller [CMi10]. If G is semisimple or more generally if G has a compact centre, then all the representations of G has an admissible metric ([MatMu63, Lemma 3.1], Proposition 2.9). NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 6
When ρ is irreducible and ρ ρ θ , thanks to the vanishing of the cohomology H • ( Z , F ) [BoW00, Theorem VII.6.7], we have the following corollary. Corollary 0.2.
Assume that dim
Z is odd and that ρ : G → GL( E ) is a finite dimen-sional complex representation of G with an admissible metric. Let ( F , g F ) be theassociated Hermitian flat vector bundle. Suppose that ρ is irreducible and ρ ρ θ .Then, R ρ ( σ ) is holomorphic at σ = so that ¯¯ R ρ (0) ¯¯ = T ( F ) .(0.11)In Section 4.5, we will show that the unitarily flat vector bundle is contained inthe class specified above. Indeed, let ρ : Γ → U( r ) be the holonomy representationof a unitarily flat vector bundle ( F , g F ). If G = G × U( r ), K = K × U( r ), and if Γ ⊂ Γ × U( r ) is the graph of ρ , then we have the identification Z = Γ \ G / K . If ρ : G → U( r ) is the projection onto the second component, it is easy to see that ρ has an admissible metric and the associated Hermitian flat vector bundle is just( F , g F ). In this way, we show : Corollary 0.3.
Assume that dim
Z is odd. If ( F , g F ) is a unitarily flat vectorbundle, and if ( F , g F ) is the Hermitian flat vector bundle as in Theorem 0.1, thenthe statements of Theorem 0.1 and Corollary 0.2 hold for F ⊗ F.Remark . The flat vector bundle F ⊗ F in Corollary 0.3 is of particular interestin study of hyperbolic volumes (see e.g. [BéDHP19]).In Section 7, we will extend all the above results to the case where Γ is nottorsion free. Then Z is an orbifold and F is a flat orbifold vector bundle. Theorem 0.5.
The statement of Theorem 0.1 and Corollaries 0.2 and 0.3 hold fororbifolds.
Proof of Theorem 0.1.
We will first show Theorem 0.1 in the case ρ θ ≃ ρ ,i.e., Theorem 4.4. Using an easy relation R ρ θ ( σ ) = R ρ ( σ ) (Proposition 3.4) andapplying Theorem 4.4 to ρ ⊕ ρ θ , we obtain Corollary 0.2. Since ρ has an admissiblemetric, ρ can be decomposed as a direct sum of irreducible representations whichare either ρ θ ≃ ρ or ρ θ ρ . In this way, we get Theorem 0.1 in full generality.Our proof of Theorem 0.1 in the case ρ θ ≃ ρ is inspired by [Sh18] and [Mü12].Let us explain main steps.0.5.1. Moscovici-Stanton’s vanishing theorem.
Let δ ( G ) ∈ N be the fundamentalrank of G , i.e., the difference between the complex ranks of G and K . Note that δ ( G ) and dim Z have the same parity.When F is unitary, by [MoSt91, Corollary 2.2, Remark 3.7], if δ ( G ) Ê
3, we have T ( F ) = R ρ ( σ ) ≡ δ ( G ) = NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 7
Selberg zeta functions.
Assume now δ ( G ) = ρ θ ≃ ρ . The proof of The-orem 0.1 in this case is based on the introduction of the Selberg zeta functions.Let us recall its definition and basic properties.Let t ⊂ k be a Cartan subalgebra of k . Let h ⊂ g be the stabiliser of t in g . By[Kn86, p. 129], h ⊂ g is a θ -invariant fundamental Cartan subalgebra of g . Let h = b ⊕ t be the Cartan decomposition of h . Note that dim 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 ,(0.13)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 .Let η = η + − η − be a virtual representation of M acting on the finite dimensionalcomplex vector spaces E η = E + η − E − η such thati) the Casimir of M acts on η ± by the same scalar;ii) the restriction of η to K M = K ∩ M lifts uniquely to a virtual representation of K .The Selberg zeta function associated to η is defined formally for σ ∈ C by Z η ( σ ) = exp µ − X [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H χ orb ¡ B [ γ ] / S ¢ m [ γ ] Tr E η s [ k − ] ¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ e − σℓ [ γ ] ¶ ,(0.14)where the sum is taken over the non elliptic conjugacy classes [ γ ] of Γ such that γ can be conjugate by element of G into the Cartan subgroup H .In [Sh18, Section 6] and [Sh20, Section 3.4], we have shown that the adjointaction of K M on p m , C lifts uniquely to a virtual representation of K . Let b η = b η + − b η − be the unique virtual representation of K such that b η | K M = Λ • ( p ∗ m , C ) b ⊗ η | K M .(0.15)The Casimir operator of g acts as a generalised Laplacian C g , Z , b η ± on the smoothsections over Z of the locally homogenous vector bundle induced by b η ± (see (3.6)).By the general theory on elliptic differential operators, the regularised determi-nant det ¡ C g , Z , b η ± + σ ¢ is holomorphic on σ ∈ C .In [Sh18, Section 7] and [Sh20, Section 5], we show that Z η ( σ ) has a meromor-phic extension to σ ∈ C . Moreover, up to a multiplication by a non zero entirefunction, Z η ( σ ) is just the graded regularised determinantdet ¡ C g , Z , b η + + σ η + σ ¢ det ¡ C g , Z , b η − + σ η + σ ¢ ,(0.16)where σ η ∈ R is some constant.One of main steps in our proof of Theorem 0.1 is to construct a family of virtual M -representations η β satisfying the above assumptions i) ii), parametrised by NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 8 finite elements β ∈ b ∗ , such that P β ∈ b ∗ e 〈 β , a 〉 Tr s £ η β ¡ k − ¢¤¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ = Tr £ ρ ¡ e a k − ¢¤ ,(0.17)and such that the following identity of virtual K -representations holds, M β ∈ b ∗ b η β = m M i = ( − i − i Λ i ¡ p ∗ C ¢ ⊗ ρ | K .(0.18)Using (0.17), we can write R ρ as an alternating product of Z η β . Thanks to (0.16),we get a relation between the Ruelle zeta function and the Casimir operator,which is the Hodge Laplacian of ( g T Z , g F ) by (0.18) (c.f. Proposition 4.2). In thisway, we get (0.8).0.5.3. Dirac cohomology.
The construction of η β is based on the Dirac cohomology[HuPa06]. Recall that in [Sh18], we have shown that ( g , z ( b )) is a symmetric pair.Let ( u , u ( b )) be the associated compact symmetric pair. The Dirac cohomology H ± D ( ρ ) of the u -representation ρ with respect to the symmetric pair ( u , u ( b )) is a( b C ⊕ m C , K M )-modules. We define η ± β to be the ( m C , K M )-modules such that H ± D ( ρ ) ≃ M β ∈ b ∗ C β ⊠ η ± β ,(0.19)where C β is the one dimensional representation of b such that a ∈ b acts as 〈 β , a 〉 ∈ R . The virtual M -representation of η β is defined by η + β − η − β . Now, theassumption i) , (0.17), and (0.18) are easy consequences of properties of the Diraccohomology. In Section 5.3, we show the assumption ii) as well, so that the Selbergzeta function of η β is well defined.Let us remark that the Dirac cohomology is more or less equivalent to the n -cohomology used in [Mü12]. However, Dirac cohomology is closer to the spin con-struction used in [Sh18, Section 6].0.5.4. Infinitesimal character and the vanishing of ( g , K ) -cohomology. The proofof (0.9) is based on a relation between the infinitesimal character and the van-ishing of ( g , K )-cohomology of a unitary Harish-Chandra ( g C , K )-module, whichis due to Vogan-Zuckermann [VZu84], Vogan [V84], and Salamanca-Riba [SR99].The idea of the proof is very similar to the one given in [Sh18, Section 8]. We referthe reader to [Sh18, Section 1H] for an introduction.0.5.5. Final remark.
In the case where G have a noncompact centre, we have n =
0. It is unnecessary to use the Dirac cohomology and the results Vogan,Zuckermann, and Salamanca-Riba. For greater clarity, we single out this casein Section 4.6. More precisely, we need assume that the Casimir of g acts on ρ as a scalar. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 9
The organisation of the article.
This article is organised as follows. InSection 1, we recall the definition of the Ray-Singer analytic torsion of a flat vectorbundle.In Section 2, we introduce the real reductive group G and the admissible met-rics on finite dimension representations of G .In Section 3, we recall the definition and the proprieties of the zeta functions ofRuelle and Selberg established in [Sh18, Sh20].In Section 4, we state our main Theorem 4.4, from which we deduce Theorem0.1, Corollaries 0.2 and 0.3. Also, we prove Theorem 4.4 when δ ( G ) Z G isnon compact.In Sections 5 and 6, we establish Theorem 4.4 when δ ( G ) = Z G is compact.Finally, in Section 7, we extends the previous results to orbifolds and we showTheorem 0.5.0.7. Notation.
Throughout the paper, we use the superconnection formalism of[Q85] and [BeGeVe04, 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 .(0.20)If B is another Z -graded algebra, we denote by A b ⊗ 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 M is a topological group, we will denote by M the connected component of theidentity in M . If V is a real vector space, we will use the notation V C = V ⊗ R C forits complexification. We make the convention that N = {
0, 1, 2, . . . } , N ∗ = {
1, 2, . . . } , R ∗+ = (0, ∞ ). 1. T HE ANALYTIC TORSION
Let Z be a closed smooth manifold of dimension m . Let ( F , ∇ F ) be a flat complexvector bundle on Z with flat connection ∇ F . Let ( Ω • ( Z , F ), d Z ) be the de Rhamcomplex of smooth sections of Λ • ( T ∗ Z ) ⊗ R F on Z . Let H • ( Z , F ) be the de Rhamcohomology.We define the Euler characteristic number χ ( Z , F ) and the derived Euler char-acteristic number χ ′ ( Z , F ) by χ ( Z , F ) = m X i = ( − i dim H i ( Z , F ), χ ′ ( Z , F ) = m X i = ( − i i dim H i ( Z , F ).(1.1)If F is trivial, we write χ ( Z ) and χ ′ ( Z ).Let g T Z be a Riemannian metric on Z . Let g F be a Hermitian metric on F . Themetrics g T Z , g F induce a scalar product 〈 , 〉 Λ • ( T ∗ Z ) ⊗ R F on Λ • ( T ∗ Z ) ⊗ R F . Let 〈 , 〉 L be an L -product on Ω • ( Z , F ) defined for s , s ∈ Ω • ( Z , F ) by 〈 s , s 〉 L = Z z ∈ Z 〈 s ( z ), s ( z ) 〉 Λ • ( T ∗ Z ) ⊗ R F dv Z ,(1.2)where dv Z is the Riemannian volume form of ( Z , g T Z ). NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 10
Let d Z , ∗ be the formal adjoint of d Z . Set (cid:3) Z = h d Z , d Z , ∗ i .(1.3)Then, (cid:3) Z is a second order self-adjoint elliptic differential operator acting on Ω • ( Z , F ). By Hodge theory, we haveker (cid:3) Z = H • ( Z , F ).(1.4)Let ¡ ker (cid:3) Z ¢ ⊥ be the orthogonal vector space to ker (cid:3) Z in Ω • ( X , F ). Then (cid:3) Z acts as an invertible operator on ¡ ker (cid:3) Z ¢ ⊥ . Let ¡ (cid:3) Z ¢ − denote the inverse of (cid:3) Z acting on ¡ ker (cid:3) Z ¢ ⊥ . If 0 É i É m , for s ∈ C and Re ( s ) > m , set θ i ( s ) = Tr s ·³ (cid:3) Z ´ − s | Ω i ( Z , F ) ¸ .(1.5)By [Se67] and [BeGeVe04, Proposition 9.35], θ i ( s ) extends to a meromorphic func-tion of s ∈ C , which is holomorphic at s =
0. The regularised determinant is definedby det ∗ ³ (cid:3) Z | Ω i ( Z , F ) ´ = exp ¡ − θ ′ i (0) ¢ .(1.6)Formally, it is the product of non zero eigenvalues counted with multiplicities. Definition 1.1.
The Ray-Singer analytic torsion [RaSi71] of F is defined by T ( F ) = m Y i = det ∗ ³ (cid:3) Z | Ω i ( Z , F ) ´ ( − i i /2 ∈ R ∗+ .(1.7)By [RaSi71] and [BZ92, Theorem 0.1], if dim Z is odd and if H • ( Z , F ) =
0, then T ( F ) does not depend on the metrics g T Z , g F . It becomes a topological invariant.For 0 É i É m , if σ >
0, the operator σ + (cid:3) Z | Ω i ( Z , F ) does not contain the zerospectrum, we denote its regularised determinant by det ³ σ + (cid:3) Z | Ω i ( Z , F ) ´ . By [Vo87]and [Sh20, Theorem 1.5], the function det ³ σ + (cid:3) Z | Ω i ( Z , F ) ´ extends to a holomorphicfunction of σ ∈ C , whose zeros are located at σ = − λ with order dim ker ³ (cid:3) Z | Ω i ( Z , F ) − λ ´ ,where λ ∈ Sp ³ (cid:3) Z | Ω i ( Z , F ) ´ .Set T ( σ ) = m Y i = det ³ σ + (cid:3) Z | Ω i ( Z , F ) ´ ( − i i .(1.8)Then, T ( σ ) is meromorphic. By (1.1), (1.7), and (1.8), as σ →
0, we have T ( σ ) = T ( F ) σ χ ′ ( Z , F ) + O ( σ χ ′ ( Z , F ) + ).(1.9)2. R EDUCTIVE GROUPS AND FINITE DIMENSIONAL REPRESENTATIONS
The purpose of this section is to recall some basic facts about real reductivegroups and their finite dimensional representations.This section is organised as follows. In Sections 2.1-2.4, we introduce the realreductive group G , its Lie algebra g , the enveloping algebra U ( g ), the Casimiroperator, the Dirac operator, as well as the semisimple elements. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 11
In Sections 2.5 and 2.6, we introduce the fundamental Cartan subalgebra of g and some related constructions. We recall a key lifting proprieties established in[Sh20, Section 3.4].In Section 2.7, we recall the definition of admissible metrics on finite dimen-sional representations of G introduced by [MatMu63]. We show that if G has acompact centre, then all the finite dimensional representations have admissiblemetrics.2.1. Real reductive groups.
Let G be a linear connected real reductive group[Kn86, p. 3], and let θ ∈ Aut( G ) be the Cartan involution. That means G is aclosed connected group of real matrices that is stable under transpose, and θ isthe composition of transpose and inverse of matrices. If g is the Lie algebra of G ,then θ acts as an automorphism on g .Let K ⊂ G be the fixed point set of θ in G . Then K is a compact connectedsubgroup of G , which is a maximal compact subgroup. If k is the Lie algebra of K , then k is the eigenspace of θ associated with the eigenvalue 1. Let p be theeigenspace of θ associated with the eigenvalue −
1, so that we have the Cartandecomposition g = p ⊕ k .(2.1)Set m = dim p , n = dim k .(2.2)By [Kn86, Proposition 1.2], we have the diffeomorphism( Y , k ) ∈ p × K → e Y k ∈ G .(2.3)Let B be a nondegenerate bilinear real symmetric form on g which is invariantunder the adjoint action Ad of G , and also under θ . Then (2.1) is an orthogonalsplitting of g with respect to B . We assume B to be positive-definite on p , andnegative-definite on k . Then, 〈· , ·〉 = − B ( · , θ · ) defines an Ad( K )-invariant scalarproduct on g such that the splitting (2.1) is still orthogonal. We denote by | · | thecorresponding norm.Let Z G ⊂ G be the centre of G with Lie algebra z g ⊂ g . By [Kn86, 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 C = g ⊗ R C be the complexification of g and let u = p− p ⊕ k be the compactform of g . By C -linearity, the bilinear form B extends to a complex symmetricbilinear form on g C . The restriction B | u to u is real and negative-definite.Let U ( g ) and U ( g C ) be the enveloping algebras of g and g C . Let Z ( g ) and Z ( g C ) be respectively the centres of U ( g ) and U ( g C ). Clearly, U ( g C ) = U ( g ) ⊗ R C , Z ( g C ) = Z ( g ) ⊗ R C .(2.6) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 12 If V is a complex vector space, and if ρ : g → End( V ) is a representation of g ,then the map ρ extends to a morphism ρ : U ( g C ) → End( V ) of algebras.If ρ : G → GL( V ) is a finite dimensional complex representation of G , then theinduced morphism ρ : U ( g C ) → End( V ) of algebras is K -equivalent. In this way, V is a finite dimensional ( g C , K )-module, i.e., a U ( g C )-module, equipped with acompatible K -action. By [KnV95, Proposition 4.46], it is equivalent to considerfinite dimensional representations of G and finite dimensional ( g C , K )-modules.In the sequel, we will not distinguish these two objets. Remark . If ρ : G → GL( V ) is a finite dimensional complex representation of G , and if W ⊂ V is a g -invariant subspace, by taking the exponential of theaction of g , we see that the group G preserves W . In particular, the set of g -subrepresentations of ρ coincides with the set of G -subrepresentations of ρ .2.2. The Casimir operator.
Let C g ∈ Z ( g ) be the Casimir element associatedto B . If e , · · · , e m is an orthonormal basis of ( p , B | p ), and if e m + , · · · , e m + n is anorthonormal basis of ( k , − B | k ), then C g = − m X i = e i + n + m X i = m + e i .(2.7)If ρ : g → End( V ) is a complex representation of g , we denote by C g , V or C g , ρ ∈ End( V ) the corresponding Casimir operator acting on V , i.e., C g , V = C g , ρ = ρ ( C g ).(2.8)Similarly, the Casimir of u (with respect to B ) acts on V , so that C u , V = C g , V .(2.9)2.3. The Dirac operator.
Let c ( p ) be the Clifford algebra of ( p , B | p ). That isan algebra over R generated by 1 ∈ R , a ∈ p with the commutation relation for a , a ∈ p , a a + a a = − B ( a , a ).(2.10)Let S p be the spinor of ( p , B | p ). If a ∈ p , the action of a on S p is denoted by c ( a ).If a ∈ k , ad( a ) | p acts as an antisymmetric endomorphism on p . It acts on S p by c ¡ ad( a ) | p ¢ = m X i , j = 〈 [ a , e i ], e j 〉 c ( e i ) c ( e j ).(2.11)Let ρ : g → End( V ) be a complex representation of g . Let D S p ⊗ V be the Diracoperator acting on S p ⊗ V , i.e., D S p ⊗ V = m X i = c ( e i ) ρ ( e i ).(2.12)Recall that k acts on p by adjoint action. The operator C k , p is defined in (2.8). Proposition 2.2.
The following identity of operators on S p ⊗ V holds, ³ D S p ⊗ V ´ = C g , V +
18 Tr h C k , p i − C k , S p ⊗ V .(2.13) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 13
Proof.
This is a consequence of [BoW00, Lemma II.6.11] and of Kostant’s strangeformula [Ko76] (see also [B11, (2.6.11), (7.5.4)]).There is another way proving this result, by imitating the proof of [B11, Theo-rem 7.2.1] which uses Kostant’s cubic Dirac operator [Ko76, Ko97]. (cid:3)
In the sequel, we will also consider the Dirac operator associated to the compactsymmetric pair ( u , k ) with respect to the positive bilinear form − B | u . Using theidentification a ∈ p → p− a ∈ p− p , we can identify c ( p ) with the Clifford algebraof the Euclidean space ( p− p , − B ). Also, S p can be identified with the spinor S p− p of ( p− p , − B ). Then, the Dirac operator D S p− p ⊗ V is just p− D S p ⊗ V . By(2.9) and (2.13), we have − ³ D S p− p ⊗ V ´ = C u , V +
18 Tr h C k , p− p i − C k , S p− p ⊗ V .(2.14)2.4. Semisimple elements. 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.Let γ ∈ G be a semisimple element, i.e., there is g γ ∈ G such that γ = g γ e a k − g − γ and a ∈ p , k ∈ K , Ad( k ) a = a .(2.15)The norm | a | depends only on the conjugacy class of γ in G . Write ℓ [ γ ] = | a | .(2.16)A semisimple element γ is called elliptic, if ℓ [ γ ] = γ is semisimple, by [Kn02, Proposition 7.25], Z ( γ ) is a (possibly non con-nected) reductive group with Cartan involution g γ θ g − γ . Let K ( γ ) ⊂ Z ( γ ) be theassociated maximal compact subgroup of Z ( γ ).2.5. The fundamental Cartan subalgebra.
Let T ⊂ K be a maximal torus of K . Let t ⊂ k be the Lie algebra of T . If N K ( T ) is the normaliser of T in K , let W ( T : K ) = N K ( T )/ T be associated Weyl group.Set b = { a ∈ p : [ a , t ] = } , h = b ⊕ t .(2.17)By [Kn86, p. 129], h is a Cartan subalgebra of g . Let H = Z ( h ) be the associ-ated Cartan subgroup of G . By [Kn86, Theorem 5.22], H is a connected abelianreductive subgroup of G , so that H = exp( b ) × T .(2.18)We will call h and H respectively the fundamental Cartan subalgebra of g and thefundamental Cartan subgroup of G .Recall that complex ranks of G and K are defined respectively by the dimen-sions of Cartan subalgebras of g C and k C . By [BSh19, Theorem 2.3], this is indeed independent of the choice of g γ . NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 14
Definition 2.3.
The fundamental rank δ ( G ) of G is defined by the difference ofcomplex ranks of G and K , i.e., δ ( G ) = dim b .(2.19)Note that m and δ ( G ) have the same parity.2.6. A splitting of g according to the b -action. By [Kn02, Proposition 7.25], Z ( b ) is a (possibly non connected) reductive subgroup of G , so that we have theCartan decomposition z ( b ) = p ( b ) ⊕ k ( b ).(2.20)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 we have the Cartan decomposition m = p m ⊕ k m .(2.21)Let M ⊂ G be the connected Lie subgroup 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 reductivesubgroup of G with maximal compact subgroup K M = M ∩ K .(2.22)Moreover, we have Z ( b ) = exp( b ) × M , z ( b ) = b ⊕ m , p ( b ) = b ⊕ p m , k ( b ) = k m .(2.23)Since h is also a Cartan subalgebra for z ( b ), we have δ ( M ) = 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 ).(2.25)And also p = b ⊕ p m ⊕ p ⊥ ( b ), k = k m ⊕ k ⊥ ( b ), g = b ⊕ m ⊕ z ⊥ ( b ).(2.26)The group K M acts trivially on b . It also acts on p m , p ⊥ ( b ), k m and k ⊥ ( b ), andpreserves the splittings (2.26). Similarly, the groups M and Z ( b ) act trivially on b , act on m , z ⊥ ( b ), and preserves the third splitting in (2.26). Remark . We can define similar objects associated to the action of p− b ⊂ u on u . Let u m and u ( b ) be the compact forms of m and z ( b ). Then, u ( b ) = p− b ⊕ u m .(2.27)Let u ⊥ ( b ) be the orthogonal space of u ( b ) in u . Then, u ⊥ ( b ) = p− p ⊥ ( b ) ⊕ k ⊥ ( b ), u = p− b ⊕ u m ⊕ u ⊥ ( b ).(2.28)Elements of b act on z ⊥ ( b ) with semisimple real eigenvalues. We fix an element f b ∈ b , called positive, such that ad( f b ) | z ⊥ ( b ) is invertible. The choice of f b is ir-relevant. Let n ⊂ z ⊥ ( b ) (resp. n ⊂ z ⊥ ( b )) be the direct sum of the eigenspaces ofad( f b ) | z ⊥ ( b ) associated to the positive (resp. negative) eigenvalues. Then, z ⊥ ( b ) = n ⊕ n , n = θ n .(2.29) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 15
Clearly, Z ( b ) acts on n , n and preserves the first decomposition in (2.29). Proposition 2.5.
The following statements hold.i) The vector spaces n , n ⊂ g are Lie subalgebras of g , which have the same evendimension.ii) The bilinear form B vanishes on n , n and induces a Z ( b ) -isomorphism, n ∗ ≃ n .(2.30) iii) The actions of M on n , n , n ∗ , n ∗ are equivalent. For É j É dim n , we haveisomorphisms of representations of M, Λ j ( n ∗ ) ≃ Λ dim n − j ( n ∗ ).(2.31) iv) The projections on p , k map n , n into p ⊥ m , k ⊥ m isomorphically.v) The actions of K M on n , n , p ⊥ m , k ⊥ m are equivalent.Proof. This is [Sh20, Proposition 3.2, Corollary 3.3] (see also [BSh19, Proposition3.10]). (cid:3)
Let R ( K ) be the representation ring of K . We can identify R ( K ) with the sub-ring of the Ad( K )-invariant smooth functions on K which is generated by thecharacters of finite dimensional complex representations of K .Similarly, we can define R ( T ). The Weyl group W ( T : K ) = N K ( T )/ T acts on R ( T ). By [BrDi85, Proposition VI.2.1], the restriction induces an isomorphism ofrings R ( K ) ≃ R ( T ) W ( T : K ) .(2.32)Since K M and K have the same maximal torus T , the restriction induces aninjective morphism R ( K ) → R ( K M ) of rings. Recall a key result established in[Sh20, Theorem 3.5, Corollary 3.6] (see also [Sh18, Theorem 6.1, Corollary 6.12]). Theorem 2.6.
For i , j ∈ N , the adjoint representations of K M on Λ i ( p ∗ m , C ) and Λ j ( n ∗ C ) have unique lifts in R ( K ) . Admissible metrics.
Let ρ : G → GL( V ) be a finite dimensional complexrepresentation of G . Set ρ θ = ρ ◦ θ .(2.33)Then ρ θ : G → GL( V ) is still a representation of G . Proposition 2.7. If δ ( G ) = , we have an isomorphism of representations of G, ρ ≃ ρ θ .(2.34) Proof.
When δ ( G ) =
0, by [Kn86, Problem XII.10.14], there is k ∈ K , such thatAd( k ) = − p and Ad( k ) = k . By (2.3), for g ∈ G , we have θ ( g ) = k gk − .Therefore, ρ ( k ) : V → V is the required isomorphism (2.34). (cid:3) Definition 2.8.
A Hermitian metric 〈 , 〉 V on V is called admissible, if for u , v ∈ V , Y ∈ p , Y ∈ k , we have ρ ( Y ) u , v ® V = u , ρ ( Y ) v ® V , ρ ( Y ) u , v ® V = − u , ρ ( Y ) v ® V .(2.35) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 16
Assume that V has an admissible metric. If ρ ∗ denotes the anti-dual represen-tation of ρ , the admissible metric induces an isomorphism of G -representations, ρ θ ≃ ρ ∗ .(2.36)If W ⊂ V is a ( g C , K )-submodule of V , then the orthogonal space W ⊥ ⊂ V is stilla ( g C , K )-submodule. Moreover, by restrictions, W and W ⊥ still have admissiblemetrics. In this way, we see that any finite dimensional G -representation with anadmissible metric is completely reducible, i.e., it can be decomposed as a directsum of irreducible G -representations.By [MatMu63, Lemma 3.1], if G is semisimple, any finite dimensional repre-sentation of G has an admissible Hermitian metric. When G is reductive and hasa compact centre, we have a similar result. Proposition 2.9.
If G has a compact centre Z G , then any finite dimensional com-plex representation ρ : G → GL( V ) has an admissible Hermitian metric.Proof. Let G ss ⊂ G be the connected Lie subgroup of G associated to the Lie alge-bra [ g , g ] ⊂ g . By [Kn02, Corollary 7.11], G ss is a closed subgroup of G , which issemisimple and G = G ss · Z G . Let U ss be the compact form of G ss . By Weyl’s The-orem [Kn02, Theorem 4.69], the universal cover e U ss of U ss is still compact. Since e U ss is simply connected, by Weyl’s unitary trick, the group e U ss acts on V which iscompatible with the action of [ g , g ]. Moreover, the e U ss -action commutes with the Z G -action. Thus, the group e U ss × Z G acts on V . Since e U ss × Z G is compact, thereis a e U ss × Z G -invariant Hermitian metric on V , which is the desired admissibleHermitian metric. (cid:3) Remark . If G has a noncompact centre, Proposition 2.9 does not hold. For ex-ample, when G = R , the representation x ∈ R → µ x ¶ ∈ GL ( C ) is not completelyreducible, so it does not have an admissible metric.3. T HE ZETA FUNCTIONS OF R UELLE AND S ELBERG
The purpose of this section is to introduce the zeta functions of Ruelle and ofSelberg on locally symmetric spaces.This section is organised as follows. In Section 3.1, we introduce the symmetricspace X = G / K , the K -principal bundle p : G → X , and a Hermitian vector bundleassociated to a finite dimensional unitary representation of K .In Section 3.2, we introduce a discrete cocompact subgroup Γ ⊂ G of G , the cor-responding locally symmetric space Z = Γ \ X , and a flat vector bundle associatedto a finite dimensional representation of Γ .Finally, in Sections 3.3 and 3.4, we introduce the zeta functions of Ruelle andof Selberg. We recall their properties established in [Sh18, Section 7] and [Sh20,Section 5].We use the notation in Section 2.3.1. Symmetric spaces.
Set X = G / K . Let p : G → X be the natural projection.Then p : G → X is a K -principal bundle. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 17
The group K acts isometrically on p . The tangent bundle of X is given by T X = G × K p .(3.1)By (3.1), the scalar product B | p on p induces a Riemannian metric g T X on X .Classically, ( X , g T X ) has a parallel and nonpositive sectional curvature.Let τ be an orthogonal (resp. unitary) representation of K acting on a finitedimensional Euclidean (resp. Hermitian) space E τ . Set E τ = G × K E τ .(3.2)Then E τ is a Euclidean (resp. Hermitian) vector bundle on X .By (3.2), we have an identification C ∞ ( X , E τ ) = C ∞ ( G , E τ ) K .(3.3)The group G acts on the left on C ∞ ( X , E τ ). Denote by C g , X , τ the Casimir elementof G on C ∞ ( X , E τ ). By (2.7), C g , X , τ is a generalised Laplacian on X in the senseof [BeGeVe04, Definition 2.2], which is self adjoint with respect to the standard L -product (c.f. (1.2)).3.2. Locally symmetric spaces.
Let Γ ⊂ G be a discrete cocompact subgroup of G . By [S60, Lemma 1] (see also [Ma19, Proposition 3.9]), Γ contains only semisim-ple elements. Let Γ e ⊂ Γ be the subset of elliptic elements, and let Γ + = Γ − Γ e bethe subset of nonelliptic elements.The group Γ acts isometrically on the left on X . Take Z = Γ \ X = Γ \ G / K .(3.4)Then, Z is a compact orbifold with Riemannian metric g T Z . We denote by b p : Γ \ G → Z and b π : X → Z the natural projections, so that the diagram G p (cid:15) (cid:15) / / Γ \ G b p (cid:15) (cid:15) X b π / / Z (3.5)commutes.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 consider the case where Γ is nottorsion free.The Γ -action on X lifts to all the homogeneous Euclidean or Hermitian vectorbundles E τ on X constructed in (3.2). Then E τ descends to a Euclidean or Hermit-ian vector bundle on Z , F τ = Γ \ E τ = Γ \ G × K E τ .(3.6)If r ∈ N ∗ , and if ρ : Γ → GL r ( C ) is a representation of Γ , let F be the associatedflat vector bundle on Z , F = Γ \ ( X × C r ).(3.7) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 18
The group Γ acts on C ∞ ( G , E τ ) K , as well as on C r by ρ . We have the identification C ∞ ( Z , F τ ⊗ F ) = ¡ C ∞ ( X , E τ ) ⊗ C r ¢ Γ .(3.8)The Casimir operator C g , X , τ ⊗ id preserves the above invariant space. Its actionon C ∞ ( Z , F τ ⊗ F ) will be denoted by C g , Z , τ , ρ . If ρ is unitary, C g , Z , τ , ρ is self-adjointwith respect to the L -product on C ∞ ( Z , F τ ⊗ F ) induced by the Hermitian metricon E τ , the standard Hermitian metric on C r , as well as the Riemannian volumeof ( Z , g T Z ) (c.f. (1.2)). When ρ is the trivial representation, we denote it by C g , Z , τ .3.3. The Ruelle zeta function.
Let us recall the definition of the Ruelle dynam-ical zeta function introduced by Fried [F87, Section 5].For γ ∈ Γ , set Γ ( γ ) = Z ( γ ) ∩ Γ .(3.9)By [S60, 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 ′ ∈ [ γ ],the locally symmetric spaces Γ ( γ ′ ) \ Z ( γ ′ )/ K ( γ ′ )(3.10)are canonically diffeomorphic, and will be denoted by B [ γ ] .By [DuKVa79, Proposition 5.15], the set of nontrivial closed geodesics on Z consists of a disjoint union a [ γ ] ∈ [ Γ + ] B [ γ ] .(3.11)If [ γ ] ∈ [ Γ + ], all the elements of B [ γ ] have the same length ℓ [ γ ] > γ ] ∈ [ Γ + ], the geodesic flow induces a locally free action of S on B [ γ ] , so that B [ γ ] / S is a closed orbifold. Let χ orb ¡ B [ γ ] / S ¢ ∈ Q be the orbifold Euler character-istic number [Sa57]. We refer the reader to [Sh18, Proposition 5.1] for an explicitformula for χ orb ¡ B [ γ ] / S ¢ . In particular, if δ ( G ) Ê
2, or if δ ( G ) = γ can not beconjugate by an element of G into the fundamental Cartan subgroup H , then χ orb ¡ B [ γ ] / S ¢ = S -action on B [ γ ] is not necessarily effective. Let m [ γ ] = ¯¯ ker ¡ S → Diff( B [ γ ] ) ¢¯¯ ∈ N ∗ (3.13)be the generic multiplicity.Recall that ρ : Γ → GL r ( C ) is a representation of Γ . By [Sh20, (4.4)], there is σ > X [ γ ] ∈ [ Γ + ] ¯¯ χ orb ¡ B [ γ ] / S ¢¯¯ m [ γ ] ¯¯ Tr £ ρ ( γ ) ¤¯¯ e − σ ℓ [ γ ] < ∞ .(3.14) The quantity ℓ [ γ ] depends only on the conjugacy class of γ in G . So they are well defined onthe conjugacy classes of Γ . NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 19
Definition 3.1.
For Re ( σ ) Ê σ , the Ruelle dynamical zeta function is defined by R ρ ( σ ) = exp ³ X [ γ ] ∈ [ Γ + ] χ orb ¡ B [ γ ] / S ¢ m [ γ ] Tr £ ρ ( γ ) ¤ e − σℓ [ γ ] ´ .(3.15) Remark . By (3.12), if δ ( G ) Ê
2, the Ruelle zeta function R ρ ( σ ) is the constantfunction 1. Moreover, if δ ( G ) =
1, then the sum on the right-hand side of (3.15)can be reduced to a sum over [ γ ] ∈ [ Γ + ] such that γ can be conjugate into H . Theorem 3.3. If dim Z is odd, the Ruelle zeta function R ρ has a meromorphicextension to σ ∈ C .Proof. This is [Sh20, Theorem 0.1 i)]. (cid:3)
Let R ρ be the meromorphic function defined for σ ∈ C by R ρ ( σ ) = R ρ ( σ ).(3.16)By [Sh20, Proposition 4.4], we have R ρ ∗ = R ρ , R ρ = R ρ .(3.17)If ρ is a finite dimensional complex representation of G , the restriction ρ | Γ is arepresentation of Γ . We write R ρ = R ρ | Γ to ease the notation. Proposition 3.4. If ρ is a finite dimensional complex representation of G withan admissible metric, then the following identity of meromorphic functions on C holds, R ρ θ = R ρ .(3.18) Proof.
This is a consequence of (2.36) and (3.17). (cid:3)
The Selberg zeta function.
In this subsection, we assume δ ( G ) = K M is defined in (2.22). We have seenthat the morphism R ( K ) → R ( K M ) of rings is injective. Assumption 3.5.
Assume that η = η + − η − is a virtual M-representation on thefinite dimensional complex vector space 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 . Following [Sh18, Definition 7.4] and [Sh20, Defintion 5.7], let us define the Sel-berg zeta function associated to η . Recall that H = exp( b ) × T is the fundamentalCartan subgroup of G . For e a k − ∈ H , we write γ ∼ e a k − ∈ H if there is g γ ∈ G such that γ = g γ e a k − g − γ . By [Sh18, (7-62)], there is σ > X [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H ¯¯ χ orb ¡ B [ γ ] / S ¢¯¯ m [ γ ] e − σ ℓ [ γ ] ¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ < ∞ .(3.19) A more general construction for the Selberg zeta function is given in [Sh20], which is associ-ated to η and to an arbitrary representation of ρ : Γ → GL r ( C ). NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 20
Definition 3.6.
For Re ( σ ) Ê σ , set Z η ( σ ) = exp µ − X [ γ ] ∈ [ Γ + ] γ ∼ e a k − ∈ H χ orb ¡ B [ γ ] / S ¢ m [ γ ] Tr E η s [ k − ] ¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ e − σℓ [ γ ] ¶ .(3.20)Recall that by Corollary 2.6, Λ • ( p ∗ m , C ) has a unique lift in R ( K ). Definition 3.7.
Let b η ∈ R ( K ) be the unique virtual representation of K on E b η = E + b η − E − b η such that the following identity in R ( K M ) holds, E b η | K M = Λ • ( p ∗ m , C ) b ⊗ E η | K M ∈ R ( K M ).(3.21)Let C g , Z , b η be the self adjoint generalised Laplacian acting on C ∞ ( Z , F b η ) intro-duced below (3.8). For λ ∈ C , set m η ( λ ) = dim ker ³ C g , Z , b η + − λ ´ − dim ker ³ C g , Z , b η − − λ ´ .(3.22)Let det gr ³ C g , Z , b η + σ ´ = det ¡ C g , Z , b η + + σ ¢ det ¡ C g , Z , b η − + σ ¢ (3.23)be a graded determinant of C g , Z , b η + σ . As in (1.8), by [Vo87] (see also [Sh20, The-orem 1.5]), the function (3.23) is meromorphic on σ ∈ C . Moreover, its zeros andpoles belong to the set © − λ : λ ∈ Sp ¡ C g , Z , b η ¢ª . If λ ∈ Sp ¡ C g , Z , b η ¢ , the order of the zeroat σ = − λ is m η ( λ ).Following [Sh18, (7-60)] and [Sh20, (5.18), (5.19)], we set σ η =
18 Tr u ⊥ ( b ) h C u ( b ), u ⊥ ( b ) i − C u m , η .(3.24) Remark . When Z G is non compact, by [Sh18, (4-52)], we have G = exp( b ) × M .(3.25)Therefore, u ⊥ ( b ) =
0. By (3.24), we have σ η = − C u m , η .(3.26)Let P η ( σ ) be the odd polynomial defined in [Sh18, (7-61)] and [Sh20, (5.20),Remark 5.9]. Theorem . The Selberg zeta function Z η ( σ ) has a meromorphic extension to σ ∈ C such that the following identity of meromorphic functions on C holds,Z η ( σ ) = det gr ³ C g , Z , b η + σ η + σ ´ exp ¡ vol( Z ) P η ( σ ) ¢ .(3.27) The zeros and poles of Z η ( σ ) belong to the set © ± i p λ + σ η : λ ∈ Sp ¡ C g , Z , b η ¢ª . If λ ∈ Sp ¡ C g , Z , b η ¢ and λ
6= − σ η , the order of zero at σ = ± i p λ + σ η is m η ( λ ) . The order ofzero at σ = is m η ( − σ η ) . Also,Z η ( σ ) = Z η ( − σ ) exp ¡ Z ) P η ( σ ) ¢ .(3.28) Proof.
This is [Sh20, Theorem 5.10] for the trivial twist (c.f. Footnote 7). (cid:3)
NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 21
4. T HE F RIED CONJECTURE AND ADMISSIBLE METRICS
In this section, we introduce a class of Hermitian flat vector bundles on locallysymmetric spaces associated to representations of G with an admissible metrics.We state the main result (Theorem 4.4) of this article, which confirms the Friedconjecture for this class of Hermitian flat vector bundles.This section is organised as follows. In Section 4.1, we introduce a Hermitianmetric on a flat vector bundle whose holonomy representation is the restriction ρ | Γ of a representation ρ of G with an admissible metric.In Section 4.2, we state Theorem 4.4.In Sections 4.3-4.5, we deduce Theorem 0.1, Corollaries 0.2 and 0.3 from Theo-rem 4.4.Finally, in Section 4.6, we show Theorem 4.4 when δ ( G ) = Z G is compact.4.1. Hermitian metrics on flat vector bundles.
Let ρ : G → GL( E ) be a finitedimensional complex representation of G with admissible Hermitian metric 〈 , 〉 E .Let F be the flat vector bundle associated to ρ | Γ defined in (3.7). Let us constructa Hermitian metric g F on F following [Mü12, Section 2.5] and [BMaZ17, Section8.1]. By the second identity of (2.35), the restriction ρ | K of ρ to K is unitary. Let E ρ | K = G × K E (4.1)be the Hermitian vector bundle on X defined in (3.2). We have a canonical G -equivariant identification[ g , v ] ∈ G × K E → ( p g , gv ) ∈ X × E .(4.2)In this way, the G -invariant Hermitian metric on E ρ | K induces a G -invariant Her-mitian metric g b π ∗ F on the trivial vector bundle b π ∗ F = X × E . It descends to aHermitian metric g F on F = Γ \ ( X × E ). Definition 4.1.
We will call such ( F , g F ) an admissible Hermitian flat vector bun-dle. The g F will be called an admissible Hermitian metric on F .By (4.2), as in (3.3) and (3.8), we have the identifications Ω • ¡ X , b π ∗ F ¢ = C ∞ ¡ G , Λ • ( p ∗ ) ⊗ R E ¢ K , Ω • ( Z , F ) = C ∞ ¡ Γ \ G , Λ • ( p ∗ ) ⊗ R E ¢ K .(4.3)Let (cid:3) X be the Hodge Laplacian on X acting on Ω • ( X , b π ∗ F ) with respect to themetrics ( g T X , g b π ∗ F ). Let (cid:3) Z be the Hodge Laplacian on Z acting on Ω • ( Z , F ) withrespect to the metrics ( g T Z , g F ). Recall that C u , ρ ∈ End( E ) is the Casimir operatorof u acting on E (see (2.9)). The following proposition is classical (see [BMaZ17,Proposition 8.4]). Proposition 4.2.
Under the identifications (4.3) , we have (cid:3) X = C g , X , Λ • ( p ∗ ) ⊗ R E | K − C u , ρ , (cid:3) Z = C g , Z , Λ • ( p ∗ ) ⊗ R E | K − C u , ρ .(4.4)Let T ( F ) be the analytic torsion of F associated to ( g T Z , g F ). Let N Λ • ( T ∗ Z ) bethe number operator, i.e. N Λ • ( T ∗ Z ) acts by multiplication by k on Ω k ( Z , F ). Theorem 4.3.
Assume δ ( G ) . For any t > , we have Tr s h³ N Λ • ( T ∗ Z ) − m ´ exp ³ − t (cid:3) Z ´i = NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 22
In particular, T ( F ) = Proof.
This is [BMaZ11] and [BMaZ17, Theorem 8.6, Remark 8.7] (see also [Ma19,Theorem 5.5]), which generalises a vanishing theorem originally due to [MoSt91,Corollary 2.2] (see also [B11, Theorem 7.9.3]) where F is assumed to be unitarilyflat. (cid:3) Recall that ρ θ = ρ ◦ θ . Clearly, 〈 , 〉 E is still an admissible metric for ρ θ . Let( F θ , g F θ ) be the associated admissible Hermitian flat vector bundle on Z . Since ρ | K = ρ θ | K , by (4.3), the Hodge Laplacians of ( F , g F ) and ( F θ , g F θ ) act on the samespace. By (4.4) and by C u , ρ = C u , ρ θ , these two Laplacians coincide. In particular, H • ( Z , F ) ≃ H • ³ Z , F θ ´ , T ( F ) = T ³ F θ ´ .(4.7)4.2. The statement of the main result.
The main result of this article is thefollowing.
Theorem 4.4.
Assume that dim
Z is odd and that ρ : G → GL( E ) is a finite di-mensional complex representation of G with an admissible metric. Let ( F , g F ) bethe associated admissible Hermitian flat vector bundle. If ρ ≃ ρ θ , then there existexplicit constants C ρ ∈ R ∗ and r ρ ∈ Z (see (4.36) and Remark 5.11) such that when σ → , we have R ρ ( σ ) = C ρ T ( F ) σ r ρ + O ( σ r ρ + ).(4.8) Moreover, if H • ( Z , F ) = , thenC ρ = r ρ = so that R ρ (0) = T ( F ) .(4.10) Proof.
Since dim Z is odd, δ ( G ) is odd. If δ ( G ) Ê
3, by Remark 3.2 and Theorem4.3, our theorem follows easily. If δ ( G ) =
1, we will consider the case where Z G isnon compact in Section 4.6 and the case where Z G is compact in Sections 5 and6. (cid:3) Remark . Assume dim Z is even. Then δ ( G ) is even as well. If δ ( G ) Ê
2, thenby Remark 3.2 and Theorem 4.3, we have R ρ (0) = T ( F ) =
1. If δ ( G ) =
0, by theTheorem of Gauss-Bonnet-Chern and by [Sh18, Proposition 4.1], we have χ ( Z , F ) = dim E · χ ( Z ) Proof of Corollary 0.2.
Let us restate Corollary 0.2.
Theorem 4.6.
Assume that dim
Z is odd and that ρ : G → GL( E ) is a finite dimen-sional complex representation of G with an admissible metric. Let ( F , g F ) be theassociated admissible Hermitian flat vector bundle. If ρ is irreducible and ρ ρ θ ,then H • ( Z , F ) = NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 23 and R ρ is regular at σ = , so that ¯¯ R ρ (0) ¯¯ = T ( F ) .(4.13) Proof.
The vanishing of the cohomology H • ( Z , F ) is a consequence of [BoW00, The-orem VII.6.7] (see (6.20) for a proof).Take ρ ′ = ρ ⊕ ρ θ .(4.14)By (3.18) and (4.14), we have R ρ ′ = R ρ R ρ .(4.15)Note that ρ ′ has an admissible metric. Let F ′ be the admissible Hermitian flatvector bundle associated to ρ ′ . By (4.7) and (4.12), we have H • ¡ Z , F ′ ¢ = H • ( Z , F ) ⊕ H • ³ Z , F θ ´ = T ( F ′ ) = T ( F ) T ³ F θ ´ = T ( F ) .(4.16)Since ρ ′ = ρ ⊕ ρ θ is invariant by θ , we can apply Theorem 4.4 to ρ ′ . Therefore, R ρ ′ is regular at σ = R ρ ′ (0) = T ( F ′ ) .(4.17)By (4.15)-(4.17), we get (4.13). (cid:3) Proof of Theorem 0.1.
Since any G -representation with an admissible met-ric can be decomposed as a direct sum of irreducible G -representations, which stillhave admissible metrics, by Theorems 4.4 and 4.6, we get Theorem 0.1. (cid:3) Proof of Corollary 0.3.
Let ρ : Γ → U( r ) be a unitary representation of Γ .Let F be the associated flat vector bundle . Since ρ is unitary, F admits a flatmetric g F . We will show that ( F , g F ) is indeed an admissible Hermitian flatvector bundle associated to a larger reductive group.Let G = G × U( r ), K = K × U( r ).(4.18)Then, G is a connected real reductive group with maximal compact subgroup K .We have an identification G / K ≃ X .(4.19)Let Γ be the graph of ρ , i.e., Γ = © ( γ , ρ ( γ )) ∈ G : γ ∈ Γ ª .(4.20)Then Γ is a discrete torsion free and cocompact subgroup of G , so that Γ \ G / K ≃ Z .(4.21)Let ρ : G → U( r ) be the representation of G defined by the projection ontothe second component. The standard Hermitian metric on C r is admissible forthe representation ρ . Also, we have an identification of Hermitian flat vectorbundles Γ \ ( G / K × C r ) ≃ F .(4.22) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 24
In particular, ( F , g F ) is an admissible Hermitian flat vector bundle associated tothe representation ρ of G .More generally, the admissible Hermitian flat vector bundle ( F , g F ) associatedto the G -representation ρ is also an admissible Hermitian flat vector bundle asso-ciated to the G -representation ρ = ρ ⊠ . Therefore, F ⊗ F with the induced Her-mitian metric is an admissible Hermitian flat vector bundle associated to ρ ⊗ ρ .By these considerations, Corollary 0.3 follows from Theorem 0.1 and Corollary0.2. (cid:3) Proof of Theorem 4.4 when δ ( G ) = and Z G is noncompact. Supposenow δ ( G ) = Z G is non compact. Let ρ be a representation of G with anadmissible metric such that ρ ≃ ρ θ . We can and we will assume that the Casimir C u acts on ρ as a scalar C u , ρ ∈ R .For β ∈ b ∗ C , denote by C β the one dimensional representation of exp( b ) such that a ∈ b acts as the scalar 〈 β , a 〉 ∈ C . Clearly, C β has an admissible metric if and onlyif β ∈ b ∗ .Recall that since Z G is not compact, we have G = exp( b ) × M (see (3.25)). Sincea representation with an admissible metric is completely reducible, we can write ρ = M β ∈ b ∗ C β ⊠ η β ,(4.23)where η β is a representation of M . Proposition 4.7.
The following statements hold.i) For β ∈ b ∗ , we have isomorphisms of representations of M, η − β ≃ η β .(4.24) ii) For β ∈ b ∗ , the representation η β of M satisfies Assumption 3.5, so thatC u m , η β = C u , ρ + | β | ∈ R ,(4.25) Proof.
Since ρ θ ≃ ρ , by (4.23), we have isomorphisms of representations of M , η θβ ≃ η − β . Since δ ( M ) =
0, by Proposition 2.7, we have isomorphisms of representationsof M , η θβ ≃ η β . From these considerations, (4.24) follows.By (3.25), we have K M = K , so Assumption 3.5 (1) is trivial. Since C u , ρ is ascalar, by (3.25) and (4.23), the Casimir of u m acts on η β as a scalar given in(4.25). In particular, η β satisfies also Assumption 3.5 (2). (cid:3) Recall that in Section 2.6, we have fixed a positive element f b ∈ b in b . Set b ∗+ = { α ∈ b ∗ : 〈 α , f b 〉 > } .(4.26)By (4.24), we can rewrite (4.23) as ρ = ⊠ η ⊕ M β ∈ b ∗+ ¡ C β ⊕ C − β ¢ ⊠ η β .(4.27)Note that η can be zero.Let Z η β ( σ ) be Selberg zeta function associated to η β . NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 25
Proposition 4.8.
The following identity of meromorphic functions on C holds,R ρ ( σ ) = ¡ Z η ( σ ) ¢ − Y β ∈ b ∗+ ³ Z η β ¡ σ + | β | ¢ Z η β ¡ σ − | β | ¢´ − .(4.28) Proof.
By (4.27), for e a k − ∈ H , we haveTr £ ρ ¡ e a k − ¢¤ = Tr £ η ¡ k − ¢¤ + X β ∈ b ∗+ Tr £ η β ¡ k − ¢¤ ³ e | β || a | + e −| β || a | ´ .(4.29)By (3.25), we have z ⊥ ( b ) =
0, so that the denominator ¯¯ det(1 − Ad( γ )) | z ⊥ ( b ) ¯¯ in(3.20) disappears. By (3.15), (3.20), and (4.29), we get (4.28). (cid:3) By (3.26), (3.27), and (4.25), we see that Z η β µ − q σ + | β | ¶ Z η β µq σ + | β | ¶ = h det gr ³ C g , Z , b η β − C u , ρ + σ ´i (4.30)is a meromorphic function on C . We have a generalisation of [Sh18, Theorem 5.6].Recall that T ( σ ) is defined in (1.8). Theorem 4.9.
The following identity of meromorphic functions on C holds,R ρ ( σ ) = T ¡ σ ¢ exp ¡ − vol( Z ) P η ( σ ) ¢ Y β ∈ b ∗+ Z η β ³ − p σ + | β | ´ Z η β ³p σ + | β | ´ Z η β ¡ σ + | β | ¢ Z η β ¡ σ − | β | ¢ .(4.31) Proof.
By (3.25), we have p = p m ⊕ b . By (3.21), we have an identity in R ( K ), b η β = m X i = ( − i − i Λ i ( p ∗ C ) ⊗ η β | K .(4.32)By (4.4), (4.27), and (4.32), we have an identity of meromorphic functions, T ( σ ) = det gr ³ C g , Z , b η − C u , ρ + σ ´ − Y β ∈ b ∗+ det gr ³ C g , Z , b η β − C u , ρ + σ ´ − .(4.33)By (3.26), (3.27), (4.25), (4.30), (4.33), we have(4.34) T ¡ σ ¢ = Z η ( σ ) − exp ¡ vol( Z ) P η ( σ ) ¢ × Y β ∈ b ∗+ µ Z η β µ − q σ + | β | ¶ Z η β µq σ + | β | ¶¶ − .By (4.28) and (4.34), we get (4.31). (cid:3) Set r η β = dim ker ³ C g , Z , b η + β − C u , ρ ´ − dim ker ³ C g , Z , b η − β − C u , ρ ´ .(4.35)Following [Sh18, (7-75)], put C ρ = Y β ∈ b ∗+ ¡ − | β | ¢ − r ηβ , r ρ = − X β ∈ { } ∪ b ∗+ r η β .(4.36)Proceeding as [Sh18, (7-76)-(7-78)], using Theorem 4.9 instead of [Sh18, Theorem7.8], we get (4.8).Let F β be the admissible Hermitian flat subbundle of F associated to the G -representation C β ⊠ η β . By (1.1), (4.4), (4.32), and (4.35), we have r η β = − χ ′ ( Z , F β ).(4.37) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 26 If H • ( Z , F ) =
0, then for all β ∈ b ∗ , H • ( Z , F β ) =
0, so r η β =
0. By (4.36), we get (4.9).The proof of Theorem 4.4 in the case when δ ( G ) = Z G is non compact iscompleted. (cid:3)
5. T
HE PROOF OF (4.8)
WHEN δ ( G ) = AND Z G IS COMPACT
The purpose of this section is to show (4.8) when δ ( G ) = Z G is compactby generalising the arguments given in Section 4.6. One of the difficulties is toconstruct virtual representations η β of M satisfying Assumption 3.5 such that ananalogue of (4.28) holds. In the case of hyperbolic manifolds, such representationsare constructed using n -cohomology [Mü12]. Here, we construct η β via Dirac co-homology. These two methods are equivalent. We adopt the latter since it is closerto certain constructions given in [Sh18, Section 6].This section is organised as follows. In Section 5.1, we recall some facts on thestructure of real reductive groups with δ ( G ) = ρ according to the action of b . We show certainrepresentations of K M obtained in this way can be lifted in R ( K ).In Section 5.3, we introduce virtual representations η β of M satisfying Assump-tion 3.5.Finally, in Section 5.4, we establish analogues of Proposition 4.8 and Theorem4.9, and we show (4.8).In this section, we assume that δ ( G ) = Z G is compact. Suppose also thatthe G -representation ( ρ , E ) has an admissible metric and is such that ρ ≃ ρ θ . Asin Section 4.6, we can and we will assume that Casimir operator C u acts as ascalar C u , ρ ∈ R .5.1. The structure of the reductive group G with δ ( G ) = . Since G has acompact centre and dim b =
1, we have b z g , so n
0. By Proposition 2.5 i), dim n is a positive even number. Set ℓ =
12 dim n ∈ N ∗ .(5.1) Proposition 5.1.
Elements of b act on n and n as a scalar, i.e., there is α ∈ b ∗ such that for a ∈ b , f ∈ n , f ∈ n , we have [ a , f ] = 〈 α , a 〉 f , h a , f i = −〈 α , a 〉 f .(5.2) In particular, £ n , n ¤ ⊂ z ( b ), [ n , n ] = £ n , n ¤ = and [ z ( b ), z ( b )] ⊂ z ( b ), £ z ( b ), z ⊥ ( b ) ¤ ⊂ z ⊥ ( b ), £ z ⊥ ( b ), z ⊥ ( b ) ¤ ⊂ z ( b ),(5.4) [ u ( b ), u ( b )] ⊂ u ( b ), £ u ( b ), u ⊥ ( b ) ¤ ⊂ u ⊥ ( b ), £ u ⊥ ( b ), u ⊥ ( b ) ¤ ⊂ u ( b ). Proof.
This is [Sh18, Propositions 6.2, 6.3, and (6-29)]. (cid:3)
Let a ∈ b be such that 〈 α , a 〉 = NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 27
By (5.4), ( u , u ( b )) is a compact symmetric pair. Recall that G has a compactcentre. Let U be the compact form of G [Kn86, Proposition 5.3]. Let U ( b ) be thecentraliser of b in U . Then, U ( b ) is a connected Lie group [Kn02, Corollary 4.51]with Lie algebra u ( b ).In [Sh18, (6-31)], we have shown that J = ad ¡ p− a ¢ ∈ End( u ⊥ ( b )) defines a U ( b )-invariant complex structure on u ⊥ ( b ). Moreover, the associated holomor-phic and anti-holomorphic subspaces are n C and n C , so that we have a U ( b )-equivariant splitting u ⊥ ( b ) ⊗ R C = n C ⊕ n C .(5.6)Let S u ⊥ ( b ) be the spinor of ( u ⊥ ( b ), − B | u ⊥ ( b ) ). Classically ([Hi74], see also [Sh18,(6-33) and (6-34)]), we have an isomorphism of U ( b )-representations, S u ⊥ ( b ) ± ≃ Λ even/odd ¡ n ∗ C ¢ ⊗ det ( n C ) − .(5.7)In the sequel, there are representations which do not always lift to U ( b ). There-fore, it is more convenient to consider S u ⊥ ( b ) as a ( b C ⊕ m C , K M )-module.For 0 É j É ℓ , let η j be the ( m C , K M )-module Λ j ( n ∗ C ). By (2.31), we have anisomorphism of ( m C , K M )-modules, η ℓ − j ≃ η ℓ + j .(5.8) Proposition 5.2.
We have an isomorphism of ( b C ⊕ m C , K M ) -modules,S u ⊥ ( b ) ≃ ℓ M j =− ℓ C j α ⊠ η ℓ − j .(5.9) For e a k − ∈ H, we have Tr S u ⊥ ( b ) s £ e a k − ¤ = ¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ .(5.10) Proof.
By (5.2), (5.7), and (5.8), we get (5.9). By (5.9) and [Sh18, Proposition 6.5],we get (5.10). (cid:3)
Let B ∗ be the bilinear form on g ∗ induced by B . Proposition 5.3.
We have
18 Tr h C u ( b ), u ⊥ ( b ) i = − ℓ B ∗ ( α , α ) .(5.11) Proof.
This is [Sh18, Proposition 6.13] with j =
0. A direct proof of this resultcan be obtained by applying Kostant’s stranger formula (see also [B11, (2.6.11),(7.5.4)]), which is left to reader. (cid:3)
A splitting of ρ according to the b -action. Recall that Z ( b ) = exp( b ) × M .Since ρ has an admissible metric and since ρ ≃ ρ θ , as in (4.23) and (4.27), we canwrite ρ | exp( b ) × M = M β ∈ b ∗ C β ⊠ ρ β = ⊠ ρ ⊕ M β ∈ b ∗+ ¡ C β ⊕ C − β ¢ ⊠ ρ β .(5.12)where ρ β are representations of M such that ρ − β ≃ ρ β .Recall that the restriction induces an injective morphism R ( K ) → R ( K M ) ofrings. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 28
Theorem 5.4.
For all β ∈ b ∗ , the restriction ρ β | K M has a unique lift in R ( K ) .Proof. By (2.32), it is enough to show that the character of ρ β | T is invariant underthe Weyl group W ( T : K ).For e a k − ∈ H , by in (5.12), as (4.29), we haveTr £ ρ ¡ e a k − ¢¤ = Tr £ ρ ¡ k − ¢¤ + X β ∈ b ∗+ ³ e | β || a | + e −| β || a | ´ Tr £ ρ β ¡ k − ¢¤ .(5.13)Let w ∈ N K ( T ). Since ρ is a G representation, for e a k − ∈ H , we haveTr £ ρ ¡ e a k − ¢¤ = Tr £ ρ ¡ we a k − w − ¢¤ = Tr h ρ ³ e Ad( w ) a wk − w − ´i .(5.14)By (2.17), Ad( w ) preserves t and b (see [Sh20, Proposition 3.4]). Since K preserves B | p , and since dim b =
1, we see that Ad( w ) a = a or − a . By (5.13), we get(5.15) Tr h ρ ³ e Ad( w ) a wk − w − ´i = Tr £ ρ ¡ wk − w − ¢¤ + X β ∈ b ∗+ ³ e | β || a | + e −| β || a | ´ Tr £ ρ β ¡ wk − w − ¢¤ .By (5.13)-(5.15), since dim b =
1, for β ∈ b ∗ , we haveTr £ ρ β ¡ wk − w − ¢¤ = Tr £ ρ β ¡ k − ¢¤ ,(5.16)i.e., the character of ρ β | T is invariant under the Weyl group W ( T : K ). (cid:3) The representation η β . In [Sh18], we have shown that η j satisfies As-sumption 3.5 and we consider the Selberg zeta function associated to η j . A naiveway to generalise the arguments in Section 4.6 is to consider the Selberg zetafunction associated to η j ⊗ ρ β . However, η j ⊗ ρ β satisfies Assumption 3.5 (1), whileAssumption 3.5 (2) fails in general. We need consider all the j together to producea virtual representation η β .Recall that ρ : G → GL( E ) is a representation of G , and that S u ⊥ ( b ) is the spinorof ¡ u ⊥ ( b ), − B | u ⊥ ( b ) ¢ . Let D S u ⊥ ( b ) ⊗ E be the Dirac operator defined in Section 2.3. Let D S u ⊥ ( b ) ⊗ E ± be the restriction of D S u ⊥ ( b ) ⊗ E to S u ⊥ ( b ) ± ⊗ E .By (2.14), we have − µ D S u ⊥ ( b ) ⊗ E ¶ = C u , ρ +
18 Tr h C u ( b ), u ⊥ ( b ) i − C u ( b ), S u ⊥ ( b ) ⊗ E .(5.17)Since u acts unitarily on ( E , 〈 , 〉 E ) , we haveker D S u ⊥ ( b ) ⊗ E = ker µ D S u ⊥ ( b ) ⊗ E ¶ .(5.18)If the u ( b )-action on E lifts to U ( b ), then ker D S u ⊥ ( b ) ⊗ E ± are representations of U ( b ).In general, ker D S u ⊥ ( b ) ⊗ E ± are ( b C ⊕ m C , K M )-modules. Note that by (5.9) and (5.12), b acts semisimplely on ker D S u ⊥ ( b ) ⊗ E ± . They are called Dirac cohomology of E (see [HuPa06]). NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 29
Definition 5.5.
Define the ( m C , K M )-modules η + β and η − β , so thatker D S u ⊥ ( b ) ⊗ E + = M β ∈ b ∗ C β ⊠ η + β , ker D S u ⊥ ( b ) ⊗ E − = M β ∈ b ∗ C β ⊠ η − β .(5.19)Set η β = η + β − η − β .(5.20)Recall that C u , ρ ∈ R is a scalar. Proposition 5.6.
The following statements hold.i) We have isomorphisms of ( m C , K M ) -modules, η ± β ≃ η ±− β .(5.21) ii) The virtual ( m C , K M ) -modules η β satisfy Assumption 3.5, so that the Casimirof u m acts on η ± β by the same scalarC u m , η β = | β | + C u , ρ +
18 Tr u ⊥ ( b ) h C u ( b ), u ⊥ ( b ) i ∈ R .(5.22) Proof.
By (5.17)-(5.19), on C β ⊠ η ± β , we have C u , ρ +
18 Tr h C u ( b ), u ⊥ ( b ) i − C u ( b ), S u ⊥ ( b ) ⊗ E = C β ⊠ η ± β , we have C u ( b ), S u ⊥ ( b ) ⊗ E = −| β | + C u m , η ± β .(5.24)By (5.23) and (5.24), we see that C u m , η ± β coincide and are given by (5.22).By (5.8), (5.9), and (5.12), we see that S u ⊥ ( b ) ± ⊗ E admits a decomposition like(5.12). By (5.17) and (5.18), ker D S u ⊥ ( b ) ⊗ E ± consists of the components on which(5.22) holds. Form these two considerations, i) follows.It remains to show each η β | K M lifts to R ( K ). By (5.19), for e a k − ∈ H , we have X β ∈ b ∗ e 〈 a , β 〉 Tr s £ η β ( k − ) ¤ = Tr ker ¡ D S u ⊥ ( b ) ⊗ E ¢ s £ e a k − ¤ .(5.25)We claim that for e a k − ∈ H , we haveTr ker ¡ D S u ⊥ ( b ) ⊗ E ¢ s £ e a k − ¤ = Tr S u ⊥ ( b ) ⊗ E s £ e a k − ¤ .(5.26)Indeed, since D S u ⊥ ( b ) ⊗ E commutes with e a k − , using the fact that the super tracevarnishes on the super commutator, we see thatTr S u ⊥ ( b ) ⊗ E s · e a k − exp µ − t µ D S u ⊥ ( b ) ⊗ E ¶ ¶¸ (5.27)does not depend on t ∈ R , from which we get (5.26).From (5.25) and (5.26), for e a k − ∈ H , we have X β ∈ b ∗ e 〈 β , a 〉 Tr s £ η β ( k − ) ¤ = Tr S u ⊥ ( b ) s £ e a k − ¤ Tr £ ρ ¡ e a k − ¢¤ .(5.28)By Theorems 2.6, 5.4, (5.9), (5.12), the right-hand side of (5.28) is a sum of prod-ucts of e 〈 β , a 〉 with elements in R ( K ) ≃ R ( T ) W ( T : K ) . Thus, η β has a lift in R ( K ). (cid:3) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 30
The following two propositions are analogues of [Sh18, Propositions 6.5, 6.10].
Proposition 5.7.
We have an isomorphism of virtual ( b C ⊕ m C , K M ) -modules, ³ S u ⊥ ( b ) + − S u ⊥ ( b ) − ´ ⊗ ρ | exp( b ) × M = M β ∈ b ∗ C β ⊠ η β = ⊠ η ⊕ M β ∈ b ∗+ ¡ C β ⊕ C − β ¢ ⊠ η β .(5.29) For e a k − ∈ H, we have (5.30) Tr s £ η ¡ k − ¢¤ + X β ∈ b ∗+ ³ e | a || β | + e −| a || β | ´ Tr s £ η β ¡ k − ¢¤ = ¯¯ det ¡ − Ad( e a k − ) ¢ | z ⊥ ( b ) ¯¯ Tr £ ρ ¡ e a k − ¢¤ Proof.
This is a consequence of (5.10), (5.21), and (5.28). (cid:3)
Proposition 5.8.
The following identity in R ( K ) holds, M β ∈ b ∗ b η β = m M i = ( − i − i Λ i ¡ p ∗ C ¢ ⊗ ρ | K .(5.31) Proof.
By (5.7) and (5.29), we have an identity in R ( K M ), M β ∈ b ∗ η β | K M = Λ • ¡ n ∗ C ¢ | K M ⊗ ρ | K M .(5.32)By (2.26), (3.21), (5.32) and Proposition 2.5 v), we get M β ∈ b ∗ b η β | K M = M β ∈ b ∗ Λ • ³ p ∗ m , C ´ b ⊗ η β = m M i = ( − i − i Λ i ¡ p ∗ C ¢ | K M ⊗ ρ | K M .(5.33)Since the restriction R ( K ) → R ( K M ) is injective, from (5.33), we get (5.31). (cid:3) The Selberg zeta function Z η β . By Proposition 5.6 ii), η β satisfies Assump-tion 3.5. Let Z η β be the associated Selberg zeta function. We have an analogue of[Sh18, Theorem 7.7] and of Proposition 4.8. Proposition 5.9.
The following identity of meromorphic function on C holds,R ρ ( σ ) = ¡ Z η ( σ ) ¢ − Y β ∈ b ∗+ ³ Z η β ¡ σ + | β | ¢ Z η β ¡ σ − | β | ¢´ − .(5.34) Proof.
This is a consequence of (3.15), (3.20), and (5.30). (cid:3)
As in (4.30), by (3.24), (3.27), and (5.22), Z η β µ − q σ + | β | ¶ Z η β µq σ + | β | ¶ = h det gr ³ C g , Z , b η β − C u , ρ + σ ´i .(5.35)is a meromorphic function on C . The following proposition is an analogue of[Sh18, Theorem 7.8] and of Theorem 4.9. Theorem 5.10.
The following identity of meromorphic functions on C holds,R ρ ( σ ) = T ¡ σ ¢ exp ¡ − vol( Z ) P η ( σ ) ¢ Y β ∈ b ∗+ Z η β ³ − p σ + | β | ´ Z η β ³p σ + | β | ´ Z η β ¡ σ + | β | ¢ Z η β ¡ σ − | β | ¢ .(5.36) Proof.
By (1.8), (4.4), (5.21), and (5.31), the statement of (4.33) still holds in thecurrent situation. The rest part of the proof is exactly the same as in the proof ofTheorem 4.9. (cid:3)
NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 31
Remark . Define r η β , C ρ , and r ρ by the same formula as in (4.35) and (4.36).Proceeding as [Sh18, (7-76)-(7-78)], using Theorem 5.10 instead of [Sh18, Theo-rem 7.8], we get (4.8) when δ ( G ) = Z G is compact.6. A COHOMOLOGICAL FORMULA FOR r η β The purpose of this section is to show (4.9) when δ ( G ) = Z G is compact. Itsproof relies on some deep results from the classification of unitary representationsof real reductive groups.This section is organised as follows. In Section 6.1, we recall the definitionof the infinitesimal characters of a U ( g C )-modules, the Harish-Chandra ( g C , K )-modules, and a relation between the infinitesimal character and the vanishingof ( g , K )-cohomology of a unitary Harish-Chandra ( g C , K )-module, which is dueto Vogan-Zuckermann [VZu84], Vogan [V84], and Salamanca-Riba [SR99]. Thelatter is our essential algebraic input in the proof of (4.9).In Section 6.2, we obtain a formula relating H • ( Z , F ) and the ( g , K )-cohomologyof certain Harish-Chandra ( g C , K )-modules.Finally, in Section 6.3, we deduce a similar formula for r η β and we prove (4.9).We use the notation in Sections 2 and 3. In Sections 6.1 and 6.2, we assumeneither δ ( G ) = Z G is compact.6.1. Some results from representation theory.
We recall some basic facts onthe representation theory of real reductive groups.6.1.1.
Infinitesimal characters.
A morphism of algebras χ : Z ( g C ) → C will becalled a character of Z ( g C ). Clearly, for a ∈ C , we have χ ( a ) = a .(6.1)By (2.6), Z ( g C ) is equipped with a complex conjugation. Moreover, the Car-tan involution θ extends to complex automorphism on Z ( g C ). Also, the anti-automorphism z → z tr of U ( g C ) [Kn02, Proposition 3.7], induced by a ∈ g → − a ∈ g ,descends to a complex automorphisms of Z ( g C ). For z ∈ Z ( g C ), set χ ( z ) = χ ( z ), χ θ ( z ) = χ ( θ z ), χ tr ( z ) = χ ¡ z tr ¢ .(6.2)Then, χ , χ θ , and χ tr are characters of Z ( g C ). Definition 6.1.
A complex representation of g C is said to have infinitesimal char-acter χ , if z ∈ Z ( g C ) acts as a scalar χ ( z ) ∈ C .A complex representation of g C is said to have generalised infinitesimal char-acter χ , if for there is i ≫ z ∈ Z ( g C ), ( z − χ ( z )) i acts like 0.If W is a complex representation of g with infinitesimal character χ W , then W , W θ (defined in an obvious way as (2.33)), and W ∗ have infinitesimal characters χ W , χ W θ , and χ W ∗ , so that χ W = χ W , χ W θ = χ θ W , χ W ∗ = χ tr W .(6.3)Therefore, if W is a unitary representation of g , we have χ tr W = χ W .(6.4) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 32 If W is a representation of g which is equipped with an admissible Hermitianmetric, then W ∗ ≃ W θ , so that χ tr W = χ θ W .(6.5)Let us recall the definition of the Harsh-Chandra parameter for a character of Z ( g C ). Let h C ⊂ g C be a Cartan subalgebra of g C . Let S ( h C ) be the symmetricalgebra of h C . If W ( h C : g C ) denotes the algebraic Weyl group, let S ( h C ) W ( h C : g C ) ⊂ S ( h C ) be the W ( h C : g C )-invariant subalgebra of S ( h C ). Let φ HC : Z ( g C ) ≃ S ( h C ) W ( h C : g C ) (6.6)be the Harish-Chandra isomorphism [Kn02, Section V.5]. For Λ ∈ h ∗ C , we canassociate to it a character χ Λ of Z ( g C ) as follows: for z ∈ Z ( g C ), χ Λ ( z ) = 〈 φ HC ( z ), Λ 〉 .(6.7)By [Kn02, Theorem 5.62], every character of Z ( g C ) is of the form χ Λ , for some Λ ∈ h ∗ C . Also, Λ is uniquely determined up to an action of W ( h C : g C ). Such anelement Λ ∈ h ∗ C is called the Harish-Chandra parameter of the character.6.1.2. Harish-Chandra ( g C , K ) -module and its ( g , K ) -cohomology. Definition 6.2.
A complex U ( g C )-module V , equipped with an action of K , iscalled a Harish-Chandra ( g C , K )-module, if(1) the space V is a finitely generated U ( g C )-module;(2) every v ∈ V is K -finite, i.e., { k · v } k ∈ K spans a finite dimensional vectorspace;(3) the actions of g C and K are compatible;(4) each irreducible K -module occurs only for a finite number of times in V .Let b G u be the unitary dual of G , that is the set of equivalence classes of complexirreducible unitary representations π of G on Hilbert spaces V π . For ( π , V π ) ∈ b G u ,let V π , K be the space of K -finite vectors. By [Kn86, Theorem 8.1, Proposition 8.5], g C acts on V π , K such that V π , K is a Harish-Chandra ( g C , K )-module.If V is a Harish-Chandra ( g C , K )-module, let H • ( g , K ; V ) be the ( g , K )-cohomologyof V [BoW00, Section I.1.2]. Theorem 6.3.
Let V be a Harish-Chandra ( g C , K ) -module with generalised infin-itesimal character χ . Let W be a finite dimensional ( g C , K ) -module with infinitesi-mal character χ W . If χ χ tr W , thenH • ( g , K ; V ⊗ W ) = Proof. If V has an infinitesimal character, the theorem is due to [BoW00, TheoremI.5.3(ii)]. If V has a generalised infinitesimal character, a proof can be found in[Sh18, Theorem 8.8]. (cid:3) The converse of (6.8) is not true in general. But it still holds if V is unitary. Theorem 6.4.
Let W be a finite dimensional ( g C , K ) -module with infinitesimalcharacter χ W . If ( π , V π ) ∈ b G u , then χ π χ tr W ⇐⇒ H • ( g , K ; V π , K ⊗ W ) = NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 33
Proof.
The direction =⇒ of (6.9) is (6.8). The direction ⇐= of (6.9) is a conse-quence of Vogan-Zuckerman [VZu84], Vogan [V84], and Salamanca-Riba [SR99].Indeed, if χ π = χ tr W , then the Harish-Chandra parameter of χ π is stronger regularin the sense of [SR99, p. 525]. Such representation π is classified in [SR99], whichhas non vanishing ( g , K )-cohomology by [VZu84, V84]. (cid:3) Root system and n -homology. We use the notation in Section 2.5. Let h = b ⊕ t be the fundamental Cartan subalgebra of g . Let R ⊂ b ∗ ⊕ p− t ∗ be a rootsystem of ( h , g ). By [Kn02, Proposition 11.16] (see also [BSh19, Proposition 3.7]),there are no real roots in R . Let R im and R c be the systems of imaginary rootsand complex roots, so that R = R im ⊔ R c .(6.10)Then, R im is a root system of ( h , z ( b )). Also, R im | t is a root system of ( t , m ).We fix a positive root system R + ⊂ R . Set R im + = R im ∩ R + , R c + = R c ∩ R + .(6.11)As explained in [BSh19, Section 3.5], we can choose R + such that R c + is stableunder complex conjugation.Set ̺ u = X α ∈ R + α ∈ b ∗ ⊕ p− t ∗ .(6.12)By Kostant’s strange formula [Ko76] or [B11, Proposition 7.5.1], we have ¯¯ ̺ u ¯¯ = −
124 Tr u £ C u , u ¤ .(6.13)Define ̺ u ( b ) ∈ b ∗ ⊕ p− t ∗ and ̺ u m ∈ p− t ∗ in the same way, which are associatedto R im + and R im +| t . Then, ̺ u ( b ) = ¡ ̺ u m ¢ , ̺ u | t = ̺ u m .(6.14)If V is a Harish-Chandra ( g C , K )-module, denote by H • ( n , V ) its n -homology. By[HeSc83, Proposition 2.24], H • ( n , V ) is a Harish-Chandra ( m C ⊕ b C , K M )-module.If V possesses an infinitesimal character with Harish-Chandra parameter Λ ∈ h ∗ C , by [HeSc83, Corollary 3.32], H • ( n , V ) can be decomposed into a finite directsum of Harish-Chandra ( m C ⊕ b C , K M )-modules whose generalised infinitesimalcharacters are given by χ w Λ + ̺ u − ̺ u ( b ) ,(6.15)for some w ∈ W ( h C : g C ).6.2. The cohomology of H • ( Z , F ) . We use the notation in Section 4.1. Recallthat Γ ⊂ G is a discrete cocompact torsion free subgroup of G and that ρ : G → GL( E ) is a representation of G with an admissible metric. Let ( F , g F ) be theassociated Hermitian flat vector bundle. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 34
By [GelGrPS69, p. 23, Theorem], we can decompose L ( Γ \ G ) into a directHilbert sum of countable irreducible unitary representations of G , L ( Γ \ G ) = Hil M π ∈ b G u n ( π ) V π ,(6.16)with n ( π ) < ∞ .For any unitary representation ( τ , E τ ) of K , since C g , Z , τ is elliptic and self-adjoint and since Z is compact, we have a finite sumker ³ C g , Z , τ − λ ´ = M π ∈ b G u , χ π ( C g ) = λ n ( π ) ¡ V π , K ⊗ E τ ¢ K .(6.17)Let X ( ρ ∗ ) be set of the infinitesimal characters of all irreducible subrepresenta-tions of ρ ∗ . Note that by Remark 2.1, the sets of all irreducible subrepresentationsof ρ ∗ of the group G and of the Lie algebra g coincide. Theorem 6.5.
We haveH • ( Z , F ) = M π ∈ b G u , χ π ∈ X ( ρ ∗ ) n ( π ) H • ¡ g , K ; V π , K ⊗ E ¢ .(6.18) If H • ( Z , F ) = , then for any π ∈ b G u such that χ π ∈ X ( ρ ∗ ) , we haven ( π ) = If ρ is irreducible such that ρ θ ρ , thenH • ( Z , F ) = Proof.
Since the G -representation with an admissible metric is completely re-ducible, we can assume that ρ is irreducible with the infinitesimal character χ ρ .By (6.3), we have χ tr ρ ( C g ) = χ ρ ( C g ) = C g , ρ . By (1.4), (4.4) and (6.17), we have H • ( Z , F ) = M π ∈ b G u , χ π ( C g ) = χ tr ρ ( C g ) n ( π ) ¡ V π , K ⊗ Λ • ( p ∗ C ) ⊗ E ¢ K .(6.21)When χ π ( C g ) = χ tr ρ ( C g ), by Hodge theory for Lie algebras [BoW00, PropositionII.3.1], we have ¡ V π , K ⊗ Λ • ( p ∗ C ) ⊗ E ¢ K = H • ( g , K ; V π , K ⊗ E ).(6.22)By (6.8), (6.21), and (6.22), we get (6.18). By (6.9) and (6.18), we get (6.19).To show (6.20), it is enough to show that if ρ θ ρ , then for all π ∈ b G u , we have χ π χ tr ρ .(6.23)Otherwise there is π ∈ b G π such that χ π = χ tr ρ . Using π ∗ ≃ π and ρ ∗ ≃ ρ θ , by (6.4)and (6.5), we have χ ρ = χ tr π = χ π = χ tr ρ = χ θρ .(6.24)Since ρ and ρ θ are irreducible and have finite dimensions, Equation (6.24) implies ρ ≃ ρ θ , which is a contradiction with our assumption. (cid:3) Remark . Equations (6.18) and (6.20) are [BoW00, Theorems VII.6.1 and VII.6.7].Equation (6.18) is originally due to Matsushima [Mat67] where ρ is supposed tobe trivial. NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 35
A formula for r η β . Assume now that δ ( G ) = Z G is compact, and that ρ : G → GL( E ) is a G -representation with an admissible metric such that ρ ≃ ρ θ and that C u , ρ ∈ R is a scalar.By [Sh18, Corollary 8.15], we have(6.25) r η β = χ ( K / K M ) X π ∈ b G u , χ π ( C g ) = C u , ρ É i É dim p m É j É ℓ ( − i + j n ( π ) ³ dim H i ¡ m , K M ; H j ( n , V π , K ) ⊗ E + η β ¢ − dim H i ¡ m , K M ; H j ( n , V π , K ) ⊗ E − η β ¢´ . Proposition 6.7.
Let ( π , V π ) ∈ b G u . Assume that χ π ( C g ) = C u , ρ and M É i É dim p m É j É ℓ s ∈ { ± } H i ¡ m , K M ; H j ( n , V π , K ) ⊗ E s η β ¢ Then, χ π ∈ X ( ρ ∗ ).(6.27) Proof.
We use the notation in Section 6.1.3. Let Λ ( π ∗ ) ∈ h ∗ C be a Harish-Chandraparameter of V π ∗ , K . We need to show that there is w ∈ W ( h C : g C ) and a Harish-Chandra parameter Λ ( ρ ) ∈ b ∗ ⊕ p− t ∗ of an irreducible g -submodule of ρ , suchthat w Λ ( π ∗ ) = Λ ( ρ ).(6.28)Recall that B ∗ is the bilinear form on g ∗ induced by B . It extends to g ∗ C in anobvious way. Since χ π ( C g ) = C u , ρ , using Harish-Chandra isomorphism (see [Kn02,Example 5.64]), we have B ∗ ¡ Λ ( π ∗ ), Λ ( π ∗ ) ¢ − B ∗ ¡ ̺ u , ̺ u ¢ = C u , ρ .(6.29)By (6.8), (6.14), (6.15), and (6.26), there exist w ∈ W ( h C : g C ), w ′ ∈ W ( t C : m C ) ⊂ W ( h C : g C ) and the highest weight µ β ∈ p− t ∗ of an irreducible ( m C , K M )-submoduleof η + β ⊕ η − β such that w Λ ( π ∗ ) | t C = w ′ ¡ µ β + ρ u m ¢ .(6.30)By Proposition 5.1 and (6.12), we have ̺ u = ̺ u ( b ) + ( ℓα , 0) .(6.31)By (5.11), (5.22), (6.14), (6.29)-(6.31), there exists w ′′ ∈ W ( h C : g C ) such that w ′′ Λ ( π ∗ ) = ¡ ± β , µ β + ρ u m ¢ = ¡ ± β , µ β ¢ + ρ u ( b ) .(6.32)In particular, Λ ( π ∗ ) ∈ b ∗ ⊕ p− t ∗ .By (5.19), ¡ β , µ β ¢ ∈ b ∗ ⊕p− t ∗ is a highest weight of an irreducible ( m C ⊕ b C , K M )-submodule of ker D S u ⊥ ( b ) ⊗ E . By [HuPa06, Theorem 4.2.2], there exists w ∈ W ( h C : g C ) and a Harish-Chandra parameter Λ ( ρ ) ∈ h ∗ C of an irreducible g -submodule of ρ , such that ¡ β , µ β ¢ = w Λ ( ρ ) − ρ u ( b ) .(6.33) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 36
By (5.19) and (5.21), ( − β , µ β ) ∈ b ∗ ⊕ p− t ∗ is also the highest weight of an ir-reducible ( m C ⊕ b C , K M )-submodule of ker D S u ⊥ ( b ) ⊗ E . As before, there exists w ∈ W ( h C : g C ) and a Harish-Chandra parameter Λ ( ρ ) ∈ h ∗ C of an irreducible g -submoduleof ρ , such that ¡ − β , µ β ¢ = w Λ ( ρ ) − ρ u ( b ) .(6.34)By (6.32)-(6.34), we get (6.28). (cid:3) Corollary 6.8.
For β ∈ b ∗ , we have (6.35) r η β = χ ( K / K M ) X π ∈ b G u , χ π ∈ X ( ρ ∗ )0 É i É dim p m É j É ℓ ( − i + j n ( π ) ³ dim H i ³ m , K M ; H j ( n , V π , K ) ⊗ E + η β ´ − dim H i ³ m , K M ; H j ( n , V π , K ) ⊗ E − η β ´ ´ . In particular, if H • ( Z , F ) = , then for all β ∈ b ∗ , we haver η β = Proof.
By (6.25) and Proposition 6.7, we get (6.35). From (6.19) and (6.35), we get(6.36). (cid:3)
Remark . By (4.36), (6.36), and by Remark 5.11, we get (4.9) in the case where δ ( G ) = Z G is compact. We finish the proof of Theorem 4.4 in full generality.7. A N EXTENSION TO ORBIFOLDS
In this section, we no longer assume Γ ⊂ G is torsion free. Then Z = Γ \ G / K isa closed Riemannian orbifold with Riemannian metric g T Z . Let us indicate theessential steps in generalising the previous results to orbifolds.If γ ∈ Γ , Γ ( γ ) is not always torsion free. The cardinality ¯¯ ker ¡ Γ ( γ ) → Diffeo( Z ( γ )/ K ( γ )) ¢¯¯ (7.1)depends only on the conjugacy class [ γ ] and will be denoted by n [ γ ] . We define B [ γ ] by the same formula as in (3.10). By [ShY17, Proposition 5.3], we havevol( Γ ( γ ) \ Z ( γ ))vol( K ( γ )) = vol ¡ B [ γ ] ¢ n [ γ ] .(7.2)By [ShY17, Remark 5.6, (5.59)], as in (3.11), the closed geodesics (see [GuHa06]or [ShY17, Remark 2.26]) on the orbifold Z with positive length are given by a [ γ ] ∈ [ Γ + ] B [ γ ] .(7.3)For [ γ ] ∈ [ Γ + ], all the elements of B [ γ ] have the same length ℓ [ γ ] > γ ] ∈ [ Γ + ], the group S acts locally freely on the orbifold B [ γ ] by rotation, sothat B [ γ ] / S is still a closed orbifold. Set m [ γ ] = n [ γ ] ¯¯ ker ¡ S → Diffeo(B [ γ ] ) ¢¯¯ ∈ N ∗ .(7.4) NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 37 If ρ : Γ → GL r ( C ) is a representation of Γ , for Re ( σ ) ≫ R ρ ( σ ) by the same formula (3.15) with m [ γ ] defined by (7.4). As before, when δ ( G ) Ê R ρ ( σ ) ≡ Z is odd, R ρ ( σ ) has a meromorphic extension to σ ∈ C .Let ρ : G → GL( E ) be a finite dimensional complex representation of G withan admissible metric. Let F be the orbifold flat vector bundle on Z associatedto ρ | Γ . As in Section 4.1, F is equipped canonically with a Hermitian metric g F .The analytic torsion of F associated to ( g T Z , g F ) is defined in [DaiY17], [ShY17,Section 4.2] (see also [Ma05]). Theorem 7.1.
The statements of Theorems 4.3, 4.4, and 4.6 still hold for orbifolds.In particular, we get Theorem 0.5.Proof.
Using the orbifold trace formula [ShY17, Theorem 5.4] and [Ma19, Theo-rem 5.4], we get the orbifold version of Theorem 4.3.The proof of Theorem 4.4 in the case of orbifold is similar as before and weneed only consider the case δ ( G ) =
1. We can define the Selberg zeta function bythe same formula (3.20) with m [ γ ] defined by (7.4). By [Sh20, Section 7.2], thestatement of Theorem 3.9 still holds for orbifolds. By exactly the same method,the statements of Proposition 4.8, Theorem 4.9, Proposition 5.9, and Theorem 5.10hold for orbifold. Using the orbifold Hodge theory (c.f. [ShY17, Theorem 4.1]), wecan deduce that the statements of Theorem 6.5 and Corollary 6.8 hold as well. Inthis way, we get Theorem 4.4 for orbifolds.As in the proof of Theorem 4.6 given in Section 4.3, the orbifold version of The-orem 4.6 is a consequence of the orbifold version of Theorem 4.4. The proof of ourTheorem is completed. (cid:3) R EFERENCES [BéDHP19] L. Bénard, J. Dubois, M. Heusener, and J. Porti,
Asymptotics of twisted Alexanderpolynomials and hyperbolic volume , arXiv:1912.12946 (2019).[BeGeVe04] 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[BMaZ11] J.-M. Bismut, X. Ma, and W. Zhang,
Opérateurs de Toeplitz et torsion analytiqueasymptotique , 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[BSh19] 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)
NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 38 [BoW00] A. Borel and N. Wallach,
Continuous cohomology, discrete subgroups, and representa-tions of reductive groups , second ed., Mathematical Surveys and Monographs, vol. 67,American Mathematical Society, Providence, RI, 2000. MR 1721403 (2000j:22015)[BWSh20] Y. Borns-Weil and S. Shen,
Dynamical zeta functions in the nonorientable case ,arXiv:2007.08043 (2020).[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)[Brö98] U. Bröcker,
Die Ruellesche Zetafunktion für G-induzierte Anosov-Flüsse , Ph.D. thesis,Humboldt-Universität Berlin, Berlin (1998).[CMi10] 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)[DaiY17] X. Dai and J. Yu,
Comparison between two analytic torsions on orbifolds , Math. Z. (2017), no. 3-4, 1269–1282. MR 3623749[DaGRSh20] N. V. Dang, C. Guillarmou, G. Rivière, and S. Shen,
The Fried conjecture in smalldimensions , Invent. Math. (2020), no. 2, 525–579. MR 4081137[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)[DyZ16] 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[DyZ17] S. Dyatlov and M. Zworski, Ruelle zeta function at zero for surfaces , Invent. Math. (2017), no. 1, 211–229. MR 3698342[Fr35] W. Franz,
Über die Torsion einer Überdeckung. , J. Reine Angew. Math. (1935),245–254 (German).[F86a] D. Fried,
Analytic torsion and closed geodesics on hyperbolic manifolds , Invent. Math. (1986), no. 3, 523–540. MR 837526 (87g:58118)[F86b] D. Fried, Fuchsian groups and Reidemeister torsion , The Selberg trace formula andrelated topics (Brunswick, Maine, 1984), Contemp. Math., vol. 53, Amer. Math. Soc.,Providence, RI, 1986, pp. 141–163. MR 853556 (88e:58098)[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)[GelGrPS69] I. M. Gel ′ fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory andautomorphic functions , Translated from the Russian by K. A. Hirsch, W. B. SaundersCo., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR 0233772 (38
Anosov flows and dynamical zeta functions ,Ann. of Math. (2) (2013), no. 2, 687–773. MR 3071508[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 Dirac operators in representation theory , Mathematics:Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2244116[Kn86] A. W. Knapp,
Representation theory of semisimple groups , Princeton MathematicalSeries, vol. 36, Princeton University Press, Princeton, NJ, 1986, An overview based onexamples. MR 855239 (87j:22022)
NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 39 [Kn02] 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)[KnV95] A. W. Knapp and D. A. Vogan, Jr.,
Cohomological induction and unitary representa-tions , Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton,NJ, 1995. MR 1330919 (96c:22023)[Ko76] B. Kostant,
On Macdonald’s η -function formula, the Laplacian and generalized expo-nents , Advances in Math. (1976), no. 2, 179–212. MR 0485661 (58 Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ -decomposition C ( g ) = End V ρ ⊗ C ( P ) , and the g -module structure of V g , Adv. Math. (1997), no. 2, 275–350. MR 1434113[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[M91] G. A. Margulis,
Discrete subgroups of semisimple Lie groups , Ergebnisse der Math-ematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825[Mat67] Y. Matsushima,
A formula for the Betti numbers of compact locally symmetric Rie-mannian manifolds , J. Differential Geometry (1967), 99–109. MR 0222908 (36 On vector bundle valued harmonic forms and au-tomorphic forms on symmetric riemannian manifolds , Ann. of Math. (2) (1963),365–416. MR 0153028[Mi68] 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
R-torsion and zeta functions for locally symmetricmanifolds , 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[Mü12] W. Müller,
The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds ,Metric and differential geometry, Progr. Math., vol. 297, Birkhäuser/Springer, Basel,2012, pp. 317–352. MR 3220447[Mü20] W. Müller,
On Fried’s conjecture for compact hyperbolic manifolds , arXiv:2005.01450(2020).[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 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 NALYTIC TORSION AND DYNAMICAL ZETA FUNCTION 40 [Se67] R. T. Seeley,
Complex powers of an elliptic operator , Singular Integrals (Proc. Sympos.Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307.MR 0237943 (38
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 .[Sh20] S. Shen,
Complex valued analytic torsion and dynamical zeta function on locally sym-metric spaces , arXiv:2009.03427 (2020).[ShY17] S. Shen and J. Yu,
Flat vector bundles and analytic torsion on orbifolds , to appear inComm. Anal. Geom., arXiv: 1704.08369 (2017).[ShY18] S. Shen and J. Yu,
Morse-Smale flow, Milnor metric, and dynamical zeta function ,arXiv: 1806.00662 (2018).[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, Functional equations of Selberg and Ruelle zeta functions for non-unitarytwists , Ann. Global Anal. Geom. (2020), no. 1, 35–77. MR 4117920[Sp20b] P. Spilioti, Twisted ruelle zeta function and complex-valued analytic torsion ,arXiv:2004.13474 (2020).[V84] D. A. Vogan, Jr.,
Unitarizability of certain series of representations , Ann. of Math. (2) (1984), no. 1, 141–187. MR 750719 (86h:22028)[VZu84] 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)[Vo87] A. Voros, Spectral functions, special functions and the Selberg zeta function , Comm.Math. Phys. (1987), no. 3, 439–465. MR 891947[Wo08] A. Wotzke,
Die Ruellsche Zetafunktion und die analytische Torsion hyperbolischerMannigfaltigkeiten , Ph.D. thesis, Bonn, Bonner Mathematische Schriften (2008),no. Nr. 389.I
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