Selberg Zeta and Theta Functions

A Differential Operator Approach

Preface

The present book is devoted to the spectral geometric investigation of locally homogeneous vector bundles over compact locally symmetric spaces of rank one. These spaces have a simple enough geometry and we achieve a rather deep level of understanding of the relation between the geometry and the spectrum of geometric operators. The spectral and geometric data are initially given as a set of eigenvalues or lenghts closed geodesics. In order to handle this information it will be encoded by meromorphic functions. While the theta function is defined by the eigenvalues of suitable operators the definition of the Selberg zeta function uses the closed geodesics of the locally symmetric space. The relation of the spectrum with the geometry becomes apparent in the fact that the singularities of the theta function contain the information needed to define the Selberg zeta function while the singularities of the Selberg zeta function carry the spectral information. In the present book we present a self-contained exposition of the theory of the theta and the Selberg zeta function for all bundles over compact locally symmetric rank one spaces. We emphasize the role played by differential operators. As major technical tools we discuss the harmonic analysis on the compact dual symmetric spaces, the fundamental solution of wave equations, and the Selberg trace formula. While much of the theory was known to the experts before there are new results:

- the construction of differential operators associated to the theta function,

- a complete understanding of the singularities of the Selberg zeta function in terms of elliptic operators including the determinant expression, and

- a functional equation and determinant representations for twisted Ruelle zeta functions. We believe that our approach in itself is original in many aspects. The reader of this book should have some background in analysis (elliptic operators, index theory, wave equation), geometry (Riemannian manifolds, vector bundles, symmetric spaces) and harmonic analysis (representation theory of compact groups).

Acknowledgements : To write this book was possible only due to the constant help and the insights of Andreas Juhl. He guided our first steps into this complicated area and some important ideas were born in discussions with him. The Sonderforschungsbereich ”Differentialgeometrie und Quantenphysik” supported by the Deutsche Forschungsgemeinschaft provided us with a surrounding needed to concentrate on research work. We thank Ulrich Bröcker, Andreas Juhl, and Andreas Nestke, and in particular Werner Hoffmann for reading parts of the manuscript and indicating necessary corrections.

Berlin, March 1995 Ulrich Bunke and Martin Olbrich

Contents

0 Introduction 9

1 Elliptic Operators, Duality, and Trace Formulas 17

1.1 Bundles and operators . . . . . . . . . . . . . . . . . . . . . . . 17 1.1.1 Symmetric and locally symmetric spaces of rank 1 … 17 1.1.2 The restriction map i^{∗} : R(K) → R(M) . . . . . . . . . 18 1.1.3 Casimir and Dirac operators . . . . . . . . . . . . . . . 27

1.2 Wave kernels and duality . . . . . . . . . . . . . . . . . . . . . 30 1.2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . 30 1.2.2 Hadamard’s analysis . . . . . . . . . . . . . . . . . . . . 32 1.2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.2.4 An equivariant proportionality principle . . . . . . . . . 36

1.3 Harmonic analysis on compact symmetric spaces . . . . . . . . 39 1.3.1 The Cartan-Helgason Theorem for vector bundles … 39 1.3.2 The G_{d}-index theorem . . . . . . . . . . . . . . . . . . . 43 1.3.3 Admissible lifts . . . . . . . . . . . . . . . . . . . . . . . 47

1.4 The trace formula . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.4.1 The general scheme . . . . . . . . . . . . . . . . . . . . 51 1.4.2 The identity contribution . . . . . . . . . . . . . . . . . 54 1.4.3 The hyperbolic contribution . . . . . . . . . . . . . . . . 58 1.4.4 The distributional trace formula . . . . . . . . . . . . . 66

2 Theta Functions 69

2.1 Theta functions for compact locally symmetric rank-one spaces 69 2.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.1.2 The main theorem . . . . . . . . . . . . . . . . . . . . . 70

2.2 Theta functions for surfaces with cusps . . . . . . . . . . . . . . 76 2.2.1 Definition of the theta function . . . . . . . . . . . . . . 77 2.2.2 The Selberg trace formula . . . . . . . . . . . . . . . . . 78 2.2.3 Meromorphic continuation of θ_{Y} across (0, −i∞) …. 81 2.2.4 The structure of θ_{Y} (t) near (0, i∞) . . . . . . . . . . . . 82 2.2.5 The meromorphic continuation of Θ_{Y} . . . . . . . . . . 93

3 The Selberg Zeta Function 95 3.1 Zeta functions and the geodesic flow . . . . . . . . . . . . . . . 95 3.2 Traces of resolvents . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.2.1 Resolvent indentities . . . . . . . . . . . . . . . . . . . . 100 3.2.2 The logarithmic derivative of Z_{S,χ}(s, σ) and resolvents . 102 3.2.3 Meromorphic continuation of the identity contribution . 105 3.2.4 Continuation and singularities of L_{S,χ}(s, σ) and L_{S,χ}^{s}(s, σ) 108

3.3 Properties of the Selberg zeta function . . . . . . . . . . . . . . 112 3.3.1 Meromorphic continuation and singularities . . . . . . . 112 3.3.2 Functional equations . . . . . . . . . . . . . . . . . . . . 116 3.3.3 Determinants and zeta functions . . . . . . . . . . . . . 118

4 The Ruelle Zeta Function 127 4.1 The functional equations . . . . . . . . . . . . . . . . . . . . . . 127 4.2 The behaviour at the central point . . . . . . . . . . . . . . . . 134

4.2.1 Odd-dimensional hyperbolic manifolds and analytic torsion134 4.2.2 Complex hyperbolic manifolds and holomorphic torsion 138

5 Examples 147 5.1 The function case . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.2 Spinors on real hyperbolic manifolds . . . . . . . . . . . . . . . 148 5.3 Differential forms on real and complex hyperbolic manifolds . . 149 5.4 One-forms on quaternionic hyperbolic manifolds . . . . . . . . 155

6 Alternative Descriptions of the Singularities 159 6.1 Dynamical Lefschetz formulas . . . . . . . . . . . . . . . . . . . 159 6.2 Group cohomology of the principal series . . . . . . . . . . . . . 162

0 Introduction

The goal of the present book is a complete investigation of the theta functions and the Selberg zeta functions associated with locally homogeneous bundles over compact locally symmetric spaces of rank one. For this rather small class of spaces our understanding of the relation between the geometry of the space and the spectrum of geometric differential operators reached a much deeper level than for broader classes of manifolds considered in spectral geometry. Most of the insights are based on a very explicit trace formula. Though the theta functions and Selberg zeta functions can be defined in more general contexts they behave particularly nice for compact locally symmetric spaces of rank one. They have meromorphic continuations to the whole complex plane. Moreover, the singularities of one of these functions contain the information needed in order to define the other. On the one hand the geometric data related to the set of closed geodesics of the space is used to define the Selberg zeta function, and the spectrum of suitable elliptic differential operators on bundles over this space determines the theta function. On the other hand, the singularities of the theta function contain the geometric information, and the singularities of the Selberg zeta function reflect spectral data. Our knowledge of the behaviour of both functions together makes the spectral geometry of compact locally symmetric rank one spaces so transparent. The spectral geometry of the Laplace-Beltrami operator on functions on finite volume locally symmetric spaces of rank one is similarly well understood [25]. While some aspects of the theory generalize to a larger class of manifolds, the whole picture becomes far less clear. By now there is a huge literature on trace formulas and the Selberg zeta function. Thus, an argument for writing or reading a new book on this subject should be in order. Almost all papers, starting with the original work of Selberg [73], treat the case of hyperbolic Riemann surfaces, i.e., the two-dimensional case (we have not tried to make our list of references complete here). Most of these papers consider even the more general finite volume case. Compared with the number of papers studying the two-dimensional case there are only a few papers dealing with more general situations. The first generalization of the Selberg zeta function to higher dimensions was obtained for the spherical case by Gangolli [24] (see also [90], [92], [91]). Scott [70] worked out the bundle case for three-dimensional hyperbolic manifolds. Wakayama [82] constructed a Selberg zeta function associated to locally homogeneous vector bundles over locally symmetric spaces (see also [81], [68], [69], [54] for special cases). In all these papers the authors followed the original approach of Selberg. They first obtained a meromorphic continuation of the logarithmic derivative of the Selberg zeta function. This led to the problem of showing that all residues were integers which caused problems in the even-dimensional case (normalization of Plancherel measures). It was only possible to show that a certain integer power of the zeta function is meromorphic. The aforementioned authors also established a functional equation for the zeta function. It has been observed that there are two types of singularities of the Selberg zeta function. The spectral singularities are essentially located on the imaginary axis and obey a symmetry with respect to zero. The remaining singularities (called topological or trivial singularities) are located on the negative real axis. In the spherical case the spectral singularities and the topological singularities have completely been understood. The former were identified with the spectrum of the Laplace-Beltrami operator. The latter are connected with the singularities of the Plancherel measure of the universal covering space. For a Selberg zeta function associated with a locally homogeneous vector bundle over a locally symmetric space all singularities up to a finite number of exceptions have been understood in a similar way as in the spherical case . The problem of the meromorphic continuation of the Selberg zeta function itself (and not of an integer power) was first solved by Fried [22]. He observed the connection with hyperbolic dynamics. His idea was to express the Selberg zeta function in terms of the Ruelle zeta function (see Ruelle [64] and below) which was known to be meromorphic before. Fried also noticed that the Selberg zeta function should rather be associated with a locally homogeneous vector bundle over the unit sphere bundle of the locally symmetric space (the phase space of the geodesic flow) than with a locally homogeneous vector bundle over the locally symmetric space itself. The price to pay in this approach is that one loses any spectral interpretation of the singularities in terms of differential operators over the locally symmetric space as well as the functional equation. The Selberg zeta function associated with a bundle over a locally symmetric space introduced by Wakayama turned out to be essentially a finite product of Selberg zeta functions considered by Fried up to a finite number of trivial factors which unfortunately may contribute further singularities.

A completely different approach to the Selberg zeta function was found by Juhl [40]. He manifestly exhibited the residues of the logarithmic derivative of the Selberg zeta function (as defined by Fried, i.e., associated to a vector bundle over the unit-sphere bundle) as integers expressing them as Euler characteristics of certain Lie algebra cohomology groups. Juhl was able to obtain the functional equation of the Selberg zeta function up to a factor which is the exponential of a polynomial. He also gave a uniform spectral (index theoretic) interpretation of all singularities of the Selberg zeta function. But it was not clear whether there is a single operator being responsible for all spectral singularities. We describe the approach of Juhl in Section 6.1. Concerning the Selberg zeta function the goal of the present paper is to complete the approach via the trace formula. Consider a Selberg zeta function associated with a locally homogeneous vector bundle over the unit sphere bundle of a locally symmetric space. We associate two elliptic operators, one over the locally symmetric space and one over its compact dual symmetric space. The spectrum of both operators together can be identified with the set of singularities of the zeta function. Moreover, the residues of the logarithmic derivative are expressed in terms of multiplicities of eigenvalues and are integer in a manifest way. We also obtain the exact functional equation. The main idea is to employ a trace formula containing the duality of the non-compact symmetric space with its compact dual in an intrinsic way. We obtain a formula for the identity contribution avoiding the use of Plancherel measures and thus circumventing the problem of normalizations. The cleanest way to relate the Selberg zeta function with the spectrum of differential operators is to express it in terms of zeta-regularized determinants of these operators. This possibility was first observed in the surface case by D’Hoker/Phong [18] (see also [80], [5], [2], [42], and many others even for the finite volume case). Corresponding formulas were established in the spherical case by Williams [92], [91], Bröcker [8]. In this book we obtain a determinant expression in the general case, and our method relies on the example worked out by Bröcker [8] (for a similar formula see the recent [15]). Very recently we found a proof of a conjecture by Patterson claiming a relation between the singularities of the Selberg zeta function and the group cohomology of the fundamental group of the locally symmetric space with coefficients in certain principal series representations of the isometry group of the noncompact symmetric space. We explain this conjecture in Subsection 6.2. Its proof requires techniques which are quite different from those employed in the present book. The functional equation satisfied by the Selberg zeta function reflects the local geometry of the underlying space as well as certain global spectral invariants. The right-hand side of the functional equation contains a Plancherel measure (depending on the local geometry of the locally homogeneous vector bundle over the unit-sphere bundle) and the Euler characteristic (in the even dimensional case) or the volume of the locally symmetric space. In the case of odd dimensions also global spectral invariants, namely the eta-invariant of a Dirac operator associated to the situation, may appear in the functional equation as well as in the value of the zeta function at zero. This observation was first made by Millson [48] for the signature operator on odd-dimensional hyperbolic spaces. Moscovici/Stanton [49] found a generalization to a much larger class of higher rank locally symmetric spaces and operators. Using the results of [49] in [40] Juhl worked out the rank one case and introduced the super zeta function. In the present book we provide direct proofs of these results for rank one. The Ruelle zeta function is another zeta function which can be naturally associated with a hyperbolic dynamical system. It is initially defined by a product of local factors labeled by the closed orbits of the system. Its meromorphic continuation was obtained by Ruelle combining the theory of Fredholm determinants with symbolic dynamics. In the special case of locally symmetric spaces of rank one the hyperbolic dynamical system is the geodesic flow on the unit sphere bundle. It was observed and employed by Fried [22] that the Ruelle zeta function can be expressed as a finite product of Selberg zeta functions in this special case. Using the same observation Juhl worked out a Lie algebra cohomology description of the singularities of the Ruelle zeta function. In the even-dimensional case he obtained a functional equation for the Ruelle zeta function up to an exponential polynomial factor. As a consequence of our functional equation for the Selberg zeta functions in the present book we obtain the exact functional equation for the Ruelle zeta function in both the even and the odd-dimensional cases. For hyperbolic and complex hyperbolic spaces we also show which operators (acting on differential forms) are responsible for the singularities of the Ruelle zeta function. We reprove the results of Fried relating the analytic and holomorphic torsion with the value of the Ruelle zeta function at zero. We admit that many results concerning the Selberg zeta function presented in this book were announced by Fried ([21], [22], [23]), although the second part of [22] has never been published. The second main topic of the present book is the theta function. It is in some sense the dual object to the Selberg zeta function, as one can either start with the zeta function and deduce the properties of the theta function in a formal way (this is the approach of Cartier/Voros [11] and Juhl [40], [39]). Or one can start with the theta function and then obtain the properties of the Selberg zeta function in a similar formal way [91]. In the present book we study the theta function in a direct manner independently of the Selberg zeta function generalizing the approach of [9]. The theta function is the trace of the (complex) one-parameter group associated to the square root of a Laplacian acting on sections of vector bundles. The restriction of the theta function to the imaginary axis is the trace of the unitary one-parameter group. It has first been studied in the case of the Laplace-Beltrami operator over general compact Riemannian manifolds by Duistermaat/Guillemin [19]. In that paper and in its successors one was particularly interested in a description of the singularities in terms of closed geodesics. The fact that in the locally symmetric rank-one case the theta function is in fact a meromorphic function on the whole complex plane was first observed by Cartier/Voros [11] in the case of two dimensions. They also discovered a functional equation and a remarkable duality with the theta function of the compact dual symmetric space. In the general case the theta function and its graded analog, the super theta function, have been introduced by Juhl [40], [39]. Instead of using the spectrum of elliptic operators he employed the singularities of the Selberg zeta function as initial data. He established a functional equation and the duality relation and also calculated the value of the super theta function at zero as a certain eta-invariant. The building block which we add to the theory of the theta function is to supply the correct differential operators which give the spectral information needed to define the theta function. We also provide an alternative method for proving the meromorphic continuation and the functional equation. We show that both results are easy consequences of the Selberg trace formula, if it is stated in the form obtained in this book. Employing the small time analysis of the wave kernels we also give an explanation of the mysterious relation between the theta functions on the locally symmetric space and its compact dual. The theory of the Selberg zeta function in the finite volume case has been worked out mainly for surfaces, but the spherical rank-one case is also understood by Gangolli/Warner [25]. The bundle case has still to be done. The necessary explicit trace formula has been worked out by Hoffmann [35], [36]. Apparently, a theta function in the finite volume case has not been studied yet. So we added a section dealing with the theta function for finite-volume Riemann surfaces. We believe that the general case can be done with the trace formula of [36]. Though the general picture is expected to be similar to the one known from surfaces a detailed understanding might be interesting.

There are some developments towards the Selberg zeta function for more general manifolds of infinite volume, in particular for convex co-compact hyperbolic spaces in the spherical case [56], [29], [57], [60], [61]. Also there are generalizations to higher rank locally symmetric spaces [49], [50], [16]. All these zeta functions depend on one complex variable. We feel that the correct generalization of the zeta function (associated to a minimal parabolic subgroup) should depend on a complex parameter of dimension equal to the rank. This is suggested by the example of a product of two rank one spaces, where the theta and the Selberg zeta functions are simply the products of the corresponding functions of the two spaces. On the way to the theta and the Selberg zeta function we developed several technical tools which might be of independent interest. The most interesting is perhaps the way we compute the terms appearing in the trace formula. First of all we consider the trace formula as a formula for the distributional trace of the fundamental solution of a wave equation. There are two types of terms (called contributions) appearing in the trace formula. We evaluate the hyperbolic contributions using a Helgason type Fourier transform for homogeneous bundles over the symmetric space in question. But note that we do neither employ a Plancherel theorem nor an explicit Plancherel measure. In fact, we deduce a formula for the Plancherel measure from our computation of the remaining term in the trace formula, the identity contribution. It turns out that the identity contribution is a distribution which only depends on the local geometry of the bundle. Up to a volume factor it is nothing else but the distributional local trace of the fundamental solution of the wave equation on the symmetric space. The main observation is that this distribution, considered as a meromorphic function, is the same as the distributional trace of the fundamental solution of the wave equation on the compact dual symmetric space for a corresponding bundle evaluated at imaginary time. To see this we employ the small-time asymptotic expansion of the fundamental solution which was introduced by Hadamard and a sort of the Hirzebruch proportionality principle. In this way we have reduced the computation of the identity contribution to the spectral decomposition of a homogeneous differential operator on the compact dual symmetric space. To solve this problem we prove a generalization of the Cartan-Helgason Theorem (the original one gave the spectral decomposition the Laplace-Beltrami operator) to bundles. It provides an explicit description of the spectrum of the operator up to a finite number of eigenvalues. Almost all eigenvalues fit into a lattice and have a multiplicity given by an explicit polynomial. The finite exceptions can be handled using the equivariant index theorem. In fact, we managed to correct the differential operator such that these exceptions disappear. It is exactly this corrected differential operator and its analog on the non-compact symmetric space which describe the singularities of the Selberg zeta function and give the correct theta function.

Chapter 1

Elliptic Operators, Duality, and Trace Formulas

1.1 Bundles and operators

1.1.1 Symmetric and locally symmetric spaces of rank 1

Our basic object will be a compact locally symmetric Riemannian manifold (Y, g) with negative sectional curvature. Its universal covering X is a Riemannian symmetric space of rank one and hence, by a classical result, known to be a real, complex, or quaternionic hyperbolic space, or the hyperbolic Cayley plane. The group of orientation preserving isometries G′ of X is a connected semisimple Lie group of real rank one. Then X = G′/K′, where K′ is the stabilizer of some fixed base point x_{0} of X and is maximal compact in G′. The fundamental group of Y acts by covering transformations on X and gives rise to a discrete, co-compact, and torsion-free subgroup Γ ⊂ G′ such that Y = Γ\G′/K′. In order to include as many locally homogeneous vector bundles on Y as possible we consider a finite covering group G of G′ such that the embedding Γ ↪ G′ lifts to an embedding Γ ↪ G. Let K be the preimage of K′ under the covering. Again, K is maximal compact in G, and we have

X = G/K and Y = Γ\G/K .

We require G to be linear in order to have the possibility of complexification. Thus, G is a quotient of one of the groups

Spin(1, n), SU(1, m), Sp(1, m), F_{4}^{−20}

by a subgroup of its center. Note that the centers of the first three groups equal ℤ_{2}, ℤ_{m+1}, and ℤ_{2}, respectively, while the center of the last group is trivial. Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian (hence one-dimensional) subspace of 𝔭, and M the centralizer of 𝔞 in K with Lie algebra 𝔪. We normalize the Ad(G)-invariant inner product (., .) on 𝔤 to restrict to the metric on T_{x_{0}}X ≅ 𝔭. Since K acts transitively on the unit vectors in 𝔭 the unit sphere bundle SX of X may be represented as the homogeneous space G/M. Hence SY = Γ\G/M. The adjoint action of 𝔞 on 𝔤 determines the root system Φ(𝔤, 𝔞). By W := W (𝔤, 𝔞) ≅ ℤ_{2} we denote its Weyl group. Fix a system of positive roots Φ^{+}(𝔤, 𝔞) ⊂ Φ(𝔤, 𝔞) and let 𝔫 = ∑_{α∈Φ^{+}(𝔤,𝔞)} 𝔫_{α} be the sum of the root spaces corresponding to elements of Φ^{+}(𝔤, 𝔞) . Then we have the Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 and the corresponding Iwasawa decomposition of the group G = KAN. Define ρ ∈ 𝔞_{ℂ}^{∗} by

ρ := \frac{1}{2} ∑_{α∈Φ^{+}(𝔤,𝔞)}dim(𝔫_{α})α

and the positive Weyl chamber 𝔞^{+} to be the half line in 𝔞 on which the positive roots take positive values. Let A^{+} := exp(𝔞^{+}) ⊂ A. The symmetric space X has a compact dual space X_{d} which is obtained in the following way. The group G, being linear can be represented as a closed subgroup of GL(n, ℝ) for some n. The corresponding representation of 𝔤 on ℝ^{n} extends by linearity to a faithful representation of 𝔤_{ℂ} on ℂ^{n}. Let G_{ℂ} and G_{d} be the analytic subgroups of GL(n, ℂ) corresponding to 𝔤_{ℂ} and 𝔤_{d} := 𝔨 ⊕ 𝔭_{d}, respectively, where 𝔭_{d} := i𝔭. Then G_{d} is the compact real form of the complexification G_{ℂ} of G, and X_{d} := G_{d}/K. We normalize the metric on X_{d} in such a way that the multiplication by i induces an isometry between 𝔭 ≅ T_{x_{0}}X and 𝔭_{d} ≅ T_{x_{0}}X_{d}. X_{d} can be considered as the compact real form of the complex symmetric space X_{ℂ} = G_{ℂ}/K_{ℂ}, where K_{ℂ} has the obvious meaning. X_{d} does not depend on the choice of G and is either a sphere, a projective space over ℂ or the quaternions, or the projective Cayley plane.

1.1.2 The restriction map i^{∗} : R(K) → R(M)

The Selberg zeta function, as it will be defined in Section 3.1, is canonically associated with the geodesic flow of Y acting on SY = Γ\G/M. In fact, it depends on the lift of that flow to a certain locally homogeneous vector bundle over SY defined by a representation σ of M and a representation χ of Γ, often called twist. Since we want to study the zeta functions by means of analysis on Y we need to construct a bundle on Y such that its pullback to SY is the bundle the zeta function is associated with. This is possible only on the level of (equivalence classes of) ℤ_{2}-graded vector bundles. So we are led to the study of the restriction map i^{∗} : R(K) → R(M) induced by the embedding i : M ↪ K, where R(K) and R(M) are the representation rings over ℤ of K and M, respectively. We first consider the case where dim Y is odd. Then X is the real hyperbolic space Hℝ^{n}, K = Spin(n), M = Spin(n−1) or K = SO(n) and M = SO(n−1) for an odd integer n ≥ 3. Let w ∈ W be the non-trivial element. Then w acts on M̂, the set of irreducible complex representations of M, by

(wσ)(m) := σ(m_{w}^{−1}mm_{w}) m ∈ M, σ ∈ M̂,

where m_{w} is a representative of w in the normalizer of 𝔞 in K. This action extends to R(M) and determines a splitting (over ℤ[\frac{1}{2}]) into the ±1-eigenspaces of w

R(M) ⊗ ℤ [\frac{1}{2}] = R(M)^{+} ⊗ ℤ [\frac{1}{2}] ⊕ R(M)^{−} ⊗ ℤ [\frac{1}{2}] .

Any Cartan subalgebra 𝔱 of 𝔪 = 𝔰𝔬(n − 1) can also be considered via the inclusion 𝔪 ↪ 𝔨 as a Cartan subalgebra of 𝔨 = 𝔰𝔬(n). We choose

𝔱 := \{T_{μ} := ( \begin{matrix}0 & −μ_{1} \\ μ_{1} & 0 \end{matrix} ; ... ; \begin{matrix}0 & −μ_{k} \\ μ_{k} & 0 \end{matrix} ) | μ_{i} ∈ ℝ, k = \frac{n − 1}{2}\}

and denote by ν = (ν_{1}, …, ν_{k}) that element in i𝔱^{∗}, which sends T_{μ} to i(ν_{1}μ_{1} + … + ν_{k}μ_{k}). The corresponding Weyl groups are given by

W_{𝔨} = {permutations of coordinates possibly with sign changes} ,
W_{𝔪} = {permutations of coordinates with an even number

of sign changes} .

For the systems of positive roots

Φ^{+}(𝔨_{ℂ}, 𝔱) := {e_{i} ± e_{j}(i < j), e_{i}} and Φ^{+}(𝔪_{ℂ}, 𝔱) := {e_{i} ± e_{j}(i < j)} ,

where e_{i} := (0, …, 0, 1, 0, …, 0) with 1 as the i’th entry, we have

ρ_{𝔨} := \frac{1}{2} ∑_{α∈Φ^{+}(𝔨_{ℂ},𝔱)}α = (k − \frac{1}{2}, k − \frac{3}{2}, …, \frac{3}{2}, \frac{1}{2}) ,

ρ_{𝔪} := \frac{1}{2} ∑_{α∈Φ^{+}(𝔪_{ℂ},𝔱)}α = (k − 1, k − 2, …, 1, 0) .

Set ϱ := ρ_{𝔨} − ρ_{𝔪} = (\frac{1}{2}, …, \frac{1}{2}). Now by the theory of highest weights we can write

K̂ = \{γ_{ν} | \begin{matrix}ν_{1} ≥ … ≥ ν_{k} ≥ 0, ν_{i} ∈ ℤ, i = 1, …, k \\ or, if K = Spin(n), ν_{i} ∈ \frac{1}{2} + ℤ, i = 1, …, k \end{matrix} \}

and

M̂ = \{σ_{ν} | \begin{matrix}ν_{1} ≥ … ≥ ν_{k−1} ≥ |ν_{k}|, ν_{i} ∈ ℤ, i = 1, …, k \\ or, if M = Spin(n − 1), ν_{i} ∈ \frac{1}{2} + ℤ, i = 1, …, k \end{matrix} \} .

In this description the action of w on M̂ turns out to be: wσ_{(ν_{1},...,ν_{k})} = σ_{(ν_{1},...,−ν_{k})}. In particular, for the half-spin representations s^{±} = σ_{(1/2,...,±1/2)} of Spin(n−1) we have ws^{±} = s^{∓}. Let s be the spin representation of Spin(n). Then we have i^{∗}(s) = s^{+} + s^{−} The first part of the following proposition can already be found in Miatello/Vargas [47].

Proposition 1.1 The map i^{∗} has the following properties :

1. i^{∗} is a bijection between R(K) and R(M)^{+}.

2. If σ ∈ R(M)^{−}, then there exists a unique element γ ∈ R(Spin(n)) with σ = (s^{+} − s^{−})i^{∗}(γ) such that s · γ ∈ R(K).

3. More explicitly, if ν_{k} > 0, then

σ_{ν} − wσ_{ν} = (s^{+} − s^{−})i^{∗}(γ_{ν−ϱ}) .  (1.1)

In addition, the K-representation s⊗γ_{ν−ϱ} splits into two representations γ^{+}(ν) and γ^{−}(ν) such that

σ_{(ν_{1},...,ν_{k})} + wσ_{(ν_{1},...,ν_{k})} = i^{∗}(γ^{+}(ν) − γ^{−}(ν)) .  (1.2)

P r o o f . Let us denote by χ_{σ} (χ_{γ}) the character of a representation σ of Spin(n − 1) (γ of Spin(n)). We introduce the Weyl denominators

Δ_{K}(exp T) := ∏_{α∈Φ^{+}(𝔨_{ℂ},𝔱)}(e^{α(T)/2} − e^{−α(T)/2}) , T ∈ 𝔱,
Δ_{M}(exp T) := ∏_{α∈Φ^{+}(𝔪_{ℂ},𝔱)}(e^{α(T)/2} − e^{−α(T)/2}) .

The Weyl character formulas for M and K, respectively, are

χ_{σ_{ν}}(exp T) = \frac{∑_{s∈W_{𝔪}} det(s)e^{s(ν+ρ_{𝔪})(T)}}{Δ_{M}(exp T)} ,
χ_{γ_{ν}}(exp T) = \frac{∑_{s∈W_{𝔨}} det(s)e^{s(ν+ρ_{𝔨})(T)}}{Δ_{K}(exp T)} .

For a highest weight ν of Spin(n) we calculate

Δ_{M}(χ_{σ_{ν+ϱ}} − χ_{wσ_{ν+ϱ}})(exp T)
= ∑_{s∈W_{𝔪}} det(s)(e^{s(ν+ϱ+ρ_{𝔪})(T)} − e^{s(ν+(1/2,...,1/2,−2ν_{k}−1/2)+ρ_{𝔪})(T)})
= ∑_{s∈W_{𝔨}} det(s)e^{s(ν+ϱ+ρ_{𝔪})(T)}
= ∑_{s∈W_{𝔨}} det(s)e^{s(ν+ρ_{𝔨})(T)} .

In the special case ν = 0 this reduces, in view of the Weyl character formula, to

Δ_{M}(χ_{s^{+}} − χ_{s^{−}})(exp T) = Δ_{K}(exp T) .

The latter identity allows us to write the result for general ν as

(χ_{σ_{ν+ϱ}} − χ_{wσ_{ν+ϱ}})(exp T) = ((χ_{s^{+}} − χ_{s^{−}})χ_{γ_{ν}})(exp T) .  (1.3)

Thus we have proved (1.1). In view of (1.3) the map

R(K) ∋ γ → (s^{+} − s^{−})i^{∗}(γ) ∈ R(Spin(n − 1))

is injective, hence i^{∗} is injective, too. Since the action of w interchanges the M-isotypic components of an element in K̂, we see that the image of i^{∗} is contained in R(M)^{+}. In order to prove assertion 1. of the proposition it remains to construct an element in the preimage in R(K) for an arbitrary σ ∈ R(M)^{+}. Since (s^{+} − s^{−})σ ∈ R(Spin(n − 1))^{−}, by 3. we find an element γ ∈ R(Spin(n)), which actually belongs to R(K), such that

(s^{+} − s^{−})σ = (s^{+} − s^{−})i^{∗}(γ) .

As a polynomial ring R(Spin(n − 1)) is integral, hence σ = i^{∗}(γ). We are left with the proof of (1.2). We view {±1}^{k} as a subgroup of GL(i𝔱^{∗}) acting by sign changes. Let us consider the characters of s

χ_{s}(exp T) = ∑_{v∈{±1}^{k}} e^{vϱ(T)}

and of s ⊗ γ_{ν}

χ_{s}χ_{γ_{ν}}(exp T) = ∑_{v∈\{±1\}^{k}} ∑_{s∈W_{𝔨}} \frac{det(s)e^{(s(ν+ρ_{𝔨})+vϱ)(T)}}{Δ_{K}(exp T)}

= ∑_{v∈\{±1\}^{k}} ∑_{s∈W_{𝔨}} \frac{det(s)e^{s(ν+vϱ+ρ_{𝔨})(T)}}{Δ_{K}(exp T)}

= ∑ ; \{v∈\{±1\}^{k}|\begin{matrix}ν+vϱ+ρ_{𝔨} \\ regular in & 𝔱^{∗} \end{matrix}\} ∑_{s∈W_{𝔨}} \frac{det(s)e^{s(ν+vϱ+ρ_{𝔨})(T)}}{Δ_{K}(exp T)} . (1.4)

In the last step we have used that for singular ξ

∑_{s∈W_{𝔨}} det(s)e^{sξ(T)} ≡ 0 .

Observe that ν + vϱ is the highest weight of an irreducible representation of K iff ν + vϱ + ρ_{𝔨} is regular. Therefore we can define

γ^{±}(ν + ϱ) := ∑_{{v∈{±1}^{k} | ν+vϱ+ρ_{𝔨}regular in 𝔱^{∗}, det(v)=±1}}γ_{ν+vϱ} ,

and (1.4) shows that s ⊗ γ_{ν} = γ^{+}(ν + ϱ) ⊕ γ^{−}(ν + ϱ). For ξ = (ξ_{1}, …, ξ_{k}) we set ξ^{−} := (ξ_{1}, …, ξ_{k−1}, −ξ_{k}). Let p be the projection p : W_{𝔨} → {±1}^{k} given by p(s)ϱ := sϱ. Then we have

(Δ_{K}(χ_{σ_{ν+ϱ}} + χ_{wσ_{ν+ϱ}}))(exp T)
= (Δ_{M}(χ_{s^{+}} − χ_{s^{−}})(χ_{σ_{ν+ϱ}} + χ_{wσ_{ν+ϱ}}))(exp T)

= ( ∑_{v∈{±1}^{k}} det(v)e^{vϱ(T)})( ∑_{s∈W_{𝔪}} det(s)(e^{s(ν+ρ_{𝔨})(T)} + e^{s((ν+ρ_{𝔨})^{−})(T)}))

= ( ∑_{v∈{±1}^{k}} det(v)e^{vϱ(T)})( ∑_{s∈W_{𝔨}} det(s) det(p(s))e^{s(ν+ρ_{𝔨})(T)})
= ∑_{v∈{±1}^{k}} ∑_{s∈W_{𝔨}} det(v) det(s)e^{s(ν+vϱ+ρ_{𝔨})(T)}
= (Δ_{K}(χ_{γ^{+}(ν)} − χ_{γ^{−}(ν)}))(exp T) .

Dividing by Δ_{K} and performing a ϱ-shift we obtain (1.2). □

We now turn to the even-dimensional symmetric spaces. The isotropy representation gives rise to a homomorphism l : K → SO(𝔭). Let K̃ be the connected component of e in p^{−1}(l(K)) ⊂ Spin(n), where p : Spin(n) → SO(n) is the canonical twofold covering. We denote the restriction of the spin and half-spin representations of Spin(n) to K̃ also by s and s^{±}, respectively. For the present case we have

Proposition 1.2 The map i^{∗} is surjective. Moreover, if γ ∈ ker i^{∗} ⊂ R(K), then there exists an element τ ∈ R(K̃) such that s ⊗ τ is defined on K and

γ = [s^{+} ⊗ τ] − [s^{−} ⊗ τ] ∈ R(K) .

P r o o f . For the assertion concerning the kernel of i^{∗} we refer to Miatello [46], Lemma 2.2. (see also [47], Lemma 2.6.). Also the surjectivity of i^{∗} was already proved by Miatello/Vargas [47], essentially by a case by case discussion. Since this surjectivity assertion is crucial for the later developments, we present here a similar discussion for the convenience of the reader. We shall show that the generators of R(M) lie in the image of i^{∗}.

The real case The two-dimensional case is clear. We may represent the real hyperbolic space X = Hℝ^{2m}, m ≥ 2 either as X = Spin(1, 2m)/Spin(2m) or as X = SO_{0}(1, 2m)/SO(2m). Let us begin with the first case. Then

K = Spin(2m) and M = Spin(2m − 1) .

Let λ_{p} := Λ^{p}ℝ^{2m} ⊗ ℂ be the exterior powers of the standard representation of SO(2m) and σ_{p} := Λ^{p}ℝ^{2m−1} ⊗ ℂ the ones of SO(2m − 1), and let s^{+} and s^{−} be the half spin representations of Spin(2m) as before, and s_{0} the spin representation of Spin(2m − 1). Then

R(K) = ℤ[λ_{1}, …, λ_{m−2}, s^{+}, s^{−}] and R(M) = ℤ[σ_{1}, …, σ_{m−2}, s_{0}] ,

and we have

i^{∗}(λ_{1}) = σ_{1}+1 , i^{∗}(λ_{p}) = σ_{p}+σ_{p−1}, p = 2, …, m−2 , i^{∗}(s^{+}) = i^{∗}(s^{−}) = s_{0} .

Now preimages are easy to find, namely

σ_{p} = i^{∗}(λ_{p} − λ_{p−1} + … + (−1)^{p−1}(λ_{1} − 1)) , p = 1, …, n − 2,
s_{0} = i^{∗}(s^{+}) = i^{∗}(s^{−}) .  (1.5)

In the second case, i.e.,

K = SO(2m) and M = SO(2m − 1) ,

the representation ring of M is given by

R(M) = ℤ[σ_{1}, …, σ_{m−2}, σ_{m−1}] .

As in (1.5) all its generators arise as restrictions of elements of R(SO(2m)).

The complex case The isometry group of the complex hyperbolic space X = Hℂ^{m}, m ≥ 2, is P U(1, m) ≅ SU(1, m)/ℤ_{m+1}, whose maximal linear covering group is SU(1, m). Thus, for any divisor k of m + 1, we may write X = G/K, where

G = SU(1, m)/ℤ_{k} , K = S(U(1) × U(m))/ℤ_{k} .

The corresponding group M is

M = {(z, B) ∈ U(1) × U(m − 1) | z^{2} det(B) = 1}/ℤ_{k} ,

where the embedding M ↪ K is given by

[(z, B)] ⟶ [(z, ( \begin{matrix}z & 0 \\ 0 & B \end{matrix} ))] .

We consider K as an \frac{m+1}{k}-fold covering of U(m) given by

π : K ∋ [(w, A)] → w^{−1}A ∈ U(n) .

Note that π(M) = U(m − 1). Let λ_{p} := Λ^{p}ℂ^{m} be the complex exterior powers of the standard representation of U(m) and σ_{p} := Λ^{p}ℂ^{m−1} the ones of U(m − 1). By means of π we consider them as representations of K and M, respectively. Then

R(K) = ℤ[λ_{1}, …, λ_{m−1}, w^{k}, w^{−k}] and R(M) = ℤ[σ_{1}, …, σ_{m−2}, z^{k}, z^{−k}] ,

where w, z denote the one-dimensional representations (w, A) → w, (z, B) → z. The restrictions of the generators of R(K) are

i^{∗}(w^{±k}) = z^{±k} ,
i^{∗}(λ_{1}) = σ_{1} + 1 ,
i^{∗}(λ_{p}) = σ_{p} + σ_{p−1}, p = 2, …, m − 2 ,
i^{∗}(λ_{m−1}) = z^{−(m+1)} + σ_{m−2} .

Hence the generators of R(M) can be written as restrictions:

z^{±k} = i^{∗}(w^{±k}) ,
σ_{p} = i^{∗}(λ_{p} − λ_{p−1} + … + (−1)^{p−1}(λ_{1} − 1)) , p = 1, …, m − 2 .

The quaternionic case We may represent the quaternionic hyperbolic space X = Hℍ^{m}, m ≥ 2, either as X = Sp(1, m)/Sp(1) × Sp(m) or as X = (Sp(1, m)/ℤ_{2})/Sp(1) · Sp(m). Let us begin with the first case. Then

K = Sp(1) × Sp(m) and M ≅ Sp(1) × Sp(m − 1) ,

where the embedding Sp(1) × Sp(m − 1) ↪ Sp(1) × Sp(m) is given by

Sp(1) × Sp(m − 1) ∋ (q, B) ⟶ (q, ( \begin{matrix}q & 0 \\ 0 & B \end{matrix} )) ∈ Sp(1) × Sp(m) .

Let λ_{p} := Λ^{p}ℂ^{2m} be the complex exterior powers of the standard representation of Sp(m) on ℍ^{m} ≅ ℂ^{2m} and σ_{p} := Λ^{p}ℂ^{2m−2} the ones of Sp(m−1). Note that for p > 1 they are not irreducible. Nevertheless they generate the representation ring of Sp(n). Extend them trivially to Sp(1) × Sp(m) and Sp(1) × Sp(m − 1), respectively. Then

R(K) = ℤ[p, λ_{1}, …, λ_{n}] and R(M) = ℤ[q, σ_{1}, …, σ_{n−1}] ,

where p (resp. q) denotes the standard representation of Sp(1) on ℍ = ℂ^{2} trivially extended to Sp(1)×Sp(m) (resp. Sp(1)×Sp(m−1)). The restrictions of the generators of R(K) are

i^{∗}(p) = q ,
i^{∗}(λ_{1}) = σ_{1} + q ,
i^{∗}(λ_{2}) = σ_{2} + qσ_{1} + 1 ,
i^{∗}(λ_{p}) = σ_{p} + qσ_{p−1} + σ_{p−2}, p = 3, …, m − 1 ,
i^{∗}(λ_{n}) = qσ_{m−1} + 2σ_{m−2} .

Preimages of the generators of R(M) can now inductively be determined by

q = i^{∗}(p) ,
σ_{1} = i^{∗}(λ_{1} − p) ,  (1.6)
σ_{p} = i^{∗}(λ_{p}) − qσ_{p−1} − σ_{p−2}, p = 2, …, m − 1 .

In the second case, i.e.,

K′ = Sp(1) · Sp(m) ≅ Sp(1) ×_{ℤ_{2}} Sp(m) and M′ ≅ Sp(1) ×_{ℤ_{2}} Sp(m − 1) ,

the representation ring of K′ is the subring of R(K) generated by

{p^{2}, λ_{i}, pλ_{j}, λ_{j}λ_{k} | i even, j, k odd, 1 ≤ i, j, k ≤ n} .

Analogously, the representation ring of M′ is the subring of R(M) generated by

{q^{2}, σ_{i}, qσ_{j}, σ_{i}σ_{j} | i even, j, k odd, 1 ≤ i, j, k ≤ n − 1} .  (1.7)

Call an element in K̂ (M̂) even, if (−1, −id) acts trivially, and odd otherwise. Then we have

K̂′ = {γ ∈ K̂ | γ even} , M̂′ = {σ ∈ M̂ | σ even} .

Thus

R(K′) = { ∑_{{γ∈K̂ | γ even}} a_{γ}γ | a_{γ} ∈ ℤ} , R(M′) = { ∑_{{σ∈M̂ | σ even}} a_{σ}σ | a_{σ} ∈ ℤ} .

From (1.6) we see that any even (odd) generator of R(M) can be expressed as the restriction of a weighted sum of even (odd) elements in K̂. We conclude that all generators in (1.7) arise as restrictions of a weighted sum of even elements in K̂, hence belong to i^{∗}(R(K′)).

The exceptional case The unique presentation of the hyperbolic Cayley plane X = Hℂa^{2} is

X = F_{4}^{−20}/Spin(9) .

Then K = Spin(9) and M is the stabilizer of a non-zero element in the spinor module S_{9} of Spin(9), which is isomorphic to the isotropy representation on T_{x_{0}}Hℂa^{2}. Note that M ≅ Spin(7). Let λ_{p} := Λ^{p}ℝ^{9} ⊗ ℂ be the exterior powers of the standard representation of SO(9) and σ_{p} := Λ^{p}ℝ^{7} ⊗ ℂ the ones of SO(7), and let s_{9} and s_{7} be the spin representations of Spin(9) and Spin(7), respectively. Then

R(K) = ℤ[λ_{1}, λ_{2}, λ_{3}, s_{9}] and R(M) = ℤ[σ_{1}, σ_{2}, s_{7}] ,

and one can show that

i^{∗}(λ_{1}) = s_{7} + 1 , i^{∗}(λ_{2}) = σ_{2} + σ_{1} + s_{7} , i^{∗}(s_{9}) = s_{7} + σ_{1} + 1 .

Preimages of the generators of R(M) are easy to find, namely

σ_{1} = i^{∗}(s_{9} − λ_{1}) , σ_{2} = i^{∗}(λ_{2} − s_{9} + 1) , s_{7} = i^{∗}(λ_{1} − 1) .
□

1.1.3 Casimir and Dirac operators

In this subsection we construct for each σ ∈ M̂ certain bundles on X, Y, and X_{d}. We have to distinguish between two cases.

Case (a): σ ∈ M̂ is invariant under the action of the Weyl group W. Note that this case includes all σ, if dim X is even. By Propositions 1.1 and 1.2 we find an element γ ∈ R(K) such that i^{∗}(γ) = σ. Moreover, γ is uniquely determined by this condition, iff dim X is odd. To γ we associate ℤ_{2}-graded homogeneous vector bundles V (γ) = V (γ)^{+} ⊕ V (γ)^{−} and V_{d}(γ) = V_{d}(γ)^{+} ⊕ V_{d}(γ)^{−} on X and X_{d}, respectively, in the following way. Represent γ by

∑ a_{i}γ_{i} ,  a_{i} ∈ ℤ, γ_{i} ∈ K̂  (1.8)

and set

V_{γ}^{±} := ∑_{sign(a_{i})=±1} ∑_{m=1}^{|a_{i}|} V_{γ_{i}} ,

where V_{γ_{i}} is the representation space of γ_{i}. We define

V (γ)^{±} := G ×_{K} V_{γ}^{±}  and  V_{d}(γ)^{±} = G_{d} ×_{K} V_{γ}^{±} .

Since we do not require that γ_{i} ≠ γ_{j} for i ≠ j, the representation (1.8) is not unique. So we have only defined an equivalence class of a ℤ_{2}-graded homogeneous vector bundle with the obvious equivalence relation. By abuse of notation V_{(d)}(γ) stands for a particular bundle in that class. Furthermore, if (χ, V_{χ}) is a finite-dimensional unitary representation of Γ, then we define

V_{Y,χ}(γ) := Γ\(V_{χ} ⊗ V (γ)) ,

which is a ℤ_{2}-graded vector bundle on Y. If χ is the trivial one-dimensional representation, then we sometimes omit the index χ. As in the beginning of Subsection 1.2 we choose a Cartan subalgebra 𝔱 of 𝔪 and a system of positive roots Φ^{+}(𝔪_{ℂ}, 𝔱). Then ρ_{𝔪} ∈ i𝔱^{∗} and the highest weight μ_{σ} ∈ i𝔱^{∗} of σ are defined. Set

c(σ) := |ρ|^{2} + |ρ_{𝔪}|^{2} − |μ_{σ} + ρ_{𝔪}|^{2} ,  (1.9)

where the norms are induced by the complex bilinear extension to 𝔤_{ℂ} of the inner product (., .) defined in Subsection 1.1. This inner product also fixes the Casimir element Ω of the complex universal enveloping algebra U(𝔤) = U(𝔤_{d}) of 𝔤. Ω acts in a natural way on sections of the bundles V (γ) and V_{d}(γ). Since Ω acts as a G-invariant differential operator on C^{∞}(X, V (γ)) it descends to C^{∞}(Y, V_{Y,χ}(γ)). We define the operators

\begin{aligned}A(γ, σ)^{2} & := & −Ω − c(σ) & : & C^{∞}(X, V (γ)) & → & C^{∞}(X, V (γ)) , \\ A_{d}(γ, σ)^{2} & := & Ω + c(σ) & : & C^{∞}(X_{d}, V_{d}(γ)) & → & C^{∞}(X_{d}, V_{d}(γ)) , \\ A_{Y,χ}(γ, σ)^{2} & := & −Ω − c(σ) & : & C^{∞}(Y, V_{Y,χ}(γ)) & → & C^{∞}(Y, V_{Y,χ}(γ)) . \end{aligned}

These operators give rise to unbounded selfadjoint operators on the corresponding Hilbert spaces of L^{2}-sections. We are particularly interested in their spectral-theoretic square roots

A(γ, σ) := \sqrt{A(γ, σ)^{2}}, A_{d}(γ, σ) := \sqrt{A_{d}(γ, σ)^{2}}, A_{Y,χ}(γ, σ) := \sqrt{A_{Y,χ}(γ, σ)^{2}} .

The spectrum of A(γ, σ)^{2}, A_{d}(γ, σ)^{2} and A_{Y,χ}(γ, σ)^{2} is bounded from below. For the possible negative values in the spectrum we take the root with positive imaginary part.

Case (b): σ ∈ M̂ is not invariant under the action of the Weyl group. In this case we have X = Hℝ^{n}, n odd. By Proposition 1.1 there is a unique element γ′ ∈ Spin(n)^{∧} and a splitting s ⊗ γ′ = γ^{+} ⊕ γ^{−}, where s is the spin representation of Spin(n) and γ^{±} are representations of K, such that for the non-trivial element w ∈ W

σ − wσ = sign(ν_{k})(s^{+} − s^{−})i^{∗}(γ′)

and

σ + wσ = i^{∗}(γ^{+} − γ^{−}) ,

where ν_{k} is the last coordinate of the highest weight of σ (see Subsection 1.1.2. Set γ := γ^{+} − γ^{−} ∈ R(K) and γ^{s} := γ^{+} + γ^{−} ∈ R(K). We define the bundles

V (γ), V_{d}(γ), V_{Y,χ}(γ), V (γ^{s}), V_{d}(γ^{s}), V_{Y,χ}(γ^{s})

and the operators

A(γ, σ), A_{d}(γ, σ), A_{Y,χ}(γ, σ), A(γ^{s}, σ), A_{d}(γ^{s}, σ), A_{Y,χ}(γ^{s}, σ)

in the same way as in the case (a). By definition, V (γ^{s}), V_{d}(γ^{s}) ,V_{Y,χ}(γ^{s}) are Clifford bundles, hence they carry Dirac operators D(σ), D_{d}(σ), D_{Y,χ}(σ). In order to make these Dirac operators unique we have to fix the sign of the Clifford multiplication c : 𝔭 ⊗ S → S, where S is the spinor module, i.e., the representation space of s. Let H be the unit vector in 𝔞^{+}. Since H is M-invariant, it respects the decomposition S = S^{+} ⊕ S^{−} with respect to Spin(n − 1). Then we fix the sign of the Clifford multiplication by requiring that H acts as multiplication by ±sign(ν_{k})i on S^{±} ⊂ S. We extend c to 𝔭 ⊗ V_{γ^{s}} (𝔭 ⊗ V_{γ^{s}} ⊗ V_{χ}), choose an orthonormal base {X_{i}} of 𝔭 and look at a section of V_{∗}(γ^{s}), ∗ = ∅, (Y, χ), as a vector valued function f on G with the corresponding invariance properties. Then we define

D_{∗}(σ)f(g) := ∑_{i=1}^{n} \frac{∂}{∂t}_{|t=0}c(X_{i})f(g exp(tX_{i})) .  (1.10)

We define D_{d}(σ) by the same formula, where we fix the sign of the Clifford multiplication c : 𝔭_{d} ⊗ S → S by the same condition as above using iH instead of H, {X_{i}} is an orthonormal base of 𝔭_{d}, and f ∈ C^{∞}(X_{d}, V_{d}(γ^{s})) should be viewed as a vector valued function on G_{d}. Note that by the Parthasarathy formula (see [55])

D(σ)^{2} = −Ω − |ρ|^{2} − |ρ_{𝔪}|^{2} + |ρ_{𝔨}|^{2} + γ′(Ω_{K})
= −Ω − |ρ|^{2} − |ρ_{𝔪}|^{2} + |μ_{γ′} + ρ_{𝔨}|^{2}

=-Ω − |ρ|^{2} − |ρ_{𝔪}|^{2} + |μ_{σ} + ρ_{𝔪}|^{2}(1.11) =A(γ^{s}, σ)^{2} ,and since the Dirac operators are self-adjoint, we have A(γ^{s}, σ) = |D(σ)| as well as A_{Y,χ}(γ^{s}, σ) = |D_{Y,χ}(σ)| and A_{d}(γ^{s}, σ) = |D_{d}(σ)|. Here Ω_{K} is the Casimir operator of K and μ_{γ′} ∈ i𝔱^{∗} denotes the highest weight of γ′. We have employed the fact that |μ_{γ′} + ρ_{𝔨}|^{2} = |μ_{σ} + ρ_{𝔪}|^{2} by (1.1).

Now we turn to the definition of the multiplicities m_{χ}(s, γ, σ), m_{d}(s, γ, σ) and m_{χ}^{s}(s, σ). They enter into the definition of the theta function and eventually describe the singularities of the Selberg zeta function. The multiplicities are weighted dimensions of eigenspaces of the operators just defined. The Hilbert spaces of L^{2}-sections of V_{∗}(γ^{(s)}) inherit ℤ_{2}-gradings from the underlying bundles. If B is a trace-class operator on a ℤ_{2}-graded Hilbert space, then we denote by Tr B the super trace corresponding to the ℤ_{2}-grading.

Definition 1.3 Let E_{A}(.) be the family of spectral projections of a normal operator A. We define for s ∈ ℂ

m_{χ}(s, γ, σ) := Tr E_{A_{Y,χ}(γ,σ)}({s}) , m_{d}(s, γ, σ) := Tr E_{A_{d}(γ,σ)}({s}) ,
m_{χ}^{s}(s, σ) := Tr(E_{D_{Y,χ}(σ)}({s}) − E_{D_{Y,χ}(σ)}({−s}) .

Note that the multiplicities do not depend on the choice of the representative (1.8) of γ, while in the case of an even-dimensional X they still depend on the lift γ of σ. In order to obtain multiplicities which are canonically associated to σ we will distinguish special lifts γ called admissible in Subsection 1.3.3.

1.2 Wave kernels and duality

1.2.1 The wave equation

We recall basic properties of the wave kernel. Most of the material can be found in standard text books like [37] and [77]. Let E(t, x, y) be the fundamental solution of the Cauchy problem

(\frac{d^{2}}{dt^{2}} + A(γ, σ)^{2}) f(t, x) = 0 ,
f(0, x) = f_{0}(x) ,
\frac{d}{dt}f(t, x)_{|t=0} = 0 .

For fixed t it is the distributional kernel of the operator cos(tA(γ, σ)). This operator has finite propagation and is well-defined on C^{∞}(X, V (γ)).

For ϕ ∈ C_{c}^{∞}(ℝ) let E_{ϕ} be defined by

E_{ϕ} := ∫_{ℝ} ϕ(t) cos(tA(γ, σ))dt .

This operator has a smooth integral kernel. By the G-invariance of A(γ, σ) the local trace tr E_{ϕ}(x, x) of this kernel is independent of x ∈ X. Here tr is the trace in the fibre of End(V(γ)). We define the distribution E(t, γ, σ) ∈ C_{c}^{∞}(ℝ)^{∗} by

E(., γ, σ) : C_{c}^{∞}(ℝ) ∋ ϕ ↦ tr E_{ϕ}(x, x) .

Then E(t, γ, σ) is independent of x ∈ X, too. X is simply connected of negative sectional curvature and A(γ, σ)^{2} is of Laplace type with real analytic coefficients. All geodesics starting at x ∈ X go to infinity. By the standard results about the propagation of singularities for hyperbolic operators the singularities of E(t, y, x) are located on the light cone {(t, y)|dist(y, x) = |t|}.

Lemma 1.4 The distribution E(t, γ, σ) is real analytic for t ≠ 0.

Moreover, E(t, γ, σ) has a singularity at t = 0 and an convergent asymptotic expansion (see Lemma 1.6)

E(t, γ, σ) ∼^{t→0} ∑_{k=−n}^{∞} b_{k}|t|^{k}

which will be studied in Subsection 1.2.2. Here n = dim(X). The distributional nature of the singularity at t = 0 and the coefficients b_{k} completely determine the distribution E(t, γ, σ). This allows us to compute E(t, γ, σ) explicitly using the analysis on the compact dual symmetric space. Let E_{d}(t, x_{d}, y_{d}) be the fundamental solution of the Cauchy problem

(\frac{d^{2}}{dt^{2}} + A_{d}(γ, σ)^{2}) f(t, x_{d}) = 0 ,
f(0, x_{d}) = f_{0}(x_{d}) ,
\frac{d}{dt}f(t, x_{d})_{|t=0} = 0 .

For fixed t it is the distributional kernel of the operator cos(tA_{d}(γ, σ)). Choose some x_{d} ∈ X_{d} and let E_{d,ϕ} be the operator

E_{d,ϕ} := ∫_{ℝ} ϕ(t) cos(tA_{d}(γ, σ)) dt .

This operator has a smooth integral kernel and we define the distribution E_{d}(t, γ, σ) by

E_{d}(., γ, σ) : C_{c}^{∞}(ℝ) ∋ ϕ ↦ tr E_{d,ϕ}(x_{d}, x_{d}) ,

where the trace is taken on End(V_{d}(γ)). This definition is independent of x_{d} ∈ X_{d}. Since X_{d} has closed geodesics the distribution E_{d}(t, γ, σ) is real analytic only on [−ϵ, ϵ] \ {0} for ϵ > 0 small enough. Moreover, it has a singularity at t = 0 and a convergent asymptotic expansion

E_{d}(t, γ, σ) ∼^{t→0} ∑_{k=−n}^{∞} b_{k,d}|t|^{k} .

1.2.2 Hadamard’s analysis

In this subsection we determine the distributional nature of the singularity of both E(t, γ, σ) and E_{d}(t, γ, σ) at t = 0. We will also obtain some information about the asymptotic coefficients b_{k}, b_{k,d}. All that will be a consequence of general local results concerning fundamental solutions of wave equations. For k ≥ 0 let τ^{k} be the regular distribution on ℝ given by the function τ^{k}(t) := t^{k}. Define τ^{−1} by

τ^{−1}(t) := \frac{d}{dt} ln |t|

and

τ^{−k−1} := −\frac{1}{k} \frac{d}{dt}τ^{−k} .

Let 0 ∈ U ⊂ ℝ^{n} be an open subset. We consider an even second order differential operator H acting on functions on U with values in some ℤ_{2}-graded vector space W. Then there are smooth End(W)^{ev}-valued functions a_{i,j}, a_{j}, a, i, j = 1, …, n, such that

H := ∑_{i,j=1}^{n} a_{i,j}D_{i}D_{j} + ∑_{j=1}^{n} a_{j}D_{j} + a ,

where D_{j} := ∂/∂x^{j}. We assume that H is formally selfadjoint and that its principal symbol is positive. More precisely there is a C < ∞ such that for all x ∈ U and all ξ = (ξ^{1}, …, ξ^{n}) ∈ ℝ^{n}

C^{−1}|ξ|^{2} ≤ − ∑_{i,j=1}^{n} a_{i,j}(x)ξ^{i}ξ^{j} ≤ C|ξ|^{2}.

One should think of H as a model for A(γ, σ) or A_{d}(γ, σ) in a local trivialization. Let E_{H}(t, x, y) be the distributional kernel of the solution operator of the Cauchy problem

(\frac{d^{2}}{dt^{2}} + H) f(t, x) = 0 ,
f(0, x) = f_{0}(x) ,  (1.12)
\frac{d}{dt}_{|t=0}f(t, x) = 0 .

If f_{0} ∈ C_{c}^{∞}(U, W) is supported near 0, then the solution of problem (1.12) exists and is unique for small |t|. One can use Hadamard’s construction in order to obtain approximations of E_{H}(t, x, y) for small |t|. There is an immediate generalization of Hörmander [37], Thm. 17.5.5, to operators acting on vectorvalued functions. The following result is deduced from this generalization.

Proposition 1.5 Let x ∈ U. If n is even, then the distribution t ↦ trE_{H}(t, x, x) has an asymptotic expansion

tr E_{H}(t, x, x) ∼^{t→0} ∑_{k=−n/2}^{∞} c_{2k}(x)τ^{2k} .

If n is odd, then we have the asymptotic expansion

tr E_{H}(t, x, x) ∼^{t→0} ∑_{k=−(n−1)/2}^{0} c_{2k}(x)δ^{(2k)}(t) + ∑_{k=0}^{∞} c_{2k+1}(x)|t|^{2k+1} .

The functions c_{k} are homogeneous differential polynomials of degree k + n in the coefficients of the operator H.

E x p l a n a t i o n. Let D_{j} have degree 1 and H have degree 2. Then the degrees of a_{i,j}, a_{i} and a are 0, 1 and 2. The function c_{k} is a polynomial in a_{i,j}, a_{i}, a and their derivatives. The total degree of a monomial in c_{k} is the sum of the degrees of the functions involved and the total number of differentiations. The c_{k} are related to the solutions of the transport equation [37], Eq. 17.4.6. Tracing back degrees we obtain that c_{k} is homogeneous of degree k+n.□

Now assume that the coefficients of the differential operator H are real analytic. For completeness we indicate a proof of the following lemma (see Hadamard [30]).

Lemma 1.6 The asymptotic expansion of the local trace tr E_{H}(t, y, y) of the fundamental solution of the problem (1.12) is convergent for small t > 0.

P r o o f . Assume that n is even. By Hadamard’s method we construct an approximate fundamental solution Ẽ_{H}(t, x, y). Then Ẽ_{H}(t, x, y) satisfies the initial conditions. Moreover,

(\frac{d^{2}}{dt^{2}} + H) Ẽ_{H}(t, x, y) = h(t, x, y)

for some h(t, x, y) which is real analytic for (t, x) inside the forward light cone near the tip (0, y) (we consider a fixed y ∈ U). Then h(t, x, y) extends to a whole neighbourhood of (0, y). Let h_{ex}(t, x, y) denote this extension. We solve the Cauchy problem (\frac{d^{2}}{dt^{2}} + H)V (t, x, y) = −h_{ex}(t, x, y) with zero initial condition. Applying the Cauchy-Kowalewskaja Theorem we obtain a unique solution V (t, x, y) which is real analytic in (t, x) near (0, y). Then E_{H}(t, x, y) = Ẽ_{H}(t, x, y) + χ(t, x, y)V (t, x, y), where χ is the characteristic function of the local forward light cone with tip (0, y). The local trace trẼ_{H}(t, y, y) is a Laurent polynomial. Hence trE_{H}(t, y, y) for small t > 0 is given by a convergent Laurent expansion. For odd n the argument is similar. □

1.2.3 Duality

In this subsection we compare E(t, γ, σ) and E_{d}(t, γ, σ). It follows that once we know E_{d}(t, γ, σ) we also understand E(t, γ, σ). It is here where we avoid much of the harmonic analysis needed in other approaches in order to compute E(t, γ, σ).

Lemma 1.7 We have the following relation between b_{k} and b_{d,k}:

b_{d,k} = i^{k+n}b_{k}.

Proof. The lemma is a consequence of the fact that A(γ, σ)^{2} and −A_{d}(γ, σ)^{2} are restrictions to real submanifolds of the same holomorphic differential operator A(γ, σ)^{2}_{ℂ}. We choose coordinates near [e] ∈ G/K = X using the exponential map. Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition of the Lie algebra 𝔤. We identify 𝔭 = ℝ^{n} by specifying a basis in 𝔭. The composition of the exponential map with the projection

f : 𝔭 ⟶^{exp} G → G/K = X

provides coordinates in a neighbourhood of [e]. Let V_{γ} be the fibre of the bundle V (γ) over [e]. We trivialize the bundle V (γ) in a neighbourhood of [e] using the radial parallel transport with respect to the invariant connection, i.e., the fibre of V (γ) over f(p), p ∈ 𝔭, is identified with V_{γ} using the parallel transport along the path f(tp), t ∈ [0, 1]. We can write the operator A(γ, σ)^{2} in the form

A(γ, σ)^{2} = ∑_{i,j=1}^{n} a_{i,j}D_{i}D_{j} + ∑_{j=1}^{n} a_{j}D_{j} + a ,

where a_{i,j}, a_{j}, a are functions on 𝔭 with values in End(V_{γ})^{ev}. The Lie algebra of G_{d} is 𝔤^{d} = 𝔨 ⊕ i𝔭. We compose the exponential map with the projection in order to obtain coordinates on a neighbourhood of [e] ∈ X_{d}:

f̃: i𝔭 ⟶^{exp} G_{d} → X_{d} .

The fibre of V_{d}(γ) over [e] is again canonically isomorphic to V_{γ}. We employ the radial parallel transport in order to trivialize the bundle V_{d}(γ) in a neighbourhood of [e]. If we multiply the fixed basis of 𝔭 by i, then we specify a basis of i𝔭. We can write the operator A_{d}(γ, σ)^{2} in the form

A_{d}(γ, σ)^{2} = ∑_{i,j=1}^{n} ã_{i,j}D̃_{i}D̃_{j} + ∑_{j=1}^{n} ã_{j}D̃_{j} + ã ,

where D̃_{i} is the derivative ∂/∂(ix^{j}) and ã_{i,j}, ã_{j} and ã are End(V_{γ})^{ev}-valued functions on i𝔭. We claim that the coefficients of A(γ, σ)^{2} and A_{d}(γ, σ)^{2} extend as holomorphic functions to a neighbourhood of 0 in the complexification 𝔭_{ℂ} and that the following relations hold:

ã_{i,j}(p) = a_{i,j}(p)
ã_{j}(p) = ia_{j}(p)
ã(p) = −a(p).

Let G_{ℂ} and K_{ℂ} be the complexifications of the Lie groups. Then X_{ℂ} := G_{ℂ}/K_{ℂ} is a complex manifold. The representation of K on V_{γ} extends to K_{ℂ}. The Casimir Ω of G_{ℂ} acts on sections of G_{ℂ} ×_{K_{ℂ}} V_{γ} =: V (γ)_{ℂ} → X_{ℂ}. Let A(γ, σ)^{2}_{ℂ} := −Ω − c(σ). We use the exponential map of G_{ℂ} in order to provide holomorphic coordinates near [e].

f^{ℂ} : 𝔭_{ℂ} ⟶^{exp} G_{ℂ} → G_{ℂ}/K_{ℂ} = X_{ℂ} .

We trivialize the bundle V (γ)_{ℂ} using the radial parallel transport. Then A(γ, σ)^{2}_{ℂ} can be written in the form

A(γ, σ)^{2}_{ℂ} = ∑_{i,j=1}^{n} a_{i,j}^{ℂ}D_{i}^{ℂ}D_{j}^{ℂ} + ∑_{j=1}^{n} a_{j}^{ℂ}D_{j}^{ℂ} + a^{ℂ},

where a_{i,j}^{ℂ} is a holomorphic, End(V_{γ})^{ev}-valued function on 𝔭_{ℂ} and D_{j}^{ℂ} = ∂/∂z^{j}, where z^{j} is the complex coordinate corresponding to x^{j}. On the one hand, by definition A(γ, σ)^{2} is the restriction of A(γ, σ)^{2}_{ℂ} to X. Thus we obtain

a_{i,j}(p) = a_{i,j}^{ℂ}(p), a_{j}(p) = a_{j}^{ℂ}(p), a(p) = a^{ℂ}(p).

On the other hand, A_{d}(γ, σ)^{2} is the restriction of −A(γ, σ)^{2}_{ℂ} to X_{d}. Thus

ã_{i,j}(ip) = a_{i,j}^{ℂ}(ip), ã_{j}(ip) = ia_{j}^{ℂ}(ip), ã(ip) = −a^{ℂ}(ip)

proving the claim. The Lemma now follows by noting that b_{k} and b_{d,k} are homogeneous differential polynomials in the coefficients of A(γ, σ)^{2} and A_{d}(γ, σ)^{2}, respectively, of degree k + n. □

Corollary 1.8 Assume that n is even. In a neighbourhood of t = 0 we have the following equality of meromorphic functions

E_{d}(t, γ, σ) = i^{n}E(it, γ, σ) .

The discussion of the corresponding relation for n odd is postponed to Subsection 1.4.2.

1.2.4 An equivariant proportionality principle

The relation between the asymptotic expansions of heat kernels of operators on X and X_{d} is very similar to the relation between the expansion of wave kernels. The former was e.g. employed in order to establish the Hirzebruch proportionality principle. This proportionality principle relates the index of a locally homogeneous differential operator on Y to that of the corresponding operator on X_{d}. The Euler characteristic is an example of such an index. In this case the proportionality principle states

\frac{χ(Y)}{χ(X_{d})} = (−1)^{\frac{dim(Y )}{2}} \frac{vol(Y)}{vol(X_{d})} .  (1.13)

In this subsection we prove a slight extension of that proportionality principle. Let dim(X) = n be even and consider a class [γ] ∈ ker(i^{∗}) ⊂ R(K). Note the difference to our normal usage of the symbol γ for an equivalence class in R(K). Let V (γ)_{ℂ} → X_{ℂ} be the associated ℤ_{2}-graded holomorphic bundle. By Proposition 1.2 we can find a representative γ such that there exists a G_{ℂ}- invariant, holomorphic Dirac operator D_{ℂ} acting on holomorphic sections of V (γ)_{ℂ}. The bundle V (γ)_{ℂ} restricts to bundles V (γ) → X and V_{d}(γ) → X_{d}, whereas D_{ℂ} restricts to Dirac operators D on V (γ) and iD_{d} on V_{d}(γ). Let L_{ℂ} be an even, G_{ℂ}-invariant, holomorphic differential operator on V (γ)_{ℂ} restricting to L on V (γ) and to i^{q}L_{d} on V_{d}(γ). Here q denotes the order of L_{ℂ}. We assume that L_{ℂ} commutes with D_{ℂ} and that L and L_{d} are formally selfadjoint. Associated to a finite-dimensional representation χ of Γ there are also induced operators D_{Y,χ}, L_{Y,χ} on V_{Y,χ}(γ).

Definition 1.9 The L-index of D_{Y,χ} is defined by

index_{L}(D_{Y,χ}, L_{Y,χ})

:= {(λ, n_{λ})|λ eigenvalue of L_{Y,χ| ker(D_{Y,χ})} with weighted multiplicity n_{λ}}.

Similarly the L-index of D_{d} is defined by

index_{L}(D_{d}, L_{d})

:= {(λ, n_{λ})|λ eigenvalue of L_{d| ker(D_{d})} with weighted multiplicity n_{λ}}.

Proposition 1.10 (The proportionality principle for the L-index)

index_{L}(D_{Y,χ}, L_{Y,χ}) = \{(i^{−q}λ, dim(χ)\frac{χ(Y)}{χ(X_{d})}n_{λ})|(λ, n_{λ}) ∈ index_{L}(D_{d}, L_{d})\} .

P r o o f . We employ the heat equation approach. For all N ∈ ℕ we have

∑_{(λ,n_{λ})∈index_{L}(D_{Y,χ},L_{Y,χ})} n_{λ}λ^{N} = Tr L_{Y,χ}^{N}E_{D_{Y,χ}}{0} ,
∑_{(λ,n_{λ})∈index_{L}(D_{d},L_{d})} n_{λ}λ^{N} = Tr L_{d}^{N}E_{D_{d}}{0} .

For any t > 0 the following holds:

Tr L_{Y,χ}^{N}E_{D_{Y,χ}}{0} = Tr L_{Y,χ}^{N}e^{−tD_{Y,χ}^{2}},
Tr L_{d}^{N}E_{D_{d}}{0} = Tr L_{d}^{N}e^{−tD_{d}^{2}} .

(1.14)

Now we argue as in Subsection 1.2.3. We use the Cartan decompositions 𝔤 = 𝔨 ⊕ 𝔭 and 𝔤^{d} = 𝔨 ⊕ i𝔭 and the exponential maps in order to provide local charts near the origins of X and X_{d}. We trivialize the bundles using the radial parallel transport. Then the operators D(σ)^{2} and D_{d}^{2} are given by

D(σ)^{2} = ∑_{i,j=1}^{n} a_{i,j}D_{i}D_{j} + ∑_{j=1}^{n} a_{j}D_{j} + a

D_{d}^{2} = ∑_{i,j=1}^{n} ã_{i,j}D̃_{i}D̃_{j} + ∑_{j=1}^{n} ã_{j}D̃_{j} + ã .

Let D_{i} have degree one and fix the degrees of the coefficients a_{.}, ã_{.} such that D(σ)^{2}, D_{d}^{2} have total degree two. Then the following relations hold :

ã_{i,j}(p) = a_{i,j}(p)
ã_{j}(p) = ia_{j}(p)
ã(p) = −a(p) .

Thus if a_{.}(p) is a coefficient of D(σ)^{2}, then ã_{.}(ip) = i^{deg(a_{.})}a_{.}(ip) is the corresponding coefficient of D_{d}^{2}. The operators L and L_{d} also have analytic coefficients and are of the form

L = ∑_{|I|≤q} l_{I}D^{I}
L_{d} = ∑_{|I|≤q} l̃_{I} D̃^{I} ,

where I runs over a set of multi-indices. We fix the degree of the coefficients of L, L_{d} in such a way that the total degree of L, L_{d} becomes q. Then it follows that:

l̃_{I}(ip) = i^{q−|I|}l_{I}(ip) = i^{deg(l_{I})}l_{I}(ip) .

The local traces of the operators on the right-hand sides of (1.14) have asymptotic expansions near t = 0 (see [26]). Let x, x_{d} denote some points in Y, X_{d}. Then

tr L_{Y,χ}^{N}e^{−tD_{Y,χ}^{2}}(x, x) ∼^{t→0} dim(χ) ∑_{k=−n/2−Nq}^{∞} t^{k}f_{k}  (1.15)

tr L_{d}^{N}e^{−tD_{d}^{2}}(x_{d}, x_{d}) ∼^{t→0} ∑_{k=−n/2−Nq}^{∞} t^{k}f_{d,k} ,

where f_{k} and f_{d,k} are the values at 0 ∈ 𝔭 of universal homogeneous differential polynomials in the coefficients a_{i,j}, a_{i}, a, l_{I} and ã_{i,j}, ã_{i}, ã, l̃_{I} of degree 2k+n+Nq. We have the relations

f_{d,k} = i^{2k+n+Nq}f_{k} .  (1.16)

The factor dim(χ) appears in (1.15) since D_{Y,χ} and L_{Y,χ} are locally isomorphic to sums of dim(χ) copies of D and L, respectively. Since the left-hand side of (1.14) is independent of t we obtain

Tr L_{Y,χ}^{N}E_{D_{Y,χ}}{0} = dim(χ)vol(Y)f_{0}
Tr L_{d}^{N}E_{D_{d}}{0} = i^{n+Nq}vol(X_{d})f_{0}

by integrating the local trace over Y and X_{d}, respectively. Applying (1.13) we obtain

Tr L_{Y,χ}^{N}E_{D_{Y,χ}}\{0\} = \frac{dim(χ)χ(Y)}{i^{Nq}χ(X_{d})}Tr L_{d}^{N}E_{D_{d}}\{0\} ,
∑_{(λ,n_{λ})∈index_{L}(D_{Y,χ},L_{Y,χ})} n_{λ}λ^{N} = \frac{dim(χ)χ(Y)}{i^{Nq}χ(X_{d})} ∑_{(λ,n_{λ})∈index_{L}(D_{d},L_{d})} n_{λ}λ^{N} .

Since this equation holds for all N ≥ 0 it implies

index_{L}(D_{Y,χ}, L_{Y,χ}) = \{(i^{−q}λ, dim(χ)\frac{χ(Y)}{χ(X_{d})}n_{λ})|(λ, n_{λ}) ∈ index_{L}(D_{d}, L_{d})\} .

□

1.3 Harmonic analysis on compact symmetric spaces

In this section we collect some facts concerning representation theory of compact Lie groups which we need to define the theta functions. Since these results are of independent interest we also treat the higher rank case.

1.3.1 The Cartan-Helgason Theorem for vector bundles

We consider a Riemannian symmetric space X of non-compact type. We represent it as a homogeneous space G/K for a connected, linear, semisimple Lie group G. Though we have dropped the rank one condition

𝔨, 𝔭, 𝔞, M, 𝔪, (., .), 𝔫, Φ^{+}(𝔤, 𝔞), G_{ℂ}, G_{d}, X_{d} etc.

can be defined in the same way as in Subsection 1.1. We choose a maximal abelian subalgebra 𝔱 of 𝔪. Then 𝔥 := 𝔱_{ℂ} ⊕ 𝔞_{ℂ} is a Cartan subalgebra of 𝔤_{ℂ}. We choose a positive root system Φ^{+}(𝔤_{ℂ}, 𝔥) having the property that, for α ∈ Φ(𝔤_{ℂ}, 𝔥), α_{|𝔞} ∈ Φ^{+}(𝔤, 𝔞) implies α ∈ Φ^{+}(𝔤_{ℂ}, 𝔥). Let δ := \frac{1}{2} ∑_{α∈Φ^{+}(𝔤_{ℂ},𝔥)} α, and set ρ_{𝔪} := δ − ρ. Define the root vector H_{α} ∈ 𝔞 for α ∈ Φ^{+}(𝔤, 𝔞) by

λ(H_{α}) := \frac{(λ, α)}{(α, α)} ∀λ ∈ 𝔞^{∗} .

Note that exp(2πiH_{α}) ∈ M. Let γ ∈ K̂ be fixed. Then we can decompose γ_{|M} = ⨁_{σ∈M̂}[γ : σ] σ. We are interested in the multiplicities [π_{Λ} : γ] for irreducible representations π_{Λ} of G_{d} with highest weight Λ ∈ 𝔥^{∗} occurring in the decomposition

L^{2}(G_{d} ×_{K} V_{γ}) = ⨁_{π_{Λ}∈Ĝ_{d}}[π_{Λ} : γ] π_{Λ}

obtained by Frobenius reciprocity.

Theorem 1.11 Let [π_{Λ} : γ] ≠ 0. Then

1. Λ_{|𝔱} is equal to the highest weight μ_{σ} of a representation σ of M with [γ : σ] ≥ [π_{Λ} : γ];

2. \frac{(Λ_{|𝔞},α)}{(α,α)} ∈ ε_{α}(σ) + ℕ ∀α ∈ Φ^{+}(𝔤, 𝔞), where ε_{α}(σ) ∈ \{0, 1/2\} is defined by

e^{2πiε_{α}(σ)} := σ(exp(2πiH_{α})) ∈ {±1} ;

3. \frac{(Λ_{|𝔞},α)}{(α,α)} ≥ max_{\{β∈Φ^{+}(𝔤_{ℂ},𝔥)|β_{|𝔞}=α\}} \frac{(μ_{σ},β)}{(α,α)} =: Φ_{α}(σ) ∀α ∈ Φ^{+}(𝔤, 𝔞) .

Moreover, one can reverse this claim to a certain extent. Assume that [γ : σ] ≠ 0 for σ ∈ M̂ with highest weight μ_{σ}. Then there exist constants C_{α}(σ, γ) ≥ Φ_{α}(σ) such that for λ ∈ 𝔞^{∗} with

\frac{(λ, α)}{(α, α)} ∈ ε_{α}(σ) + ℕ ,  (1.17)
\frac{(λ, α)}{(α, α)} ≥ C_{α}(σ, γ) ∀α ∈ Φ^{+}(𝔤, 𝔞)

the following equation holds

[π_{Λ} : γ] = [γ : σ] ,

where Λ := λ + μ_{σ} ∈ 𝔥^{∗}.

P r o o f . Note that by holomorphic continuation π_{Λ} extends to an irreducible representation of G_{ℂ}, hence of G, acting on the same vector space V_{π_{Λ}}. Let v_{Λ} be a highest weight vector of V_{π_{Λ}}. Then, up to multiplication by scalars, v_{Λ} is the unique highest weight vector for the action of M in the 𝔫-invariants of V_{π_{Λ}}, denoted by V_{π_{Λ}}^{𝔫}. Thus the representation (π_{Λ}_{|M}, V_{π_{Λ}}^{𝔫}) of M is irreducible and defines an element σ ∈ M̂. The Iwasawa decomposition implies that v_{Λ} is a cyclic vector for the action of K in V_{π_{Λ}}. Therefore the mapping

F : Hom_{K}(V_{π_{Λ}}, V_{γ}) ⟶ Hom_{M}(V_{σ}, V_{γ})  (1.18)

defined by

F (T) := T ∘ i ; i : V_{σ} → V_{π_{Λ}}^{𝔫} ,

where V_{σ} and V_{γ} are representation spaces for σ and γ, respectively, is injective. This proves 1. In order to prove 2., we recall that (see Helgason [33], Ch. VII, Thm. 8.5.)

K ∩ exp i𝔞 = exp{2πi ∑_{α∈Φ^{+}(𝔤,𝔞)}n_{α}H_{α} | n_{α} ∈ ℤ} ⊂ M .  (1.19)

All these elements are of order at most 2. Hence

±v_{Λ} = σ(exp(2πiH_{α}))v_{Λ} = π_{Λ}(exp(2πiH_{α}))v_{Λ} = e^{2πi \frac{(Λ_{|𝔞},α)}{(α,α)}} v_{Λ} .

Item 3. is a simple consequence of the Φ^{+}(𝔤_{ℂ}, 𝔥)-dominance of Λ and the fact that for β ∈ Φ^{+}(𝔤_{ℂ}, 𝔥) with β_{|𝔞} ≠ 0 the root −θβ is in Φ^{+}(𝔤_{ℂ}, 𝔥), too. Here, θ is the Cartan involution of 𝔤. We turn to the revers direction of the theorem. Given a pair (σ, λ) ∈ M̂ × 𝔞^{∗} with \frac{(λ,α)}{(α,α)} ≥ Φ_{α}(σ) ∀α ∈ Φ^{+}(𝔤, 𝔞) we form the Φ^{+}(𝔤_{ℂ}, 𝔥)-dominant element Λ := λ + μ_{σ} ∈ 𝔥^{∗}. We claim that if λ also satisfies the condition (1.17), then Λ is integral. We have to show that for H = H_{𝔞} + H_{𝔱} ∈ i𝔞 ⊕ 𝔱 with exp(H) = e also e^{Λ(H)} = 1. Indeed, since exp(H_{𝔞}) = exp(H_{𝔱})^{−1} ∈ M, we obtain for a highest weight vector v_{σ} ∈ V_{σ}

e^{Λ(H)}v_{σ} = e^{λ(H_{𝔞})}e^{μ_{σ}(H_{𝔱})}v_{σ} = σ(exp(H_{𝔞}))σ(exp(H_{𝔱}))v_{σ} = v_{σ} .

Thus, Λ is the highest weight of a representation (π_{Λ}, V_{π_{Λ}}) of G_{d} such that the irreducible representation of M on V_{π_{Λ}}^{𝔫} has highest weight μ_{σ}. This shows that this representation coincides with σ on the connected component M_{0} of e ∈ M. Using (1.17) and the fact that

M = (K ∩ exp i𝔞) · M_{0} ([33], Ch. IX, ex. A3)

we see that this representation is given by σ on the whole group M. In fact, in view of (1.19) it is enough to compute for v ∈ V_{π_{Λ}}^{𝔫}

π_{Λ}(exp(2πiH_{α}))v = e^{2πiλ(H_{α})}v
= e^{2πiε_{α}(σ)}v = σ(exp(2πiH_{α}))v .

We are done, if we can show that the map F defined in (1.18) is surjective for λ large enough. The dual map

F^{∗} : Hom_{M}(V_{γ}, V_{σ}) ⟶ Hom_{K}(V_{γ}, V_{π_{Λ}})

applied to t ∈ Hom_{M}(V_{γ}, V_{σ}) is given by

F^{∗}(t) = ∫_{K} π_{Λ}(k^{−1})i ∘ tγ(k) dk .

Indeed, for T ∈ Hom_{K}(V_{π_{Λ}}, V_{γ}) we have

tr(T ∘ F^{∗}(t)) = tr(T ∫_{K} π_{Λ}(k^{−1})i ∘ tγ(k) dk)
= tr( ∫_{K} γ(k^{−1})T ∘ i ∘ tγ(k) dk) = tr(F(T) ∘ t) .

In order to decide upon the injectivity of F^{∗}, we consider for (σ, λ) ∈ M̂ × 𝔞^{∗}, (λ, α) > 0 ∀α ∈ Φ^{+}(𝔤, 𝔞) the c-function

c_{γ}(σ, λ) := P_{σ} ∫\overline{N} a(n̄)^{−(λ+ρ)}γ(κ(n̄)) dn̄ P_{σ} ∈ End_{M}V_{γ} ,

where P_{σ} is the projection onto the σ-isotypic component of V_{γ}, \overline{N} := θN and n̄ = κ(n̄)a(n̄)n(n̄) is decomposed into its Iwasawa components. There exist constants C′_{α}(σ, λ) such that

det c_{γ}(σ, λ) ≠ 0 ,  whenever \frac{(λ, α)}{(α, α)} ≥ C′_{α}(σ, γ) ∀α ∈ Φ^{+}(𝔤, 𝔞) . (1.20)

This can be seen using the expression of det c_{γ}(σ, λ) in terms of the Γ-function given for example in Wallach [87], Cor.10.4.7. Alternatively one could argue that the integrand concentrates near the identity for large λ.

Let ⟨., .⟩ be a G_{d}-invariant hermitian scalar product on V_{π_{Λ}}, v ∈ V_{γ},v_{λ} ∈ V_{π_{Λ}}^{𝔫}, t ∈ Hom_{M}(V_{γ}, V_{σ}). We compute

⟨F^{∗}(t)v, v_{λ}⟩ = ⟨∫_{K} π_{Λ}(k^{−1})i ∘ tγ(k)v dk, v_{λ}⟩
= ⟨∫_{\overline{N}M}a(n̄)^{−2ρ}π_{Λ}(κ(n̄)^{−1})π_{Λ}(m)^{−1}i ∘ tγ(m)γ(κ(n̄))v dm dn̄, v_{λ}⟩
= ∫\overline{N}⟨a(n̄)^{−2ρ}i ∘ tγ(κ(n̄))v, π_{Λ}(κ(n̄))v_{λ}⟩ dn̄
= ∫\overline{N}⟨i ∘ ta(n̄)^{−(λ+2ρ)}γ(κ(n̄))v, π_{Λ}(n̄)v_{λ}⟩ dn̄
= ⟨i ∘ tc_{γ}(σ, λ + ρ)v, v_{λ}⟩ .

In the last step we have used that with respect to ⟨., .⟩

π_{Λ}(n̄)^{∗} = π_{Λ}(θ(n̄)^{−1})

and that θ(n̄)^{−1} ∈ N acts trivially on i(V_{σ}). We see that F^{∗} is injective iff c_{γ}(σ, λ + ρ) is injective. This together with (1.20) implies the assertion. □

If γ is trivial, then the c-function is explicitly known. In this case for all α ∈ Φ^{+}(𝔤, 𝔞) we have C_{α} = Φ_{α} = ε_{α} = 0. Inserting these data into the previous theorem we get the original Cartan-Helgason Theorem (see Helgason [34], Ch. V, Thm. 4.1.).

1.3.2 The G_{d}-index theorem

We assume in addition to the situation in the previous subsection that

rank(G_{d}) = rank(K) .

Otherwise the index of any G_{d}-homogeneous elliptic operator on X_{d} would vanish (see Bott [7]). Note that the rank condition implies that dim X is even. It is satisfied by all symmetric spaces of rank one and even dimension. Our goal is to represent any (irreducible) representation of G_{d} as the index of a homogeneous Dirac operator operator on X_{d}. That this is possible, even for general compact homogeneous spaces G/H satisfying the equal rank condition, was already shown by Bott ([7]) under the additional hypothesis that π_{1}(G) has no 2-torsion. Instead of referring to that theorem we prefer to include an elementary proof here relying only on the Weyl character formula and having the advantage of explicitly constructing the desired Dirac operator.

First, we need some more root systems and Weyl groups. We choose a Cartan algebra 𝔥_{𝔨} ⊂ 𝔨 being also a Cartan algebra of 𝔤_{d}. Let H be the corresponding maximal torus. Let Φ^{+}(𝔤_{ℂ}, 𝔥_{𝔨}) be a system of positive roots for 𝔤_{d} and Φ^{+}(𝔨_{ℂ}, 𝔥_{𝔨}) be the subsystem of positive roots of 𝔨. Note that #(Φ^{+}(𝔤_{ℂ}, 𝔥_{𝔨}) \ Φ^{+}(𝔨_{ℂ}, 𝔥_{𝔨})) = n/2, where n is the dimension of X_{d}. We set

ρ_{𝔤} := \frac{1}{2} ∑_{α∈Φ^{+}(𝔤_{ℂ},𝔥_{𝔨})} α, ρ_{𝔨} := \frac{1}{2} ∑_{α∈Φ^{+}(𝔨_{ℂ},𝔥_{𝔨})} α .

Let W_{𝔤} := W (Φ^{+}(𝔤_{ℂ}, 𝔥_{𝔨})) and W_{𝔨} := W (Φ^{+}(𝔨_{ℂ}, 𝔥_{𝔨})) be the corresponding Weyl groups and set

W^{1} = {w ∈ W_{𝔤} | Φ^{+}(𝔨_{ℂ}, 𝔥_{𝔨}) ⊂ w(Φ^{+}(𝔤_{ℂ}, 𝔥_{𝔨}))}.

The following considerations and computations are completely analogous to those in Parthasarathy [55] for the noncompact case. Let S := S^{+}⊕S^{−} = S(𝔭_{d}) be the spinor module of 𝔭_{d} and c : 𝔭_{d} → End(S) the Clifford multiplication, which in the even dimensional case is uniquely determined. The Spinor module S is a Spin(𝔭_{d})-representation. Let s : Spin(𝔭_{d}) → GL(S) and p : Spin(𝔭_{d}) → SO(𝔭_{d}) be the canonical maps. The isotropy representation induces a homomorphism l : K → SO(𝔭_{d}). Let K̃ be the connected component of e in p^{−1}(l(K)) ⊂ Spin(𝔭_{d}). We will denote the restriction of s to K̃ by the same letter. Let Λ ∈ i𝔥_{𝔨}^{∗} be the highest weight of an irreducible representation π_{Λ} of G_{d}. It is not difficult to see (compare [55]) that for all w ∈ W^{1} the functional w(Λ + ρ_{𝔤}) − ρ_{𝔨} is dominant and integral with respect to K̃, hence it is the highest weight of an irreducible representation γ_{Λ,w} of K̃ acting on a vector space V_{Λ,w}. Furthermore, if the non-trivial element of p^{−1}(e) is contained in K̃, then it acts on S as well as on V_{Λ,w} as multiplication by −1. Indeed, this element is central in Spin(𝔭_{d}), hence in K̃, and, thus, of the form exp(H) for some H ∈ 𝔥_{𝔨}. Therefore we have

γ_{Λ,w}(exp(H)) = e^{(w(Λ+ρ_{𝔤})−ρ_{𝔨})(H)} · id = e^{(wρ_{𝔤}−ρ_{𝔨})(H)} · id = −id ,

since wρ_{𝔤} − ρ_{𝔨} is a weight of S. Thus, the representation

s ⊗ γ_{Λ,w} : K → GL(S ⊗ V_{Λ,w})

is well-defined as a representation of K. We extend Clifford multiplication to S ⊗ V_{Λ,w} and consider the homogeneous vector bundle

E_{Λ,w} := G_{d} ×_{K} (S ⊗ V_{Λ,w}) .

Now we can define the Dirac operator D_{Λ,w} associated to the Clifford bundle structure of E_{Λ,w} as in formula (1.10). The grading S = S^{+} ⊕ S^{−} induces a corresponding grading E_{Λ,w} = E_{Λ,w}^{+} ⊕ E_{Λ,w}^{−}. The operator D_{Λ,w} splits as

D_{Λ,w}^{±} : C^{∞}(E_{Λ,w}^{±}) → C^{∞}(E_{Λ,w}^{∓}) .

Note that D^{±} are G_{d}-invariant differential operators. The G_{d}-index of D_{Λ,w} is the element in the representation ring R(G_{d}) over ℤ of G_{d} given by

index_{G_{d}}D_{Λ,w} := [kerD_{Λ,w}^{+}] − [kerD_{Λ,w}^{−}] .

Theorem 1.12 The G_{d}-index of the Dirac operator D_{Λ,w} is equal to

(−1)^{n/2} det(w)[π_{Λ}] ∈ R(G_{d}) .

Hence, any π ∈ R(G_{d}) can be realized as the index of a homogeneous Dirac operator.

Proof. Let ψ ∈ C^{∞}(G_{d}) be a G_{d}-finite function. Then ψ acts on C^{∞}(E_{Λ,w}) as a finite dimensional operator l_{ψ} via convolution. Considering f ∈ C^{∞}(E_{Λ,w}) as a function on G_{d} we may write this action as follows

l_{ψ}(f)(g) = ∫_{G_{d}} ψ(x)f(x^{−1}g) dx
= ∫_{G_{d}} ψ(gx^{−1})f(x) dx
= ∫_{G_{d}} ψ(gkx^{−1})(s ⊗ γ_{Λ,w})(k)f(x) dx (for any k ∈ K)
= ∫_{G_{d}} ∫_{K} ψ(gkx^{−1})(s ⊗ γ_{Λ,w})(k) dk f(x) dx .  (1.21)

We assume all Haar measures to be normalized to have total mass 1. It is enough to show that the supertrace of l_{ψ} with respect to the grading of E_{Λ,w} is given by integration of ψ against the character χ_{Λ} of π_{Λ} with the appropriate sign. In fact, it is enough to show this for each class function ψ. In view of (1.21) l_{ψ} is an integral operator with kernel given by

l_{ψ}(x, y) = ∫_{K} ψ(xky^{−1})(s ⊗ γ_{Λ,w})(k) dk .

Thus, for a class function ψ the super trace can be obtained in the following way:

Tr(l_{ψ}) = ∫_{X_{d}} ∫_{K} ψ(xkx^{−1})tr(s ⊗ γ_{Λ,w})(k) dk dx
= ∫_{K} ψ(k)tr(s ⊗ γ_{Λ,w})(k) dk .

Let us denote by χ_{s^{±}} and χ_{γ} the characters of S^{±} and γ_{Λ,w}, respectively. They are uniquely defined as functions on K̃, but their product pushes down to K. We introduce the Weyl denominators

Δ_{G_{d}}(exp X) := ∏_{α∈Φ^{+}(𝔤_{ℂ},𝔥_{𝔨})}(e^{α(X)/2} − e^{−α(X)/2}) ,
Δ_{K}(exp X) := ∏_{α∈Φ^{+}(𝔨_{ℂ},𝔥_{𝔨})}(e^{α(X)/2} − e^{−α(X)/2}) , X ∈ 𝔥_{𝔨},
Δ_{n} := Δ_{G_{d}}/Δ_{K} .

Now, using Weyl’s integral and character formulas for K and G_{d} and

(χ_{s^{+}} − χ_{s^{−}})_{|H} = Δ_{n} (see [55]) ,

we can compute

T r(l_{ψ}) = ∫_{K} ψ(k)tr(s ⊗ γ_{Λ,w})(k) dk
= ∫_{K} ψ(k)(χ_{s^{+}} − χ_{s^{−}})(k)χ_{γ}(k) dk
= \frac{1}{|W_{𝔨}|} ∫_{H} ψ(h)|Δ_{K}(h)|^{2}(χ_{s^{+}} − χ_{s^{−}})(h)χ_{γ}(h) dh
= \frac{1}{|W_{𝔨}|} ∫_{H} ψ(h)|Δ_{K}(h)|^{2}Δ_{n}(h) ∑_{s∈W_{𝔨}} \frac{det(s)e^{sw(Λ+ρ_{𝔤})(log(h))}}{Δ_{K}(h)} dh

= \frac{1}{|W_{𝔨}|} ∫_{H} ψ(h)|Δ_{G_{d}}(h)|^{2} ∑_{s∈W_{𝔨}} \frac{det(s)e^{sw(Λ+ρ_{𝔤})(log(h))}}{Δ_{K}(h)\overline{Δ_{n}(h)}} dh

= \frac{(−1)^{n/2}}{|W_{𝔨}||W_{𝔤}|} ∫_{H} ψ(h)|Δ_{G_{d}}(h)|^{2} ∑_{t∈W_{𝔤}} ∑_{s∈W_{𝔨}} \frac{det(s)e^{tsw(Λ+ρ_{𝔤})(log(h))}}{Δ_{G_{d}}(t^{−1}h)} dh

= \frac{(−1)^{n/2} det(w)}{|W_{𝔤}|} ∫_{H} ψ(h)|Δ_{G_{d}}(h)|^{2} ∑_{t∈W_{𝔤}} \frac{det(t)e^{t(Λ+ρ_{𝔤})(log(h))}}{Δ_{G_{d}}(h)} dh

= \frac{(−1)^{n/2} det(w)}{|W_{𝔤}|} ∫_{H} ψ(h)|Δ_{G_{d}}(h)|^{2}χ_{Λ}(h) dh
= (−1)^{n/2} det(w) ∫_{G_{d}} ψ(g)χ_{Λ}(g) dg .

The theorem follows. □

1.3.3 Admissible lifts

Now we want to see more explicitly what Theorem 1.11 tells us about the spectral decomposition of L^{2}(G_{d} ×_{K} V_{γ}) with respect of the action of the Casimir operator Ω if G_{d}/K has rank one. The root system Φ^{+}(𝔤, 𝔞) is of the form Φ^{+}(𝔤, 𝔞) = {α} or Φ^{+}(𝔤, 𝔞) = {α/2, α}. We call α the long root. We identify

𝔞^{∗} ≅ ℝ, 𝔞_{ℂ}^{∗} ≅ ℂ

sending α to |α|. Now we fix the necessary bookkeeping for the action of Ω on L^{2}(G_{d} ×_{K} V_{γ}).

Definition 1.13 Let α be the long root as before and set T := |α|. For σ ∈ M̂ we define ϵ_{σ} ∈ {0, 1/2} by

ϵ_{σ} ≡ |ρ|/T + ε_{α}(σ) mod ℤ ,

and the lattice L(σ) ⊂ ℝ ≅ 𝔞^{∗} by

L(σ) := T (ϵ_{σ} + ℤ) .

Finally, for λ ∈ 𝔞_{ℂ}^{∗} ≅ ℂ we set

P_{σ}(λ) := ∏_{β∈Φ^{+}(𝔤_{ℂ},𝔥)} \frac{(λ + μ_{σ} + ρ_{𝔪}, β)}{(δ, β)} .

For dim X even the root system Φ^{+}(𝔤_{ℂ}, 𝔥) contains a root which vanishes on 𝔱. Hence, in this case, P_{σ}(0) = 0, and we can define the polynomial

Q_{σ}(λ) := \frac{P_{σ}(λ)}{λ} .

We remark that P_{σ}(−λ) = (−1)^{dim X−1}P_{σ}(λ). This can be easily deduced from the fact that for β ∈ Φ^{+}(𝔤_{ℂ}, 𝔥) with β_{|𝔞} ≠ 0 the root −θβ is also contained in Φ^{+}(𝔤_{ℂ}, 𝔥). Here θ is the Cartan involution of 𝔤. In particular, Q_{σ} is an even polynomial. Furthermore, for the non-trivial element w ∈ W we have

P_{wσ}(λ) = (−1)^{dim X−1}P_{σ}(−λ) = P_{σ}(λ) .

Recall the definition of Φ_{α}(σ) from Theorem 1.11 and let

Φ(σ) := max{|α|Φ_{α}(σ) | α ∈ Φ^{+}(𝔤, 𝔞)} + ρ .

Consider for λ ∈ L(σ), λ ≥ Φ(σ) and Λ := λ − ρ + μ_{σ} the representation π_{Λ} of G_{d} with highest weight Λ. Then Weyl’s dimension formula states that

dim(π_{Λ}) = P_{σ}(λ) .

Recall the definition of the shift constant

c(σ) := |ρ|^{2} + |ρ_{𝔪}|^{2} − |μ_{σ} + ρ_{𝔪}|^{2} = |δ|^{2} − |μ_{σ} + ρ_{𝔪}|^{2} .

Then we have

π_{Λ}(Ω) = (|Λ + δ|^{2} − |δ|^{2})id = (λ^{2} − c(σ))id .

Now, in our situation Theorem 1.11 reads

Proposition 1.14 The spectrum of Ω acting on L^{2}(G_{d} ×_{K} V_{γ}) is contained in

⋃_{{σ∈M̂| [γ:σ]≠0}}{λ^{2} − c(σ) | λ ∈ L(σ), λ ≥ Φ(σ)} .

Moreover, each eigenspace E_{ν} of Ω splits into a direct sum of subspaces

E_{ν} = ⨁_{(σ,λ)∈R_{γ}(ν)} E_{σ,λ} ,

where

R_{γ}(ν) := {(σ, λ) | [γ : σ] ≠ 0, λ ∈ L(σ), λ ≥ Φ(σ), λ^{2} − c(σ) = ν} .

E_{σ,λ} is isomorphic to a certain multiple of π_{λ−ρ+μ_{σ}} as a representation of G_{d}, and there exist constants C(σ, γ) ≥ Φ(σ) such that

dim E_{σ,λ} = [γ : σ]P_{σ}(λ) ∀λ ∈ L(σ), λ ≥ C(σ, γ) .

Now we return to the multiplicities m_{d}(s, γ, σ) and m_{χ}(s, γ, σ) defined in Subsection 1.1.3. We have the following

Corollary 1.15 Let σ ∈ M̂ be a Weyl invariant element and γ ∈ R(K) such that i^{∗}(γ) = σ. Then m_{d}(s, γ, σ) = P_{σ}(s) for all but finitely many 0 ≤ s ∈ L(σ). Moreover, there is only a finite number of s ∈ ℂ \ (L(σ) ∩ ℝ^{+}) with m_{d}(s, γ, σ) ≠ 0.

The exceptional set, where m_{d}(s, γ, σ) ≠ P_{σ}(s), depends on γ. The next corollary will be improved in Subsection 1.4.2.

Corollary 1.16 Let σ ∈ M̂ be not invariant under the action of W and γ ∈ R(K) such that i^{∗}(γ) = σ + wσ. Then m_{d}(s, γ, σ) = P_{σ}(s) + P_{wσ}(s) = 2P_{σ}(s) for all but finitely many 0 ≤ s ∈ L(σ), and there is only a finite number of s ∈ ℂ \ (L(σ) ∩ ℝ^{+}) with m_{d}(s, γ, σ) ≠ 0.

We now come to the crucial definition.

Definition 1.17 Let dim X be even and σ ∈ M̂. Then γ ∈ R(K) is called σ-admissible if i^{∗}(γ) = σ and m_{d}(s, γ, σ) = P_{σ}(s) for all 0 ≤ s ∈ L(σ).

Thus, for σ-admissible γ the weighted multiplicities of the spectrum of A_{d}(γ, σ) have a particular nice regularity.

Lemma 1.18 Let dim X be even and σ ∈ M̂. Then there exists a σ-admissible γ ∈ R(K).

P r o o f . Let λ ∈ L(σ) be regular in the sense that

(λ + μ_{σ} + ρ_{𝔪}, β) ≠ 0 for all β ∈ Φ^{+}(𝔤_{ℂ}, 𝔥) .

Then there exist an element w^{λ} ∈ W (𝔤_{ℂ}, 𝔥) having the property

(w^{λ}(λ + μ_{σ} + ρ_{𝔪}), β) > 0 for all β ∈ Φ^{+}(𝔤_{C}, 𝔥) .

Note that w^{λ} = e for λ large enough, and that

dim π_{w^{λ}(λ+μ_{σ}+ρ_{𝔪})−δ} = det(w^{λ})P_{σ}(λ) .

If λ is singular, then P_{σ}(λ) = 0. Now we start with an arbitrary γ ∈ R(K) such that i^{∗}(γ) = σ. By Proposition 1.14 the following difference of virtual representations of G_{d} (one has to take the grading of V_{d}(γ) into account)

∑_{λ∈L(σ) regular, λ≥0} det(w^{λ})π_{w^{λ}(λ+μ_{σ}+ρ_{𝔪})−δ} − L^{2}(V_{d}(γ))

defines an element in R(G_{d}) and, hence, can be represented by Theorem 1.12 as the index of a homogeneous Dirac operator acting on a graded Clifford bundle V_{d}(γ′). By construction we have i^{∗}(γ′) = 0 and

L^{2}(V_{d}(γ′)) = ∑_{λ∈L(σ) regular, λ≥0} det(w^{λ})π_{w^{λ}(λ+μ_{σ}+ρ_{𝔪})−δ} − L^{2}(V_{d}(γ)) .

We conclude that γ′′ := γ + γ′ is σ-admissible. □

Again, there is no unique choice of an admissible lift of σ ∈ M̂, since there are in general Clifford bundles on X_{d} carrying a homogeneous Dirac operator with zero index. But, fortunately, we have the following

Proposition 1.19 The multiplicities m_{χ}(s, γ, σ) do not depend on the choice of the σ-admissible γ.

P r o o f . This proposition is a consequence of the equivariant proportionality principle (Theorem 1.10). Let γ, γ̃ ∈ R(K) be two σ-admissible elements. Then i^{∗}(γ − γ̃) = 0. By Proposition 1.2 we may represent γ_{0} := γ − γ̃ in such a way that there exist odd Dirac operators D_{d} and D_{Y,χ} acting on sections of V_{d}(γ_{0}) and V_{Y,χ}(γ_{0}), respectively. We denote by B_{d} and B_{Y,χ} the action of the Casimir operator on the corresponding bundle. D_{∗} and B_{∗} commute. The eigenspace of B_{Y,χ} corresponding to λ splits as H(λ) = H(λ)^{0} ⊕ H(λ)^{⊥}, where H(λ)^{0} is in the kernel of D_{Y,χ} and H(λ)^{⊥} is orthogonal to the kernel of D_{Y,χ}. On H(λ)^{⊥} the operator D_{Y,χ} induces an odd isomorphism. Thus Tr1_{H(λ)^{⊥}} = 0 (note that we take the super trace). Let index_{L}(D_{Y,χ}, B_{Y,χ}) = {(λ, m_{λ})}. Then Tr 1_{H(λ)^{0}} = m_{λ}. We must show that index_{L}(D_{Y,χ}, B_{Y,χ}) = 0. Theorem 1.10 states that

index_{L}(D_{Y,χ}, B_{Y,χ}) = \{(−λ, dim(χ)\frac{χ(Y)}{χ(X_{d})}n_{λ})|(λ, n_{λ}) ∈ index_{L}(D_{d}, B_{d})\} .

Thus it is enough to show that index_{L}(D_{d}, B_{d}) = 0, i.e., all n_{λ} vanish. Let (λ, n_{λ}) ∈ index_{L}(D_{d}, B_{d}). The eigenspace H_{d}(λ) of B_{d} splits into H_{d}(λ)^{0} ⊕ H_{d}(λ)^{⊥} with respect to D_{d}. On H_{d}(λ)^{⊥}, D_{d} induces an odd isomorphism. Thus Tr 1_{H_{d}(λ)} = Tr 1_{H_{d}(λ)^{0}} = n_{λ}. Since γ as well as γ̃ are σ-admissible, we conclude that n_{λ} = 0. Hence, index_{L}(D_{d}, B_{d}) = 0 and the proposition follows.

□

In Subsection 1.4.2 we shall see that for dim(X) odd the unique element in R(K) corresponding to σ ∈ M̂ is a priori admissible in some sense.

1.4 The trace formula

The aim of the present section is to establish an explicit formula for the trace of operators like cos(tA_{Y,χ}(γ, σ)). The final result is stated in Proposition 1.27. Though it would be possible to prove it using the far developed general theory of the Selberg trace formula which is well documented in the literature (see, e.g., [73], [86]), we present here a self-contained approach avoiding the use of deep results in harmonic analysis on G like the Harish-Chandra-Plancherel theorem or character theory.

1.4.1 The general scheme

We fix a finite-dimensional unitary representation (χ, V_{χ}) of Γ and consider the operators cos(tA_{Y,χ}(γ, σ)) and, in case (b), D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)). For ϕ ∈ C_{c}^{∞}(ℝ) we introduce the averaged operators

E_{Y,ϕ} := ∫_{ℝ} ϕ(t) cos(tA_{Y,χ}(γ, σ)) dt

and, in case (b),

E_{Y,ϕ}^{s} := ∫_{ℝ} ϕ(t)D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)) dt .

They have smooth integral kernels and, hence, are of trace class. The linear map

C_{c}^{∞}(ℝ) ∋ ϕ → Tr E_{Y,ϕ}^{(s)}

defines a distribution on ℝ formally written as

Tr cos(tA_{Y,χ}(γ, σ)) or

Tr D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)) , respectively. A trace formula is an expression for these distributions as a sum of contributions, each associated with a conjugacy class of Γ. Since Γ ⊂ G is co-compact and torsion-free, there are only two types of conjugacy classes, the class of the identity e ∈ Γ and classes of hyperbolic elements, where g ∈ G is called hyperbolic if Ad(g) is semisimple and has dim X − 1 eigenvalues of modulus greater than one. Let Ω_{<g>} ⊂ X be a fundamental domain of the group < g > generated by g. Recall the definition of the distribution E(t, γ, σ) from Section 1.2. In the case (b) we define a distribution E^{s}(t, γ, σ) in an analogous manner replacing cos(tA_{Y,χ}(γ, σ)) by D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)). Namely, for ϕ ∈ C_{c}^{∞}(ℝ) let E_{ϕ}^{s}(x, y) be the smooth integral kernel of the operator

∫_{ℝ} ϕ(t)D(σ) cos(tA(γ^{s}, σ)) dt .

Its local trace tr E_{ϕ}^{s}(x, x) does not depend on x ∈ X, and we define the distribution E^{s}(., γ, σ) by

E^{s}(., γ, σ) : C_{c}^{∞}(ℝ) ∋ ϕ → tr E_{ϕ}^{s}(x, x) .

Definition 1.20 The distribution vol(Y)E^{(s)}(., γ, σ) is called the contribution of the identity. For a hyperbolic g ∈ G we define the hyperbolic contribution to be the distribution

C_{c}^{∞}(ℝ) ∋ ϕ → I_{ϕ}^{(s)}(g, σ) := ∫_{Ω_{<g>}} tr E_{ϕ}^{(s)}(x, gx)) dx ,

where we have identified V (γ)_{x} with V (γ)_{gx} using the action of g in order to define the local (super) trace.

For g ∈ Γ let n_{Γ}(g) := #(Γ_{g}/ < g >), where Γ_{g} is the centralizer of g in Γ . By CΓ we denote the set of conjugacy classes of Γ.

Proposition 1.21 We have

Tr E_{Y,ϕ}^{(s)} = dim(χ)vol(Y)⟨E^{(s)}(., γ, σ), ϕ⟩

+ ∑_{[g]∈CΓ,[g]≠[e]} \frac{tr χ(g)}{n_{Γ}(g)}I_{ϕ}^{(s)}(g, σ) .

P r o o f . Note that cos(tA_{Y,χ}(γ, σ)) solves the Cauchy problem of the wave equation in V_{Y,χ}(γ). A solution pulled back to X solves the Cauchy problem in V (γ) ⊗ V_{χ} with the pull back of the initial condition. Hence we have for a section f ∈ C^{∞}(Y, V_{Y,χ}(γ)) ≅ (C^{∞}(X, V (γ)) ⊗ V_{χ})^{Γ}

∫_{Y} E_{Y,ϕ}(., y)f(y) dy = ∫_{X}(E_{ϕ}(., ŷ) ⊗ id)f(ŷ) dŷ =∑_{g∈Γ} ∫_{Ω_{Γ}} (E_{ϕ}(., gŷ) ⊗ χ(g))f(ŷ) dŷ ,

where Ω_{Γ} ⊂ X is a fundamental domain of Γ. We conclude that

E_{Y,ϕ}(x, y) = ∑_{g∈Γ} E_{ϕ}(x̂, gŷ) ⊗ χ(g) ,

where x̂, ŷ ∈ Ω_{Γ} are lifts of x, y with respect to the covering X → Y. Analogously

E_{Y,ϕ}^{s}(x, y) = ∑_{g∈Γ} E_{ϕ}^{s}(x̂, gŷ) ⊗ χ(g) .

For g ∈ Γ we identify the fibre V (γ^{(s)})x̂ with V (γ^{(s)})_{gx̂} using the homogeneous bundle structure of V (γ^{(s)}). Since ϕ has compact support and cos(tA(γ^{(s)}, σ)) has unit propagation speed, the operator E_{ϕ}^{(s)} has finite propagation, and thus the sum is uniformly finite, if x and y vary in Ω_{Γ}. We compute

Tr E_{Y,ϕ}^{(s)} = ∫_{Ω_{Γ}} ∑_{g∈Γ} tr(E_{ϕ}^{(s)}(x, gx) ⊗ χ(g)) dx =∫_{Ω_{Γ}} ∑_{[g]∈CΓ} ∑_{h∈[g]} tr(E_{ϕ}^{(s)}(x, hx) ⊗ χ(h)) dx

=∑_{[g]∈CΓ} ∑_{[h]∈Γ_{g}\Γ} ∫_{Ω_{Γ}} tr(E_{ϕ}^{(s)}(x, h^{−1}ghx) ⊗ χ(h^{−1}gh)) dx

=∑_{[g]∈CΓ} ∑_{[h]∈Γ_{g}\Γ} ∫_{hΩ_{Γ}} tr(E_{ϕ}^{(s)}(x, gx) ⊗ χ(g)) dx .

If h runs through a set of representatives for Γ_{g}\Γ, then Ω_{Γ_{g}} = ⋃ hΩ_{Γ} is a fundamental domain for Γ_{g}. Thus

Tr E_{Y,ϕ}^{(s)} =

∑_{[g]∈CΓ} ∫_{Ω_{Γg}} tr(E_{ϕ}(x, gx) ⊗ χ(g)) dx . The term corresponding to [g] = 1 is nothing else but

⟨dim(χ)vol(Y)E^{(s)}(., γ, σ), ϕ⟩ .

Since Ω_{<g>} is a disjoint union of n_{Γ}(g) translates of Ω_{Γ_{g}}, the terms corresponding to [g] ≠ 1 are equal to \frac{tr χ(g)}{n_{Γ}(g)}I_{ϕ}^{(s)}(g). The proposition follows.

□

1.4.2 The identity contribution

In this subsection we compute the distribution E(t, γ, σ) explicitly and show that E^{s}(t, γ, σ) vanishes identically. We use the results of Subsection 1.2.3 in order to reduce the computation of E(t, γ, σ) to that of E_{d}(t, γ, σ). Recall the definition of the polynomials P_{σ} and Q_{σ} and of the multiplicities m_{d}(s, γ, σ) from the Definitions 1.13 and 1.3, respectively.

Proposition 1.22 Let dim X be odd. We set r = 1 in the case (a) and r = 2 in case (b). Then we have

E_{d}(t, γ, σ) = \frac{rπ}{T vol(X_{d})}P_{σ}(i\frac{d}{dt})δ(t)

and

E(t, γ, σ) = \frac{rπ}{T vol(X_{d})}P_{σ}(\frac{d}{dt})δ(t) .

The multiplicities m_{d}(s, γ, σ) are given by

m_{d}(s, γ, σ) = rP_{σ}(s)  for all 0 < s ∈ L(σ),
m_{d}(0, γ, σ) = rP_{σ}(0)/2 , if ϵ_{σ} = 0,
m_{d}(s, γ, σ) = 0  otherwise .

P r o o f . Note that

vol(X_{d})E_{d}(t, γ, σ) = ∑_{s∈specA_{d}(γ,σ)} m_{d}(s, γ, σ) cos(sT t) .

Set

Ẽ_{d}(t, γ, σ) := \frac{r}{vol(X_{d})} ∑_{k=0}^{∞} P_{σ}((k + ϵ_{σ})T) cos((k + ϵ_{σ})T t) .

First we compute Ẽ_{d}(t, γ, σ) for the case ϵ_{σ} = 0. Note that

∑_{k=0}^{∞} cos(kT t) = \frac{1}{2} + \frac{1}{2}

∑_{k=−∞}^{∞} e^{ikTt} = \frac{1}{2} + \frac{π}{T}δ(t) . Since P_{σ} is an even polynomial, we obtain

Ẽ_{d}(t, γ, σ) = \frac{r}{2vol(X_{d})}P_{σ}(0) + \frac{rπ}{T vol(X_{d})}P_{σ}(i\frac{d}{dt})δ(t) .

Using

∑_{k=0}^{∞} cos((k + \frac{1}{2})T t) = e^{\frac{1}{2} Tt} \frac{1}{2}

∑_{k=−∞}^{∞} e^{iktT} = \frac{π}{T}δ(t) . we obtain for ϵ_{σ} = \frac{1}{2}

Ẽ_{d}(t, γ, σ) = \frac{rπ}{T vol(X_{d})}P_{σ}(i\frac{d}{dt})δ(t) .

The asymptotic expansion of E_{d}(t, γ, σ) for t ↓ 0 (see Proposition 1.5) consists entirely of terms b_{d,2k+1}|t|^{2k+1}, k ∈ ℕ_{0}. But by Corollary 1.16 the difference Ẽ_{d}(t, γ, σ) − E_{d}(t, γ, σ) is a sum of a finite number of terms of the form a_{i} cos(b_{i}t).

We conclude that b_{d,2k+1} = 0, k ∈ ℕ_{0}, and

E_{d}(t, γ, σ) = \{ \begin{matrix}Ẽ_{d}(t, γ, σ) − \frac{r}{2vol(X_{d})}P_{σ}(0) , & ϵ_{σ} = 0 \\ Ẽ_{d}(t, γ, σ) , & ϵ_{σ} = \frac{1}{2} . \end{matrix}

Thus the multiplicities have the form claimed in the proposition. We now combine Proposition 1.5 and Lemma 1.7 in order to obtain the formula for E(t, γ, σ). □

Proposition 1.23 The distribution E^{s}(t, γ, σ) vanishes identically.

P r o o f . By an argument analogous to the one used to obtain Proposition 1.22 it is enough to show that the distribution Tr(D_{d}(σ) cos(tA_{d}(γ^{s}, σ))) vanishes. This is equivalent to the symmetry of the spectrum of D_{d}(σ). Let θ be the lift of the Cartan involution to G_{d}. Then θ acts on V_{d}(γ^{s}) = G_{d} ×_{K} V_{γ^{s}} by sending [g, v] to [θ(g), v]. From the definition (1.10) of D_{d}(σ) it is easy to see that D_{d}(σ) ∘ θ = −θ ∘ D_{d}(σ). Thus θ defines an isomorphism between the eigenspaces of D_{d}(σ) to the eigenvalues λ and −λ. The proposition follows.

□

We remark that the above argument shows the symmetry of the spectrum of any Dirac operator on a compact symmetric space.

Proposition 1.24 Let n = dim X be even and γ ∈ K̂ be σ-admissible. Then we have in the case ϵ_{σ} = 0

E_{d}(t, γ, σ) = −\frac{T}{4vol(X_{d})}Q_{σ}(i\frac{d}{dt})\frac{1}{sin^{2}(T t/2)}

E(t,γ, σ) = (−1)^{n/2} \frac{T}{4vol(X_{d})}Q_{σ}(\frac{d}{dt})\frac{1}{sinh^{2}(T t/2)}

and in the case ϵ_{σ} = 1/2 E_{d}(t, γ, σ) = −\frac{T}{4vol(X_{d})}Q_{σ}(i\frac{d}{dt})\frac{cos(T t/2}{sin^{2}(T t/2)}

E(t,γ, σ) = (−1)^{n/2} \frac{T}{4vol(X_{d})}Q_{σ}(\frac{d}{dt})\frac{cosh(T t/2}{sinh^{2}(T t/2)} .

Here we interpret

\frac{1}{sinh^{2}(T t/2)} and

cosh(T t/2)_{\overline{sinh^{2}(T t/2)}} as distributions on ℝ writing them as derivatives of locally integrable functions:

\frac{1}{sinh^{2}(T t/2)} = −\frac{4}{T^{2}} \frac{d^{2}}{dt^{2}} ln(| sinh(T t/2)|) ,

cosh(T t/2)_{\overline{sinh^{2}(T t/2)=− \frac{4}{T^{2}} \frac{d^{2}}{dt^{2}} ln(|tanh(T t/4)|) .}}

We interpret the singularity of E_{d}(t, γ, σ) at t = 0 in the analogous way.

P r o o f . By Corollary 1.8, it is enough to compute E_{d}(t, γ, σ). Since P_{σ} is antisymmetric, it is not possible to do this in the straightforward way used in the proof of Proposition 1.22. First we consider for Re(t) > 0 the function Tr e^{−tA_{d}(γ,σ)} = ∑_{0≤λ∈L} P_{σ}(λ)e^{−tλ}. In the case ϵ_{σ} = 0 we have

∑_{0≤λ∈L} P_{σ}(λ)e^{−tλ} = ∑_{k=0}^{∞} -d_{\overline{dtQ_{σ}( \frac{d}{dt})e^{−kTt}}}

=-Q_{σ}(\frac{d}{dt})\frac{d}{dt} \frac{1}{1−e^{−T t}} (1.22)

=T_{\overline{4Q_{σ}( \frac{d}{dt}) \frac{1}{sinh^{2}(T t/2)} .}}

For ϵ_{σ} = 1/2 we obtain

∑_{0≤λ∈L} P_{σ}(λ)e^{−tλ} = -Q_{σ}(\frac{d}{dt})\frac{d}{dt} \frac{e^{−Tt/2}}{1−e^{−T t}} =T_{\overline{4Q_{σ}( \frac{d}{dt}) \frac{cosh(T t/2)}{sinh^{2}(T t/2)} .}}

(1.23)

Tr e^{−(t_{0}+it)A_{d}(γ,σ)} converges for t_{0} ↓ 0 as a distribution in t, and we obtain for t ∉ \{\frac{2kπ}{T} |k ∈ ℤ\}

E_{d}(t, γ, σ) = \frac{1}{2vol(X_{d})}(Tr e^{itA_{d}(γ,σ)} + Tr e^{−itA_{d}(γ,σ)}) = 1_{\overline{vol(X_{d})Tr e^{itA_{d}(γ,σ)}}}

=- T_{\overline{4vol(X_{d})Q_{σ}(i \frac{d}{dt})\{}}

\frac{1}{sin^{2}(T t/2)} , if ϵ_{σ} = 0,
\frac{cos(T t/2)}{sin^{2}(T t/2)} , if ϵ_{σ} = 1/2 .

Proposition 1.5 shows how the singularity of E_{d}(t, γ, σ) has to be interpreted in a distributional sense.

□

1.4.3 The hyperbolic contribution

Let g ∈ G be hyperbolic. Then there is an Iwasawa decomposition G = NAK such that g = am ∈ A^{+}M. As usual, M is the centralizer of the Lie algebra 𝔞 of A in K.

Definition 1.25 We define

l(g) := | log(a)|

and for σ ∈ M̂ and a finite-dimensional unitary representation χ of γ

C(g, σ) := (−1)^{n−1} \frac{l(g)e^{ρl(g)}tr σ(m)}{2 det(1 − Ad(ma)_{𝔫})} , C_{χ}(g, σ) := tr(χ(g))C(g, σ) .

Note that for g ∈ Γ the number l(g) is nothing else but the length of the closed geodesic on Y defined by g. For a discussion of the geometric meaning of C(g, σ) and C_{χ}(g, σ) we refer to Section 3.1. The main result of this subsection is

Proposition 1.26 The hyperbolic contributions I_{ϕ}^{(s)}(g, σ) are given by

I_{ϕ}(g) = C(g, σ)(ϕ(l(g)) + ϕ(−l(g))) , case (a), I_{ϕ}(g) = (C(g, σ) + C(g, wσ))(ϕ(l(g)) + ϕ(−l(g))) , case (b), I_{ϕ}^{s}(g) = i(C(g, σ) − C(g, wσ))(ϕ′(l(g)) − ϕ′(−l(g))) ,

where ϕ′(t) denotes the derivative of ϕ with respect to t.

P r o o f . We first identify the domain Ω_{<g>}. We use the adapted Iwasawa decomposition G = ANK and identify X with AN. Then it is easy to see that one can choose Ω_{<g>} := [1, a] × N ⊂ AN. Moreover, we trivialize the bundle

V(γ^{(s)}) = AN × V_{γ^{(s)}} (1.24) identifying V_{γ^{(s)}} = V (γ^{(s)})_{x_{0}}. The kernel E_{ϕ}^{(s)}(x, y) becomes an End(V_{γ^{(s)}})-valued function. We first carry out the integration with respect to A.

I_{ϕ}^{(s)}(g, σ) = ∫_{0}^{a} ∫_{N} tr

(E_{ϕ}^{(s)}(bn, abα_{m}(n))γ^{(s)}(m)) dbdn

=—log(a)| ∫_{N} tr (E_{ϕ}^{(s)}(n, aα_{m}(n))γ^{(s)}(m)) dn

=l(g)∫_{N} tr (E_{ϕ}^{(s)}(α_{am}(n^{−1})n, a)γ^{(s)}(m)) dn .

The map h(ma) : n → α_{am}(n^{−1})n is a diffeomorphism of N and the Haar measure dn transforms as (compare Helgason [34] 1.5.4)

dh(ma)(n) = (−1)^{n−1} det(1 − Ad(ma)_{𝔫})dn .

We continue the evaluation of I_{ϕ}^{(s)}(g, σ) obtaining

I_{ϕ}^{(s)}(g, σ) = (−1)^{n−1} \frac{l(g)}{det(1−Ad(ma)_{𝔫})} ∫_{N} tr (E_{ϕ}^{(s)}(a^{−1}n, e)γ^{(s)}(m)) dn . (1.25)

In order to evaluate the N-integral we employ the Fourier transform of Helgason-type

F : C_{c}^{∞}(X, V (γ^{(s)})) → C^{∞}

(ia^{∗} × K, V_{γ^{(s)}}) , which is defined by

⟨v, F (f)(λ, k)⟩ := ∫_{X}⟨f(x), ϕ_{−λ,v}(k^{−1}x)⟩_{x}dx.

Here v ∈ V_{γ^{(s)}}^{∗} and ϕ_{λ,v} ∈ C^{∞}(X, V (γ^{(s)})) is defined by ϕ_{λ,v}(an) := a^{λ+ρ}v with respect to the trivialization (1.24). The Fourier transform obviously extends to distributions with compact support f ∈ C_{c}^{−∞}(X, V (γ^{(s)}))

F (f)(λ, k) := ⟨f, l_{k}ϕ_{−λ,v}⟩ ,

where l_{k} is the left action by k ∈ K. For λ ∈ i𝔞^{∗} we have

-Ωϕ_{λ,v} = ϕ_{λ,(|λ|^{2}+|ρ|^{2}−Ω_{M})v} , (1.26)

where Ω_{M} is the Casimir operator of M. This can be seen by using the decomposition 𝔤 = 𝔪 ⊕ 𝔞 ⊕ (𝔫 + 𝔫̄). By the very definition of A(γ^{(s)}, σ) formula (1.26) implies for f ∈ C_{c}^{∞}(X, V)

F (A(γ^{(s)}, σ)^{2}f) = (|λ|^{2} + |ρ|^{2} − Ω_{M} − c(σ))F (f) .

We first derive the formula for I_{ϕ}(g, σ). We claim that

F(cos(tA(γ, σ))f) = cos(t\sqrt{|λ|^{2} + |ρ|^{2} − Ω_{M} − c(σ)})F (f) .(1.27) In fact, h(t) := cos(tA(γ, σ))f satisfies the wave equation

(\frac{∂^{2}}{∂t^{2}} + A(γ, σ)^{2})h = 0, h(0) = f, \frac{∂}{∂t}h_{t=0} = 0 .

Thus, F (h(t)) must satisfy the equation

(\frac{∂^{2}}{∂t^{2}} +|λ|^{2} +|ρ|^{2} −Ω_{M} −c(σ))F (h(t)) = 0, F (h(0)) = F (f), \frac{∂}{∂t}F (h(t))_{t=0} = 0 .

This equation uniquely determines F (h(t)) and we have

F (h(t)) = cos(t\sqrt{|λ|^{2} + |ρ|^{2} − Ω_{M} − c(σ)})F (f)

proving the claim. Let I label the M-types σ′ occurring in V_{γ}, i.e.,

V_{γ} = ⨁_{σ′∈I} V_{γ}(σ′) ,

and let P_{σ′} be the corresponding projections. Abbreviate

d(λ, σ′) := \sqrt{|λ|^{2} + |ρ|^{2} − σ′(Ω_{M}) − c(σ)} .

Integrating (1.27) against ϕ(t) we obtain

F(E_{ϕ}f) = ∑_{σ′∈I} \frac{1}{2}(ϕ̂(d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′}F (f) , where

ϕ̂(s) = ∫_{ℝ} ϕ(t)e^{its}dt .

This equation extends to distributions f ∈ C_{c}^{−∞}(X, V (γ)). Inserting for f the delta distribution located in the origin e ∈ AN ≅ X with values in V_{γ}^{∗}, we obtain

F (E_{ϕ}(., e))(λ, e) = ∑_{σ′∈I} \frac{1}{2}(ϕ̂(d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′} ∈ End(V_{γ}) .

Now we can continue the evaluation of the hyperbolic contribution. Note that E_{ϕ}(a^{−1}n, e)γ(m) is a smooth End(V_{γ})-valued function with compact support. Using i^{∗}(γ) = σ we compute for case (a)

a^{−ρ} ∫_{N} tr (E_{ϕ}(a^{−1}n, e)γ(m)) dn

=1_{\overline{2πi ∫_{i𝔞^{∗}} ∫_{A} e^{λ(log(a)+log(b))}b^{ρ} ∫_{N} tr(E_{ϕ}(bn,e)γ(m)) dn db dλ}}

=1_{\overline{2πi ∫_{i𝔞^{∗}} e^{λ(log(a))}tr(F(E_{ϕ}(.,e))(−λ,e)γ(m)) dλ}} =1_{\overline{2πi}} ∫_{i𝔞^{∗}} e^{λ(log(a))}

∑_{σ′∈I} \frac{1}{2}tr ((ϕ̂(d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′}

γ(m) dλ(1.28) =1_{\overline{2πi ∫_{i𝔞^{∗}} e^{λ(log(a))} \frac{1}{2} (ϕ̂(−|λ|)+ϕ̂(|λ|))tr σ(m) dλ}}

=1_{\overline{2(ϕ(l(g))+ϕ(−l(g)))tr σ(m) .}} (1.29)

The contribution of the M-types different from σ cancel in (1.28), and σ contributes with multiplicity one. We obtain the expression for I_{ϕ}(g, σ) claimed in the proposition by inserting (1.29) into (1.25). The desired formula for case (b) follows from r(γ) = σ + wσ by the analogous computation. The computation of I_{ϕ}^{s}(g, σ) proceeds similarly. First we need to know the Fourier transform of D(σ)f for f ∈ C_{c}^{∞}(X, V (γ^{s})). Choose an orthonormal basis (X_{0} = H, X_{i}, i = 1, …, n − 1) of 𝔭 with H ∈ 𝔞^{+}, where 𝔤 = 𝔨 ⊕ 𝔭 is the Cartan decomposition and Y_{i} ∈ 𝔨 such that X_{i} + Y_{i} ∈ 𝔫. We compute for λ ∈ 𝔞_{ℂ}^{∗}, v ∈ V_{γ^{s}}

D(σ)ϕ_{λ,v}(an) = ∑_{i=0}^{n−1} c(X_{i})\frac{d}{dt}_{|t=0}ϕ_{λ,v}(an exp tX_{i})

= (λ + ρ)ϕ_{λ,c(H)v}(an) + ∑_{i=1}^{n−1} ϕ_{λ,c(X_{i})γ^{s}(Y_{i})v}(an) .

Now, by the explicit formulas for the spin representation of 𝔨 = 𝔰𝔬(n) we have s(Y_{i}) = −\frac{1}{2}c(H)c(X_{i}). Hence, using V_{γ^{s}} = S ⊗ V_{γ′}, where S denotes the spinor module, we obtain

c(X_{i})γ^{s}(Y_{i}) = (c(X_{i}) ⊗ id_{γ′}) ∘ (−\frac{1}{2}c(H)c(X_{i}) ⊗ id_{γ′} + id_{s} ⊗ γ′(Y_{i})) =-1_{\overline{2c(H)⊗id_{γ′}+c(X_{i})γ′(Y_{i}) .}}

64 Chapter 1. Elliptic Operators, Duality, and Trace Formulas

Since ρ = (n − 1)/2 we end up with

D(σ)ϕ_{λ,v} = ϕ_{λ,(λc(H)+∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i}))v} ,

which implies that for f ∈ C_{c}^{∞}(X, V (γ^{s}))

F (Df) = (λc(H) + ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i}))F (f) .

Arguing as above we find

F(D(σ) cos(tA(γ^{s}, σ))f) = (λc(H) + ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i})) cos(t\sqrt{|λ|^{2} + ρ^{2} − Ω_{M} − c(σ)})F (f) and F(E_{ϕ}^{s}f) = (λc(H) + ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i})) ∑_{σ′∈J} \frac{1}{2}(ϕ̂(d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′}F (f) , where J labels the M-types in V_{γ^{s}}. Hence, inserting the delta distribution for f, we obtain the following equation in End(V_{γ}^{s}):

F (E_{ϕ}^{s}(., e))(λ, e) =(λc(H) + ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i})) ∑_{σ′∈I} \frac{1}{2}(ϕ̂ (d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′} . Again, we compute a^{−ρ} ∫_{N} tr (E_{ϕ}^{s}(a^{−1}n, e)γ(m)) dn

=1_{\overline{2πi ∫_{i𝔞^{∗}} e^{λ(log(a))}tr(F(E_{ϕ}^{s}(.,e))(−λ,e)γ^{s}(m)) dλ}}

=1_{\overline{2πi}} ∫_{i𝔞^{∗}} e^{λ(log(a))}tr ((−λc(H) + ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i}))

∑_{σ′∈J} \frac{1}{2}(ϕ̂(d(λ, σ′)) + ϕ̂(−d(λ, σ′)))P_{σ′}

γ^{s}(m) dλ

Note that r(γ^{s}) = (s^{+} ⊗ r(γ′)) ⊕ (s^{−} ⊗ r(γ′)) as an M-module and that ∑_{i=1}^{n−1} c(X_{i})γ′(Y_{i}) intertwines these two summands. Thus this sum does not contribute to the trace. We have chosen the Clifford structure of S in such a way that c(H) acts by ±i on s^{+} iff (s^{+} − s^{−}) ⊗ r(γ′) = ±(σ − wσ). We obtain

a^{−ρ} ∫_{N} tr (E_{ϕ}^{s}(a^{−1}n, e)γ(m)) dn

=-1_{\overline{2π ∫_{i𝔞^{∗}} λe^{λ(log(a))} \frac{1}{2} (ϕ̂(−|λ|)+ϕ̂(|λ|))tr(σ(m)−wσ(m)) dλ}}

=i_{\overline{2(ϕ′(l(g))−ϕ′(−l(g)))tr(σ(m)−wσ(m)),}}

where ϕ′(t) = \frac{d}{dt}ϕ(t). The contribution of the M-types different from σ and wσ cancels and σ, wσ contribute with multiplicity 1 and −1, respectively. We find the formula for I_{ϕ}^{s}(g, σ) by inserting the last line into (1.25).

□

1.4.4 The distributional trace formula

Now we are able to state the distributional trace formula in its final form.

Theorem 1.27 The following equations of distributions on ℝ hold :

Tr cos(tA_{Y,χ}(γ, σ)) = dim(χ)vol(Y)E(t, γ, σ)
+ ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ)}{n_{Γ}(g)}(δ(t − l(g)) + δ(t + l(g)))

in case (a) and

Tr cos(tA_{Y,χ}(γ, σ)) = dim(χ)vol(Y)E(t, γ, σ)

+ ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ) + C_{χ}(g, wσ)}{n_{Γ}(g)}(δ(t − l(g)) + δ(t + l(g))) ,

Tr D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)) =

−i ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)}(δ′(t − l(g)) − δ′(t + l(g)))

in case (b). The distribution E(t, γ, σ) is given by Propositions 1.22 and 1.24, respectively. Set L := max({|Imλ| | λ ∈ spec A_{Y,χ}(γ^{(s)}, σ)} ∪ {ρ}), and denote by C_{L,n}(ℝ) the space of all functions ϕ ∈ C^{n+1}(ℝ) such that there is an ϵ > 0 with

||ϕ||_{ϵ} := sup_{t∈ℝ} max_{k∈{0,...,n+1}}(|ϕ^{(k)}(t)| e^{(L+ϵ)t}) < ∞ .

Then all distributions appearing in the above equations extend continuously to C_{L,n}(ℝ), where n = dim(Y). Hence these equations can be read as equations of linear functionals on C_{L,n}(ℝ).

P r o o f . Inserting Propositions 1.22, 1.23, 1.24, and 1.26 into Proposition 1.21 the equations follow. If |ϕ(t)| ≤ Ce^{−L+ϵ}, then the operators

∫_{ℝ} ϕ(t) cos(tA_{Y,χ}(γ, σ)) dt and ∫_{R} ϕ(t)D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)) dt

exist. Using Weyl’s asymptotic law

#{λ ∈ spec A_{Y,χ}(γ, σ) | |λ| ≤ r} ∼ Cr^{n}

we see that they are of trace class for ϕ ∈ C_{L,n}(ℝ). Moreover, if ϕ_{l} ⊂ C_{c}^{∞}(ℝ) is a sequence converging to ϕ in the norm ||.||_{ϵ}, then the traces of the corresponding operators also converge. The identity contribution is a derivative of order n of a locally integrable function, which grows at most linearly for |t| → ∞ (see Propositions 1.22 and 1.24). Hence the sequence of its applications to ϕ_{l} also converges for l → ∞. There is some constant C such that

C_{χ}(g, σ) ≤ Ce^{−ρl(g)} .  (1.30)

Using the volume growth of X

vol(B(x_{0}, R)) ∼ C′e^{2ρR}

and the co-compactness of Γ we obtain

#{[g] ∈ CΓ | l(g) ≤ R} ≤ #{g ∈ Γ | l(g) ≤ R} ≤ C′′e^{2ρR} .  (1.31)

Hence the hyperbolic contribution extends continuously to C_{L,n}(ℝ), too. □ We conclude this subsection with a slight variant of the trace formula, which we shall need in Subsection 3.2.2.

Corollary 1.28 For ϕ ∈ C_{L,n}(ℝ) the operators

∫_{−∞}^{∞} ϕ(t)\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt , ∫_{−∞}^{∞} ϕ(t)D_{Y,χ}(σ)\frac{sin(tA_{Y,χ}(γ^{s}, σ))}{A_{Y,χ}(γ^{s}, σ)} dt (1.32)

exist and are of trace class. Their traces are given by

Tr ∫_{−∞}^{∞} ϕ(t)\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt

= (−1)^{n−1}d\frac{dim(χ)vol(Y)}{T vol(X_{d})} ∫_{−∞}^{∞} F (t) (P_{σ}(\frac{d}{dt})ϕ) (t) dt  (1.33)

+ ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ)}{n_{Γ}(g)} ∫_{l(g)}^{∞}(ϕ(t) − ϕ(−t)) dt ,

Tr ∫_{−∞}^{∞} ϕ(t)D_{Y,χ}(σ)\frac{sin(tA_{Y,χ}(γ^{s}, σ))}{A_{Y,χ}(γ^{s}, σ)} dt

= −i ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)}(ϕ(l(g) − ϕ(−l(g))) ,

where d = −(1)^{n/2} if n is even, d = 1 otherwise, and

F (t) := \{ \begin{matrix}ln(| sinh(T t/2)|), & case ϵ_{σ} = 0, n even \\ ln(|tanh(T t/4)|), & case ϵ_{σ} = 1/2, n even \\ sign(t)π/2, & case (a), n odd \\ sign(t)π, & case (b), n odd . \end{matrix}

P r o o f . The argument at the beginning of the proof of the preceding theorem also shows the existence and traceability of the operators (1.32). In order to compute their traces one has to insert ϕ into the trace formulas in Theorem 1.27, to use the explicit formulas for the identity contribution from Propositions 1.22 and 1.24, and to integrate the resulting distribution. The integration constant is determined by the fact that

Tr \frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} and Tr D_{Y,χ}(σ)\frac{sin(tA_{Y,χ}(γ^{s}, σ))}{A_{Y,χ}(γ^{s}, σ)}

are odd distributions on ℝ. □

Theta Functions

2.1 Theta functions for compact locally symmetric rank-one spaces

2.1.1 History

Though its properties are easy consequences of the trace formula the theta function for locally symmetric spaces of rank one has only recently been introduced. The case of Riemann surfaces was studied by Cartier/Voros in [11]. Their method is different from that used in the present book as they start with the Selberg zeta function and obtain the theta function by a path integral encircling a certain set of singularities of the Selberg zeta function. The properties of the theta function are derived using the functional equation and the Euler product representation of the Selberg zeta function. The idea of this method can be traced back to Cramér [14] (see also the recent paper [38]). The results of [11] were generalized to theta functions associated to M-types over compact rank-one locally symmetric spaces by Juhl ([40], [39], and the appendix to [9]) who also introduced the super theta function. In [9] we gave a simple direct proof of the meromorphic continuation of the theta function in case σ = 1 using the wave equation of the Laplace-Beltrami operator. The main motivation for introducing the notion of σ-admissible lifts was to extend the method of [9] to general σ ∈ M̂. The most important step was to find the correct elliptic operators A_{Y,χ}(γ, σ). From other points of view the theta function, in particular its restriction to the imaginary axis, has been an object of interest for a long time (there is a whole bunch of papers starting with [19]). In these works a general compact Riemannian manifold Y is considered, and A_{Y,χ}(γ, σ)^{2} is replaced by the Laplace-Beltrami operator. One is interested in the description of the singularities of the theta function on the imaginary axis in terms of the geodesic flow of Y. In the locally symmetric rank-one situation we, of course, reproduce the results known from that general theory, but our understanding in this special case is much deeper. In particular, one does not expect a meromorphic continuation of the theta function to all of ℂ for non-locally symmetric spaces. If the curvature is not strictly negative (e.g. on a locally symmetric, but higher rank space), one should consider theta functions with multi-dimensional time as one can learn it from the case of a product of two rank-one locally symmetric spaces. Let γ be an admissible lift of the M-type σ ∈ M̂. The theta function θ_{χ}(t, σ) of A_{Y,χ}(γ, σ) encodes all spectral information about this operator. Initially defined of the right half-plane it extends to a meromorphic function on all of ℂ. An interesting observation is that from its singularities one can read off all information needed to define the Selberg zeta function, namely the hyperbolic contributions. The theta function satisfies a functional equation which in even dimensions reveals an intimate link with the dual theta function θ_{d}(t, σ) associated to A_{d}(γ, σ). From the general theory developed in [19] one expects that the singularities of θ_{χ}(t, σ) on the imaginary axis are related with the closed geodesics of Y. But surprisingly, the singularities of θ_{χ}(t, σ) on the negative real axis connected with the closed orbits of the ’analytic continuation’ of the geodesic flow to X_{d}. This observation was first made by Cartier/Voros [11]. It is obvious from the equivariant proportionality principle how θ_{χ}(t, σ) changes if one replaces γ by a non-admissible lift. If n is odd in case (b) we also consider the super theta function θ_{χ}^{s}(t, σ) associated to D_{Y,χ}(σ).

2.1.2 The main theorem

Let σ ∈ M̂. If n is even, then let γ be an admissible lift of σ. If n is odd, then in case (a) let γ be a lift of σ, and in case (b) let γ be a lift of σ + wσ and γ^{s} be defined as in Subsection 1.1.3.

Definition 2.1 The theta functions θ_{χ}(t, σ), θ_{χ}(t, σ)_{d} are defined by

θ_{χ}(t, σ) := Tr e^{−tA_{Y,χ}(γ,σ)} = ∑_{s∈ℂ}m_{χ}(s, γ, σ)e^{−ts}, Re(t) > 0
θ_{d}(t, σ) := Tr e^{−tA_{d}(γ,σ)} = ∑_{s∈ℂ}m_{d}(s, γ, σ)e^{−ts}, Re(t) > 0 .

If n is odd in case (b) we define the super theta function θ_{χ}^{s}(t, σ) by

θ_{χ}^{s}(t, σ) := Tr sign(D_{Y,χ}(σ))e^{−tA_{Y,χ}(γ^{s},σ)} = ∑_{s∈ℝ,s>0}m_{χ}^{s}(s, σ) e^{−ts} , Re(t) > 0 ,

where sign(x) = 1 if x > 0, sign(x) = −1 if x < 0 and sign(0) = 0.

By Weyl’s asymptotic law the theta functions are holomorphic functions on the right half-plane. An immediate consequence of the formulas (1.22) and (1.23) is

Corollary 2.2 For even n the dual theta functions are explicitly given by

θ_{d}(t, σ) = vol(X_{d})E_{d}(it, γ, σ)
= \frac{T}{4} Q_{σ}(i\frac{d}{dt}) \{ \begin{matrix}\frac{1}{sinh^{2}(T t/2)}, & ϵ_{σ} = 0 \\ \frac{cosh(T t/2)}{sinh^{2}(T t/2)}, & ϵ_{σ} = 1/2 \end{matrix} .

It is a simple exercise to obtain the corresponding formulas for odd n. It follows from the relation between the polynomial P_{σ} and the Borel-Weil-Bott Theorem that the definition of the dual theta function given here coincides with that of [40]. We are interested in a meromorphic continuation of the theta functions. Consider, e.g., θ_{χ}(t, σ). Since the eigenvalues of the elliptic operator A_{Y,χ}(γ, σ) satisfy Weyl’s asymptotic law, it is easy to see that

θ_{χ}(is + ϵ, σ), s ∈ ℝ ϵ > 0 ,

converges to a tempered distribution as ϵ → 0. The trace formula gives an explicit expression for the symmetrization

θ_{χ}(is + 0, σ) + θ_{χ}(is − 0, σ) .

We read off a candidate for a continuation of θ_{χ}(t, σ) to the left half-plane and then verify that both pieces in fact glue nicely. This is the philosophy behind the proof of the following theorem.

Theorem 2.3 The theta functions θ_{χ}(t, σ), θ_{χ}^{s}(t, σ) admit meromorphic continuations to the whole complex plane. They satisfy the functional equations

θ_{χ}(t, σ) + θ_{χ}(−t, σ) = 2 dim(χ)vol(Y)E(it, γ, σ)
= \{ \begin{matrix}2\frac{dim(χ)χ(Y)}{χ(X_{d})}θ_{d}(it, σ) & n even \\ 0 & n odd \end{matrix}
θ_{χ}^{s}(t, σ) − θ_{χ}^{s}(−t, σ) = 0

Here we consider the distributions E(t, γ, σ), E_{d}(t, γ, σ) as meromorphic functions. If n is even, then the singularities of θ_{χ}(t, σ) are

• first order poles at ±il(g), g ∈ CΓ, with residue \frac{C_{χ}(g,σ)}{πn_{Γ}(g)},

• poles of order n at 2πk/T, k = −1, −2, … with the same singular part as 2 dim(χ)vol(Y)E(it, γ, σ),

• a ”big” pole of order n at 0.

[figure: closed geodesics of X_{d}; closed geodesics of Y; ”big” singularity; _{2π/T}; t-plane; singularity picture of θ_{χ}(t, σ), n even; Fig. 1]

If n is odd, then then the singularities of θ_{χ}(t, σ), θ_{χ}^{s}(t, σ) are first order poles at ±il(g), g ∈ CΓ, with residue

\frac{C_{χ}(g, σ)}{πn_{Γ}(g)} , case (a) ,
\frac{C_{χ}(g, σ) + C_{χ}(g, wσ)}{πn_{Γ}(g)} , case (b), θ_{χ}(t, σ) ,

±\frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{πn_{Γ}(g)} , case (b), θ_{χ}^{s}(t, σ) .

Moreover, θ_{χ}(t, σ) has a pole of order n at t = 0 while

θ_{χ}^{s}(0, σ) = η(D_{Y,χ}(σ)) ,

where η(D_{Y,χ}(σ)) is the eta-invariant of the Dirac operator D_{Y,χ}(σ) (see the appendix to this subsection).

It is natural to consider the locus −\frac{2π}{T} ℕ of the singularities of the theta function on the negative real axis as the lenghts of the closed geodesics of X_{d}. P r o o f . We consider θ_{χ}(t, σ) and use the proposed functional equation in order to define θ_{χ}(t, σ) on the left half-plane. For Re(t) < 0 set θ_{χ}(t, σ) := −θ_{χ}(−t, σ)+2 dim(χ)vol(Y)E(it, γ, σ). Then θ_{χ}(t, σ) is meromorphic on ℂ\iℝ and has singularities as claimed in the theorem. We have to show that the two pieces on the left and the right half-planes glue nicely at the imaginary axis to give a global meromorphic function. To approach this problem we define θ_{χ}(t, σ) as a distribution on U := ℂ \ \{\frac{−2π}{T} ℕ_{0}\}. Let ϕ ∈ C_{c}^{∞}(U). Then by definition

⟨θ_{χ}(., σ), ϕ⟩ = lim_{ϵ→0} ∫_{|Re(v)|≥ϵ} θ_{χ}(v, σ)ϕ(v)dv .

It follows from Weyl’s asymptotic law that this limit indeed defines a distribution. We compute the distributional derivative ¯∂θ and find by partial integration

⟨¯∂θ_{χ}(., σ), ϕ⟩ = ⟨θ_{χ}(., σ), ¯∂^{∗}ϕ⟩

= lim_{ϵ→0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} \frac{1}{2}[−\frac{∂}{∂u} − \frac{∂}{∂(it)}](ϕ(it + u) + ϕ(−it − u)) θ_{χ}(it + u, σ)du dt
− lim_{ϵ→0} dim(χ)vol(Y) ∫_{ϵ}^{∞} ∫_{−∞}^{∞}[−\frac{∂}{∂u} − \frac{∂}{∂(it)}]ϕ(−it − u) E(t − iu, γ, σ)dt du
= ⟨\frac{1}{2}(θ_{χ}(it + 0, σ) + θ_{χ}(−it + 0, σ)) − dim(χ)vol(Y)E(t, γ, σ), ϕ(it)⟩ .

Here we have employed the fact that the theta function is holomorphic on the right half-plane. The distributional trace formula Theorem 1.27 states

⟨θ_{χ}(it + 0, σ) + θ_{χ}(−it + 0, σ), ϕ(it)⟩
= 2⟨Tr cos(tA_{Y,χ}(γ, σ)), ϕ(it)⟩
= ⟨2 dim(χ)vol(Y)E(t, γ, σ), ϕ(it)⟩

+2 ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ)}{n_{Γ}(g)}⟨(δ(t − l(g)) + δ(t + l(g))), ϕ(it)⟩ .

It follows that

⟨¯∂θ_{χ}(., σ), ϕ⟩ = ∑_{[g]∈CΓ,[g]≠[e]} \frac{C_{χ}(g, σ)}{n_{Γ}(g)}(ϕ(il(g)) + ϕ(−il(g))) .

Since ¯∂\frac{1}{z} = πδ(z) we obtain that θ_{χ}(t, σ) is holomorphic near the imaginary axis except at t = ±il(g), g ∈ CΓ, g ≠ 1, where it has first order poles with residues as claimed in the Theorem. The pole at t = 0 can be discussed as in Duistermaat-Guillemin [19]. This finishes the proof of the theorem for θ_{χ}(t, σ). We now consider θ_{χ}^{s}(t, σ). We first obtain a meromorphic continuation of the derivative (θ_{χ}^{s})′(t, σ). It turns out that this derivative has second order poles with vanishing residues and can thus be integrated to give θ_{χ}^{s}(t, σ). Since D_{Y,χ}(σ) = sign(D_{Y,χ}(σ))A_{Y,χ}(γ^{s}, σ) we have

(θ_{χ}^{s})′(t, σ) = −Tr D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)}, Re(t) > 0 .

We define (θ_{χ}^{s})′(t, σ) on the left half-plane proposing the functional equation

(θ_{χ}^{s})′(t, σ) + (θ_{χ}^{s})′(−t, σ) = 0 .  (2.1)

In order to show that both pieces glue nicely at the imaginary axis we again consider (θ_{χ}^{s})′(t, σ) as a distribution on ℂ. Let ϕ ∈ C_{c}^{∞}(ℂ). Then by definition

⟨(θ_{χ}^{s})′(., σ), ϕ⟩ := lim_{ϵ→0} ∫_{|Re(v)|≥ϵ}(θ_{χ}^{s})′(v, σ)ϕ(v)dv .

We compute the distributional derivative ¯∂(θ_{χ}^{s})′(t, σ) and find

⟨¯∂(θ_{χ}^{s})′(., σ), ϕ⟩

= ⟨(θ_{χ}^{s})′(., σ), ¯∂^{∗}ϕ⟩
= lim_{ϵ→0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} \frac{1}{2}[−\frac{∂}{∂u} − \frac{∂}{∂(it)}](ϕ(it + u) + ϕ(−it − u))(θ_{χ}^{s})′(it + u, σ)du dt
= ⟨\frac{1}{2}((θ_{χ}^{s})′(it + 0, σ) + (θ_{χ}^{s})′(−it + 0)), ϕ(it)⟩ .

The distributional trace formula Theorem 1.27 states

⟨((θ_{χ}^{s})′(it + 0, σ) + (θ_{χ}^{s})′(−it + 0, σ)), ϕ(it)⟩

= ⟨−2Tr D_{Y,χ}(σ) cos(tA_{Y,χ}(γ^{s}, σ)), ϕ(it)⟩
= −2i ∑_{[g]∈CΓ,[g]≠[1]} \frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)} (\frac{d}{dt}_{|t=l(g)}ϕ(it) − \frac{d}{dt}_{|t=−l(g)}ϕ(it)) .

Thus,

⟨¯∂(θ_{χ}^{s})′(., σ), ϕ⟩ = −i ∑_{[g]∈CΓ,[g]≠[1]} \frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)}

(\frac{d}{dt}_{|t=l(g)}ϕ(it) − \frac{d}{dt}_{|t=−l(g)}ϕ(it)) .

Write z = u + it. Since

−¯∂\frac{i}{z^{2}} = \frac{∂}{∂t} ¯∂\frac{1}{z} = π \frac{∂}{∂t}δ(z)

we obtain that (θ_{χ}^{s})′(t, σ) is holomorphic near the imaginary axis except at t = ±il(g), g ∈ CΓ, where it has second order poles with the singular part

±\frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{πn_{Γ}(g)} \frac{1}{(t ∓ il(g))^{2}} .

Thus we have provided a meromorphic continuation of (θ_{χ}^{s})′(t, σ) with vanishing residues as required. Note that (θ_{χ}^{s})′(t, σ) → 0 exponentially as t → ∞. Since the same holds for θ_{χ}^{s}(t, σ) itself, we have

θ_{χ}^{s}(t, σ) := ∫_{∞}^{t}(θ_{χ}^{s})′(u, σ)du .

Hence θ_{χ}^{s}(t, σ) is meromorphic and has first order poles with residues as claimed in the theorem. We now evaluate θ_{χ}^{s}(0, σ). Integrating (2.1) we obtain the functional equation

θ_{χ}^{s}(t, σ) − θ_{χ}^{s}(−t, σ) = 0 .

Moreover,

θ_{χ}^{s}(0, σ) = − ∫_{0}^{∞}(θ_{χ}^{s})′(u, σ)du .  (2.2)

We we identify the integral (2.2) with η(D_{Y,χ}(σ)). The η-invariant of D_{Y,χ}(σ) is defined as the value at s = 0 of the η-function

η(D_{Y,χ}(σ))(s) = Tr\frac{sign(D_{Y,χ}(σ))}{|D_{Y,χ}(σ)|^{s}} .

Initially defined for large Re(s) this function has a meromorphic extension and is regular at s = 0. We have

η(D_{Y,χ}(σ))(s) = Tr\frac{sign(D_{Y,χ}(σ))}{|D_{Y,χ}(σ)|^{s}}
= \frac{1}{Γ(s + 1)} ∫_{0}^{∞} u^{s}Tr D_{Y,χ}(σ) e^{−uA_{Y,χ}(γ^{s},σ)}du .

Since −Tr D e^{−uA_{Y,χ}(γ^{s},σ)} = (θ_{χ}^{s})′(u, σ) is regular at u = 0 we can put s = 0 and obtain

− ∫_{0}^{∞}(θ_{χ}^{s})′(u, σ)du = η(D_{Y,χ}(σ)) .

This finishes the proof of the theorem for θ_{χ}^{s}(t, σ). □

Appendix : For the convenience of the reader we recall some facts about the eta invariant [1]. Let D be a Dirac operator over a compact manifold Y. For Re(s) > dim(Y) we can form the eta function

η(D)(s) := Tr\frac{D}{|D|^{s+1}} .

Using the Mellin transform and the asymptotic expansion of Tr De^{−tD^{2}} for t → 0 one obtains a meromorphic continuation of η(D)(s) to all of ℂ. It turns out that this continuation is regular at s = 0 and the eta invariant of D is defined by η(D) := η(D)(0) (see [26]). It was shown by Bismut/Freed [4] that Tr De^{−tD^{2}} is integrable at t = 0 (for D associated to a Clifford bundle) and that

η(D) = \frac{1}{\sqrt{π}} ∫_{0}^{∞} Tr De^{−tD^{2}} \frac{dt}{t} .

2.2 Theta functions for surfaces with cusps

This section has a supplementary character. We use the example of hyperbolic surfaces with cusps in order to demonstrate how the theory of the theta function should be generalized to locally symmetric spaces of rank one and finite volume. In this book we try to produce a rather self-contained exposition of the theory of zeta and theta functions for compact quotients. But in the present section we use the properties of the scattering matrix and the trace formula for finite volume surfaces which we are not going to prove in this book. For hyperbolic Riemann surfaces of finite volume these topics have already been known for a long time starting with the original work of Selberg [73]. Good references for all the facts we need are the books of Lax/Phillips [45], Hejhal [32] and Venkov [78], [79]. We concentrate on the theta function and omit a discussion of the Selberg zeta function. The properties of the latter are well documented in the literature cited above. A theory of theta and Selberg zeta functions has not been worked out yet for non-trivial M-types in higher dimensional rank-one cases. However, we do not expect that completely new phenomenons will appear.

2.2.1 Definition of the theta function

We consider the case σ = 1, use the lift γ = 1, i.e., we consider the function case, and we choose the trivial twist χ. We will also employ a simplified notation deleting all references to σ , γ and χ. Let Y be a complete Riemann surface of constant negative curvature −1 and finite volume. Consider the unique selfadjoint extension Δ_{Y} of the Laplacian and form A_{Y} := A_{Y,1}(1, 1) = \sqrt{Δ_{Y} − \frac{1}{4}}, where we take the square root with positive imaginary part. It turns out that the correct theta function for A_{Y} is the sum of two partial theta functions, namely one coming from the discrete spectrum of A_{Y} and the other related to the singularities of the scattering matrix. Let P_{d} be the projection onto the subspace of L^{2}(Y) spanned by the L^{2}- eigenfunctions of Δ_{Y}, i.e., onto the discrete subspace. The eigenvalue theta function of Y is defined by

Definition 2.4

θ_{p}(t) := Tr P_{d}e^{−tA}P_{d} , Re(t) > 0 .

It is known, that the number of eigenvalues of Δ_{Y} (counted with multiplicity) being smaller than λ is bounded by Cλ^{2} for some C < ∞. Hence, θ_{p}(t) is a well-defined holomorphic function on the right half-plane. It is an outstanding conjecture that for generic surfaces the discrete spectrum is finite. This would imply that θ_{p} has a holomorphic continuation. But, certainly, there are arithmetic examples with infinite discrete spectrum. In general we are not able to provide a meromorphic continuation of θ_{p}(t) alone. The point is that the eigenvalue theta function enters into the trace formula in a combination with another (scattering) theta function obtained from the singularities of the scattering determinant. Let C(z) be the scattering matrix. It is a meromorphic family of r×r-matrices, where r is the number of cusps of Y. By S(z) := det C(z) we denote the scattering determinant of Y. S(z) is a meromorphic function of finite order, satisfying

S(z) = S(−z)^{−1}, \overline{S(z)} = S(z̄)^{−1}, z ∈ ℂ .  (2.3)

Let Σ be the set of those singularities σ of S(z) with Re(σ) ≥ 0, Im(σ) > 0. Let m_{σ} (−m_{σ}, 2m_{σ}, −2m_{σ}) be the order of the pole with Re(σ) > 0 (zero with Re(σ) > 0, pole with Re(σ) = 0, zero with Re(σ) = 0). There is a C < ∞ such that all σ ∈ Σ satisfy Im(σ) < C. Moreover, there are at most finitely many zeros on Σ and the total multiplicity of all singularities with Re(σ) < λ is bounded by Cλ^{2} for some C < ∞. Thus

Definition 2.5

θ_{s}(t) := ∑_{σ∈Σ} m_{σ}e^{−tσ}, Re(t) > 0

is a well-defined holomorphic function on the right half-plane.

Definition 2.6 The theta function associated to a hyperbolic surface of finite volume is defined for Re(t) > 0 by

θ_{Y} (t) := θ_{p}(t) + θ_{s}(t) .

The plan of the remainder of the present section is as follows. In Subsection 2.2.2 we state the version of the Selberg trace formula suitable for our purpose. In Subsection 2.2.3 we provide a meromorphic continuation of θ_{Y} (t) to the left half-plane across (0, −i∞). Here the trace formula can be applied immediately. In Subsection 2.2.4 we discuss the necessary modifications of the trace formula needed to provide a meromorphic continuation of the theta function across the positive imaginary axis. Finally, in Subsection 2.2.5 we combine all results together and state our main theorem about the meromorphic continuation of the theta function to the Riemann surface of the logarithm and describe the singularities in detail.

2.2.2 The Selberg trace formula

Let ϕ̂ be the Fourier transform

ϕ̂(λ) = ∫_{−∞}^{∞} e^{iλt}ϕ(t) dt

of an even function ϕ satisfying for some δ > 0, C < ∞

|ϕ(t)| < Ce^{−(1/2+δ)|t|}
|\frac{dϕ̂}{dλ}| < \frac{C}{(1 + |λ|)^{3+δ}} .

We introduce the hyperbolic contribution.

Definition 2.7

c_{h}(ϕ) := ∑_{c−closed geodesic} \frac{l_{c}}{2n_{c} sinh(l_{c}/2)}ϕ(l_{c}) ,

where l_{c} is the length of the geodesic c and n_{c} is its multiplicity.

We employ here the language of closed geodesics for simplicity. In fact there is a one-to-one correspondence between non-trivial conjugacy classes in Γ = π_{1}(Y) and closed geodesics of Y. The hyperbolic contribution defined above is compatible with the hyperbolic contribution introduced in Definition 1.20. The contribution of the identity is given by

Definition 2.8

c_{e}(ϕ) := −\frac{vol(Y)}{4π} ∫_{−∞}^{∞} \frac{\frac{d}{dt}ϕ(t)}{sinh(t/2)} dt .

Note that the integral is well-defined at zero since ϕ is even. Let r denote the number of cusps of Y. The Selberg trace formula (see [73], [45] (note that the final formula in [45] contains a mistake), [79]) then has the form

∑_{λ eigenvalue of A_{Y}} ϕ̂(λ) = c_{e}(ϕ) + c_{h}(ϕ)

−r ln(2)ϕ(0) + \frac{q}{2}ϕ̂(0)

−\frac{r}{2π} ∫_{−∞}^{∞} ϕ̂(λ)\frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ

−\frac{i}{4π} ∫_{−∞}^{∞} ϕ̂(λ)\frac{Ṡ(λ)}{S(λ)} dλ ,

where q = \frac{1}{2}(r −trC(0)). On the left-hand side in the sum over the eigenvalues the latter are counted with their multiplicities. Set ϕ(t) = ψ(t)+ψ(−t) for some ψ ∈ C_{c}^{∞}(0, ∞). We can write all terms of the trace formula as applications of distributions to ψ. Let

Definition 2.9

θ̃_{p}(t) := T rP_{d}e^{tA}P_{d}, Re(t) < 0 .

Then θ̃_{p}(t) is a holomorphic function on the left half-plane. The left-hand side of the trace formula can be written as

⟨θ_{p}(−it + 0) + θ̃_{p}(−it − 0), ψ⟩ .

The hyperbolic contribution has the form

⟨ ∑_{c−closed geodesic} \frac{l_{c}}{2n_{c} sinh(l_{c}/2)}δ(t − l_{c}), ψ⟩

and the contribution of the identity is

⟨−\frac{vol(Y)}{4π} \frac{cosh(t/2)}{sinh^{2}(t/2)}, ψ⟩ .

The last two terms of the trace formula are less trivial. We first consider the term

−\frac{r}{2π} ∫_{−∞}^{∞} ψ̂(λ)\frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ .

Since Γ(1 + iλ) is of finite order and ψ̂ decreases exponentially in the upper half-plane, we can apply Cauchy’s integral formula. Γ(1 + iλ) has poles of first order at λ = ik, k = 1, 2, . . .. Thus

−\frac{r}{2π} ∫_{−∞}^{∞} ψ̂(λ)\frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ = r ∑_{k=1}^{∞} ψ̂(ik)

= ⟨\frac{re^{−t}}{1 − e^{−t}} , ψ⟩ .

In an analogous manner

−\frac{r}{2π} ∫_{−∞}^{∞} ψ̂_{−}(λ)\frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ = 0 ,

where ψ_{−}(t) := ψ(−t). In order to rewrite the term involving the scattering determinant we employ a formula sometimes called the Mini Trace Formula ([45], Thm. 9.11 or [52], Eq. (5.1) ):

\frac{i}{4π} ∫_{−∞}^{∞} ϕ̂(λ)\frac{Ṡ(λ)}{S(λ)} dλ = ∑_{σ∈Σ} m_{σ}(ψ̂(σ) + ψ̂(−σ̄)) .

Let Definition 2.10

θ̃_{s}(t) := \overline{θ_{s}(−t̄)}, Re(t) < 0 .

θ̃_{s}(t) is a holomorphic function on the left half-plane. We can write

\frac{i}{4π} ∫_{−∞}^{∞} ϕ̂(λ)\frac{Ṡ(λ)}{S(λ)} dλ = ⟨θ_{s}(−it + 0) + θ̃_{s}(−it − 0), ψ⟩ .

Let θ̃_{Y} := θ̃_{p} + θ̃_{s}. The following Lemma is now obvious.

Lemma 2.11 The following equation of distributions holds on (0, ∞):

θ_{Y} (−it + 0) + θ̃_{Y} (−it − 0) = −\frac{vol(Y)}{4π} \frac{cosh(t/2)}{sinh^{2}(t/2)}

+ ∑_{c−closed geodesic} \frac{l_{c}}{2n_{c} sinh(l_{c}/2)}δ(t − l_{c})

+q + \frac{r}{e^{t} − 1} .

2.2.3 Meromorphic continuation of θ_{Y} across (0, −i∞)

Lemma 2.12 θ_{Y} (t) has a meromorphic continuation across (0, −i∞) to the left half-plane

θ_{Y} (t) = −θ̃_{Y} (t) + \frac{vol(Y)}{4π} \frac{cos(t/2)}{sin^{2}(t/2)} + q + \frac{r}{e^{−it} − 1} , Re(t) < 0.  (2.4)

The singularities are

• first order poles at t = −il_{c}, c a closed geodesic, with residue \frac{l_{c}}{4πn_{c} sinh(l_{c}/2)} (If there are different closed geodesics of the same length, then the residues add up.) and

• second order poles at t = −2πk, k = 1, 2, … with the same singular part

as

\frac{vol(Y)}{4π} \frac{cos(t/2)}{sin^{2}(t/2)} + \frac{r}{e^{−it} − 1} .

P r o o f . We use (2.4) in order to define θ_{Y} (t) as a distribution on U := ℂ \ ([0, i∞) ∪ −2πℕ). Then we compute ¯∂θ_{Y}. The interesting region is near (0, −i∞), where we glue the theta function defined on the right half-plane together with the function defined by (2.4) on the left half-plane. For ϕ ∈ C_{c}^{∞}(U) we define the value of the distribution θ_{Y} on ϕ as

⟨θ_{Y} , ϕ⟩ = lim_{ϵ↓0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} θ_{Y} (it + u)ϕ(it + u) dt du

+ lim_{ϵ↓0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} θ_{Y} (it − u)ϕ(it − u) dt du .

In view of the polynomial bounds on the discrete spectrum and the singularities of the scattering determinant this defines a distribution. We compute, using Lemma 2.11,

⟨¯∂θ_{Y} , ϕ⟩ = ⟨θ_{Y} , ¯∂^{∗}ϕ⟩

= lim_{ϵ↓0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} θ_{Y} (it + u)\frac{1}{2} [−\frac{∂}{∂u} − \frac{∂}{∂(it)}] ϕ(it + u) dt du

+ lim_{ϵ↓0} ∫_{ϵ}^{∞} ∫_{−∞}^{∞} θ_{Y} (it − u)\frac{1}{2} [\frac{∂}{∂u} − \frac{∂}{∂(it)}] ϕ(it − u) dt du

= lim_{ϵ↓0} ∫_{−∞}^{∞} θ_{Y} (it + ϵ)\frac{1}{2}ϕ(it + ϵ) dt

− lim_{ϵ↓0} ∫_{−∞}^{∞} θ_{Y} (it − ϵ)\frac{1}{2}ϕ(it − ϵ) dt

= \frac{1}{2}⟨θ_{Y} (it + 0) + θ̃_{Y} (it − 0) + \frac{vol(Y)}{4π} \frac{cosh(t/2)}{sinh^{2}(t/2)} − q − \frac{r}{e^{t} − 1}, ϕ(it)⟩
= ⟨  ∑_{c−closed geodesic} \frac{l_{c}}{4n_{c} sinh(l_{c}/2)}δ(t + il_{c}), ϕ(it)⟩ .

Thus θ_{Y} (t) is holomorphic on U with the exception of the points −il_{c}, c a closed geodesic. Note that ¯∂\frac{1}{t} = πδ(t). Thus θ_{Y} (t) has a first-order pole at t = −il_{c} with the residue as claimed in the lemma. Since θ̃_{Y} (t) is holomorphic for Re(t) < 0 the singularities of θ_{Y} (t) in the left half-plane come from the additional terms

\frac{vol(Y)}{4π} \frac{cos(t/2)}{sin^{2}(t/2)} + \frac{r}{e^{−it} − 1} .

□

2.2.4 The structure of θ_{Y} (t) near (0, i∞)

In this subsection we obtain a modification of the trace formula expressing the distribution θ_{Y} (it + 0) + θ̃_{Y} (it − 0) on (0, ∞). The result will then be applied in a similar fashion as Lemma (2.11) in the proof of Lemma 2.12 in order to obtain the meromorphic continuation of θ_{Y} (t) across (0, i∞). Note that the trace formula 2.11 involves the distributions

θ_{s}(it + 0) + θ̃_{s}(it − 0), θ_{p}(it + 0) + θ̃_{p}(it − 0) .

Obviously we have

θ_{p}(it + 0) + θ̃_{p}(it − 0) = θ_{p}(−it + 0) + θ̃_{p}(−it − 0) .

But unfortunately

θ_{s}(it + 0) + θ̃_{s}(it − 0) ≠ θ_{s}(−it + 0) + θ̃_{s}(−it − 0) .

We will apply the method of Cramer [14], Jorgenson/Lang [38] in order to provide a meromorphic continuation of

V (t) := θ̃_{s}(it) − θ_{s}(−it) ,

which is initially defined for Im(t) > 0. V (t) will serve as a correction term to the trace formula similar to 2.11 but now formulated for distributions on (0, i∞). In terms of the singularities of the scattering determinant we have

V (t) = ∑_{σ∈Σ} m_{σ}(e^{itσ̄} − e^{itσ}) .

We factor the scattering determinant as S(λ) = G(λ)L(λ), where

G(λ) := e^{a+bλ} [\sqrt{π} \frac{Γ(iλ)}{Γ(1/2 + iλ)}]^{r} .

( r is the number of cusps). By Hejhal, [32], Ch.8, 3.35, L(λ) has a representation as a certain Dirichlet series of the form

L(λ) := 1 + ∑_{q∈Q} \frac{c_{q}}{q^{iλ}} .

Here, Q is a certain set of positive real numbers > 1 and the c_{q} are real ’multiplicities’. In order to obtain the ”1” in front of the sum one has to adjust a, b ∈ ℝ appropriately. The Dirichlet series converges for Im(λ) small enough. Let R := {(n, q_{1}, …, q_{n})|n ∈ ℕ, q_{i} ∈ Q, i = 1, …, n}. For p = (n, q_{1}, …, q_{n}) ∈ R we define the multiplicity

c(p) := \frac{(−1)^{n+1}}{n} ∏_{j=1}^{n} c_{q_{j}}

and

|p| := ∏_{j=1}^{n} q_{i} .

Then ln L(λ) also has a Dirichlet series representation

ln L(λ) := ∑_{p∈R} \frac{c(p)}{|p|^{iλ}} ,

which converges uniformly and absolutely in each half-plane Im(λ) ≤ −A for some ∞ > A > 0. We fix such an A for the remainder of this section. Since S and G are meromorphic functions of finite order, L is also a meromorphic function of finite order. As a consequence of (2.3) L satisfies the functional equation

L(λ) = \frac{1}{G(−λ)G(λ)} \frac{1}{L(−λ)} .  (2.5)

In [38] it was shown that V (t) can be written as a contour integral. Let γ be the polygonal path going from iA + ∞ to iA, then down to −iA and back to −iA + ∞. The down path should surround the singularities of S(λ) with Re(λ) = 0 on the positive real part side. By [38], Thm. 2.3, we have

2πiV (t) = ∫_{γ} e^{itλ} \frac{Ṡ(λ)}{S(λ)} + 2πi ∑_{σ∈Σ,Re(σ)=0} m_{σ}(e^{−itσ} − e^{itσ}) .

We now carry out a detailed study of the integral above. As a result we will obtain a meromorphic continuation of V and its singularities. Note that

\frac{Ṡ(λ)}{S(λ)} = \frac{Ġ(λ)}{G(λ)} + \frac{L̇(λ)}{L(λ)} .

Since G has no singularities in the region encircled by γ, we have

∫_{γ} e^{itλ} \frac{Ṡ(λ)}{S(λ)} = ∫_{γ} e^{itλ} \frac{L̇(λ)}{L(λ)} .

Let h_{1} be the global holomorphic function

h_{1}(t) := −i ∫_{−A}^{A} e^{−tλ} \frac{L̇(iλ)}{L(iλ)} dλ

2.2. Theta functions for surfaces with cusps 85

(the path of integration surrounds the singularities as above). Then

∫_{γ} e^{itλ} \frac{L̇(λ)}{L(λ)} = h_{1}(t)
−e^{−At} ∫_{0}^{∞} e^{itλ} \frac{L̇(iA + λ)}{L(iA + λ)} dλ  (2.6)

+e^{At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA + λ)}{L(−iA + λ)} dλ .

We employ now the functional equation (2.5) in order to rewrite (2.6).

−e^{−At} ∫_{0}^{∞} e^{itλ} \frac{L̇(iA + λ)}{L(iA + λ)} dλ = −e^{−At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA − λ)}{L(−iA − λ)} dλ
−e^{−At} ∫_{0}^{∞} e^{itλ} \frac{Ġ(−iA − λ)}{G(−iA − λ)} dλ (2.7)

+e^{−At} ∫_{0}^{∞} e^{itλ} \frac{Ġ(iA + λ)}{G(iA + λ)} dλ .

In order to evaluate the integrals involving L we use the representation of ln L by a Dirichlet series.

−e^{−At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA − λ)}{L(−iA − λ)} dλ

= −e^{−At} ln L(−iA) − e^{−At}it ∫_{0}^{∞} e^{itλ} ln L(−iA − λ) dλ
= −e^{−At} ln L(−iA) − e^{−At}it ∫_{0}^{∞} e^{itλ} ∑_{p∈R} \frac{c(p)}{|p|^{A−iλ}} dλ
= −e^{−At} ln L(−iA) − e^{−At}it ∑_{p∈R} c(p) ∫_{0}^{∞} e^{itλ}e^{(−A+iλ) ln |p|} dλ
= −e^{−At} ln L(−iA) + e^{−At}t ∑_{p∈R} \frac{c(p)}{|p|^{A}(t + ln |p|)} .

Analogously we get

e^{At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA + λ)}{L(−iA + λ)} dλ

= −e^{At} ln L(−iA) − e^{At}it ∫_{0}^{∞} e^{itλ} ln L(−iA + λ) dλ

= −e^{At} ln L(−iA) − e^{At}it ∫_{0}^{∞} e^{itλ} ∑_{p∈R} \frac{c(p)}{|p|^{A+iλ}} dλ

= −e^{At} ln L(−iA) − e^{At}it ∑_{p∈R} c(p) ∫_{0}^{∞} e^{itλ}e^{(−A−iλ) ln |p|} dλ

= −e^{At} ln L(−iA) + e^{At}t ∑_{p∈R} \frac{c(p)}{|p|^{A}(t − ln |p|)} .

We introduce the global holomorphic function

h_{2}(t) := −e^{−At} ln L(−iA) − e^{At} ln L(−iA)

and the global meromorphic function

W (t) := e^{−At}t ∑_{p∈R} \frac{c(p)}{|p|^{A}(t + ln |p|)} + e^{At}t ∑_{p∈R} \frac{c(p)}{|p|^{A}(t − ln |p|)} .  (2.8)

Then for a piece of (2.6) we obtain

−e^{−At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA − λ)}{L(−iA − λ)} dλ + e^{At} ∫_{0}^{∞} e^{itλ} \frac{L̇(−iA + λ)}{L(−iA + λ)} dλ = h_{2}(t) + W(t) .

Note that

W (t) = W (−t) , h_{2}(t) = h_{2}(−t) .  (2.9)

In order to evaluate the right-hand side of (2.6) completely it remains to study the integrals in (2.7) involving the G-factors. We have

\frac{Ġ(λ)}{G(λ)} = b + ir (\frac{Γ̇(iλ)}{Γ(iλ)} − \frac{Γ̇(1/2 + iλ)}{Γ(1/2 + iλ)}) .

The terms involving b of (2.7) will eventually cancel. Thus we can set b := 0 in the following. We consider

J_{1}(t) := −e^{−At} ∫_{0}^{∞} e^{itλ} \frac{Ġ(−iA − λ)}{G(−iA − λ)} dλ .

Then J_{1} = I_{1} + I_{2} with

I_{1}(t) := −ie^{−At} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(A − iλ)}{Γ(A − iλ)} dλ
I_{2}(t) := ie^{−At} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1/2 + A − iλ)}{Γ(1/2 + A − iλ)} dλ .

We first evaluate I_{1}. We define the global holomorphic function

h_{3}(t) := −e^{−At} ∫_{1−A}^{0} e^{−tλ} \frac{Γ̇(A + λ)}{Γ(A + λ)} dλ .

Since the singularities of Γ(A − iλ) are at λ = −iA − ik, k = 0, 1, …, by a change of the path of integration we obtain

I_{1}(t) = h_{3}(t) − ie^{−t} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 − iλ)}{Γ(1 − iλ)} dλ .

Assume Re(t) > 0, Im(t) > 0 for a moment and define

F (t) := \frac{1}{t} ∫_{0}^{∞} \frac{λ}{e^{λ} − 1} \frac{dλ}{λ + t}

and let Eu be the Euler constant. It was shown by Cramér [14], eq. (12), that

i ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 − iλ)}{Γ(1 − iλ)} dλ = t ∫_{0}^{∞} e^{itλ} ln Γ(1 − iλ)d(iλ) = \frac{Eu}{t} − F(t) .

Moreover,

M(t) := F (t) − \frac{ln t}{e^{−t} − 1} ,

is a global meromorphic function, where we choose the branch of the logarithm such that ln 1 = 0 and the cut is at (−∞, 0]. The poles of M(t) are at t ∈ 2πiℤ with residue π/2sign(k) at t = 2πik, k ∈ ℤ, k ≠ 0. Now I_{1} has the form

I_{1}(t) = h_{3}(t) − \frac{e^{−t}Eu}{t} + e^{−t}M(t) − \frac{ln t}{e^{t} − 1} .

Hence I_{1} is a holomorphic function on ℂ \ (−∞, 0]. The jump on the negative real axis is given by

I_{1}(t − i0) − I_{1}(t + i0) = \frac{2πi}{e^{t} − 1}, t ∈ (−∞, 0) .

We now evaluate I_{2}(t). We define the global holomorphic function

h_{4}(t) := e^{−At} ∫_{1/2−A}^{0} e^{−tλ} \frac{Γ̇(A + 1/2 + λ)}{Γ(A + 1/2 + λ)} dλ .

Since the singularities of Γ(A+1/2−iλ) are at λ = −i(A+1/2)−ik, k = 0, 1, …, by a change of the path of integration we obtain

I_{2}(t) = h_{4}(t) + ie^{−t/2} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 − iλ)}{Γ(1 − iλ)} dλ .

Again I_{2} can be written in terms of M. We have

I_{2}(t) = h_{4}(t) + \frac{e^{−t/2}Eu}{t} − e^{−t/2}M(t) + \frac{e^{t/2} ln t}{e^{t} − 1} .

I_{2} is a holomorphic function on ℂ \ (−∞, 0]. The jump on the negative real axis is given by

I_{2}(t − i0) − I_{2}(t + i0) = −2πi\frac{e^{t/2}}{e^{t} − 1}, t ∈ (−∞, 0) .

Now we consider

J_{2}(t) := e^{−At} ∫_{0}^{∞} e^{itλ} \frac{Ġ(iA + λ)}{G(iA + λ)} dλ .

Then J_{2} = I_{3} + I_{4} with

I_{3}(t) := ie^{−At} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(−A + iλ)}{Γ(−A + iλ)} dλ
I_{4}(t) := −ie^{−At} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(−(A − 1/2) + iλ)}{Γ(−(A − 1/2) + iλ)} dλ .

We evaluate I_{3}. We can assume that A ∉ \frac{1}{2}ℤ and define the global holomorphic function

h_{5}(t) := e^{−At} ∫_{0}^{A+1} e^{tλ} \frac{Γ̇(−A + λ)}{Γ(−A + λ)} dλ ,

where the path surrounds singularities on the side with positive imaginary part. The singularities of Γ(−A + iλ) are at λ = −iA + ik, k = 0, 1, . . .. By a change of the path of integration we obtain

I_{3}(t) = h_{5}(t) + ie^{t} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ .

Assuming Re(t) < 0 for a moment we can write

ie^{t} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ = e^{t} ∫_{0}^{∞} e^{tλ} \frac{Γ̇(1 + λ)}{Γ(1 + λ)} dλ
= −te^{t} ∫_{0}^{∞} e^{tλ} ln Γ(1 + λ) dλ
= te^{t} ∫_{0}^{−∞} e^{−tλ} ln Γ(1 − λ) dλ
= e^{t}(\frac{Eu}{t} + F(t)) ,

where we again have applied a formula of Cramér (p. 114, top 3) in order to write I_{3} in terms of F. We get

I_{3}(t) = h_{5}(t) + \frac{e^{t}Eu}{t} + e^{t}M(−t) − \frac{ln(−t)}{e^{−t} − 1} .

I_{3} is a holomorphic function on ℂ \ [0, ∞). The jump on the positive real axis is given by

I_{3}(t + i0) − I_{3}(t − i0) = \frac{2πi}{e^{−t} − 1}, t ∈ (0, ∞) .

We now evaluate I_{4}(t). We define the global holomorphic function

h_{6}(t) := −e^{−At} ∫_{0}^{A+1/2} e^{tλ} \frac{Γ̇(−(A − 1/2) + λ)}{Γ((A − 1/2) + λ)} dλ ,

where the path surrounds the singularities on the side with positive imaginary part. The singularities of Γ(−(A − 1/2) + iλ) are at λ = −i(A − 1/2) + ik, k = 0, 1, . . .. By a change of the path of integration we obtain

I_{4}(t) = h_{6}(t) − ie^{t/2} ∫_{0}^{∞} e^{itλ} \frac{Γ̇(1 + iλ)}{Γ(1 + iλ)} dλ .

Now we can again apply the formula of Cramér in order to write I_{4} in terms of M.

I_{4}(t) = h_{6}(t) − \frac{e^{t/2}Eu}{t} − e^{t/2}M(−t) + \frac{e^{−t/2} ln(−t)}{e^{−t} − 1} .

I_{4} is a holomorphic function on ℂ \ [0, ∞). The jump on the real positive axis is given by

I_{4}(t + i0) − I_{4}(t − i0) = −2πi\frac{e^{−t/2}}{e^{−t} − 1}, t ∈ (0, ∞) .

Now we put all formulas together. Define the global holomorphic function

h(t) := h_{1}(t) + h_{2}(t) + rh_{3}(t) + rh_{4}(t) + rh_{5}(t) + rh_{6}(t)

+2πi ∑_{σ∈Σ,Re(σ)=0} m_{σ}(e^{−itσ} − e^{itσ}) .

Lemma 2.13 If Im(t) > 0, then

2πiV (t) := 2πi(θ̃_{s}(it) − θ_{s}(−it))
= h_{1}(t) + h_{2}(t) + W (t) + rJ_{1}(t) + rJ_{2}(t)

= h(t) + W (t)

+r (−\frac{e^{−t}Eu}{t} + \frac{e^{−t/2}Eu}{t/2} + \frac{e^{t}Eu}{t} − \frac{e^{t/2}Eu}{t})

+r (e^{−t}M(t) − e^{−t/2}M(t) + e^{t}M(−t) − e^{t/2}M(−t))

−r (\frac{ln t}{e^{t} − 1} − \frac{e^{t/2} ln t}{e^{t} − 1} + \frac{ln(−t)}{e^{−t} − 1} − \frac{e^{−t/2} ln(−t)}{e^{−t} − 1}) .

□

Lemma 2.14

h(t) − h(−t) = 0 .

Proof. We start rewriting the integral defining h_{1} using the functional equation for L. We have

h_{1}(t) = −i ∫_{−A}^{A} e^{−t(λ−i0)} \frac{L̇(i(λ − i0))}{L(i(λ − i0))} dλ
= −i ∫_{0}^{A} e^{−t(λ−i0)} \frac{L̇(i(λ − i0))}{L(i(λ − i0))} dλ
−i ∫_{−A}^{0} e^{−t(λ−i0)} \frac{L̇(i(λ − i0))}{L(i(λ − i0))} dλ ,  (2.10)

where we have indicated how the path of integration should surround the singularities of the integrand. We rewrite (2.10) as

−i ∫_{0}^{A} e^{t(λ+i0)} \frac{L̇(−i(λ + i0))}{L(−i(λ + i0))} dλ

and apply the functional equation

\frac{L̇(−i(λ + i0))}{L(−i(λ + i0))} = \frac{L̇(i(λ + i0))}{L(i(λ + i0))} + \frac{Ġ(i(λ + i0))}{G(i(λ + i0))} − \frac{Ġ(−i(λ + i0))}{G(−i(λ + i0))}

in order to obtain h_{1} = h_{7} + rh_{8} with

h_{7}(t) = −i ∫_{0}^{A} (e^{−t(λ−i0)} \frac{L̇(i(λ − i0))}{L(i(λ − i0))} + e^{t(λ+i0)} \frac{L̇(i(λ + i0))}{L(i(λ + i0))}) dλ

rh_{8}(t) = −i ∫_{0}^{A} e^{t(λ+i0)} (\frac{Ġ(iλ − 0)}{G(iλ − 0)} + \frac{Ġ(−iλ + 0)}{G(−iλ + 0)}) dλ .  (2.11)

The anti-symmetrization h_{7}(t) − h_{7}(−t) can be expressed in terms of the singularities of L on the interval i(0, A). The scattering matrix S as well as the factor G contribute to these singularities. We obtain

h_{7}(t) − h_{7}(−t) = −2πi ∑_{σ∈Σ,Re(σ)=0} 2m_{σ}(e^{−itσ} − e^{itσ})

−2πir ∑_{0<k<A}(e^{−tk} − e^{tk}) + 2πir ∑_{0<k<A−1/2}(e^{−t(k+1/2)} − e^{t(k+1/2)}) .

Inserting the definition of G into (2.11) we obtain

h_{8}(t) = ∫_{0}^{A} e^{t(λ+i0)}(\frac{Γ̇(−λ − i0)}{Γ(−λ − i0)} − \frac{Γ̇(1/2 − λ − i0)}{Γ(1/2 − λ − i0)}

−\frac{Γ̇(λ + i0)}{Γ(λ + i0)} + \frac{Γ̇(1/2 + λ + i0)}{Γ(1/2 + λ + i0)}).

We also have

h_{3}(t) = − ∫_{1}^{A} e^{−tλ} \frac{Γ̇(λ)}{Γ(λ)} dλ
h_{4}(t) = ∫_{1/2}^{A} e^{−tλ} \frac{Γ̇(1/2 + λ)}{Γ(1/2 + λ)} dλ
h_{5}(t) = ∫_{−1}^{A} e^{−tλ} \frac{Γ̇(−λ + i0)}{Γ(−λ + i0)} dλ
h_{6}(t) = − ∫_{−1/2}^{A} e^{−tλ} \frac{Γ̇(1/2 − λ + i0)}{Γ(1/2 − λ + i0)} dλ .

Thus defining

h_{9}(t) := h_{8}(t) − h_{3}(−t) − h_{4}(−t) − h_{5}(−t) − h_{6}(−t)

and anti-symmetrizing we obtain

h_{9}(t) − h_{9}(−t) = 2πi ∑_{0<k<A}(e^{−tk} − e^{tk}) − 2πi ∑_{0<k<A−1/2}(e^{−t(k+1/2)} − e^{t(k+1/2)}) .

Since h_{2} is symmetric,

h(t) − h(−t)

= h_{7}(t) − h_{7}(−t) + r(h_{9}(t) − h_{9}(−t)) + 2πi ∑_{σ∈Σ,Re(σ)=0} 2m_{σ}(e^{−itσ} − e^{itσ}) = 0

and the lemma follows. □ We can now write down a distributional trace formula on (0, i∞) containing θ_{Y} (t + 0) + θ̃_{Y} (t − 0). In fact, for t > 0

2πi(θ_{s}(it + 0) + θ̃_{s}(it − 0))
= 2πi(θ_{s}(−it + 0) + θ̃_{s}(−it + 0) + V (t + i0) − V (−t + i0))
= 2πi(θ_{s}(−it + 0) + θ̃_{s}(−it − 0))

+W (t + i0) − W (−t + i0)

+2πir(\frac{1}{e^{−t} − 1} − \frac{e^{−t/2}}{e^{−t} − 1}) .

Equation (2.8) implies

W (t + i0) − W (−t + i0) = −2πi ∑_{p∈R} ln |p|c(p)δ(t − ln |p|) .

If we combine these two equations with the distributional trace formula, then we obtain

Lemma 2.15 The following equation of distributions on (0, ∞) holds :

θ_{Y} (it + 0) + θ̃_{Y} (it − 0)
= q − r + r \frac{e^{t/2}}{e^{t} − 1}

−\frac{vol(Y)}{4π} \frac{cosh(t/2)}{sinh^{2}(t/2)}

+ ∑_{c−closed geodesic} \frac{l_{c}}{2n_{c} sinh(l_{c}/2)}δ(t + l_{c})

− ∑_{p∈R} ln |p|c(p)δ(t + ln |p|).

We can now apply exactly the same technique as in the proof of Lemma 2.12 to prove

Lemma 2.16 θ_{Y} (t) has a meromorphic continuation from the left half-plane to another sheet of the right half-plane across (0, i∞). For Re(t) > 0 this continuation denoted by θ_{1} is given by

θ_{1}(t) = θ_{Y} (t) − r \frac{e^{it/4}}{2 cos(t/4)} .

The singularities on the positive imaginary axis are first order poles at t = il_{c}, c a closed geodesic, with residue \frac{l_{c}}{4πn_{c} sinh(l_{c}/2)} and at t = i ln |p|, p ∈ R,

with residue \frac{− ln |p|c(p)}{2π} (if two such points coincide, then the residues add up). Moreover, it has first-order poles at t = (4k − 2)π, k = 1, 2, …, with residue 2ir.

2.2.5 The meromorphic continuation of Θ_{Y}

We have seen that θ_{Y} (t) extends meromorphically to the Riemann surface of the logarithm. The difference between two sheets is

θ_{1}(t) − θ_{Y} (t) = −r \frac{e^{it/4}}{2 cos(t/4)} .

Thus the modified theta function

Θ_{Y} (t) := θ_{Y} (t) − \frac{r}{4πi} \frac{e^{it/4}}{cos(t/4)} ln(it)

admits a meromorphic continuation to the complex plane. The final theorem of this section is the following:

Theorem 2.17 The meromorphic continuation of the modified theta function has the following singularities:

• First-order poles at t = −il_{c}, c a closed geodesic, with residue \frac{l_{c}}{4πn_{c} sinh(l_{c}/2)},

• first-order poles at t = il_{c}, c a closed geodesic, with residue \frac{l_{c}}{4πn_{c} sinh(l_{c}/2)}

and at t = i ln |p|, p ∈ R, with residue \frac{− ln |p|c(p)}{2π} (if two such points coincide, then the residues add up)

• first-order poles at t = (4k−2)π, k = 1, 2, …, with residue \frac{ir}{2} +\frac{r}{π} ln(π(4k− 2))

• second-order poles at t = −(4k − 2)π, k = 1, 2, …, with residue \frac{ir}{2} + \frac{r}{π} ln(π(4k − 2)) and second Laurent coefficient −\frac{vol(Y)}{π}

• and second order poles at t = −4kπ, k = 1, 2, …, with residue ir and second Laurent coefficient \frac{vol(Y)}{π}.

Θ_{Y} has a pole of at most second order at t = 0.

□

If one compares this result with the compact case the following differences become obvious. First of all it is necessary to consider a theta function which takes the scattering poles into account. Due to the contribution of the cusps this theta function does only extend to the Riemann surface of the logarithm. With the exception of the real axis in one of the sheets the continuation also has poles on the positive real axis. The poles on the negative imaginary axis are related to the closed geodesics as expected from the compact case. On the positive imaginary axis there are poles coming from the closed geodesics and additional poles related to the Dirichlet series giving the scattering matrix. Tracing back the construction of the scattering matrix given in [32] one can find explicit formulas for the location and the residues of these additional poles in terms of the fundamental group of Y.

The Selberg Zeta Function

The present chapter is the heart of this book. In its first section we introduce certain holomorphic functions defined on some half right plane canonically associated with the geodesic flow of Y: the Ruelle and Selberg zeta functions. The remainder of this chapter is devoted to the analytic continuation of the Selberg zeta function to all of ℂ, the description of its singularities, its functional equation and its representation via regularized determinants. The main step is to obtain the analytic continuation of its logarithmic derivative by relating it with the trace of (products of) resolvents of A_{Y,χ}(γ, σ)^{2} using the trace formula and to observe the integrality of its residues. In the next chapter we will derive the properties of the Ruelle zeta function as a corollary of the theory developed here.

3.1 Zeta functions and the geodesic flow

We have already seen in Subsection 1.1.1 that the unit sphere bundle of Y can be identified with the homogeneous space

SY = Γ\G/M .  (3.1)

There is a canonical dynamical system on SY, the geodesic flow φ determined by the metric of Y. In the representation (3.1) it is given by

φ : ℝ × SY ∋ (t, ΓgM) → Γg exp(−tH)M ∈ SY ,  (3.2)

where H is the unit vector in 𝔞^{+}. If V_{χ}(σ) := Γ\(G ×_{M} V_{σ} ⊗ V_{χ}) is the vector bundle corresponding to finite-dimensional unitary representations (σ, V_{σ}) of M and (χ, V_{χ}) of Γ, then we define a lift φ_{χ,σ} of φ to V_{χ}(σ) by

φ_{χ,σ} : ℝ × V_{χ}(σ) ∋ (t, [g, v ⊗ w]) → [g exp(−tH), v ⊗ w] ∈ V_{χ}(σ) . (3.3)

We are particularly interested in the closed orbits of φ, i.e., the closed geodesics on Y. It is a fundamental fact that the set of closed geodesics on a manifold Z of negative sectional curvature corresponds bijectively to the set of free homotopy classes of closed loops or, in other words, to the set of conjugacy classes in π_{1}(Z) (see, e.g., [27] ). In our situation this can be directly read off from (3.2). If [e] ≠ [g] ∈ CΓ, then the corresponding closed orbit is

c := {Γg′ exp(−tH)M | t ∈ ℝ} ,

where g′ is chosen such that g^{′−1}gg′ := ma ∈ MA^{+}, while its length l(c) is given by l(g) = | log(a)| (see Definition 1.25). In fact,

Γg′ exp(−l(g)H)M = Γg^{−1}g′M = Γg′M .

The lift of c to V_{χ}(σ) induces a linear transformation μ_{χ,σ}(c) on the fibre over Γg′M, the monodromy of c. It acts as follows

μ_{χ,σ}(c)([g′, v ⊗ w]) = [g′a^{−1}, v ⊗ w] = [g′a^{−1}m^{−1}, σ(m)v ⊗ w]
= [g^{−1}g′, σ(m)v ⊗ w] = [g′, σ(m)v ⊗ χ(g)w] .

A closed orbit c through y ∈ SY corresponding to [g] ∈ CΓ is called prime if t = l(g) is the smallest time with φ(t, y) = y. In this case, i.e. if n_{Γ}(g) = 1 (see Subsection 1.4.1), we call the class [g] primitive. Now we encode the data corresponding to φ_{χ,σ} by a single function.

Definition 3.1 For s ∈ ℂ, Re(s) > 2ρ, we define the Ruelle zeta function for the flow φ_{χ,σ} by the infinite product

Z_{R,χ}(s, σ) := ∏_{c prime} det (1 − μ_{χ,σ}(c)e^{−sl(c)})^{(−1)^{n−1}}

= ∏_{1≠[g]∈CΓ ; primitive} det (1 − (σ(m) ⊗ χ(g))e^{−sl(g)})^{(−1)^{n−1}} ,

where n is the dimension of Y.

In fact, using the growth estimate (1.31) for Γ and

ln Z_{R,χ}(s, σ) = (−1)^{n−1} ∑_{1≠[g]∈CΓ ; primitive} tr log (1 − (σ(m) ⊗ χ(g))e^{−sl(g)})

= (−1)^{n} ∑_{1≠[g]∈CΓ ; primitive} ∑_{k=1}^{∞} tr \frac{((σ(m) ⊗ χ(g))e^{−sl(g)})^{k}}{k}

= (−1)^{n} ∑_{1≠[g]∈CΓ} \frac{tr σ(m) tr χ(g)}{n_{Γ}(g)} e^{−sl(g)}  (3.4)

we see that the infinite product defining Z_{R,χ}(s, σ) converges for Re(s) > 2ρ and, hence, defines a holomorphic function without zeroes there. Ruelle introduced this zeta function for arbitrary Anosov flows and showed its meromorphy under certain conditions in [64]. While it is perhaps the most natural zeta function associated with φ_{χ,σ}, we introduce another one, the Selberg zeta function, which is well suited for an investigation with the help of the trace formula. Its definition explicitly uses the Anosov property of φ. The Anosov property is the existence of a dφ-invariant splitting

T SY = T^{s}SY ⊕ T^{0}SY ⊕ T^{u}SY ,  (3.5)

where T^{0}SY consists of vectors tangential to the orbits, while the vectors in T^{s}SY (T^{u}SY) shrink (grow) exponentially with respect to the metric for t → ∞, when transported with dφ. In our situation this splitting is given by

T SY ≅ Γ\G ×_{M} (𝔫̄ ⊕ 𝔞 ⊕ 𝔫) ,

where, as usual, 𝔫̄ = θ𝔫 is the sum of negative root spaces of 𝔞. The monodromy P_{c} in T SY of a closed orbit c, called the Poincaré map, decomposes according to this splitting

P_{c} = P_{c}^{s} ⊕ id ⊕ P_{c}^{u} .

Definition 3.2 For s ∈ ℂ, Re(s) > ρ, we define the Selberg zeta function for the flow φ_{χ,σ} by the infinite product

Z_{S,χ}(s, σ) := ∏_{c prime} ∏_{k=0}^{∞} det (1 − μ_{χ,σ}(c) ⊗ S^{k}(P_{c}^{s})e^{−(s+ρ)l(c)})

= ∏_{1≠[g]∈CΓ ; primitive} ∏_{k=0}^{∞} det (1 − (σ(m) ⊗ χ(g) ⊗ S^{k}(Ad(ma)𝔫̄))e^{−(s+ρ)l(g)}) ,

where S^{k} denotes the k^{t}h symmetric power of an endomorphism. In case (b) we also define

S_{χ}(s, σ) := Z_{S,χ}(s, σ)Z_{S,χ}(s, wσ)

and the super zeta function

S_{χ}^{s}(s, σ) := Z_{S,χ}(s, σ)/Z_{S,χ}(s, wσ) ,

where w ∈ W is the non-trivial element.

Taking the estimates (1.30) and (1.31) into account the following lemma shows that the infinite product defining Z_{S,χ}(s, σ) indeed converges for Re(s) > ρ. It also makes the connection with the trace formula apparent.

Lemma 3.3 In case (a) the logarithmic derivative of Z_{S,χ}(s, σ) is given by

L_{S,χ}(s, σ) := \frac{d}{ds} ln Z_{S,χ}(s, σ) = 2 ∑_{1≠[g]∈CΓ} \frac{C_{χ}(g, σ)}{n_{Γ}(g)} e^{−sl(g)} .

In case (b) we have

L_{S,χ}(s, σ) := \frac{d}{ds} ln S_{χ}(s, σ) = 2 ∑_{1≠[g]∈CΓ} \frac{C_{χ}(g, σ) + C_{χ}(g, wσ)}{n_{Γ}(g)} e^{−sl(g)}

and

L_{S,χ}^{s}(s, σ) := \frac{d}{ds} ln S_{χ}^{s}(s, σ) = 2 ∑_{1≠[g]∈CΓ} \frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)} e^{−sl(g)} .

P r o o f . For an endomorphism W with eigenvalues of modulus lower than one the following identity holds:

∑_{k=0}^{∞} tr S^{k}(W) = det(1 − W)^{−1} .

We are going to use this equation for W = Ad(ma)𝔫̄ and compute

ln Z_{S,χ}(s, σ)
= ∑_{1≠[g]∈CΓ ; primitive} ∑_{k=0}^{∞} tr log (1 − (σ(m) ⊗ χ(g) ⊗ S^{k}(Ad(ma)𝔫̄)) e^{−(s+ρ)l(g)})

= − ∑_{1≠[g]∈CΓ ; primitive} ∑_{k=1}^{∞} ∑_{j=0}^{∞} tr \frac{((σ(m) ⊗ χ(g) ⊗ S^{k}(Ad(ma)𝔫̄))e^{−(s+ρ)l(g)})^{j}}{j}

= − ∑_{1≠[g]∈CΓ} ∑_{k=1}^{∞} \frac{tr σ(m) tr χ(g) tr S^{k}(Ad(ma)𝔫̄)}{n_{Γ}(g)} e^{−(s+ρ)l(g)}

= − ∑_{1≠[g]∈CΓ} \frac{tr σ(m) tr χ(g)}{det(1 − Ad(ma)𝔫̄)n_{Γ}(g)} e^{−(s+ρ)l(g)}

= (−1)^{n} ∑_{1≠[g]∈CΓ} \frac{tr σ(m) tr χ(g) e^{(ρ−s)l(g)}}{det(1 − Ad(ma)_{𝔫})n_{Γ}(g)} .  (3.6)

In the last step we have used that

det(1 − Ad(ma)𝔫̄) = (−1)^{n−1}a^{−2ρ} det(1 − Ad(ma)_{𝔫}) .

The lemma now follows by differentiation and Definition 1.25. □

We conclude this Subsection with the observation due to Fried [22] that the Ruelle zeta function can be expressed in terms of Selberg zeta functions. In the next chapter we shall use this representation in order to discuss properties of the Ruelle zeta function. Let 𝔫_{ℂ} be the complexification of the Lie algebra of N. For p ≥ 0 consider Λ^{p}𝔫_{ℂ} as a representation of MA. For λ ∈ ℂ ≅ 𝔞_{ℂ}^{∗} let ℂ_{λ} denote the one-dimensional representation of A given by A ∋ a → a^{λ}. There are sets

I_{p} = {(τ, λ)|τ ∈ M̂, λ ∈ ℝ}

such that Λ^{p}𝔫_{ℂ} decomposes with respect to MA as

Λ^{p}𝔫_{ℂ} = ∑_{(τ,λ)∈I_{p}} V_{τ} ⊗ ℂ_{λ},

where V_{τ} is the space of the representation τ. We define

Z_{p,χ}(s, σ) := ∏_{(τ,λ)∈I_{p}} Z_{S,χ}(s + ρ − λ, τ ⊗ σ) .  (3.7)

Proposition 3.4 The Ruelle zeta function has the representation

Z_{R,χ}(s, σ) = ∏_{p=0}^{n−1} Z_{p,χ}(s, σ)^{(−1)^{p}} .

Proof. Combine the contributions of [g] to all Selberg zeta functions occurring in Z_{p,χ}(s, σ). Then the contribution of [g] to (−1)^{n} ln Z_{p,χ}(s, σ) is (see (3.6))

L(s, g, p) := \frac{tr σ(m) tr χ(g) e^{−sl(g)}}{det(1 − Ad(ma)_{𝔫})n_{Γ}(g)} ∑_{(τ,λ)∈I_{p}} e^{λl(g)}tr τ(m) .

We can also write

L(s, g, p) = \frac{tr σ(m) tr χ(g) e^{−sl(g)}}{det(1 − Ad(ma)_{𝔫})n_{Γ}(g)} tr Λ^{p}Ad(ma)_{𝔫} .

We want to form the alternating sum over p. But for any endomorphism W we have

∑_{p=0}^{∞}(−1)^{p}tr Λ^{p}W = det(1 − W) .

Hence

∑_{p=0}^{n−1}(−1)^{p}L(s, g, p) = \frac{tr σ(m) tr χ(g) e^{−sl(g)}}{n_{Γ}(g)} .

The assertion follows from (3.4). □

3.2 Traces of resolvents

3.2.1 Resolvent indentities

The aim of this purely algebraic subsection is to show how a product of resolvents of some operator A can be expressed as a linear combination of resolvents. This generalizes the well known identity 3.8. We set

R(s) := \frac{1}{s + A} .

The resolvent set of A is the set of all s ∈ ℂ such that A − s is invertible.

Lemma 3.5 Let s_{1}, …, s_{N} ∈ ℂ be in the resolvent set of −A and s_{i} ≠ s_{j} for all i ≠ j. Then

∏_{i=1}^{N} R(s_{i}) = ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N} \frac{1}{s_{j} − s_{i}} ) R(s_{i}) .

Proof. We prove this equation by induction on N using the resolvent identity

(b − a)R(a)R(b) = R(a) − R(b) .  (3.8)

For N = 1 the assertion of the lemma is obvious. Assume that the assertion holds for N. Then

∏_{i=1}^{N+1} R(s_{i}) = ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{n} \frac{1}{s_{j} − s_{i}} ) R(s_{i})R(s_{N+1})

= ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N} \frac{1}{(s_{j} − s_{i})}) \frac{1}{(s_{N+1} − s_{i})}(R(s_{i}) − R(s_{N+1}))

= ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}} ) R(s_{i}) − ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}} ) R(s_{N+1}) .

It remains to show that

∑_{i=1}^{N} ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}} = − ∏_{j=1}^{N} \frac{1}{s_{j} − s_{N+1}}

or, equivalently,

∑_{i=1}^{N+1} ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}} = 0.

This is a special case of the Lemma 3.6. □

Lemma 3.6 For any N-tuple s_{1}, …, s_{N} ∈ ℂ of complex numbers with s_{i} ≠ s_{j} for all i ≠ j and l = 0, 1, …, N − 2 we have

∑_{i=1}^{N+1} s_{i}^{l} ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}} = 0.

P r o o f . Consider

f(s_{1}) := ∑_{i=1}^{N+1} s_{i}^{l} ∏_{j=1,j≠i}^{N+1} \frac{1}{s_{j} − s_{i}}

as a rational function of s_{1}. It might have poles of first order at s_{1} = s_{j}, j = 2, …, N. But the residues of f vanish. In fact

res_{s_{1}=s_{j}}f(s_{1}) = lim_{s_{1}→s_{j}} f(s_{1})(s_{1} − s_{j})

= (−s_{1}^{l} ∏_{k=2,k≠j}^{N+1} \frac{1}{s_{k} − s_{1}} + s_{j}^{l} ∏_{k=2,k≠j}^{N+1} \frac{1}{s_{k} − s_{j}} )_{|s_{1}=s_{j}}
= 0 .

Since for l ≤ N − 2 the function f(s_{1}) is bounded and tends to zero at infinity it must vanish identically. □ We adopt the following convention: If f(s) is a function of one complex variable s, then we let Ψ(f(s)) be the function of N complex variables s_{1}, …, s_{N} defined by

Ψ(f(s))(s_{1}, …, s_{N}) := ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N} \frac{1}{s_{j}^{2} − s_{i}^{2}} ) f(s_{i}) .

Lemma 3.7 Let a_{0} + ta_{1} + t^{2}a_{2} + … be the Taylor expansion of Ψ(f(st)) as a function of t at t = 0. Then a_{2n} = 0 for n = 0, …, N − 1.

P r o o f . Let f(st) = ∑_{m=0}^{∞} \frac{b_{m}s^{m}t^{m}}{m!} be the Taylor expansion of f at t = 0. Then

Ψ(f(st))(s_{1}, …, s_{N}) = ∑_{m=0}^{∞} \frac{b_{m}t^{m}}{m!} ∑_{i=1}^{N} s_{i}^{m} ( ∏_{j=1,j≠i}^{N} \frac{1}{s_{j}^{2} − s_{i}^{2}} ) .

The assertion now follows from Lemma 3.6 applied to {s_{i}^{2}} . □

3.2.2 The logarithmic derivative of Z_{S,χ}(s, σ) and resolvents

In this subsection we express the logarithmic derivative L_{S,χ}(s, σ) of the Selberg zeta function in terms of the trace of the resolvent

R(s^{2}) := (s^{2} + A_{Y,χ}(γ, σ)^{2})^{−1}

(Proposition 3.8). Since the resolvent is not of trace class we need a regularization. Employing the results of Subsection 3.2.1 we can express the product of resolvents R(s_{i}^{2}) at different points s_{i} as a linear combination of the R(s_{i}^{2}).

If we take sufficiently many factors this product will be of trace class. We obtain an expression for the corresponding linear combination of the values L_{S,χ}(s, σ)(s_{i}) in terms of the trace of the product of resolvents. This leads to the meromorphic continuation of L_{S,χ}(s, σ) and to a description of its singularities. A similar idea was employed by Parnovskij [54]. The operator A_{Y,χ}(γ, σ) may have a finite number of imaginary eigenvalues. Let L > ρ be such that −L^{2} is a lower bound of A_{Y,χ}(γ, σ)^{2}, i.e, −L^{2} < A_{Y,χ}(γ, σ)^{2}. For Re(s) > L we have

R(s^{2}) := \frac{1}{s^{2} + A_{Y,χ}(γ, σ)^{2}} = ∫_{0}^{∞} e^{−st} \frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt .

Let s_{1}, …, s_{N} ∈ ℂ be complex numbers with s_{i} ≠ s_{j} for all i ≠ j and Re(s_{i}) > L. By Lemma 3.5 we can write

∏_{i=1}^{N} R(s_{i}^{2}) = ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N} \frac{1}{s_{j}^{2} − s_{i}^{2}} ) R(s_{i}^{2})

= ∫_{0}^{∞} ∑_{i=1}^{N} ( ∏_{j=1,j≠i}^{N} \frac{1}{s_{j}^{2} − s_{i}^{2}} ) e^{−s_{i}t} \frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt

= ∫_{0}^{∞} Ψ(e^{−st})\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt .

Recall that n = dim(X). For N > n/2 by Weyl’s asymptotic law the product

∏_{i=1}^{N} R(s_{i}^{2})

is of trace class. We introduce the following regular distributions on ℝ:

F (t) := \{ \begin{matrix}ln(| sinh(T t/2)|), & case ϵ_{σ} = 0, n even \\ ln(|tanh(T t/4)|), & case ϵ_{σ} = 1/2, n even \\ π/2, & case (a), n odd \\ π, & case (b), n odd . \end{matrix}

If n is even or in case (a) if n is odd L_{S,χ}(s, σ) will denote the logarithmic derivative of Z_{S,χ}(s, σ). In case (b) if n is odd L_{S,χ}(s, σ) will denote the logarithmic derivative of S_{χ}(s, σ). The discussion of the logarithmic derivative of the super zeta function in case (b) will be postponed to the end of Subsection 3.2.4. Let d := −(−1)^{n/2} if n is even and d := 1 if n is odd.

Proposition 3.8 Let γ be an admissible lift of σ. If N > n/2 and s_{1}, …, s_{N} ∈ ℂ with Re(s_{i}) > L and s_{i} ≠ s_{j} for all i ≠ j, then

Tr ∏_{i=1}^{N} R(s_{i}^{2}) = \frac{d dim(χ)vol(Y)}{Tvol(X_{d})} ∫_{0}^{∞} Ψ(P_{σ}(s)e^{−st})F(t)dt+Ψ(\frac{L_{S,χ}(s, σ)}{2s}) . (3.9)

P r o o f . Consider the function h(t) := h(t, s_{1}, …, s_{N}) being equal to \frac{1}{2}Ψ(e^{−st}) for t ≥ 0 and to −\frac{1}{2}Ψ(e^{st}) for t ≤ 0. By Lemma 3.7 we can find approximations h_{ϵ}(t) := h_{ϵ}(t, s_{1}, …, s_{N}) such that h_{ϵ}(t) = −h_{ϵ}(−t), ∀t ∈ ℝ, h(t) = h_{ϵ}(t) for |t| ≥ 1 and such that h_{ϵ}(., s_{1}, …, s_{N}) ↦ h(., s_{1}, …, s_{N}) in C_{loc}^{2N−1}(ℝ) if ϵ ↦ 0. We claim that the limit

∫_{0}^{∞} Ψ(e^{−pt})\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt = lim_{ϵ→0} ∫_{−∞}^{∞} h_{ϵ}(t)\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)} dt

exists in the sense of trace class operators. Let Q be the projection onto the orthogonal complement of ker(A_{Y,χ}(γ, σ)). It is enough to show the trace class convergence of the integral restricted to the range of Q. Integrating partially 2R-times, we obtain

∫_{−∞}^{∞}(−1)^{R}h_{ϵ}(t)\frac{sin(A_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)}Q dt = ∫_{−∞}^{∞} h_{ϵ}(t)^{(2R)} \frac{sin(tA_{Y,χ}(γ, σ))}{(A_{Y,χ}(γ, σ))^{2R+1}} Q dt .

If N > R ≥ n/2, then the limit ϵ ↦ 0 of the right-hand side exists in the trace class. This proves the claim. We apply the trace formula (1.33) to

∫_{−∞}^{∞} h_{ϵ}(t)\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)}

and obtain

Tr ∫_{−∞}^{∞} h_{ϵ}(t)\frac{sin(tA_{Y,χ}(γ, σ))}{A_{Y,χ}(γ, σ)}  (3.10)
= (−1)^{n−1} \frac{2d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{∞}(P_{σ}(\frac{d}{dt})h_{ϵ})(t)F (t) dt

+ ∑_{[g]∈CΓ,[g]≠1} \frac{2C_{χ}(g, σ)}{n_{Γ}(g)} ∫_{l(g)}^{∞} h_{ϵ}(t) dt .

Now we take the limit ϵ ↦ 0. Using the explicit expression of L_{S,χ}(s, σ) (Lemma 3.3) we derive

Tr ∏_{i=1}^{N} R(s_{i}^{2}) = (−1)^{n−1} \frac{d dim(χ)vol(Y)}{Tvol(X_{d})} ∫_{0}^{∞} Ψ(P_{σ}(\frac{d}{dt})e^{−st})F(t) dt

+Ψ(\frac{L_{S,χ}(s, σ)}{2s}) .

The convergence of the first term of (3.10) is assured by the C^{2N−1}-convergence of h_{ϵ} while for the second term we use Re(s_{i}) > ρ and the growth estimate (1.31). Equation (3.9) follows by performing the differentiation. □

3.2.3 Meromorphic continuation of the identity contribution

The integral on the right-hand side of (3.9) is called the identity contribution and will be denoted by I. We will show that it extends to a meromorphic function on ℂ^{N} and compute its singularities. If n is odd the identity contribution is easily evaluated :

I(s_{1}, …, s_{N}) := \frac{d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{∞} Ψ(P_{σ}(\frac{d}{dt})e^{−st})F(t) dt
= \frac{rπd dim(χ)vol(Y)}{2T vol(X_{d})}Ψ(\frac{P_{σ}(s)}{s}),

where r = 1 in case (a) and r = 2 in case (b). Thus if n is odd, then the identity contribution I is meromorphic with singularities at the planes s_{i} = s_{j}, i ≠ j coming from the Ψ-construction and possibly at s = 0. The identity contribution in the even-dimensional case is much more involved. Thus let n be even and consider the integral

I(s_{1}, …, s_{N}) := \frac{d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{∞} Ψ(P_{σ}(s)e^{−st})F(t) dt .

This integral converges for Re(s_{i}) > 0, i = 1, …, N, and defines a holomorphic function on the complement of the planes s_{i} = s_{j}, i ≠ j. Integrating by parts we obtain

I = \frac{d dim(χ)vol(Y)}{2vol(X_{d})} ∫_{0}^{∞} Ψ(Q_{σ}(s)e^{−st}) \{ \begin{matrix}\frac{cosh(tT/2)}{sinh(tT/2)}, & ϵ_{σ} = 0 \\ \frac{1}{sinh(tT/2)}, & ϵ_{σ} = 1/2 \end{matrix} \} dt .

(3.11)

For justification note that by Lemma 3.7 Ψ(Q_{σ}(s)e^{−st}) vanishes of second order at t = 0. In case ϵ_{σ} = 0 we use coth(x) = 1 + \frac{e^{−x}}{sinh(x)} in order to write

I = \frac{d dim(χ)vol(Y)}{T vol(X_{d})}Ψ(\frac{Q_{σ}(s)}{s})  (3.12)

+\frac{d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{∞} Ψ(Q_{σ}(s)e^{−t(s+T/2)})\frac{1}{sinh(tT/2)} dt .

We see that I is a meromorphic function in the region where Re(s_{i}) > −T/2, i = 1, …, N. Since we are not able to evaluate I explicitly we proceed with a qualitative discussion. We decompose I as a sum of an even and an odd part which are defined initially for |Re(s_{i})| < T/2, i = 1, …, N.

I_{ev} := \frac{1}{2}(I(s_{1}, …, s_{N}) + I(−s_{1}, …, −s_{N})),
I_{odd} := \frac{1}{2}(I(s_{1}, …, s_{N}) − I(−s_{1}, …, −s_{N})) .

We can evaluate the odd part.

Lemma 3.9 I_{odd} has a meromorphic continuation to ℂ^{N} given by

I_{odd} = −\frac{d dim(χ)vol(Y)}{vol(X_{d})} \frac{π}{2T} Ψ (Q_{σ}(s) \{ \begin{matrix}tan(πs/T), & ϵ_{σ} = 1/2 \\ −cot(πs/T), & ϵ_{σ} = 0 \end{matrix} \}) .

P r o o f . We first consider the case ϵ_{σ} = 1/2. We have

I_{odd} = −\frac{d dim(χ)vol(Y)}{2vol(X_{d})} ∫_{0}^{∞} Ψ(Q_{σ}(s) sinh(st))\frac{1}{sinh(tT/2)} dt .

Assuming |s_{i}| < T/2, i = 1, …, N, for a moment we can apply the formula [65], 3.311, to obtain

I_{odd} = −\frac{d dim(χ)vol(Y)}{vol(X_{d})} \frac{π}{2T} Ψ(Q_{σ}(p)tan(πs/T)) .

This formula provides the desired meromorphic continuation of I_{odd} in the case ϵ_{σ} = 1/2. Now we consider the case ϵ_{σ} = 0. We must compute the odd part of

\frac{d dim(χ)vol(Y)}{2vol(X_{d})} ∫_{0}^{∞} Ψ(Q_{σ}(s)e^{−t(s+T/2)})\frac{1}{sinh(tT/2)} dt .

Using the formula [65], 3.331, we obtain

\frac{d dim(χ)vol(Y)}{2vol(X_{d})} ∫_{0}^{∞} e^{−Tt/2}Ψ(Q_{σ}(s) sinh(st))\frac{1}{sinh(tT/2)} dt
= −\frac{d dim(χ)vol(Y)}{2vol(X_{d})}Ψ(\frac{Q_{σ}(s)}{s}) + \frac{d dim(χ)vol(Y)}{2vol(X_{d})} \frac{π}{2T} Ψ(Q_{σ}(s)cot(πs/T)) .

Substituting this into equation (3.12) we obtain the desired result for ϵ_{σ} = 0.

□

Now we study the even part I_{ev}. It is here where the compact dual symmetric space enters the discussion of the Selberg zeta function.

Lemma 3.10 The even part I_{ev} has a meromorphic continuation to ℂ^{N}. It can be written as the trace of a meromorphic family of trace class operators:

I_{ev}(s_{1}, …, s_{N}) = −\frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} − A_{d}(γ, σ)^{2}} ,

where γ is an admissible lift of σ.

P r o o f . Note that

I_{ev} = (\frac{d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{∞} Ψ(\frac{d^{2}}{dt^{2}} Q_{σ}(\frac{d}{dt})e^{−st}/s)F(t) dt)_{ev} .

We want to employ partial integration in order to move the differential operator \frac{d^{2}}{dt^{2}} Q_{σ}(\frac{d}{dt}) to the place in front of F. Thus we write

I_{ev} = lim_{r→0} (\frac{d dim(χ)vol(Y)}{T vol(X_{d})} ∫_{r}^{∞} Ψ(\frac{d^{2}}{dt^{2}} Q_{σ}(\frac{d}{dt})e^{−st}/s)F(t) dt)_{ev}
= lim_{r→0} (−\frac{d dim(χ)vol(Y)}{vol(X_{d})} ∫_{r}^{∞} Ψ(\frac{e^{−st}}{s})θ_{d}(t, σ) dt)_{ev}  (3.13)
= lim_{r→0} \frac{d dim(χ)vol(Y)}{vol(X_{d})} ∫_{r}^{∞} tΨ(\frac{sinh(st)}{st})θ_{d}(t, σ) dt
= \frac{d dim(χ)vol(Y)}{vol(X_{d})} ∫_{0}^{∞} tΨ(\frac{sinh(st)}{st})Tr e^{−tA_{d}(γ,σ)} dt ,  (3.14)

where |Re(s_{i})| < T, i = 1, … N. In order to justify the equality (3.13) note that by Lemma 3.7 Ψ((\frac{e^{−st}}{s})_{ev}) = −tΨ(\frac{sinh(st)}{st}) vanishes of order 2N at t = 0. Moreover, we have employed Corollary 2.2 in order to write the theta function in this place. The integral in (3.14) converges in the trace class sense. Thus we can interchange Tr and the integral and obtain

I_{ev} = −\frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr Ψ (\frac{1}{2s}(\frac{1}{A_{d}(γ, σ) + s} − \frac{1}{A_{d}(γ, σ) − s}))
= −\frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr Ψ(\frac{1}{s^{2} − A_{d}(γ, σ)^{2}} )

= −\frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} − A_{d}(γ, σ)^{2}} .

This formula provides the meromorphic continuation of I_{ev}. □

3.2.4 Continuation and singularities of L_{S,χ}(s, σ) and L_{S,χ}^{s}(s, σ)

The following two propositions are the main results of the present section. They are based on the observation that

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}}

is a meromorphic function on ℂ^{N} with singularities at

{(s_{1}, …, s_{N}) ∈ ℂ^{N} | ∃i ∈ {1, …, N}with − s_{i}^{2} ∈ spec(A_{Y,χ}(γ, σ)^{2})} .

All proofs of meromorphic continuations of Selberg zeta functions via the trace formula employ this observation in some way or another.

Proposition 3.11 Let γ be an admissible lift of σ and d := −(−1)^{n/2} if n is even and d = 1 otherwise. The following formula provides a meromorphic continuation of the logarithmic derivative L_{S,χ}(s, σ) of the Selberg zeta function:

Ψ(\frac{L_{S,χ}(s, σ)}{2s})  (3.15)
= Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}}

+\frac{d dim(χ)vol(Y)}{T vol(X_{d})} \frac{π}{2} Ψ (\frac{P_{σ}(s)}{s} \{ \begin{matrix}\frac{tan(πs/T)}{s} & case ϵ_{σ} = 1/2, n even \\ −\frac{cot(πs/T)}{s} & case ϵ_{σ} = 0, n even \\ 1 & case (a), n odd \\ 2 & case (b), n odd \end{matrix} \})

\frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} − A_{d}(γ, σ)^{2}} n even
0  n odd .

In the even-dimensional case L_{S,χ}(s, σ) has the following singularities :

• first-order poles at ±is with residue m_{χ}(s, γ, σ) if s ≠ 0 is an eigenvalue of A_{Y,χ}(γ, σ),

• a first-order pole at s = 0 with residue 2m_{χ}(0, γ, σ) if 0 is an eigenvalue of A_{Y,χ}(γ, σ), • first-order poles at −s, s ∈ T (ℕ−ϵ_{σ}), with residue 2d\frac{dim(χ)vol(Y)}{vol(X_{d})}m_{d}(s, γ, σ).

Then s > 0 is an eigenvalue of A_{d}(γ, σ).

If two such points coincide, then the residues add up. In the odd-dimensional case L_{S,χ}(s, σ) has the following singularities :

• first-order poles at ±is with residue m_{χ}(s, γ, σ) if s ≠ 0 is an eigenvalue of A_{Y,χ}(γ, σ),

• a first-order pole at s = 0 with residue 2m_{χ}(0, γ, σ) if 0 is an eigenvalue of A_{Y,χ}(γ, σ).

Proof. We know from Proposition 3.8 (and the Lemmas 3.9, 3.10 if n is even) that equation (3.15) holds on an open subset of ℂ^{N}. Moreover, the right-hand side of this equation is meromorphic on all of ℂ^{N}. This provides a meromorphic continuation of Ψ(\frac{L_{S,χ}(s,σ)}{2s}). We obtain the meromorphic continuation of L_{S,χ}(s, σ) by fixing s_{2}, …, s_{N} and viewing Ψ(\frac{L_{S,χ}(s,σ)}{2s}) as a function of s_{1}. It remains to verify the assertion concerning the singularities. We first consider the even-dimensional case. Let s := s_{1}. Choose s_{i}, i = 2, …, N, pairwise distinct with non-vanishing real and imaginary parts. The points s ∈ T (ℕ−ϵ_{σ}) are eigenvalues of A_{d}(γ, σ) with multiplicity m_{d}(s, γ, σ) = sQ_{σ}(s). In fact, A_{d}(γ, σ) may have more eigenvalues, but the weighted multiplicities of these additional eigenvalues are zero. Because of m_{d}(0, γ, σ) = 0 we have that I_{ev} is regular at s = 0. Since by (3.12) and (3.11) I is regular for s > 0 the poles of I_{ev} and I_{odd} cancel each other on the positive real axis and add up on the negative real axis. Thus the poles of the identity contribution I are located at −T (ℕ − ϵ_{σ}) and s_{i}, i = 2, …, N. The poles at s_{i} come from the regularization using Ψ. It is easy to see that L_{S,χ}(s, σ) is regular in these points. Consider the case ϵ_{σ} = 0. Then I_{odd} has poles at s = −kT, k = 1, 2, …, with the residue

\frac{d dim(χ)vol(Y)}{vol(X_{d})} ∏_{i=2}^{N} (\frac{1}{s_{i}^{2} − (kT)^{2}} ) \frac{m_{d}(kT, γ, σ)}{2kT} .

The term of Ψ(\frac{L_{S,χ}(s,σ)}{2s}) containing L_{S,χ}(s, σ) has the form

\frac{L_{S,χ}(s, σ)}{2s} ∏_{i=2}^{N} \frac{1}{s_{i}^{2} − s^{2}} .

Thus L_{S,χ}(s, σ) has a pole at s = −kT and I contributes to the residue by

2\frac{d dim(χ)vol(Y)}{vol(X_{d})}m_{d}(kT, γ, σ) .

In case ϵ_{σ} = 1/2 the poles of I_{odd} on (−∞, 0) are located at (−k + 1/2)T, k = 1, 2, . . .. I contributes to the residue of L_{S,χ}(s, σ) at s = (−k + 1/2)T by

2\frac{d dim(χ)vol(Y)}{vol(X_{d})}m_{d}((k − 1/2)T, γ, σ) .

Let now t ≠ 0 be an eigenvalue of A_{Y,χ}(γ, σ) with m_{χ}(t, γ, σ) ≠ 0. Then the term of

T r ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}}

which is singular at s = ±it is

m_{χ}(t, σ, γ)\frac{1}{s^{2} + t^{2}} ∏_{i=2}^{N} \frac{1}{s_{i}^{2} + t^{2}} .

It contributes to the residue of L_{S,χ}(s, σ) at s = ±it by m_{χ}(t, σ, γ). The case t = 0 is discussed similarly. This finishes the proof of the proposition for even n. The argument in the case of odd n is similar. Since the identity contribution is regular one has only to consider the spectral singularities at ±is, s ∈ spec(A_{Y,χ}(γ, σ)). □ A formula similar to (3.15) appeared in [83], where the trace of the dual resolvent is replaced by an explicit expression in terms of the eigenvalues of A_{d}(γ, σ). An evaluation of that trace in terms of special functions can be found in [44]. We now consider the logarithmic derivative L_{S,χ}^{s}(s, σ) of the super zeta function S_{χ}^{s}(s, σ) in the case (b) for odd n. Let

R(s^{2}) = (s^{2} + A_{Y,χ}(γ^{s}, σ)^{2})^{−1} .

Proposition 3.12 For N > n/2 + 1

Tr D_{Y,χ}(σ) ∏_{i=1}^{N} R(s_{i}^{2}) = −\frac{i}{2}Ψ(L_{S,χ}^{s}(s, σ))  (3.16)

provides a meromorphic continuation of L_{S,χ}^{s}(s, σ) to all of ℂ. Moreover, the singularities of L_{S,χ}^{s}(s, σ) are first-order poles at is, s ∈ spec(D_{Y,χ}(σ)), with residue m_{χ}^{s}(s, σ).

Proof. Let s_{1}, …, s_{N} ∈ ℂ with Re(s_{i}) > ρ, i = 1, …, N and s_{i} ≠ s_{j} for all i ≠ j. Note that the spectrum of A_{Y,χ}(γ^{s}, σ) is real since A_{Y,χ}(γ^{s}, σ) = |D_{Y,χ}(σ)|. The integral

D_{Y,χ}(σ) ∏_{i=1}^{N} R(s_{i}^{2}) = ∫_{0}^{∞} D_{Y,χ}(σ)Ψ(e^{−st})\frac{sin(tA_{Y,χ}(γ^{s}, σ))}{A_{Y,χ}(γ^{s}, σ)} dt

converges in operator norm. We apply the trace formula Corollary 1.28 in the same way as in the proof of Proposition 3.8. Note that the identity contribution vanishes. We obtain

Tr D_{Y,χ}(σ) ∏_{i=1}^{N} R(s_{i}^{2}) = ∑_{[g]∈CΓ,[g]≠1} −i\frac{C_{χ}(g, σ) − C_{χ}(g, wσ)}{n_{Γ}(g)}Ψ(e^{−sl(g)})
= −iΨ(L_{S,χ}^{s}(s, σ)).

Now we argue exactly as in the proof of Proposition 3.11. □

The functional equations for L_{S,χ}(s, σ) and L_{S,χ}^{s}(s, σ) are immediate consequences of Equations (3.15) and (3.16). Recall that d := −(−1)^{n/2} if n is even and d = 1 otherwise.

Corollary 3.13 The logarithmic derivatives of the Selberg zeta functions satisfy the functional equations

L_{S,χ}(s, σ) + L_{S,χ}(−s, σ)  (3.17)

= \frac{2dπ dim(χ)vol(Y)}{T vol(X_{d})}P_{σ}(s) \{ \begin{matrix}\frac{tan(πs/T)}{s} & case ϵ_{σ} = 1/2, n even \\ −\frac{cot(πs/T)}{s} & case ϵ_{σ} = 0, n even \\ 1 & case (a), n odd \\ 2 & case (b), n odd \end{matrix}

L_{S,χ}^{s}(s, σ) − L_{S,χ}^{s}(−s, σ) = 0, case (b), n odd .

P r o o f . We first consider the non-super case. Fix s_{2}, …, s_{N} ∈ ℂ, pairwise distinct, and subtract equation (3.15) at s_{1} := −s from (3.15) at s_{1} := s. Then we multiply by 2s. The left-hand side of the resulting equation becomes that of (3.17). On the right hand side the terms containing the resolvents cancel. Since that P_{σ}(s) is an even polynomial for odd n and an odd polynomial for even n the remaining term is odd. It is easy to see that one can remove the Ψ-construction. In fact Ψ can only annihilate odd terms. In the super case of (b) we again fix s_{2}, …, s_{N} ∈ ℂ, pairwise distinct, and subtract equation (3.16) at s_{1} := −s from (3.16) at s_{1} := s. Since the left-hand side is even, the assertion follows. □

3.3 Properties of the Selberg zeta function

3.3.1 Meromorphic continuation and singularities

In this subsection we obtain the meromorphic continuation and the description of the singularities of the Selberg zeta function. Note that the logarithmic derivatives L_{S,χ}(s, σ) and L_{S,χ}^{s}(s, σ) decrease exponentially as Re(s) → ∞. The logarithms of the zeta functions can be obtained from the logarithmic derivatives by integration as follows:

\begin{matrix}ln(Z_{S,χ}(s, σ)) \\ ln(S_{χ}(s, σ)) \end{matrix} \} = − ∫_{s}^{∞} L_{S,χ}(t, σ) dt, \{ \begin{matrix}n even \\ n odd, cases (a),(b) \end{matrix}(3.18)
ln(S_{χ}^{s}(s, σ)) = − ∫_{s}^{∞} L_{S,χ}^{s}(t, σ) dt, n odd, case (b) .  (3.19)

Since the logarithmic derivatives have singularities the integrals above depend on the choice of the path. But in order to get the zeta functions themselves we have to exponentiate the right hand sides of (3.19) and (3.18). We would obtain meromorphic continuations of the Selberg zeta functions if we showed that all residues of the logarithmic derivatives are integers. It was a longstanding problem to show that in the case of even n the residues of the logarithmic derivative L_{S,χ}(s, σ) on the negative real axis are integers. Within the harmonic analysis approach previous to the present book for general σ ∈ M̂ it was only possible to show that these residues are rational. This implied the meromorphy of a certain power of Z_{S,χ}(s, σ) [24],[82],[90],[92],[91]. But Fried [22] showed that Z_{S,χ}(s, σ) itself is meromorphic. Fried expressed Z_{S,χ}(s, σ) in terms of the Ruelle zeta function which was known to be meromorphic from the theory of hyperbolic dynamical systems. The first direct verification of the fact that the residues are integers appeared in the work of Juhl [40]. We prove here directly using the trace formula that the residues are integers thus completing the original harmonic analysis approach. The main point is that we expressed the identity contribution in terms of the spectrum of A_{d}(γ, σ) (Lemma 3.10) and avoided the explicit use of the Plancherel measure for X.

Proposition 3.14 At all singularities of L_{S,χ}(s, σ) and L_{S,χ}^{s}(s, σ) the residues are integer.

P r o o f . If n is odd this follows immediately from Proposition 3.11 (cases (a) and (b)) and from Proposition 3.12 (for the super zeta function in case (b)). If n is even, then we employ again Proposition 3.11. It is sufficient to show that

\frac{vol(Y)}{vol(X_{d})} ∈ ℤ .

We use Hirzebruch’s proportionality principle. It is enough to exhibit an elliptic homogeneous (with respect to G_{d}) differential operator D_{d} on X_{d} with index(D_{d}) = 1. Then there is a corresponding homogeneous operator on X inducing D on Y. By the proportionality principle (which is the special case of Theorem 1.10 for L := 1)

\frac{vol(Y)}{vol(X_{d})} = |\frac{index(D)}{index(D_{d})}| = |index(D)|

is an integer. We showed in Theorem 1.12 that any representation of G_{d} can be realized as the G_{d}-index of a homogeneous Dirac operator. In particular, the operator realizing the trivial representation has index one. □

Theorem 3.15 The Selberg zeta functions Z_{S,χ}(s, σ), S_{χ}(s, σ), and S_{χ}^{s}(s, σ) have meromorphic continuations to all of ℂ. If n is even and γ is σ-admissible, then the singularities (zeros and poles) of Z_{S,χ}(s, σ) are the following:

• at ±is of order m_{χ}(s, γ, σ) if s ≠ 0 is an eigenvalue of A_{Y,χ}(γ, σ),

• at s = 0 of order 2m_{χ}(0, γ, σ) if 0 is an eigenvalue of A_{Y,χ}(γ, σ),

• at −s, s ∈ T (ℕ − ϵ_{σ}) of order 2\frac{d dim(χ)vol(Y)}{vol(X_{d})}m_{d}(s, γ, σ). Then s > 0 is an eigenvalue of A_{d}(γ, σ).

If two such points coincide, then the orders add up. If n is odd, then the singularities of Z_{S,χ}(s, σ) (case (a)) and of S_{χ}(s, σ) (case (b)) are

• at ±is of order m_{χ}(s, γ, σ) if s ≠ 0 is an eigenvalue of A_{Y,χ}(γ, σ),

• at s = 0 of order 2m_{χ}(0, γ, σ) if 0 is an eigenvalue of A_{Y,χ}(γ, σ).

In case (b) (n odd) the singularities of S_{χ}^{s}(s, σ) are at is and have order m_{χ}^{s}(s, γ^{s}) if s ∈ ℝ an eigenvalue of D_{Y,χ}(σ). For odd n in case (b) the zeta function Z_{S,χ}(s, σ) has singularities at is, ±s ∈ spec(A_{Y,χ}(γ^{s}, σ)) of order \frac{1}{2}(m_{χ}(|s|, γ, σ) + m_{χ}^{s}(s, σ)) if s ≠ 0 and m_{χ}(0, γ, σ) if s = 0.

P r o o f . With the exception of the last assertion the theorem follows from Propositions 3.11, 3.14 and 3.12. Let n be odd and consider the case (b). Then for Re(s) > ρ we have Z_{S,χ}(s, σ) = \sqrt{S_{χ}(s, σ)S_{χ}^{s}(s, σ)}. In order to obtain a meromorphic continuation of Z_{S,χ}(s, σ) we must show that all the singularities of S_{χ}(s, σ)S_{χ}^{s}(s, σ) have even order. Recall that we can choose the representative γ of the lift of σ + wσ such that V (γ) = V (γ^{s}) if one forgets the ℤ_{2}-gradings. Then A_{Y,χ}(γ^{s}, σ) = A_{Y,χ}(γ, σ) up to the grading. Thus m_{χ}^{s}(s, σ) ≡ m_{χ}(|s|, γ, σ)(mod2). The product S_{χ}(s, σ)S_{χ}^{s}(s, σ) has its singularities at is, s ∈ spec(A_{Y,χ}(γ, σ)), of even order m_{χ}^{s}(s, σ) + m_{χ}(|s|, γ, σ). Hence Z_{S,χ}(s, σ) is meromorphic and its singularities are as claimed. □

[figure: topological; spectral; ρ; −ρ; _{T}; s-plane; singularity picture of Z_{S,χ}(s, σ); Fig. 2]

If n is even, then the singularities of Z_{S,χ}(s, σ) at −s, s ∈ T (ℕ − ϵ_{σ}), are called topological. They depend on Γ through the volume vol(Y) or, equivalently, through the Euler characteristic χ(Y), only. The remaining singularities are related to the spectrum of A_{Y,χ}(γ, σ) and therefore called spectral. In fact there may be an overlap of the topological and the spectral singularities at finitely many points. At those points we consider the portion

−2\frac{dim(χ)χ(Y)}{χ(X_{d})}m_{d}(s, γ, σ)

of the order as topological and the remainder as spectral. The spectral singularities can be characterized by their symmetry with respect to zero. It is due the choice of an σ-admissible γ that we have such a clean distinction of the spectral from the topological singularities. We consider now what happens for general lifts γ of σ.

Lemma 3.16 The formula (3.15) holds for all lifts γ of σ.

P r o o f . Since the lemma is trivial for n odd we assume that n is even. In this case the lemma is a consequence of the equivariant proportionality principle (Proposition 1.10). Let γ̃ be an arbitrary lift of σ. Then we can choose representatives of γ and γ̃ in R(K) denoted by the same symbol such that V := V_{Y,χ}(γ) ⊕ V_{Y,χ}(γ̃)^{op} carries an odd Dirac operator D. Analogously, there is a Dirac operator D_{d} on V_{d} := V_{d}(γ)⊕V_{d}(γ̃)^{op}. There are finitely many s ∈ ℂ with m_{χ}(s, γ, σ) ≠ m_{χ}(s, γ̃, σ). Such an s with Im(s), Re(s) ≥ 0, contributes to the A_{Y,χ}^{2}(γ, σ) ⊕ A_{Y,χ}^{2}(γ̃, σ)-index of D by (s^{2}, m_{χ}(s, γ, σ) − m_{χ}(s, γ̃, σ)). It follows from the proportionality principle (Proposition 1.10) that if (−s^{2}, (m_{d}(s, γ, σ)− m_{d}(s, γ̃, σ))) is the contribution of −s^{2} to the A_{d}^{2}(γ, σ) + A_{d}^{2}(γ̃, σ)-index of D_{d}, then

m_{χ}(s, γ, σ) − m_{χ}(s, γ̃, σ) = −\frac{dvol(Y)}{vol(X_{d})}(m_{d}(s, γ, σ) − m_{d}(s, γ̃, σ)) .

Hence the combination

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}} + \frac{d dim(χ)vol(Y)}{vol(X_{d})}Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} − A_{d}(γ, σ)^{2}}

is independent of the choice of the lift γ of σ. □ We see that for a non-admissible lift γ̃ of σ some of the former topological singularities may now appear as spectral singularities and vice versa. If n is odd there are only spectral singularities. While in case (a) the singularities of Z_{S,χ}(s, σ) have the reflection symmetry this is not true in case (b). In the latter case the singularities are related to the spectrum of a Dirac operator. In fact the asymmetry is the reason for the appearance of the eta invariant of D_{Y,χ}(σ) in the functional equation for Z_{S,χ}(s, σ) below.

3.3.2 Functional equations

The Selberg zeta function Z_{S,χ}(s, σ) satisfies a functional equation relating its value at s with that at −s. We first consider case (a) and S_{χ}(s, σ) in case (b). Recall that d := −(−1)^{n/2} if n is even and d = 1 otherwise.

Theorem 3.17 The Selberg zeta functions satisfy the following functional equations:

\frac{Z_{S,χ}(s, σ)}{Z_{S,χ}(−s, σ)} ; \frac{S_{χ}(s, σ)}{S_{χ}(−s, σ)} \} = exp (\frac{2dπ dim(χ)vol(Y)}{T vol(X_{d})}

∫_{0}^{s} P_{σ}(t) \{ \begin{matrix}\frac{tan(πt/T)}{s} & case ϵ_{σ} = 1/2, n even \\ −\frac{cot(πt/T)}{s} & case ϵ_{σ} = 0, n even \\ 1 & case (a), n odd \\ 2 & case (b), n odd \end{matrix} \} dt) .

P r o o f . Note that the right-hand side does not depend on the choice of the path of integration. The theorem is a consequence of the functional equation for the logarithmic derivative Corollary 3.13. □ We now consider the super case.

Theorem 3.18 In case (b) the Selberg zeta functions satisfy the functional equations :

S_{χ}^{s}(s, σ)S_{χ}^{s}(−s, σ) = e^{2πiη(D_{Y,χ}(σ))}  (3.20)
\frac{Z_{S,χ}(s, σ)}{Z_{S,χ}(−s, wσ)} = e^{πiη(D_{Y,χ}(σ))}
exp \{\frac{2π dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt\} , (3.21)

where η(D_{Y,χ}(σ)) is the eta invariant of D_{Y,χ}(σ) (see the appendix of Subsection 2.1.2). In particular, S_{χ}^{s}(0, σ) = e^{πiη(D_{Y,χ}(σ))}.

P r o o f . We start with (3.20). Write

R(s^{2}) = ∫_{0}^{∞} e^{−s^{2}t}e^{−tA_{Y,χ}(γ^{s},σ)^{2}} dt .

If Re(s_{i}) > 0, ∀i = 1, …, N, then by Proposition 3.12

Ψ(L_{S,χ}^{s}(s, σ)) = 2iTr D_{Y,χ}(σ) ∏_{i=1}^{N} R(s_{i}^{2})
= 2iTr D_{Y,χ}(σ)Ψ(R(s^{2}))
= 2iTr ∫_{0}^{∞} Ψ(e^{−s^{2}t})D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)^{2}} dt .

Since Tr D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)^{2}} is regular near t = 0 (and vanishes at t = 0) and decreases exponentially as Re(t) → ∞ we obtain

L_{S,χ}^{s}(s, σ) = 2i ∫_{0}^{∞} e^{−s^{2}t}Tr D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)^{2}} dt ,  (3.22)
ln(S_{χ}^{s}(s, σ)) = 2i ∫_{0}^{∞} \frac{\sqrt{π}}{2\sqrt{t}}(1 − Φ(t^{1/2}s))Tr D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)^{2}} dt ,

where Φ(x) := \frac{2}{\sqrt{π}} ∫_{0}^{x} e^{−x^{2}}dx is the error integral. In fact, if we fix s_{2}, …, s_{N}, then Ψ(L_{S,χ}^{s}(s, σ)) as a function of s = s_{1} determines L_{S,χ}^{s}(s, σ) up to an even polynomial. We obtain (3.22) using the condition that L_{S,χ}^{s}(s, σ) decreases exponentially as Re(s) → ∞. Hence

ln(S_{χ}^{s}(s, σ)) + ln(S_{χ}^{s}(−s, σ)) = 2i ∫_{0}^{∞} \begin{matrix}√ & \overline{π} \\ √ & \overline{t} \end{matrix} Tr D_{Y,χ}(σ)e^{−tA_{Y,χ}(γ^{s},σ)^{2}} dt
= 2πiη(D_{Y,χ}(σ)) .

The functional equation (3.20) follows. The equation (3.21) follows from

\frac{Z_{S,χ}(s, σ)}{Z_{S,χ}(−s, wσ)} = \sqrt{\frac{S_{χ}(s, σ)S_{χ}^{s}(s, σ)}{S_{χ}(−s, wσ)S_{χ}^{s}(−s, wσ)}}

= \sqrt{\frac{S_{χ}(s, σ)}{S_{χ}(−s, σ)}S_{χ}^{s}(s, σ)S_{χ}^{s}(−s, σ)}
= e^{πiη(D_{Y,χ}(σ))} exp (\frac{2π dim(χ)vol(Y)}{Tvol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt) ,

where we have employed (3.20) and Theorem 3.17. □ For even n the functional equation of the Selberg zeta function was obtained by Juhl [40] up to a factor of the form e^{F(s)}, where F (s) is some polynomial. It was shown in Bröcker [8], that F = 0. A similar functional equation was established by Wakayama for his zeta functions associated to K-types [82].

The appearance of the eta invariant in the functional equation of the super zeta function for odd n was first observed by Millson [48]( see also Wakayama [85] in the special case of the signature operator on hyperbolic manifolds. This observation was generalized even to higher rank spaces by Moscovici/Stanton [49]. Juhl [40] obtained equation (3.20) by specializing the results of [49].

3.3.3 Determinants and zeta functions

In this subsection we show how the Selberg zeta functions can be expressed in terms of regularized determinants of elliptic differential operators on Y and X_{d}. For the convenience of the reader we recall some basic facts about regularized determinants in the appendix to this subsection.

Theorem 3.19 If n is even, then the Selberg zeta function has the representations

Z_{S,χ}(s, σ) = det(s^{2} + A_{Y,χ}(γ, σ)^{2})(det(A_{d}(γ, σ)^{2} − s^{2}))^{− \frac{dim(χ)χ(Y )}{χ(X_{d})}}  (3.23)
exp (−\frac{dim(χ)χ(Y)}{χ(X_{d})} \frac{π}{T} ∫_{0}^{s} P_{σ}(t) \{ \begin{matrix}tan(πt/T) & ϵ_{σ} = 1/2 \\ −cot(πt/T) & ϵ_{σ} = 0 \end{matrix} \} dt)

for any lift γ of σ and

Z_{S,χ}(s, σ) = det(s^{2} + A_{Y,χ}(γ, σ)^{2}) det(A_{d}(γ, σ) + s)^{− \frac{2 dim(χ)χ(Y )}{χ(X_{d})}}  (3.24)
exp (\frac{dim(χ)χ(Y)}{χ(X_{d})} ∑_{m=1}^{n/2} c_{−m} \frac{s^{2m}}{m!} (∑_{r=1}^{m−1} \frac{1}{r} − 2 ∑_{r=1}^{2m−1} \frac{1}{r}))

for σ-admissible lifts γ. The coefficients c_{k} are defined by the asymptotic expansion

Tr e^{−tA_{d}(γ,σ)^{2}} ∼^{t→0} ∑_{k=−n/2}^{∞} c_{k}t^{k} .

If Q_{σ}(x) = ∑_{i≥0} q_{2i}x^{2i}, then c_{−(i+1)} = q_{2i}i! / 2T. If n is odd, then the Selberg zeta functions have the representation

Z_{S,χ}(s, σ) = det(s^{2} + A_{Y,χ}(γ, σ)^{2})  case (a)  (3.25)

exp (π \frac{dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt) ,

S_{χ}(s, σ) = det(s^{2} + A_{Y,χ}(γ, σ)^{2}) case (b)

exp (2π \frac{dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt) .

P r o o f . First let n be even and γ be σ-admissible. For |Re(s_{i})| >> 0, i = 1, …, N, we have

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}} = TrΨ (∫_{0}^{∞} e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} dt)

= TrΨ (∫_{0}^{∞} −\frac{d}{ds^{2}} e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} \frac{dt}{t})
= Ψ (−\frac{d}{ds^{2}} P.F. ∫_{0}^{∞} Tr e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} \frac{dt}{t}) ,

where P.F. ∫_{0}^{∞} … is the finite part of ∫_{ϵ}^{∞} . . .. Applying (3.36) (c.f. [74], Theorem 5.1.1) we obtain

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}} = Ψ (\frac{d}{ds^{2}} [ln(det(s^{2} + A_{Y,χ}(γ, σ)^{2})) + Eu a_{Y}(s^{2})]) ,

(3.26)

where a_{Y} (s^{2}) is the constant coefficient in the asymptotic expansion

Tr e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})}, t → 0

and Eu is the Euler constant. Equation (3.26) extends analytically to all of ℂ. In an analogous manner for |Im(s_{i})| >> 0, i = 1, …, N, we have

−Tr ∏_{i=1}^{N} \frac{1}{A_{d}(γ, σ)^{2} − s_{i}^{2}} = Ψ (\frac{d}{ds^{2}} [ln det(s^{2} − A_{d}(γ, σ)^{2})) + Eu a_{d}(s^{2})]) ,

(3.27)

where a_{d}(s^{2}) is the constant term in the asymptotic expansion

Tr e^{−t(A_{d}(γ,σ)^{2}−s^{2})}, t → 0 .

It follows from (1.16) that a_{Y} (s^{2}) = \frac{dim(χ)χ(Y)}{χ(X_{d})}a_{d}(s^{2}). In the following equation derived from (3.15) the terms involving a_{Y} and a_{d} cancel.

Ψ(\frac{L_{S,χ}(s, σ)}{2s}) = −\frac{dim(χ)χ(Y)}{χ(X_{d})} \frac{π}{2T} Ψ (Q_{σ}(s) \{ \begin{matrix}tan(πs/T) & ϵ_{σ} = 1/2 \\ −cot(πs/T) & ϵ_{σ} = 0 \end{matrix} \})

+Ψ (\frac{d}{ds^{2}} ln(det(s^{2} + A_{Y,χ}(γ, σ)^{2})))

−\frac{dim(χ)χ(Y)}{χ(X_{d})}Ψ (\frac{d}{ds^{2}} ln(det(A_{d}(γ, σ)^{2} − s^{2}))) .

Removing the Ψ we obtain

L_{S,χ}(s, σ) = F (s)′ − \frac{dim(χ)χ(Y)}{χ(X_{d})} \frac{π}{T} P_{σ}(s) \{ \begin{matrix}tan(πs/T) & ϵ_{σ} = 1/2 \\ −cot(πs/T) & ϵ_{σ} = 0 \end{matrix} \}

+\frac{d}{ds} ln(det(A_{Y,χ}(γ, σ)^{2} + s^{2}))

−\frac{dim(χ)χ(Y)}{χ(X_{d})} \frac{d}{ds} ln(det(A_{d}(γ, σ)^{2} − s^{2})) ,

where F (s)′ is a certain odd polynomial. Integrating once and exponentiating we obtain

Z_{S,χ}(s, σ) = e^{F(s)} det(s^{2} + A_{Y,χ}(γ, σ)^{2})(det(A_{d}(γ, σ)^{2} − s^{2}))^{− \frac{dim(χ)χ(Y)}{χ(X_{d})}} (3.28)

exp (−\frac{dim(χ)χ(Y)}{χ(X_{d})} \frac{π}{T} ∫_{0}^{s} P_{σ}(t) \{ \begin{matrix}tan(πt/T) & ϵ_{σ} = 1/2 \\ −cot(πt/T) & ϵ_{σ} = 0 \end{matrix} \} dt) .

Note that the exponential of the integral on the right-hand side of (3.28) is well defined by Lemma 3.14. It remains to show that F (s) = 0. The following argument was suggested to us by U. Bröcker. We define the zeta functions Z^{±}(s, λ) := Tr(A_{d}(γ, σ) ± λ)^{−s} for Re(s) >> 0 and ±λ > 0. Because of the asymptotic expansion

θ_{d}(t, σ) = Tr e^{−tA_{d}(γ,σ)} ∼^{t→0} ∑_{k=−n}^{∞} d_{k}t^{k}

(note that d_{k} = 0 for k odd) the functions Z^{±}(s, λ) have meromorphic continuations with respect to s, λ and the regularized determinants

D^{±}(λ) := exp (−\frac{d}{ds}_{|s=0}Z^{±}(s, −λ)) = det(A_{d}(γ, σ) ∓ λ)

are well defined entire functions of finite order having zeros at ±λ, λ ∈ spec(A_{d}(γ, σ)), of multiplicity m_{d}(λ, γ, σ). The Weierstraß products

Δ^{±}(λ) := ∏_{μ∈spec(A_{d}(γ,σ))} [(1 ∓ \frac{λ}{μ}) exp (∑_{r=1}^{n} \frac{(±λ)^{r}}{rμ^{r}} )]^{m_{d}(μ,γ,σ)}

are also entire functions of the same order and with the same zeros as D^{±}. Thus Δ^{±} and D^{±} coincide up to a factors which can be written as the exponential of polynomials of order n. These polynomials were calculated by Voros [80], Eq. 4.12. Set Z^{±}(s) := Z^{±}(s, 0). Then

D^{±}(λ) = Δ^{±}(λ)  (3.29)

exp (−(Z^{±})′(0) − ∑_{m=1}^{n}(P.F.Z^{±})(m)\frac{(±λ)^{m}}{m} − ∑_{m=1}^{n} d_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{λ^{m}}{m!} ) .

Now

Δ(λ) := Δ^{+}(λ)Δ^{−}(λ)  (3.30)
= ∏_{μ∈spec(A_{d}(γ,σ))} [(1 − \frac{λ^{2}}{μ^{2}} ) exp (2 ∑_{r=2,even}^{n} \frac{λ^{r}}{rμ^{r}} )]^{m_{d}(μ,γ,σ)}

= ∏_{μ∈spec(A_{d}(γ,σ))} [(1 − \frac{z}{μ^{2}} ) exp (∑_{r=1}^{n/2} \frac{z^{r}}{rμ^{2r}} )]^{m_{d}(μ,γ,σ)}

is the Weierstraß product corresponding to the eigenvalues of A_{d}(γ, σ)^{2}, where z = λ^{2}. Define the zeta function Z(s, z) := Tr(A_{d}(γ, σ)^{2} + z)^{−s} and the regularized determinant

D(z) := e^{− \frac{d}{ds}_{|s=0}Z(s,−z)} = det(A_{d}(γ, σ)^{2} − z) .

Then again by the computation of Voros

D(z) = Δ(z)  (3.31)

exp (−Z′(0) − ∑_{m=1}^{n/2}(P.F.Z)(m)\frac{z^{m}}{m} − ∑_{m=1}^{n/2} c_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{z^{m}}{m!} ) .

Note that Z^{+}(s) = Z^{−}(s) = Z(s/2). Combining (3.30), (3.29) and (3.31) we obtain

\frac{D(λ^{2})}{D^{+}(λ)D^{−}(λ)} = exp (− ∑_{m=1}^{n/2} c_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{λ^{2m}}{m!}  (3.32)

+2 ∑_{m=1,even}^{n} d_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{λ^{m}}{m!} ) .

We use the reflection formula ([8], Lemma 4, [80] for n = 2 ):

\frac{D^{+}(λ)}{D^{−}(λ)} = exp (−\frac{π}{T} ∫_{0}^{λ} P_{σ}(s) \{ \begin{matrix}tan(πs/T) & ϵ_{σ} = 1/2 \\ −cot(πs/T) & ϵ_{σ} = 0 \end{matrix} \} ds) . (3.33)

Combining (3.32) and (3.33) we obtain

D(λ^{2}) = D^{−}(λ)^{2} exp (−\frac{π}{T} ∫_{0}^{λ} P_{σ}(s) \{ \begin{matrix}tan(πs/T) & ϵ_{σ} = 1/2 \\ −cot(πs/T) & ϵ_{σ} = 0 \end{matrix} \} ds

− ∑_{m=1}^{n/2} c_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{λ^{2m}}{m!}

+2 ∑_{m=1,even}^{n} d_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{λ^{m}}{m!} ) .

We now substitute this into (3.28) and obtain

Z_{S,χ}(s, σ) = det(s^{2} + A_{Y,χ}(γ, σ)^{2}) det(A_{d}(γ, σ) + s)^{−2 \frac{dim(χ)χ(Y )}{χ(X_{d})}} (3.34)
exp (F (s) + \frac{dim(χ)χ(Y)}{χ(X_{d})} ∑_{m=1}^{n/2} c_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{s^{2m}}{m!}

−2\frac{dim(χ)χ(Y)}{χ(X_{d})} ∑_{m=1,even}^{n} d_{−m} (∑_{r=1}^{m−1} \frac{1}{r}) \frac{s^{m}}{m!} ) .

We determine F (s) from the asymptotic expansion of the logarithm of both sides of (3.34) as s → ∞. In fact, ln Z_{S,χ}(s, σ) decreases exponentially as s → ∞. Let the coefficients e_{k} be defined by

Tr e^{−tA_{Y,χ}(γ,σ)^{2}} ∼^{t→0} ∑_{k=−n/2}^{∞} e_{k}t^{k} .

Then the e_{k} are related to the c_{k} via the proportionality principle :

e_{k} = (−1)^{k} \frac{dim(χ)χ(Y)}{χ(X_{d})}c_{k} .

Using (3.37) we get the asymptotic expansions

ln det(A_{Y,χ}(γ, σ)^{2} + s^{2}) ∼^{s→∞} ∑_{k=n/2}^{1} e_{−k} (ln(s^{2}) − (∑_{r=1}^{k} \frac{1}{r})) (−1)^{k} \frac{s^{2k}}{k!}

+e_{0} ln(s^{2}) + o(1)
ln det(A_{d}(γ, σ) + s) ∼^{s→∞} ∑_{k=n,even}^{1} d_{−k} (ln(s) − (∑_{r=1}^{k} \frac{1}{r})) (−1)^{k} \frac{s^{k}}{k!}
+d_{0} ln(s) + o(1) .

Substituting these expansions into (3.34) we obtain

−F (s) = \frac{dim(χ)χ(Y)}{χ(X_{d})}( c_{0} ln(s^{2}) − 2d_{0} ln(s)

+ ∑_{m=1}^{n/2} c_{−m}(ln(s^{2}) − \frac{1}{m})\frac{s^{2m}}{m!} − 2 ∑_{m=1,even}^{n} d_{−m}(ln(s) − \frac{1}{m})\frac{s^{m}}{m!} ) .

Now we claim \frac{c_{−m}}{m!} = \frac{d_{−2m}}{(2m)!}. Assuming the claim for the moment we immediately conclude F (s) = 0. To see the claim note that Z(s) = Z^{±}(2s). The coefficients c_{k}, d_{k} can be expressed in terms of the residues of Z, Z^{+}, respectively. In fact \frac{c_{−m}}{(m−1)!} = res_{s=−m}Z(s) and \frac{d_{−2m}}{(2m−1)!} = res_{s=−2m}Z^{+}(s). This proves the claim. Equation (3.24) now follows from (3.34). Next we study the relation between the coefficients c_{−i} and the polynomial Q_{σ}. In fact by (1.22), (1.23)

Tr e^{−tA_{d}(γ,σ)} = −Q_{σ}(\frac{d}{dt})\frac{d}{dt}e^{−ϵ_{σ}Tt} \frac{1}{1 − e^{−Tt}}
= −Q_{σ}(\frac{d}{dt})\frac{d}{dt} \frac{1}{T t} + O(1)
= \frac{1}{T} ∑_{i≥0} \frac{q_{2i}(2i + 1)!}{t^{2i+2}} + O(1) .

We obtain d_{−(2i+2)} = \frac{q_{2i}(2i+1)!}{T}. It follows c_{−(i+1)} = \frac{q_{2i}i!}{2T}. If γ is any not necessarily admissible lift of σ we apply Lemma 3.16 in order to see that (3.23) remains true. This finishes the proof of Theorem 3.19 in the case of even n. For odd n the proof is much simpler. Let r := 1 in the case (a) and r := 2 in the case (b). For |Re(s_{i})| sufficiently large, i = 1, …, N, we have

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}} = TrΨ(∫_{0}^{∞} e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} dt)
= TrΨ(∫_{0}^{∞} −\frac{d}{ds^{2}} e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} \frac{dt}{t})

= Ψ(−\frac{d}{ds^{2}} P.F. ∫_{0}^{∞} Tr e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})} \frac{dt}{t}) .

Since the constant term in the asymptotic expansion of

Tr e^{−t(s^{2}+A_{Y,χ}(γ,σ)^{2})}, t → 0

vanishes we can write by (3.36) :

Tr ∏_{i=1}^{N} \frac{1}{s_{i}^{2} + A_{Y,χ}(γ, σ)^{2}} = Ψ(\frac{d}{ds^{2}} ln(det(s^{2} + A_{Y,χ}(γ, σ)^{2}))) .  (3.35)

Equation (3.35) extends analytically to all of ℂ. Substituting (3.35) into the representation of L_{S,χ}(s, σ) given in Proposition 3.11 we obtain

Ψ(\frac{L_{S,χ}(s, σ) − \frac{rπ dim(χ)vol(Y)}{T vol(X_{d})}P_{σ}(s)}{2s}) = Ψ(\frac{d}{ds^{2}} ln(det(s^{2} + A_{Y,χ}(γ, σ)^{2}))) .

Removing Ψ we get

L_{S,χ}(s, σ) − \frac{rπ dim(χ)vol(Y)}{T vol(X_{d})}P_{σ}(s) = R′(s) + \frac{d}{ds} ln(det(A_{Y,χ}(γ, σ)^{2} + s^{2})) ,

where R′(s) is a certain odd polynomial. Integrating once and exponentiating we obtain

\begin{matrix}case (a) & Z_{S,χ}(s, σ) \\ case (b) & S_{χ}(s, σ) \end{matrix} \} = e^{F (s)} det(s^{2} + A_{Y,χ}(γ, σ)^{2})

exp (\frac{rπ dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt) .

We claim that F (s) = 0. Note that ln L_{S,χ}(s, σ) → 0 exponentially as s → 0. Define the numbers c_{(2k−n)/2} by the asymptotic expansion

Tr e^{−tA_{Y,χ}(γ,σ)^{2}} ∼^{t→0} ∑_{k=0}^{∞} c_{(2k−n)/2}t^{(2k−n)/2} .

Then by (3.37) we obtain

− ln det(s^{2} + A_{Y,χ}(γ, σ)^{2}) ∼^{s→∞} ∑_{k=0}^{∞} c_{(2k−n)/2}Γ((2k − n)/2)s^{n−2k} .

This asymptotic expansion containing only odd powers of s must cancel with the Plancherel term, i.e.,

\frac{rπ dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{σ}(t) dt = ∑_{k=0}^{(n−1)/2} c_{(2k−n)/2}Γ((2k − n)/2)s^{n−2k} ,

and we have c_{(2k−n)/2} = 0 for 2k > n. We also conclude the claim : F (s) = 0. This finishes the proof of Theorem 3.19 in the case of odd n. □ The determinant representation (3.24) of the Selberg zeta function was first obtained in the case of hyperbolic surfaces by D’Hoker/Phong [18], Voros [80] (see also [11]), Bolte [5] and others. In the spherical case σ = 1 (3.24) was established in this form by Williams [91] and Bröcker [8]. The case of general σ was also considered by Deitmar [15]. He obtained (3.24) up to the exponent of a certain polynomial (that is trivial by the above discussion). In rudimentary form the determinant formula appears in papers concerning the special values of the Selberg zeta function at positive integer points (Beilinson/Manin [2], Takase [76]). A generalization of the determinant representation of the Selberg zeta function to the finite volume case in two and three dimensions has been worked out by Sarnak [66], Efrat [20], Koyama [42], [43], Müller [52]. Appendix : Let A be a selfadjoint positive elliptic operator on a compact manifold Y. Then ζ(A)(s) = Tr A^{−s} is holomorphic for Re(s) > \frac{dim(Y)}{ord(A)}. Using the Mellin transform and the asymptotic expansion

Tr e^{−tA} ∼^{t→0} ∑_{α≥ \frac{−n}{ord(A)}} a_{α}t^{α}

one obtains a meromorphic continuation of ζ(A)(s) to all of ℂ. In particular, ζ(A) is regular at s = 0 and we define the regularized determinant of A by

det(A) := e^{−ζ′(A)(0)} .

Let P.F. ∫_{0}^{∞} Tr e^{−tA} \frac{dt}{t} denote the finite part of ∫_{ϵ}^{∞} Tr e^{−tA} \frac{dt}{t}. Then

det(A) = −P.F. ∫_{0}^{∞} Tr e^{−tA} \frac{dt}{t} − Eu a_{0} ,  (3.36)

where Eu is the Euler number. On can show that det(s + A) extends to an entire function with zeros at s ∈ R with −s ∈ spec(A) of order dim ker(s + A). Moreover, ln(det(s+A)) has an asymptotic expansion ([80], Eq. 5.1) for s → ∞

− ln(det(s + A)) ∼^{s→∞} ∑_{α∉−ℕ_{0}} a_{α}Γ(α)s^{−α}  (3.37)

− ∑_{α∈−ℕ_{0}} a_{α} [ln(s) − (∑_{r=1}^{−α} \frac{1}{r})] \frac{(−s)^{−α}}{(−α)!} .

If the operator A acts on a Z_{2}-graded bundle by diag(A^{+}, A^{−}), then we define

det(A) := det(A^{+}) det(A^{−})^{−1} .

The Ruelle Zeta Function

In view of the results of the previous chapter the representation of the Ruelle zeta functions in terms of Selberg zeta functions (Proposition 3.4) provides the meromorphic continuation of the former to all of ℂ. Using this representation we derive the functional equations for the Ruelle zeta functions and relate their behaviour at the central point s = 0 with certain topological invariants of Y like the Euler characteristic and the analytic torsion. We reproduce the main results of Fried’s papers [21] and [23] in Theorem 4.8 and 4.14.

4.1 The functional equations

We employ the functional equations for the Selberg zeta functions in order to provide functional equations for the Ruelle zeta functions. In contrast to the previous chapter we have to consider Selberg zeta functions associated with not necessarily irreducible representations of M. This causes no difficulties since for a finite dimensional representation σ of M we have

Z_{S,χ}(s, σ) = ∏_{σ′∈M̂} Z_{S,χ}(s, σ′)^{[σ:σ′]} .

We begin with the case n = dim(Y) even. First we write

Z_{R,χ}(s, σ) = ∏_{p=0}^{n/2−1} ( ∏_{(τ,λ)∈I_{p}} \frac{Z_{S,χ}(s + ρ − λ, σ ⊗ τ)}{Z_{S,χ}(s + λ − ρ, σ ⊗ τ)})^{(−1)^{p}} .

Here we employed Proposition 3.4 and Poincaré duality, i.e. for p ≤ n/2 − 1

I_{n−1−p} = {(τ, 2ρ − λ)|(τ, λ) ∈ I_{p}} .

Using the functional equations for the Selberg zeta functions (Theorem 3.17) we obtain

Z_{R,χ}(s, σ)Z_{R,χ}(−s, σ)

= ∏_{p=0}^{n/2−1} ( ∏_{(τ,λ)∈I_{p}} \frac{Z_{S,χ}(s + ρ − λ, σ ⊗ τ)Z_{S,χ}(−s + ρ − λ, σ ⊗ τ)}{Z_{S,χ}(−s + λ − ρ, σ ⊗ τ)Z_{S,χ}(s + λ − ρ, σ ⊗ τ)})^{(−1)^{p}}

= exp (\frac{2dπ dim(χ)vol(Y)}{T vol(X_{d})} ∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}} ∫_{s−(ρ−λ)}^{s+(ρ−λ)} P_{σ⊗τ} (r)

\{ \begin{matrix}tan(πr/T) & ϵ_{τ} ≢ ϵ_{σ} + ρ/T mod ℤ \\ −cot(πr/T) & ϵ_{τ} ≡ ϵ_{σ} + ρ/T mod ℤ \end{matrix} \} dr) ,  (4.1)

where P_{σ⊗τ} (r) := ∑_{σ′∈M̂}[(σ ⊗ τ) : σ′] P_{σ′}(r) and d = −(1)^{n/2}. Let h(s) denote the derivative of the exponent (up to the prefactor), i.e.,

h(s) = ∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}}(P_{σ⊗τ} (s + (ρ − λ)) − P_{σ⊗τ} (s − (ρ − λ)))

\{ \begin{matrix}tan(π(s + (ρ − λ))/T) & ϵ_{τ} ≢ ϵ_{σ} + ρ/T mod ℤ \\ −cot(π(s + (ρ − λ))/T) & ϵ_{τ} ≡ ϵ_{σ} + ρ/T mod ℤ \end{matrix} \} ,

where we have used the periodicity of tan and cot and that ρ, λ ∈ \frac{1}{2}T ℤ. Since Λ^{p}𝔫_{ℂ} is a representation of MA_{d}, where A_{d} := exp(i𝔞) ⊂ G_{d}, we have using the definition of ϵ_{τ}

ϵ_{τ} ≡ \frac{λ + ρ}{T} mod ℤ, ∀(τ, λ) ∈ I_{p} .  (4.2)

Hence,

h(s) = ∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}}(P_{σ⊗τ} (s + (ρ − λ)) − P_{σ⊗τ} (s − (ρ − λ)))

\{ \begin{matrix}tan(πs)/T) & ϵ_{σ} ≢ ρ/T mod ℤ \\ −cot(πs)/T) & ϵ_{σ} ≡ ρ/T mod ℤ \end{matrix} \} .

Let H(s) be the polynomial

H(s) := ∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}}(P_{σ⊗τ} (s + (ρ − λ)) − P_{σ⊗τ} (s − (ρ − λ))) .

Lemma 4.1 The polynomial H(s) is a constant

H(s) = \frac{n}{2} dim(σ)χ(X_{d}) .

P r o o f . The claim is a consequence of relations between the polynomials P_{σ′} for different σ′. In order to establish them we employ a geometric argument. It is enough to evaluate H(s) at a large number of points. Extend ℂ_{λ} to a representation of A_{d} for λ ∈ \frac{1}{2}T ℤ. Let σ′ ∈ M̂. If λ ∈ ℝ is such that T ϵ_{σ′} + ρ + λ ∈ T ℤ, then the representation V_{σ′} ⊗ ℂ_{λ} exists as a representation of MA_{d}. Form the holomorphic vector bundle

W (λ, σ′) = G_{d} ×_{MA_{d}} (V_{σ′} ⊗ ℂ_{λ}) → G_{d}/MA_{d} =: B

over the Kähler manifold B, which can be considered as the manifold of oriented geodesics of X_{d}. Recall the Borel-Weil-Bott Theorem (see, e.g., [88]). It asserts that the representation of G_{d} with the highest weight Λ = μ_{σ′} + λ (see Subsection 1.3.1 for conventions) can be realized as the space of holomorphic sections of W (λ, σ′) and all higher Dolbeault cohomology groups of W (λ, σ) vanish. A consequence of the theorem of Borel-Weil-Bott is

P_{σ′}(λ + ρ) = χ_{a}(B, W (λ, σ′)) , λ + ρ + T ϵ_{σ′} ∈ T ℤ ,

where χ_{a} is the analytic genus of the bundle W (λ, σ′), i.e., the Euler characteristic of the complex given by the Dolbeault resolution of W (λ, σ′). Hence for σ, τ ∈ M̂ we have

P_{σ⊗τ} (λ + ρ) = χ_{a}(B, W (λ, σ ⊗ τ)), λ + T (ϵ_{σ} + ϵ_{τ}) ∈ T ℤ ,  (4.3)

where W (λ, σ ⊗ τ) has the obvious meaning. Thus

P_{σ⊗τ} (λ + ρ) = index(¯∂ + ¯∂^{∗}) ,

i.e., P_{σ⊗τ} (λ + ρ) is the index of the Dirac type operator ¯∂ + ¯∂^{∗} on a ℤ_{2}-graded vector bundle Λ^{0,∗}T^{∗}B ⊗ W (λ, σ ⊗ τ) (the grading is given by even and odd form degree). For p ≤ n/2 − 1 let

H_{p}(s) := ∑_{(τ,λ)∈I_{p}} P_{σ⊗τ} (s + ρ − λ) .

Using (4.2) we have for s ∈ T ℤ

H_{p}(s) = χ_{a}(B, ⨁_{(τ,λ)∈I_{p}} W (s − λ, σ ⊗ τ)) .

But

G_{d} ×_{MA_{d}} ( ⨁_{(τ,λ)∈I_{p}} V_{σ} ⊗ V_{τ} ⊗ ℂ_{s−λ}) = G_{d} ×_{MA_{d}} (V_{σ} ⊗ Λ^{p}(𝔫_{ℂ})^{∗} ⊗ ℂ_{s}) .

Now

G_{d} ×_{MA_{d}} Λ^{p}(𝔫_{ℂ})^{∗} = Λ^{p,0}T^{∗}B ,

and we obtain for p < n/2 − 1

H_{p}(s) = χ_{a}(B, Λ^{p,0}T^{∗}B ⊗ W (s, σ)) .

For p > n/2 − 1 let

H_{p}(s) := ∑_{(τ,λ)∈I_{n−p−1}} P_{σ⊗τ} (s − ρ + λ) .

Applying equation (4.3) we obtain

H_{p}(s) = χ_{a}(B, ⨁_{(τ,λ)∈I_{n−p−1}} W (λ + s − 2ρ, σ ⊗ τ)) .

But

G_{d} ×_{MA_{d}} ( ⨁_{(τ,λ)∈I_{n−p−1}} V_{σ} ⊗ V_{τ} ⊗ ℂ_{s+λ−2ρ})

= G_{d} ×_{MA_{d}} (V_{σ} ⊗ Λ^{n−p−1}𝔫_{ℂ} ⊗ ℂ_{s−2ρ})
= G_{d} ×_{MA_{d}} (V_{σ} ⊗ Λ^{p}(𝔫^{c})^{∗} ⊗ ℂ_{s})
= Λ^{p,0}T^{∗}B ⊗ W (s, σ) .

Since H(s) = ∑_{p=0}^{n−1}(−1)^{p}H_{p}(s) we obtain that H(s) is the index of (¯∂ + ¯∂^{∗}) on the bundle

Λ^{∗,∗}T^{∗}B ⊗ W (s, σ) = Λ^{∗}T^{∗}B ⊗ W (s, σ).

Now the differential operator (¯∂+ ¯∂^{∗}) can be deformed to D = d_{∇} + d_{∇}^{∗}, where d_{∇} is the exterior differential twisted by the connection ∇ on W (s, σ) induced by the structure of a Hermitian vector bundle. The deformation is given by

D_{t} = (¯∂ + ¯∂^{∗}) + t(∂_{∇} + ∂_{∇}^{∗})

and stays inside the elliptic operators. The index of D_{t} is independent of t. But index(D_{1}) can be computed as follows (see, e.g., [3])

index(D_{1}) = ∫_{B} χ_{TB} ∧ ch(W(s, σ)) ,

where χ_{T B} denotes Euler class and ch the Chern character. χ_{T B} is concentrated in the top degree. Thus index(D_{1}) = dim(σ)χ(B). That the Euler characteristic of B can be expressed in terms of X_{d} was observed by Juhl [40]. In fact, for compact homogeneous spaces L/H with rank(L) =rank(H) the Euler characteristic of L/H is equal to the quotient of the orders of Weyl groups (see for example [7])

χ(B) = \frac{|W (G_{d})|}{|W (MA_{d})|} = \frac{|W (G_{d})|}{|W (M)|} = \frac{|W (G_{d})|}{|W (K)|} \frac{|W (K)|}{|W (M)|} = \frac{|W (K)|}{|W (M)|}χ(X_{d}) .

A case-by-case computation shows that

\frac{|W (K)|}{|W (M)|} = \frac{n}{2} .

Hence H(s) = index(D_{1}) = \frac{n}{2} dim(σ)χ(X_{d}). □ Now we come back to discuss the functional equation. Note that h(s) is the derivative of the multivalued function

C − \frac{T n}{2π} dim(σ)χ(X_{d}) \{ \begin{matrix}ln(cos(πs/T)) & ϵ_{σ} ≢ ρ/T mod ℤ \\ ln(sin(πs/T)) & ϵ_{σ} ≡ ρ/T mod ℤ \end{matrix}

for any constant C. Hence

∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}} ∫_{s−(ρ−λ)}^{s+(ρ−λ)} P_{σ⊗τ} (r) \{ \begin{matrix}tan(πr/T) & ϵ_{τ} ≢ ϵ_{σ} + ρ/T mod ℤ \\ −cot(πr/T) & ϵ_{τ} ≡ ϵ_{σ} + ρ/T mod ℤ \end{matrix} \} dr

= C_{σ} − \frac{T n}{2π} dim(σ)χ(X_{d}) \{ \begin{matrix}ln(cos(πs/T)) & ϵ_{σ} ≢ ρ/T mod ℤ \\ ln(sin(πs/T)) & ϵ_{σ} ≡ ρ/T mod ℤ \end{matrix} . (4.4)

We argue that C_{σ} = 0. We consider the asymptotics of both sides of this equation for s → i∞. The r.h.s. behaves as C_{σ} + \frac{n}{2} dim(σ)χ(X_{d})is + O(e^{ics}) for some c > 0. An approximation for large Im(s) of the l.h.s. is

∑_{p=0}^{n/2−1}(−1)^{p} ∑_{(τ,λ)∈I_{p}} ∫_{s−(ρ−λ)}^{s+(ρ−λ)} P_{σ⊗τ} (r) dr + O(e^{ics}) .

The integral

∫_{s−(ρ−λ)}^{s+(ρ−λ)} P_{σ⊗τ} (r) dr

is an odd polynomial in s since P_{σ⊗τ} is odd. In particular, its constant term vanishes. Thus, the constant term C_{σ} of the r.h.s. of (4.4) must vanish, too.

Inserting (4.4) into (4.1) and using

\frac{vol(Y)}{vol(X_{d})} = (−1)^{n/2} \frac{χ(Y)}{χ(X_{d})}

we obtain

Theorem 4.2 For n even the Ruelle zeta function satisfies the functional equation

Z_{R,χ}(s, σ)Z_{R,χ}(−s, σ) = \{ \begin{matrix}cos(πs/T)^{n dim(χ) dim(σ)χ(Y)} & ϵ_{σ} ≢ ρ/T mod ℤ \\ sin(πs/T)^{n dim(χ) dim(σ)χ(Y)} & ϵ_{σ} ≡ ρ/T mod ℤ \end{matrix} .

If we specialize this theorem to the case that σ and χ are trivial we obtain for Z_{R}(s) := Z_{R,χ}(s, σ)

Corollary 4.3 For n even the following functional equation holds

Z_{R}(s)Z_{R}(−s) = sin(πs/T)^{nχ(Y)} .

In particular, the order of the singularity at s = 0 is \frac{n}{2}χ(Y) while the order of the singularities at s < −2ρ, s ∈ T ℤ, is nχ(M).

Up to an exponential polynomial this functional equation was already obtained by Juhl [40]. For different approaches to the order of the singularity at s = 0 see Juhl [41] and Seifarth [72]. Now we turn to the case of odd dimension.

Theorem 4.4 The Ruelle zeta function associated with a Weyl-invariant representation σ ∈ M̂ satisfies the functional equation

\frac{Z_{R,χ}(s, σ)}{Z_{R,χ}(−s, σ)} = exp (\frac{2π(n + 1) dim(χ) dim(σ)vol(Y)}{T vol(X_{d})}s) .

P r o o f . Write σ_{p} for the representation of M on Λ^{p}𝔫_{ℂ}. This coincides with the notation used in Subsection 1.1.2. Using the MA-module equivalence

Λ^{p}𝔫_{ℂ} ≅ σ_{p} ⊗ ℂ_{pT} ,

Proposition 3.4, Theorem 3.17 and Poincaré duality we obtain

\frac{Z_{R,χ}(s, σ)}{Z_{R,χ}(−s, σ)} = (\frac{Z_{S,χ}(s, σ_{\frac{n−1}{2}} ⊗ σ)}{Z_{S,χ}(−s, σ_{\frac{n−1}{2}} ⊗ σ)})^{(−1)^{\frac{n−1}{2}}}

∏_{p=0}^{(n−3)/2} (\frac{Z_{S,χ}(s + ρ − pT, σ_{p} ⊗ σ)Z_{S,χ}(s − (ρ − pT), σ_{p} ⊗ σ)}{Z_{S,χ}(−s + ρ − pT, σ_{p} ⊗ σ)Z_{S,χ}(−s − (ρ − pT), σ_{p} ⊗ σ)})^{(−1)^{p}}

= exp (\frac{2π dim(χ)vol(Y)}{T vol(X_{d})}[(−1)^{\frac{n−1}{2}} ∫_{0}^{s} P_{σ_{\frac{n−1}{2}} ⊗σ}(q) dq+  (4.5)

+ ∑_{p=0}^{(n−3)/2}(−1)^{p}(∫_{0}^{s+ρ−pT} P_{σ_{p}⊗σ}(q) dq + ∫_{0}^{s−(ρ−pT)} P_{σ_{p}⊗σ}(q) dq)]) .

Let h(s) denote the derivative of the exponent (up to the prefactor), i.e., the polynomial

h(s) = (−1)^{\frac{n−1}{2}} P_{σ_{\frac{n−1}{2}} ⊗σ}(s)

+ ∑_{p=0}^{\frac{n−3}{2}}(−1)^{p}(P_{σ_{p}⊗σ}(s + ρ − pT) + P_{σ_{p}⊗σ}(s − (ρ − pT)) .

As in the proof of Lemma 4.1 one sees that

h(s) ≡ dim(σ)χ(B) ,

where B = SO(n + 1)/SO(n − 1) × SO(2). The Euler characteristic of B is equal to the quotient of the orders of Weyl groups (see for example [7])

χ(B) = \frac{|W (SO(n + 1))|}{|W (SO(n − 1) × SO(2))|} = \frac{|W (SO(n + 1))|}{|W (SO(n − 1))|} = \frac{2^{\frac{n−1}{2}} (\frac{n+1}{2})!}{2^{\frac{n−3}{2}} (\frac{n−1}{2})!} = n + 1 .

We conclude that the r.h.s. of (4.5) is the exponential of

\frac{2π dim(χ)vol(Y)}{T vol(X_{d})}(dim(σ)(n + 1)s + C)

for some constant C. But the l.h.s. evaluated ar s = 0 is 1, hence C = 0. The theorem follows. □ In the case that σ ∈ M̂ is not Weyl-invariant we define a super Ruelle zeta function by

Z_{R,χ}^{s}(s, σ) := Z_{R,χ}(s, σ)/Z_{R,χ}(s, wσ) .

Theorem 4.5 If σ ∈ M̂ is not Weyl-invariant, we have

Z_{R,χ}^{s}(s, σ)Z_{R,χ}^{s}(−s, σ) = e^{2πiη({ \atop D_{Y,χ}(σ)})} ,

where { \atop D_{Y,χ}(σ)} acts on sections of the graded bundle

V_{Y,χ}(γ^{s}) ⊗ (∑_{k=0}^{n−1}(−1)^{k}(n − k)Λ^{k}T^{∗}Y )

as the twist of D_{Y,χ}(σ) with the Levi-Civita connection on the second factor. In particular,

Z_{R,χ}^{s}(0, σ) = e^{πiη({ \atop D_{Y,χ}(σ)})} .

Moreover, the following functional equation holds:

\frac{Z_{R,χ}(s, σ)}{Z_{R,χ}(−s, wσ)} = e^{πiη({ \atop D_{Y,χ}(σ)})} exp (\frac{2π(n + 1) dim(χ) dim(σ)vol(Y)}{Tvol(X_{d})}s) .

P r o o f . The functional equation for Z_{R,χ}^{s} easily follows from Proposition 3.4, the functional equation for S_{χ}^{s} (3.21), and the fact that in the notation of Subsection 1.1.2

i^{∗}(∑_{k=0}^{n−1}(−1)^{k}(n − k)λ_{k}) = ∑_{p=0}^{n−1}(−1)^{p}σ_{p} .

In order to obtain the functional equation for Z_{R,χ} one derives a functional equation for the product Z_{R,χ}(., σ)Z_{R,χ}(., wσ) by the same reasoning as in the proof of Theorem 4.4, and combines this with the functional equation of Z_{R,χ}^{s}.

□

4.2 The behaviour at the central point

In the case of even dimension Theorem 4.2 and Corollary 4.3 show that the behaviour of Z_{R,χ}(s, σ) at the central point s = 0 carries topological information about Y. This section is devoted to a closer study of that phenomenon in the cases of odd dimensional and complex hyperbolic manifolds, respectively, by means of determinant representations of the Ruelle zeta functions.

4.2.1 Odd-dimensional hyperbolic manifolds and analytic torsion

Throughout this subsection we assume that Y is an odd-dimensional hyperbolic manifold. First we derive a determinant representation of Z_{R,χ}(s, σ).

We define

A_{Y,χ}(σ_{p} ⊗ σ) := ⨁_{[σ′]∈M̂/W} ⨁_{i=1}^{[(σ_{p}⊗σ):σ′]} A_{Y,χ}(γ, σ′) ,

where γ ∈ R(K) is the unique lift of σ′, if σ′ is Weyl-invariant, or of σ′ + wσ′ otherwise.

Proposition 4.6 The Ruelle zeta function has the representation

Z_{R,χ}(s, σ) = ∏_{p=0}^{n−1} det((s + ρ − pT)^{2} + A_{Y,χ}(σ_{p} ⊗ σ)^{2})^{(−1)^{p}}

exp (\frac{π(n + 1) dim(χ) dim(σ)vol(Y)}{T vol(X_{d})}s)

in case (a) and

Z_{R,χ}(s, σ)Z_{R,χ}(s, wσ) = ∏_{p=0}^{n−1} det((s + ρ − pT)^{2} + A_{Y,χ}(σ_{p} ⊗ σ)^{2})^{(−1)^{p}}

exp (\frac{2π(n + 1) dim(χ) dim(σ)vol(Y)}{T vol(X_{d})}s)

in case (b).

Proof. We prove the assertion for case (a). The proof for case (b) is essentially the same. In our situation we have by Proposition 3.4

Z_{R,χ}(s, σ) = ∏_{p=0}^{n−1} Z_{S,χ}(s + ρ − pT, σ_{p} ⊗ σ)^{(−1)^{p}} .

Inserting the determinant representations of the Selberg zeta functions (Theorem 3.19) we obtain

Z_{R,χ}(s, σ) = ∏_{p=0}^{n−1} det((s + ρ − pT)^{2} + A_{Y,χ}(σ_{p} ⊗ σ)^{2})^{(−1)^{p}}

∏_{p=0}^{n−1} exp (\frac{π dim(χ)vol(Y)}{T vol(X_{d})} ∫_{0}^{s+ρ−pT} P_{σ_{p}⊗σ}(t) dt)^{(−1)^{p}} .

The second factor has already been evaluated in the proof of Proposition 4.4 and is equal to

exp (\frac{π(n + 1) dim(χ) dim(σ)vol(Y)}{T vol(X_{d})}s) .

The Proposition follows. □

Corollary 4.7 Assume that A_{Y,χ}(σ_{p} ⊗ σ)^{2} + (ρ − pT)^{2} has no kernel for p = 0, …, n − 1. Then Z_{R,χ}(s, σ) is regular at s = 0, and we have

Z_{R,χ}(0, σ) = ∏_{p=0}^{n−1} det(A_{Y,χ}(σ_{p} ⊗ σ)^{2} + (ρ − pT)^{2})^{(−1)^{p}} ,

in case (a) and

Z_{R,χ}(0, σ) = e^{\frac{πi}{2} η({ \atop D_{Y,χ}(σ)})} ∏_{p=0}^{n−1} det(A_{Y,χ}(σ_{p} ⊗ σ)^{2} + (ρ − pT)^{2})^{(−1)^{p}/2}

in case (b), where { \atop D_{Y,χ}(σ)} is defined in Theorem 4.5.

P r o o f . For case (a) the corollary is immediate. In case (b) one uses

Z_{R,χ}(s, σ) = \sqrt{Z_{R,χ}(s, σ)Z_{R,χ}(s, wσ)Z_{R,χ}^{s}(s, σ)}

and Theorem 4.5. □ Now we specialize to the case that σ is trivial. Let λ_{p} be the p^{th} exterior power of the standard representation of SO(n). Then for p = 0, …, n − 1 we have i^{∗}(λ_{p}) = σ_{p}+σ_{p−1}. Setting γ_{p} := ∑(−1)^{k}λ_{p−k} we obtain i^{∗}(γ_{p}) = σ_{p}, p = 0, …, n − 1. It is easy to see that c(σ_{p}) = (ρ − pT)^{2}. Thus the actions of A_{Y,χ}(σ_{p})^{2} + (ρ − pT)^{2} and −Ω on C^{∞}(Y, V_{Y,χ}(γ_{p})) coincide. But on C^{∞}(Y, V_{Y,χ}(λ_{k})) the Casimir operator Ω acts by the negative form Laplacian −Δ_{χ,k}. We obtain

∏_{p=0}^{n−1} det((s + ρ − pT)^{2} + A_{Y,χ}(σ_{p})^{2})^{(−1)^{p}}

= ∏_{k=0}^{n−1} ∏_{p=k}^{n−1} det(s(s + 2(ρ − pT)) + Δ_{χ,k})^{(−1)^{k}} .

Furthermore, by Hodge theory the operators A_{Y,χ}(σ_{p})^{2} + (ρ − pT)^{2}, p = 0, …, n − 1, have no kernel iff χ is acyclic, i.e., H^{k}(Γ, V_{χ}) = 0 for all k. In this case the analytic torsion T_{χ}, as introduced by Ray/Singer [62], is defined as the square root of

∏_{k=1}^{n} det(Δ_{χ,k})^{k(−1)^{k−1}} = ∏_{k=0}^{n−1} det(Δ_{χ,k})^{(n−k)(−1)^{k}} .  (4.6)

The famous result of Cheeger [13] and Müller [51] asserts that T_{χ} is equal to the Reidemeister torsion which is a topological invariant of Y. Summarizing the above discussion we obtain for the Ruelle zeta function Z_{R,χ}(s) associated with the trivial M-representation

Theorem 4.8 The Ruelle zeta function Z_{R,χ} has the representation

Z_{R,χ}(s) = ∏_{k=0}^{n−1} ∏_{p=k}^{n−1} det(s(s + (n − 1 − 2p)T) + Δ_{χ,k})^{(−1)^{k}}

exp (\frac{π(n + 1) dim(χ)vol(Y)}{T vol(X_{d})}s) .

Let b_{k,χ} := dim H^{k}(Γ, V_{χ}). Then the singularity of Z_{R,χ} at s = 0 is of order

∑_{k=0}^{(n−1)/2}(n + 1 − 2k)(−1)^{k}b_{k,χ} .

If the twist χ is acyclic, then Z_{R,χ} is regular at s = 0 and we have

Z_{R,χ}(0) = T_{χ}^{2} .

The identification of the value of Z_{R,χ} at the central point with the analytic torsion is due to Fried [21]. A generalization of this result to higher rank situations has been obtained by Moscovici/Stanton [50] (fundamental rank one) and Deitmar [16] (higher fundamental rank). We conclude this subsection with a determinant formula for even-dimensional hyperbolic manifolds which can be proved along the same lines as the formula for odd-dimensional ones appearing in the above theorem.

Proposition 4.9 For even-dimensional real hyperbolic manifolds Y the Ruelle zeta function Z_{R,χ} can be written in terms of determinants of the form Laplacians Δ_{χ,k} and Δ_{d,k} of Y and X_{d} = S^{n}, respectively, as follows

Z_{R,χ}(s) = ∏_{k=0}^{n−1} ∏_{p=k}^{n−1} det(s(s + (n − 1 − 2p)T) + Δ_{χ,k})^{(−1)^{k}}
det(Δ_{d,k} − s(s + (n − 1 − 2p)T)^{(−1)^{k+1} \frac{1}{2} dim(χ)χ(Y)}
sin(πs/T)^{\frac{n}{2} dim(χ)χ(Y)} .

4.2.2 Complex hyperbolic manifolds and holomorphic torsion

We consider the Ruelle zeta function Z_{R,χ} associated with the trivial representation of M for complex hyperbolic manifolds, i.e., a quotient of Hℂ^{m}. For the concrete form of the groups G, K and M in this situation see Subsection 1.1.2 (The complex case). The starting point has to be a good understanding of the decomposition of Z_{R,χ} into a product of Selberg zeta functions (Proposition 3.4). We have the MA-module isomorphism

𝔫_{ℂ} ≅ Λ^{1,0}𝔫_{α} ⊕ Λ^{0,1}𝔫_{α} ⊕ 𝔫_{2α} ≅ σ_{1} ⊗ ℂ_{α} ⊕ σ̄_{1} ⊗ ℂ_{α} ⊕ ℂ_{2α} ,  (4.7)

where σ_{1} is the standard representation of U(m − 1) as in Subsection 1.1.2, α is the short root in Φ^{+}(𝔤, 𝔞), and the bar denotes the complex conjugate representation. Let σ_{q} denote the complex exterior powers of σ_{1} and σ_{p,q} the primitive part of σ_{p}⊗σ̄_{q}, i.e., the kernel of the contraction (multiplication) with the invariant bivector in σ_{1} ⊗ σ̄_{1} for p + q ≤ m + 1 (p + q ≥ m + 1). The representations σ_{p,q} are irreducible. Poincaré duality gives σ_{p,q} ≅ σ_{m−1−q,m−1−p}. We use the convention that σ_{p,q} = 0 whenever p > m − 1, q > m − 1, p < 0, or q < 0, and that Z_{S,χ}(s, 0) ≡ 1.

Lemma 4.10 Let I_{p,q} = {(τ, λ) ∈ M̂ × 𝔞_{ℂ}^{∗}} be such that

Λ^{p,q}𝔫_{α} ⊕ Λ^{p−1,q}𝔫_{α} ⊗ 𝔫_{2α} ≅ ∑_{(τ,λ)∈I_{p,q}} τ ⊗ ℂ_{λ}

and define partial Ruelle zeta functions

R_{p,χ}(s) := ∏_{q=0}^{m−1} ∏_{(τ,λ)∈I_{p,q}} Z_{S,χ}(s + mα − λ, τ)^{(−1)^{q}} .

Then Z_{R,χ} has the following representation

Z_{R,χ}(s) = ∏_{p=0}^{m} R_{p,χ}(s)^{(−1)^{p}} .

Furthermore, we have

R_{p,χ}(s) = ∏_{q=0}^{m−1−p} Z_{S,χ}(s + (m − (p + q))α, σ_{p,q})^{(−1)^{q}}

∏_{q=m−p+1}^{m} Z_{S,χ}(s + (m − (p + q))α, σ_{p−1,q−1})^{(−1)^{q−1}} .

P r o o f . The first assertion is a simple consequence of Proposition 3.4, formula (4.7) and the fact that ρ = mα. The second assertion follows from the following computation in the representation ring of MA. Observe that

Λ^{p,q}𝔫_{α} − Λ^{p−1,q−1}𝔫_{α} ⊗ 𝔫_{2α} = \{ \begin{matrix}σ_{p,q} ⊗ ℂ_{(p+q)α} & if p + q ≤ m − 1 , \\ 0 & if p + q = m , \\ −σ_{p−1,q−1} ⊗ ℂ_{(p+q)α} & if p + q ≥ m + 1 . \end{matrix}

We obtain

∑_{q=0}^{m−1}(−1)^{q}(Λ^{p,q}𝔫_{α} + Λ^{p−1,q}𝔫_{α} ⊗ 𝔫_{2α})

= ∑_{q=0}^{m}(−1)^{q}(Λ^{p,q}𝔫_{α} − Λ^{p−1,q−1}𝔫_{α} ⊗ 𝔫_{2α})

= ∑_{q=0}^{m−1−p}(−1)^{q}σ_{p,q} ⊗ ℂ_{(p+q)α} + ∑_{q=m−p+1}^{m}(−1)^{q−1}σ_{p−1,q−1} ⊗ ℂ_{(p+q)α} .
□

The partial Ruelle zeta functions R_{p,χ} were introduced by Fried [23] in order to relate holomorphic torsion with the closed geodesics of Y. We will turn to this point at the end of this subsection. Now we want to express R_{χ} and R_{p,χ} via determinants of form Laplacians. Thus we have to find lifts in R(K) of σ_{p,q} and to compute the shift constants c(σ_{p,q}). Let λ_{p} be the p^{th} exterior power of the standard representation of U(m), Λ_{p,q} := λ_{p} ⊗ λ̄_{q}, and λ_{p,q} be its primitive part. Then the following lemma is fairly standard.

Lemma 4.11 We have i^{∗}(λ_{p,q}) = σ_{p,q} ⊕ σ_{p−1,q} ⊕ σ_{p,q−1} ⊕ σ_{p−1,q−1} .

The lemma implies that

σ_{p,q} = i^{∗} (∑_{r,s}(−1)^{r+s}λ_{p−s,q−r})  (4.8)

as well as

σ_{p,q} = i^{∗} (∑_{r,s}(−1)^{r+s}λ_{p+1+s,q+1+r}) .

In addition, we have

λ_{p,q} = \{ \begin{matrix}Λ_{p,q} − Λ_{p−1,q−1} & if p + q ≤ m , \\ Λ_{p,q} − Λ_{p+1,q+1} & if p + q ≥ m . \end{matrix}

Hence we obtain that σ_{p,q} is equal to the restriction of

Λ_{p,q} + ∑_{r≥1}(−1)^{r}(Λ_{p,q−r} + Λ_{p−r,q})  if p + q ≤ m ,
Λ_{p+1,q+1} + ∑_{r≥1}(−1)^{r}(Λ_{p+1,q+1+r} + Λ_{p+1+r,q+1}) if p + q ≥ m − 1 . (4.9)

Lemma 4.12 For p + q ≤ m − 1 the constants c(σ_{p,q}) defined in Subsection 1.1.3, equation (1.9), are given by

c(σ_{p,q}) = |(m − (p + q))α|^{2} .

Proof. Let 𝔱 ⊂ 𝔪 ⊂ 𝔰𝔲(1, m) be the standard Cartan algebra of 𝔪 given by diagonal matrices of the form diag(q, q, x_{1}, …, x_{m−1}), q, x_{i} ∈ iℝ, 2q+∑_{i=1}^{m−1} x_{i} = 0. We can write weights as m + 1-tuples of complex numbers such that the first two entries coincide and the total sum vanishes. The positive roots of 𝔪 are {E_{ij}}, 2 ≤ i < j ≤ m, where the only non-trivial entries of E_{ij} are 1 at the i^{th} and −1 at in j^{th} place. The half-sum ρ_{𝔪} of the positive roots is

ρ_{𝔪} = \frac{1}{2}(0, 0, m − 2, m − 4, …, 4 − m, 2 − m) .

The weight of the i^{th} coordinate z^{i} of σ_{1} is

(−1/2, −1/2, 0, …, 0, 1, 0, …, 0) ,

where 1 is at the (i + 2)^{th} entry. Analogously the weight of z̄^{i} is

(1/2, 1/2, 0, …, 0, −1, 0, …, 0) .

For p + q ≤ m − 1 the highest weights of σ_{p} ⊗ σ̄_{q} and σ_{p,q} coincide and are given by

μ_{σ_{p,q}} = (\frac{q − p}{2}, \frac{q − p}{2}, \underbrace{1, …, 1}_{p times}, 0, …, 0, \underbrace{−1, …, −1}_{q times}) .

If we use the invariant scalar product on 𝔤 given by the trace norm (X, Y) := tr XY, then the metric induced on 𝔱^{∗} is simply |(x_{0}, …, x_{m})|^{2} = ∑_{i=0}^{m} |x_{i}|^{2}, and |α|^{2} = 1/2. Thus for an arbitrary norm |.| we obtain

c(σ_{p,q}) = |ρ|^{2} + |ρ_{𝔪}|^{2} − |μ_{σ_{p,q}} + ρ_{𝔪}|^{2}
= m^{2}|α|^{2} − (μ_{σ_{p,q}} + 2ρ_{𝔪}, μ_{σ_{p,q}})
= (m^{2} − ((p + q)^{2} − 2m(p + q))) |α|^{2} = |(m − (p + q))α|^{2} .
□

Now we have provided all necessary tools we need to prove the main results of this subsection. The first result is

Theorem 4.13 For complex hyperbolic manifolds Y the Ruelle zeta function Z_{R,χ}(s) can be written in terms of determinants of the form Laplacians Δ_{χ,k} and Δ_{d,k} of Y and X_{d} = P ℂ^{m}, respectively, as follows:

∏_{k=0,k≠m}^{2m}(det(s(s + (m − k)T) + Δ_{χ,k}) det(Δ_{d,k} − s(s + (m − k)T))^{− dim(χ)χ_{a}(Y)}

∏_{l=1}^{|m−k|−1} det(s(s + sign(m − k)lT) + Δ_{χ,k})^{2}

det(Δ_{d,k} − s(s + sign(m − k)lT))^{−2 dim(χ)χ_{a}(Y)})^{sign(m−k)(−1)^{k}}

sin(πs/T)^{m(m+1) dim(χ)χ_{a}(Y)} ,

where χ_{a}(Y) is the analytic genus of Y.

P r o o f . By the proportionality principle we have χ_{a}(Y) = χ(Y)/χ(X_{d}). −Ω acts on (p, q)-forms with values in V_{Y,χ}(1) via the form Laplacian Δ_{χ,p,q}. Though V_{Y,χ}(Λ_{p,q}) is isomorphic to the bundle of (q, p)-forms with values in V_{Y,χ}(1), we will associate with Λ_{p,q} the Laplacian Δ_{χ,p,q} which has the same spectrum as Δ_{χ,q,p}. Combining formula (4.9) and Lemma 4.12 with Theorem 3.19 we obtain for p + q ≤ m − 1

Z_{S,χ}(s + (m − (p + q))α, σ_{p,q})
= ∏_{r≥0} det(s(s + 2(m − (p + q))α) + Δ_{χ,p,q−r})^{(−1)^{r}}
det(Δ_{d,p,q−r} − s(s + 2(m − (p + q))α))^{(−1)^{r+1} dim(χ)χ_{a}(Y)}
∏_{r≥1} det(s(s + 2(m − (p + q))α) + Δ_{χ,p−r,q})^{(−1)^{r}}  (4.10)

det(Δ_{d,p−r,q} − s(s + 2(m − (p + q))α))^{(−1)^{r+1} dim(χ)χ_{a}(Y)}

exp (dim(χ)χ_{a}(Y)\frac{π}{T} ∫_{0}^{s+(m−(p+q))α} P_{σ_{p,q}}(t) cot(πt/T) dt) .

Analogously, for p + q ≥ m + 1 we have

Z_{S,χ}(s + (m − (p + q))α, σ_{p−1,q−1})^{−1}
= ∏_{r≥0} det(s(s + 2(m − (p + q))α) + Δ_{χ,p,q+r})^{(−1)^{r−1}}
det(Δ_{d,p,q+r} − s(s + 2(m − (p + q))α))^{(−1)^{r} dim(χ)χ_{a}(Y)}
∏_{r≥1} det(s(s + 2(m − (p + q))α) + Δ_{χ,p+r,q})^{(−1)^{r−1}}  (4.11)
det(Δ_{d,p+r,q} − s(s + 2(m − (p + q))α))^{(−1)^{r} dim(χ)χ_{a}(Y)}

exp (dim(χ)χ_{a}(Y)\frac{π}{T} ∫_{0}^{s+(m−(p+q)α)} P_{σ_{p−1,q−1}}(t) cot(πt/T) dt) .

By Lemma 4.10 we have to form the alternating product over p and q of these expressions (p, q = 0, …, m; p + q ≠ m) in order to obtain Z_{R,χ}. The calculations in Subsection 4.1 show that the factors involving the polynomials P_{σ_{p,q}} contribute the factor sin(πs/T)^{m(m+1) dim(χ)χ_{a}(Y)}. The remaining terms give

∏_{k=0}^{m−1} ( det(s(s + 2(m − k)α) + Δ_{χ,k}) det(Δ_{d,k} − s(s + 2(m − k)α))^{− dim(χ)χ_{a}(Y)}

∏_{l=1}^{m−k−1} det(s(s + 2lα) + Δ_{χ,k})^{2} det(Δ_{d,k} − s(s + 2lα))^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{k}}

∏_{k=m+1}^{2m}(det(s(s+2(m−k)α) + Δ_{χ,k}) det(Δ_{d,k} − s(s+2(m−k)α))^{− dim(χ)χ_{a}(Y)}

∏_{l=1}^{k−m−1}det(s(s − 2lα) + Δ_{χ,k})^{2} det(Δ_{d,k} − s(s − 2lα))^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{k−1}}.

Replacing α by T/2 the theorem follows. □

Now we discuss the relation between R_{p,χ} and the p^{th} holomorphic torsion T_{p,χ}. For a Hermitian manifold Y of complex dimension m and a unitary representation χ of π_{1}(Y) one defines T_{p,χ} according to Ray/Singer [63] as the square root of

∏_{q=1}^{m} det(□′_{χ,p,q})^{q(−1)^{q−1}} ,

where □_{χ,p,q} := ¯∂^{∗} ¯∂ + ¯∂¯∂^{∗} is the ¯∂-Laplacian on twisted (p, q)-forms, and ′ means restriction to the orthogonal complement of the kernel. For Kähler manifolds one has □_{χ,p,q} = \frac{1}{2}Δ_{χ,p,q} or, in other words,

□_{χ,p,q}^{g} = Δ_{χ,p,q}^{2g} ,  (4.12)

where g is the metric on Y. The following theorem is essentially due to Fried [23].

Theorem 4.14 The order of the singularity of R_{p,χ} at s = 0 is equal to

(−1)^{p}(m + p) dim(χ)χ_{a}(Y) + ∑_{q=1}^{m} q(−1)^{q−1}h_{p,q}(χ) .

There are constants c_{p} and β_{p,q}, which are independent of Γ such that the leading coefficient of Laurent expansion of R_{p,χ} at s = 0 is given by

±T_{p,χ}^{2}c_{p}^{dim(χ)χ_{a}(Y)} ∏_{q=0}^{m} β_{p,q}^{h_{p,q}(χ)} ,

where h_{p,q}(χ) := dim(ker Δ_{χ,p,q}) are the Hodge numbers.

P r o o f . If we express Z_{S,χ}(s + (m − (p + q))α, σ_{p,q}) and Z_{S,χ}(s + (m − (p + q))α, σ_{p−1,q−1})^{−1} via determinants using formula (3.24) instead of (3.23) (note that (4.9) defines an admissible lift of σ_{p,q}) and form the alternating product over q ≠ m − p, then we obtain the following formula for R_{p,χ}(s) (compare (4.10) and (4.11))

∏_{q=0}^{m−1−p} ( ∏_{l=1}^{m−(p+q)} det(s(s + 2lα) + Δ_{χ,p,q})

det(\sqrt{Δ_{d,p,q} + (lα)^{2}} + s + lα)^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{q}}

∏_{q=0}^{m−1−p} ∏_{j=0}^{p−1} ( det(s(s + 2(m − (p + q))α) + Δ_{χ,j,q})

det(\sqrt{Δ_{d,j,q} + ((m−(p + q))α)^{2}} +s+ (m−(p + q))α)^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{p+q−j}}

∏_{q=m−p+1}^{m} ( ∏_{l=1}^{p+q−m} det(s(s − 2lα) + Δ_{χ,p,q})

det(\sqrt{Δ_{d,p,q} + (lα)^{2}} + s − lα)^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{q−1}}

∏_{q=m−p+1}^{m} ∏_{j=p+1}^{m} ( det(s(s + 2(m − (p + q))α) + Δ_{χ,j,q})

det(\sqrt{Δ_{d,j,q} + ((m−(p + q))α)^{2}} +s+ (m−(p + q))α)^{−2 dim(χ)χ_{a}(Y)})^{(−1)^{p+q−j−1}}

exp(G_{p}(s)) ,

where G_{p} is a certain polynomial, which can be expressed by the polynomials P_{σ_{p,q}} and P_{σ_{p−1,q}} and the analytic genus of Y. We discuss the order of the singularity of R_{p,χ} at s = 0. First we observe that the contributions involving Δ_{χ,j,q} for j ≠ p cancel by Poincaré duality. In addition we have

dim(ker Δ_{d,p,q}) = δ_{p,q}

and, by the proportionality principle,

∑_{q=0}^{m}(−1)^{q}h_{p,q}(χ) = (−1)^{p} dim(χ)χ_{a}(Y) .

Thus we obtain

n_{p} := ord_{s=0}R_{p,χ}(s)
= ∑_{q=0}^{m−1−p}(m − (p + q))(−1)^{q}h_{p,q}(χ) + ∑_{q=m−p+1}^{m} ((p + q) − m)(−1)^{q−1}h_{p,q}(χ)

+ \{ \begin{matrix}0 & p ≤ m/2 \\ (−1)^{p}(4p − 2m) dim(χ)χ_{a}(Y) & p ≥ (m + 1)/2 \end{matrix} \}

+ \{ \begin{matrix}(−1)^{p}2p dim(χ)χ_{a}(Y) & p ≤ m/2 \\ (−1)^{p}(2m − 2p) dim(χ)χ_{a}(Y) & p ≥ (m + 1)/2 \end{matrix} \}

= ∑_{q=1}^{m} q(−1)^{q−1}h_{p,q}(χ) + (−1)^{p}(m + p) dim(χ)χ_{a}(Y) .

Now we determine the coefficient a_{p} at s^{n_{p}} in the Laurent expansion of R_{p,χ}(s) at s = 0 modulo all terms only involving the Hodge numbers h_{p,q}(χ) and the polynomials P_{σ_{p−1,q}}, P_{σ_{p,q}}. So we may disregard the polynomial G_{p}(s) as well as all contributions coming from the dual Laplacians, since they are expressible by the coefficients of P_{σ_{p−1,q}}, P_{σ_{p,q}} and derivatives of the Riemann zeta function at integer points. The leading coefficient of

det(s(s + 2lα) + Δ_{χ,p,q})

at s = 0 is

(2lα)^{h_{p,q}(χ)} det(Δ′_{χ,p,q}) .

Again, the contributions from Δ_{χ,j,q}, j ≠ p, cancel and give ±1. We obtain

a_{p} = α′_{p} ∏_{q=0}^{m−1−p} det(Δ′_{χ,p,q})^{(m−(p+q))(−1)^{q}} ∏_{q=m−p+1}^{m} det(Δ′_{χ,p,q})^{((p+q)−m)(−1)^{q−1}}

= α′′_{p} ∏_{q=0}^{m} det(Δ′_{χ,p,q})^{q(−1)^{q−1}} ,

where α′_{p} and α′′_{p} only depend on the Hodge numbers and the polynomials P_{σ_{p−1,q}}, P_{σ_{p,q}}. Let R̃_{p,χ} be the zeta function associated with the metric 2g, where g is the metric on Y. Then by (4.12) its leading coefficient at s = 0 is given by

ã_{p} = α̃′′_{p} T_{p,χ}^{2} .

But since R̃_{p,χ}(s) = R_{p,χ}(2s) we obtain

a_{p} = 2^{−n_{p}} ã_{p} = α_{p}T_{p,χ}^{2} .

The theorem follows. □ A first attempt towards a generalization of results of this kind to higher rank situations has been undertaken by Deitmar [17].

Examples

In this chapter we give a more explicit description of the singularities of the Selberg zeta functions associated with natural bundles on SY like spinors ore differential forms. For simplicity we restrict our considerations to the case of the trivial twist χ = 1. The only ingredient we have to provide in addition to Theorem 3.15 is the relation between the operators A_{Y,χ}(γ, σ), A_{d}(γ, σ) and natural defined operators on Y and X_{d} like form Laplacians and Dirac operators. We also give the complete singularity pattern for the Ruelle zeta functions associated with the trivial M-representation for real and complex hyperbolic manifolds.

5.1 The function case

We consider the Selberg zeta function Z_{S,χ}(s, σ) =: Z_{S}(s) associated to the trivial M-type σ = 1 and the trivial twist χ. Then the trivial K-type γ = 1 is an admissible lift of σ. We have A(γ, σ)^{2} = Δ_{X} − ρ^{2}, where Δ_{X} is the Laplace Beltrami operator of X. Hence the spectral singularities of Z_{S}(s) are given by the eigenvalues of the Laplace-Beltrami operator Δ_{Y} of Y. On the dual side we have A_{d}(γ, σ)^{2} = Δ_{X_{d}} + ρ^{2}. From Theorem 3.15 we immediately obtain

Corollary 5.1 For n even the singularities of Z_{S}(s) are at

• λ ∈ (−ρ, ρ] \ {0}, ρ^{2} − λ^{2} ∈ spec(Δ_{Y}) ,of order dim ker(Δ_{Y} − ρ^{2} + λ^{2}),

• at 0 of order 2 dim ker(Δ_{Y} − ρ^{2}),

• 0 ≠ ±iλ, ρ^{2} + λ^{2} ∈ spec(Δ_{Y}), of order dim ker(Δ_{Y} − ρ^{2} − λ^{2}),

• λ = −ρ − T k, k = 1, 2, 3, …, of order −2\frac{χ(Y)}{χ(X_{d})} dim ker(Δ_{X_{d}} + ρ^{2} − λ^{2}),

• −ρ of order 1 − 2\frac{χ(Y)}{χ(X_{d})}.

For odd n the singularities of Z_{S}(s) are at

• λ ∈ (−ρ, ρ] \ {0}, ρ^{2} − λ^{2} ∈ spec(Δ_{Y}) of order dim ker(Δ_{Y} − ρ^{2} + λ^{2}),

• at 0 of order 2 dim ker(Δ_{Y} − ρ^{2}),

• 0 ≠ ±iλ, ρ^{2} + λ^{2} ∈ spec(Δ_{Y}), of order dim ker(Δ_{Y} − ρ^{2} − λ^{2}),

This result was first obtained by Gangolli [24]. In [9] we gave a derivation of this result employing explicit solution formulas of the wave equation on X. The determinant formula Theorem 3.19 specializes as follows.

Corollary 5.2 For even n we have

Z_{S}(s) = det(s^{2} + Δ_{Y} − ρ^{2})(det(Δ_{X_{d}} + ρ^{2} − s^{2}))^{− \frac{χ(Y )}{χ(X_{d})}}

exp (−\frac{χ(Y)}{χ(X_{d})} \frac{π}{T} ∫_{0}^{s} P_{1}(t) \{ \begin{matrix}tan(πt/T) & X_{d} ≠ P ℂ^{n/2}, n/2even \\ −cot(πt/T) & X_{d} = P ℂ^{n/2}, n/2even \end{matrix} \} dt) .

For odd n the follwing formula holds:

Z_{S}(s) = det(s^{2} + Δ_{Y} − ρ^{2}) exp (\frac{πvol(Y)}{T vol(X_{d})} ∫_{0}^{s} P_{1}(t)dt) .

5.2 Spinors on real hyperbolic manifolds

In order to consider the Dirac operator we take G = Spin(n, 1), K = Spin(n) and M = Spin(n − 1). If Γ ⊂ G is a discrete co-compact subgroup without elliptic elements, then Y = Γ\G/K is a closed spin manifold. We consider the Selberg zeta function Z_{S,1}(s, S_{M}) =: Z_{S}(s, S_{M}) associated to the spinor module S_{M} (n even) or S_{M} := S_{M}^{+} (n odd) of M. Let n be even. The spinor module S_{K} of K is reducible and splits S_{K} = S_{K}^{+} ⊕ S_{K}^{−}. We have i^{∗}(S_{K}^{±}) = S_{M}. It is known that discrete series representations of G can not occur in L^{2}(X, V (S_{K}^{±})) (see e.g. Seifarth [71]). Hence S_{K}^{±} are S_{M}-admissible. Let D_{±}^{2} be the restrictions of D^{2} to C^{∞}(X, V (S_{K}^{±})). Then by the Parthasarathy formula (1.11) we obtain D_{±}^{2} = A^{2}(S_{K}^{±}, S_{M}). Let D_{Y} be the Dirac operator on Y. Note that the non-zero spectrum of D_{Y} is real and symmetric. The Hirzebruch proportionality principle implies index(D_{Y,+}) = 0. On the dual side let D_{X_{d}} be the Dirac operator of S^{n}. From Theorem 3.15 and the computation of the spectrum of D_{X_{d}} by Sulanke [75] we obtain Corollary 5.3 For even n the singularities of Z_{S}(s, S_{M}) are at

• iλ ∈ iℝ, λ ∈ spec(D_{Y}), of order dim ker(D_{Y} − λ),

• −(\frac{n}{2} + j)T, j = 0, 1, 2, …, of order −χ(Y) ( \begin{matrix}n − 1 − j \\ j \end{matrix} ) 2^{n/2−1}.

If n is odd, then S_{K} is irreducible and i^{∗}(S_{K}) = S_{M}^{+} ⊕S_{M}^{−}. Here wS_{M}^{+} = S_{M}^{−} and we are in case (b). The lifts of S_{M}^{+} are γ = γ^{s} = S_{K}. Let D_{Y} := D_{Y,1}(S_{M}) be the Dirac operator on Y with the Clifford module structure on S_{K} distinguished in Subsection 1.1.3.

Corollary 5.4 For odd n the singularities of Z_{S}(s, S_{M}) are at iλ ∈ iℝ, λ ∈ spec(D_{Y}), of order dim ker(D_{Y} − λ).

From Theorem 3.18 we obtain

S_{S}(0, S_{M}^{+}) = lim_{s→0} \frac{Z_{S}(s, S_{M}^{+})}{Z_{S}(s, S_{M}^{−})} = e^{πiη(D_{Y})} .

5.3 Differential forms on real and complex hyperbolic manifolds

We first consider the Selberg zeta function Z_{S}(s, σ_{p}) := Z_{S,1}(s, σ_{p}), p = 0, …, [\frac{n−1}{2}] for n-dimensional real hyperbolic manifolds Y, where σ_{p} is the p^{th} exterior power of the standard representation of SO(n − 1). σ_{p} is irreducible unless n is odd and p = (n − 1)/2. In this case we have a splitting into two irreducible components σ_{\frac{n−1}{2}} = σ^{+}⊕σ^{−}, where in the notation of Subsection 1.1.2 σ^{±} have highest weight (1, 1, …, 1, ±1). So we are also interested in the zeta functions Z_{S}(s, σ^{±}). We have (compare Subsections 1.1.2 and 4.2.1)

σ_{p} = i^{∗} (∑_{k=0}^{p}(−1)^{k}λ_{p−k}) ,

where λ_{p} is the p^{th} exterior power of the standard representation of SO(n), and

c(σ_{p}) = (\frac{n − 1}{2} − p)^{2}T^{2} .

Now we express the singularities of Z_{S}(s, σ_{p}) in terms of the form Laplacians Δ_{p} and Δ_{d,p} on Y and X_{d} = S^{n}, respectively.

Proposition 5.5 For real hyperbolic manifolds the Selberg zeta function Z_{S}(s, σ_{p}), p < n/2, has the following singularities:

• a zero at 0 ≠ s = iλ ∈ iℝ ∪ (−\frac{n−1−2p}{2}T, \frac{n−1−2p}{2}T) of order

dim\{Δ_{p}ω = ((λ^{2} + (\frac{n − 1 − 2p}{2}T)^{2})ω , δω = 0\} ,

• if p ≠ \frac{n−1}{2} a zero at s = 0 of order

2 dim\{Δ_{p}ω = (\frac{n − 1 − 2p}{2}T)^{2}ω , δω = 0\} ,

• a singularity at s = \frac{n−1−2p}{2}T of order

∑_{k=0}^{p}(−1)^{p−k}b_{k} ,

• a singularity at s = −\frac{n−1−2p}{2}T of order

∑_{k=0}^{n−p−1}(−1)^{k+1}b_{k} ,

if n is even, and

∑_{k=0}^{p}(−1)^{p−k}b_{k} ,

if n is odd,

(If p = \frac{n−1}{2} the latter two singularities coincide, and the orders add up.)

• if n is even, a singularity at ℝ ∋ s = λ < −\frac{n−1−2p}{2}T of order

−χ(Y) dim\{Δ_{d,p}ω = (λ^{2} − (\frac{n − 1 − 2p}{2}T)^{2})ω , δω = 0\} .

Here δ denotes the codifferential and b_{k} the k^{th} Betti number of Y.

Note that T = 1 if we normalize the metric on Y to be of sectional curvature −1. There are related results on the Selberg zeta function for differential forms on real hyperbolic manifolds by Schuster [68], Fried [21], Juhl [39], Parnovskij [54]. P r o o f . We employ Theorem 3.15. The action of the Casimir operator −Ω on k-forms on Y coincides with the form Laplacian Δ_{k}. Thus A_{Y} (λ_{k}, σ_{p})^{2} = Δ_{k} −(\frac{n−1−2p}{2}T)^{2}. The order of the singularity at s ∈ iℝ∪(−\frac{n−1−2p}{2}T, \frac{n−1−2p}{2}T] is given by

∑_{k=0}^{p}(−1)^{p−k} dim\{Δ_{k}ω = (−s^{2} + (\frac{n − 1 − 2p}{2})^{2})ω\} .

The differential d induces isomorphisms of eigenspaces corresponding to nonzero eigenvalues of different Laplacians. Taking this into account we obtain the assertion about the spectral singularities. A similar argument works for the topological singularities at s < −\frac{n−1−2p}{2}T in the even-dimensional case. The points s = ±\frac{n−1−2p}{2}T correspond to harmonic forms. Here d vanishes. Hence the result for s = \frac{n−1−2p}{2}T follows. At s = −\frac{n−1−2p}{2}T we also have the contribution of the 0^{th} Betti number of the dual space, if n is even. □ We are left with the discussion of the singularities of Z_{S}(s, σ^{±}). Since wσ^{+} = σ^{−}, case (b) applies here. It is easy to see that γ′ is equal to the spinor module s. Then γ^{s} = s ⊗ s can be identified with ∑_{k=0}^{(n−1)/2} λ_{2k} as well as with ∑_{k=0}^{(n−1)/2} λ_{2k+1}. If n ≡ 3 (mod4), then we identify D_{Y,1}(σ^{+}) with the signature operator D on odd forms, defined on C^{∞}(Y, V_{Y} (λ_{2k+1})) by

D := (−1)^{\frac{n−3}{4} +k}(∗d + d∗) ,

where ∗ is the Hodge operator. The lift γ of σ^{+} + σ^{−} is given by

∑_{k=0}^{(n−1)/2}(−1)^{\frac{n−3}{4} +k}λ_{2k+1} .

If n ≡ 1 (mod4), then we use the identification with even forms, hence

γ = ⨁_{k=0}^{(n−1)/2}(−1)^{\frac{n−1}{4} +k}λ_{2k} ,

and the operator D corresponding to D_{Y,1}(σ^{+}) is given on C^{∞}(Y, V_{Y} (λ_{2k})) by

D := (−1)^{\frac{n−1}{4} +k}i(∗d + d∗) .

For σ^{−} one has to take −D. Using the isomorphisms induced by ∗d between the eigenspaces of D in different form degrees corresponding to non-zero eigenvalues we obtain from Theorem 3.15 Proposition 5.6 The order of the singularity of the zeta function Z_{S}(s, σ^{±}) at s = iλ ≠ 0 is

dim \{ω ∈ C^{∞}(Y, V_{Y} (λ_{\frac{n−1}{2}} )) | δω = 0, \begin{matrix}± ∗ dω = λω & n ≡ 3 (mod4) \\ ± ∗ dω = ∓iλω & n ≡ 1 (mod4) \end{matrix} \} ,

while at s = 0 we have a singularity of order

∑_{k=0}^{(n−1)/2}(−1)^{\frac{n−1}{2} −k}b_{k} .

From Theorem 3.18 we recover the result of Millson [48]

S_{S}(0, σ^{+}) = lim_{s→0} \frac{Z_{S}(s, σ^{+})}{Z_{S}(s, σ^{−})} = e^{πiη(D)} .

Now we turn to the zeta functions Z_{S}(s, σ_{p,q}), p + q ≤ m − 1, for complex hyperbolic manifolds of complex dimension m. The representations σ_{p,q} and λ_{p,q} of U(m − 1) and U(m), respectively, were introduced in Subsection 4.2.2. Let Δ_{p,q}^{∘} be the restriction of Δ_{p,q} to the primitive (p, q)-forms and ω be the Kähler form of Y. By h_{p,q}^{∘} we denote the dimension of the primitive (p, q)- cohomology of Y, i.e.,

h_{p,q}^{∘} = dim(H^{p,q}(Y)/ωH^{p−1,q−1}(Y)) = dim(ker Δ_{p,q}^{∘}) .

Combining (4.8) with Lemma 4.12 and Theorem 3.15 we obtain

Proposition 5.7 For complex hyperbolic manifolds the Selberg zeta function Z_{S}(s, σ_{p,q}), p + q ≤ m − 1, has the following singularities:

• a zero at 0 ≠ s = iλ ∈ iℝ ∪ (−\frac{m−(p+q)}{2}T, \frac{m−(p+q)}{2}T) of order

dim\{Δ_{p,q}^{∘}η = (λ^{2} + (\frac{m − (p + q)}{2}T)^{2})η, δη = 0\} ,

• a zero at s = 0 of order

2 dim\{Δ_{p}^{∘}η = (\frac{m − (p + q)}{2}T)^{2}η , δη = 0\} ,

• a singularity at s = \frac{m−(p+q)}{2}T of order

∑_{k,l}(−1)^{k+l}h_{p−k,q−l}^{∘} ,

• a singularity at s = −\frac{m−(p+q)}{2}T of order

∑_{k,l}(−1)^{k+l}h_{p−k,q−l}^{∘} + (1)^{p+q−1}2χ_{a}(Y) ,

• a singularity at ℝ ∋ s = λ < −\frac{m−(p+q)}{2}T of order

−2χ_{a}(Y) dim\{Δ_{d,p}^{∘}η = (λ^{2} − (\frac{m − (p + q)}{2}T)^{2})η , δη = 0\} .

We have employed the isomorphisms induced by d of the non-zero eigenspaces of the different Laplacians on the bundles corresponding to the K-types occurring in formula 4.8. Note that if η is co-closed and primitive, then dη is primitive, too. If η is primitive, then so is δη (see Wells [89], 4.3.10.). If we normalize the metric on Y such that the sectional curvature varies between −1 and −4, then T = 2.

[figure: 0; −2T; −4T; −6T; 2T; 4T; 6T; b_{0} − b_{1}; b_{0}; b_{0} −b_{1} +b_{2}; Thm. 4.8; b_{0} − b_{1}; b_{0}; b_{0} −b_{1} +b_{2}; −1; −1; −1; −1; −1; −1; contribution of Betti numbers; contribution of an eigenvalue λ >> 0 of Δ_{1}; singularities of the Ruelle zeta function for Y = Γ\H^{7}; −1; −1; −1; −1; −1; −1; Fig. 3]

[figure: −7T; −6T; −5T; −4T; −3T; −2T; −T; T; 3T; 5T; 6χ(Y); 6χ(Y); 3χ(Y); b_{1}−b_{0}; −b_{0}; −1; −1; −1; −1; −1; −1; −1; −1; −1; −1; singularities of the Ruelle zeta function for Y = Γ\H^{6}; Fig. 4]

In principle, it is possible to combine the results of Proposition 5.5 and Proposition 5.7, respectively, for all form degrees in order to obtain the singularity pattern for the Ruelle zeta function Z_{R}(s) = Z_{R,1}(s, 1). Another possibility is to use the determinant formulas of Theorem 4.8, Proposition 4.9 and Theorem 4.13. We want to demonstrate how the singularity patterns look like with the help of figures 3-5. The vertical lines are the axes where the non-real singularities lie. We give the location and the orders of the topological singularities and the contribution of a large eigenvalue of Δ_{k}. The pictures as well as the determinant formulas show that there is a qualitative difference between the singularity patterns in the real and the complex case. It would be very interesting to see explicitly what happens in the remaining cases. For this one needs, at least, a good understanding of the sets I_{p} appearing in formula (3.7), which does not seem to be available, if p is large. The next section shows that already the Selberg zeta function corresponding to one-forms on the sphere bundle of quaternionic hyperbolic manifolds shows a rather complex behaviour.

[figure: −3T; −2T; −T; T; −2T; 3T; −4T; 6χ(Y); 3χ(Y); −b_{2}+2b_{1}−2b_{0}; b_{1}−2b_{0}; −b_{0}; −2; −1; −2; −1; singularities of the Ruelle zeta function for Y = Γ\Hℂ^{3}; Fig. 5]

5.4 One-forms on quaternionic hyperbolic manifolds

In this section we study the Selberg zeta function for quaternionic hyperbolic manifolds associated with the adjoint representation of M on the root space (𝔫_{α})_{ℂ} of the short root. In the notation of Subsection 1.1.2 (The quaternionic case) which we are going to use throughout this subsection this representation is isomorphic to qσ_{1}. Formula (1.6) shows that

qσ_{1} = i^{∗}(pλ_{1} − λ′ − 1) ,

where λ′ is the second symmetric power of p. The representation pλ_{1} is nothing else but the isotropy representation of K on T_{x}^{∗}Hℍ^{m}, hence V_{Y} (pλ_{1}) ≅ T^{∗}Y. The bundles V (λ′) and V_{Y} (λ′) can be viewed as subbundles of Λ^{2}T^{∗}Hℍ^{m} and Λ^{2}T^{∗}Y, respectively, where the fibre consists of the Kähler forms distinguished by the quaternionic Kähler structure.

The shift constant of qσ_{1} is c(qσ_{1}) = (mT)^{2}. Hence, A_{Y} (1, qσ_{1})^{2} = Δ_{0}−(mT)^{2}, where Δ_{0} is the function Laplacian, A_{Y} (pλ_{1}, qσ_{1})^{2} = Δ_{1} − (mT)^{2}, where Δ_{1} is the one-form Laplacian and A_{Y} (λ′, qσ_{1})^{2} = Δ_{2|V_{Y} (λ′)} − (mT)^{2}, where Δ_{2|V_{Y} (λ′)} is the restriction of the two-form Laplacian to V_{Y} (λ′). We have used the fact that the G-Casimir operator on differential forms coincides with the Laplacian up to the sign. In the normalization of the metric with sectional curvatures in [−4, −1] the period T is equal to 2. We define the differential operator

B : C^{∞}(Hℍ^{m}, Λ^{1}T^{∗}Hℍ^{m}) → C^{∞}(Hℍ^{m}, V (λ′))

by Bη = πdη, where π : Λ^{2}T^{∗}Hℍ^{n} → V (λ′) is the orthogonal projection. We denote the corresponding operator on Y by the same symbol. Its adjoint has the following remarkable property

Lemma 5.8 ker B^{∗} ⊂ {ω ∈ C^{∞}(Y, V_{Y} (λ′)) | Δ_{2}ω = 4(m − 1)T^{2}ω}

P r o o f . First we discuss the behaviour of B^{∗} on Hℍ^{m}. The details of the following argument can be found in [53], Kap. 5. Let H^{σ′,z} be the hyperfunction globalization of the principal series representation of G associated to the M-representation i^{∗}(λ′) and z ∈ ℂ ≅ 𝔞_{ℂ}^{∗}. We consider the Poisson transforms

P_{σ′,z}^{pλ_{1}} : H^{σ′,z} → \{η ∈ C^{∞}(Hℍ^{m}, Λ^{1}T^{∗}Hℍ^{n})|Δ_{1}η = ((\frac{2m + 1}{2}T)^{2} +2T^{2} −z^{2})η\}

and

P_{σ′,z}^{λ′} : H^{σ′,z} → \{η ∈ C^{∞}(Hℍ^{m}, V (λ′)) | Δ_{1}η = (\frac{2m + 1}{2}T)^{2} + 2T^{2} − z^{2})η\}.

We have

B^{∗}P_{σ′,z}^{λ′} = −\frac{2z + (2m − 3)T}{4m} P_{σ′,z}^{λ_{1}}.

For Re(z) ≥ 0 the Poisson transform P_{σ′,z}^{λ′} is surjective. Hence all eigenforms of Δ_{2|V (λ′)} arise as Poisson transforms. The only point with non-negative real part, where P_{σ′,z}^{λ_{1}} is not injective, is z = \frac{2m−3}{2}T. Therefore, if ω is an eigenform of Δ_{2} and B^{∗}ω = 0, then Δ_{2}ω = 4(m − 1)T^{2}ω. In particular

{ω ∈ C^{∞}(Hℍ^{m}, V (λ′)) | δω = dω = 0} = {0} .  (5.1)

Note that C^{∞}(Y, V_{Y} (λ′)) is the space of Γ-invariant sections of C^{∞}(Hℍ^{m}, V (λ′)). Any form in C^{∞}(Y, V_{Y} (λ′)) can be decomposed into eigenforms of Δ_{2}. The claim of the lemma follows. □

Proposition 5.9 The singularities of the Selberg zeta function Z_{S}(s, qσ_{1}) are

• a zero at s = iλ ∈ iℝ ∪ (−mT, mT), s ∉ {−(m − 2)T, 0, (m − 2)T } of order

dim{η ∈ C^{∞}(Y, Λ^{1}T^{∗}Y) | Δ_{1}η = (λ^{2} + (mT)^{2})η, δη = 0, Bη = 0} ,

• a zero at s = 0 of order

2 dim{η ∈ C^{∞}(M, Λ^{1}T^{∗}M) | Δ_{1}η = (λ^{2} + (mT)^{2})η, δη = 0, Bη = 0} ,

• a singularity at s = ±(m − 2)T of order

dim{η ∈ C^{∞}(M, Λ^{1}T^{∗}M) | Δ_{1}η = 4(m − 1)T^{2}η, δη = 0}
− dim{ω ∈ C^{∞}(M, V_{M}(λ′)) | Δ_{2}ω = 4(m − 1)T^{2}ω} ,

• a pole at s = mT of order b_{0}(Y) − b_{1}(Y) = 1 ,

• a zero at s = −mT of order b_{1}(Y) − b_{0}(Y) + \frac{2χ(Y)}{m+1} = \frac{2χ(Y)}{m+1} − 1 ,

• poles at s = −(m + 2)T, −(m + 3)T, −(m + 4)T … of order

2\frac{χ(Y)}{m + 1} dim \{η ∈ C^{∞}(P ℍ^{m}, Λ^{1}T^{∗}P ℍ^{m}) | \begin{matrix}Δ_{1}η = (λ^{2} − (mT)^{2})η, \\ δη = 0, Bη = 0 \end{matrix} \} .

P r o o f . We first discuss the spectral singularities at s ∈ iℝ ∪ (−mT, mT). The operator d : C^{∞}(Y) → C^{∞}(Y, Λ^{1}T^{∗}Y) induces an isomorphic embedding of the eigenspaces of A_{Y} (1, qσ_{1}) into the corresponding eigenspaces of A_{Y} (λ_{1}, qσ_{1}) for all eigenvalues λ ≠ imT. The kernel of d is the space of constant functions and corresponds to the eigenvalue imT of A_{Y} (1, qσ_{1}). The orthogonal complement of the image of d is characterized by δη = 0. By Lemma 5.8 the operator B^{∗} induces an isomorphic embedding of the eigenspaces of A_{Y} (λ′, qσ_{1}) into the corresponding eigenspaces of A_{Y} (λ_{1}, qσ_{1}) for all eigenvalues λ ≠ i(m − 2)T of A_{Y} (λ′, σ_{1}). At λ = i(m − 2)T one has to subtract

dim{ω ∈ C^{∞}(Y, V_{Y} (λ′)) | Δ_{2}ω = 4(m − 1)T^{2}ω}

by hand. Since im(B^{∗}) ⊥ im(d) in L^{2}(Y, Λ^{1}T^{∗}Y) the assertion about the spectral singularities follows easily. The point s = mT corresponds to harmonic forms. Here d fails to be injective and one has to subtract 1 = b_{0}(Y). On V_{Y} (λ′) there are no harmonic forms (see (5.1)). Thus the order of the singularity at s = mT is b_{1}(M)−1. Furthermore, combining Borel/Wallach [6], II.7.8 and VII.6.1 it follows that b_{1}(Y) = 0. Thus the Selberg zeta function has a pole of first order at s = mT. At s = −mT there is also a contribution from the spectrum of A_{d}(1, qσ_{1}). Since there are no harmonic one- and two-forms on P ℍ^{m}, we have m_{d}(mT, qσ_{1}) = −1. The order of the singularity of Z_{S}(s, qσ_{1}) at s = −mT follows. The argument for the remaining singularities on the negative real axis is similar to the one given for the spectral singularities. □

Alternative Descriptions of the Singularities

In this short last chapter we sketch without any proof two intimately related ways how the singularities of the Selberg zeta function can be described using suitable cohomology theories which have, at a first glance, nothing to do with spectral theory of elliptic operators on Y. In these theories all singularities can be explained in a uniform way. So the distinction between topological and spectral singularities which one might consider as artificial disappears. A further advantage of the approach via the Γ-cohomology of principal series representations is that the resulting formulas for the orders of the singularities (Theorem 6.3) suggest interesting generalizations to situations where Γ is not necessarily co-compact. In fact, the setting of Patterson’s original conjecture [58] was fairly general. While the idea of handling the Selberg zeta functions in such a cohomological way came out of the dynamical approach via Markov partitions (see Patterson [59]) all rigorous results obtained in this direction still rest on one form or another of the Selberg trace formula.

6.1 Dynamical Lefschetz formulas

In this section we shortly describe the approach of Juhl [40] to the theory of the Selberg zeta function. This theory can be understood as a kind of quantization of the dynamical system given by the geodesic flow on T^{∗}Y (or SY). While defined by the data of the classical system (lengths of closed geodesics, determinants of Poincaré maps) the Selberg zeta function gives the energy levels of the quantum system (eigenvalues of A_{Y,χ}(γ, σ)^{2}) by the location of its singularities (Theorem 3.15). However, according to the philosophy of geometric quantization (see, e.g., [28]) any polarization of the phase space T^{∗}Y should lead to the same quantum numbers. Polarization means that one has to take into account for quantization only functions or sections which depend on a half of the variables of the phase space. The approach chosen in the present book can be interpreted as quantization by means of functions/sections on the configuration space Y. However, there is another natural polarization of SY provided by the Anosov property of the geodesic flow. The stable bundle T^{s}SY (see formula (3.5)) is integrable. Thus one could also consider all functions/sections which are constant along its integral manifolds, the stable leaves. However, since these leaves are dense in SY, such functions do not exist, and one is led to consider the higher cohomology groups of the following complex

0 → C^{∞}(SY, V_{χ}(σ)) ⟶^{d_{s}} C^{∞}(SY, V_{χ}(σ) ⊗ Λ^{1}(T^{s}Y)^{∗}) ⟶^{d_{s}} … (6.1)

… ⟶^{d_{s}} C^{∞}(SY, V_{χ}(σ) ⊗ Λ^{n−1}(T^{s}Y)^{∗}) → 0 ,

where V_{χ}(σ) has the same meaning as in Subsection 3.1, and d_{s} is the exterior differential differentiating only along stable directions. In the following we consider only the trivial twist χ = 1. There are many analytical difficulties which obstruct a direct investigation of (6.1). For instance, it is still not known whether d_{s} has closed range. (However, if one replace C^{∞} by C^{ω}, then it has closed range as shown in [10].) The way out proposed by Juhl is to decompose L^{2}(Γ\G) into a Hilbert sum of irreducible unitary representations π of G with finite multiplicities N_{Γ}(π)

L^{2}(Γ\G) ≅ ⨁_{π∈Ĝ} N_{Γ}(π)π ,

and to consider the K- finite vectors for each summand of this decomposition seperatly. The subcomplex of (6.1) corresponding to the K-finite vectors of a single summand is isomorphic to

0 → [V_{π,K} ⊗ V_{σ}]^{M} ⟶^{d𝔫̄} [V_{π,K} ⊗ V_{σ} ⊗ Λ^{1}(𝔫̄)^{∗}]^{M} ⟶^{d𝔫̄} …  (6.2)

… ⟶^{d𝔫̄} [V_{π,K} ⊗ V_{σ} ⊗ Λ^{n−1}(𝔫̄)^{∗}]^{M} → 0 ,

where V_{π,K} is the space of K-finite vectors in the representation space of π, and V_{σ} is the representation space of σ. (6.2) is nothing else but the complex of M-invariants of the tensor product of V_{σ} with the standard Lie algebra cohomology complex of 𝔫̄ with values in V_{π,K}. Hence its cohomology groups are isomorphic to

[H^{∗}(𝔫̄, V_{π,K}) ⊗ V_{σ}]^{M} .

The cohomology groups H^{∗}(𝔫̄, V_{π,K}) are known to be finite-dimensional MAmodules ([12]). The bridge to the theory of the Selberg zeta function is provided by the following variant of the Selberg trace formula (Juhl [40], Satz 4.1.1) which is called the dynamical Lefschetz formula because of its analogy with Lefschetz fixed point formulas in the sense of Atiyah/Bott:

Proposition 6.1 Let H ∈ 𝔞^{+} be the unit vector. For suitable test functions ϕ ∈ C^{∞}(A^{+}) we have in the notation of Subsection 3.1

∑_{p=0}^{n−1}(−1)^{p} ∑_{π∈Ĝ} N_{Γ}(π) ∫_{A^{+}} ϕ(a)tr(a, [H^{p}(𝔫̄, V_{π,K}) ⊗ V_{σ}]^{M}) da

= (−1)^{n−1} ∑_{c closed geodesic} \frac{l′(c) tr μ_{1,σ}(c)}{det(1 − P_{c}^{s})} ϕ(exp(l(c)H)) ,

where l′(c) is the length of the corresponding prime geodesic.

This formula can be obtained if one combines the Selberg trace formula in representation theoretic form as given in Wallach [86] with a theorem due to Hecht/Schmid [31], also known as the Osborne conjecture, which expresses the value of the character of a unitary representation V_{π} of G at g ∈ MA^{+} by the traces of g on H^{∗}(𝔫̄, V_{π,K}). For details of the proof of Proposition 6.1 see also Schubert [67]. In [40] one can also find analogous formulas for higher rank situations. Now standard arguments give the final result ([40], Satz 7.2.1)

Theorem 6.2 Let Z_{S}(s, σ) be the Selberg zeta function associated with a representation σ of M and the trivial twist χ = 1, and H ∈ 𝔞^{+} the unit vector. For π ∈ Ĝ and λ ∈ ℂ define

χ(π, σ, λ) :=∑_{p=0}^{n−1}(−1)^{p} dim {v ∈ [H^{p}(𝔫̄, V_{π,K}) ⊗ V_{σ}]^{M} | ∃k s.th. (H − λ)^{k}v = 0} .

(6.3)

Then we have

ord_{s=λ}Z_{S}(s, σ) = (−1)^{n−1} ∑_{π∈Ĝ} N_{Γ}(π)χ(π, σ, λ) .  (6.4)

The number k appearing in formula (6.3) can be chosen to be 1 for almost all representations π ∈ Ĝ and 2 for the remaining ones. For fixed λ there are only finitely many π which contribute to the sum (6.4), namely those having the infinitesimal character determined by the pair (σ, λ). This sum should be considered as a regularization of the Euler characteristic of the λ-eigenspace of the cohomology of (6.1).

6.2 Group cohomology of the principal series

In this section we describe the singularities of the Selberg zeta function Z_{S,χ}(s, σ) in terms of the group cohomology of the principal series representations. This description was conjectured by Patterson [58] and proved in [10]. We take the trivial twist χ = ℂ. Let Z_{S}(s, σ) be the Selberg zeta function associated to the M-type σ and the trivial twist. It turns out to be convenient to consider Z_{S}(s, σ) as a function on 𝔞_{ℂ}^{∗}. We are interested in the order ord_{s=λ}Z_{S}(s, σ) of the zeta function at λ ∈ 𝔞_{ℂ}^{∗}. If Z_{S}(s, σ) is regular and nonzero at λ, then this order is zero by definition. If there is a zero of order m, then Z_{S}(s, σ) has the order m while the order of Z_{S}(s, σ) at a pole of order m is defined to be −m. Thus ord_{s=λ}Z_{S}(s, σ) is an integer parametrized by the pair (σ, λ), σ ∈ M̂, λ ∈ 𝔞_{ℂ}^{∗}. The pairs (σ, λ) are exactly the parameters of the principal series representations (π^{σ,λ}, H^{σ,λ}) of G. The Hilbert space H^{σ,λ} is given by

H^{σ,λ} := {f : G → V_{σ}|f(gman) = a^{λ−ρ}σ(m)^{−1}f(g), ∀man ∈ MAN, f_{|K} ∈ L^{2}} ,

where G acts by the left regular representation

(π^{σ,λ}(g)f)(h) := f(g^{−1}h) .

By H_{−ω}^{σ,λ} we denote the space of its hyperfunction vectors which is defined as follows. Let ({ \atop H^{σ,λ}})_{ω} be the space of analytic vectors of the dual representation of H^{σ,λ} with its natural locally convex topology. Then H_{−ω}^{σ,λ} is defined to be the topological dual of ({ \atop H^{σ,λ}})_{ω}. Restricting the action of G to Γ ⊂ G we obtain a representation of Γ. By H^{∗}(Γ, H_{−ω}^{σ,λ}) we denote the group cohomology of Γ with coefficients in H_{−ω}^{σ,λ}. At the point λ = 0 we need a slightly modified principal series representation. Since A is connected, in the definition of H^{σ,λ} we could require the invariance condition f(ga) = a^{λ−ρ}f(g) infinitesimally, i.e.,

d/dt_{|t=0} f(g exp(tH)) = (λ − ρ)(H)f(g)

for all H ∈ 𝔞. Let

Ĥ^{σ,0} := {f : G → V_{σ} | f(gmn) = σ^{−1}(m)f(g), ∀mn ∈ MN,

\frac{d^{2}}{dt^{2}}_{|t=0}f(g exp(tH)) = 0, f_{|K} ∈ L^{2}\} .

Then Ĥ^{σ,0} is an extension of H^{σ,0} with itself. As above let Ĥ_{−ω}^{σ,0} be the space of its hyperfunction vectors. Then the group cohomological description of the order of the Selberg zeta function is given by the following theorem:

Theorem 6.3 The cohomology groups H^{p}(Γ, H_{−ω}^{σ,λ}), H^{p}(Γ, Ĥ_{−ω}^{σ,0}) are finite dimensional for all p ≥ 0,

χ(Γ, H_{−ω}^{σ,λ}) = ∑_{p=0}^{∞}(−1)^{p} dim H^{p}(Γ, H_{−ω}^{σ,λ}) = 0 ,  (6.5)

χ(Γ, Ĥ_{−ω}^{σ,0}) = ∑_{p=0}^{∞}(−1)^{p} dim H^{p}(Γ, Ĥ_{−ω}^{σ,0}) = 0 ,  (6.6)

and the order of Z_{Γ}(s, σ) at s = λ ∈ 𝔞_{ℂ}^{∗} can be expressed in terms of the group cohomology of Γ with coefficients in H_{−ω}^{σ,λ}, Ĥ_{−ω}^{σ,0} as follows:

ord_{s=λ≠0}Z_{Γ}(s, σ) = − ∑_{p=0}^{∞}(−1)^{p}p dim H^{p}(Γ, H_{−ω}^{σ,λ}) ,  (6.7)

ord_{s=0}Z_{Γ}(s, σ) = − ∑_{p=0}^{∞}(−1)^{p}p dim H^{p}(Γ, Ĥ_{−ω}^{σ,0}) .  (6.8)

Previous to [10] almost nothing was known about the Γ-cohomology of such infinite dimensional representations like H_{−ω}^{σ,λ}. In order to prove the above theorem in [10] we carry out a detailed study of the properties of the group cohomology of Γ and the Lie algebra cohomology of 𝔫 with coefficients in the analytic and hyperfunction vectors of admissible representations of G. In particular we found an expression of H^{p}(Γ, H_{−ω}^{σ,λ}) in terms of the 𝔫-cohomology of admissible Harish-Chandra modules. Now Theorem 6.2 provides the bridge to the Selberg zeta function.

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Index

A, 18 ; A^{+}, 18 ; A(γ, σ), 28 ; A_{d}(γ, σ), 28 ; A_{Y,χ}(γ, σ), 28 ; b_{k}, 31 ; b_{d,k}, 31 ; C(z), 71 ; CΓ, 51 ; C_{χ}(g, σ), 55 ; ℂ_{λ}, 93 ; c(σ), 27 ; D(σ), 29 ; D_{d}(σ), 29 ; D_{Y,χ}(σ), 29 ; d, 97 ; E(t, γ, σ), 30 ; E^{s}(t, γ, σ), 50 ; E_{d}(t, γ, σ), 31 ; F (t), 62 ; G, 17 ; G(λ), 77 ; G_{d}, 18 ; I_{ϕ}^{s}, 50 ; I_{p}, 93 ; I_{ϕ}, 50 ; i^{∗}, 19 ; K, 17 ; L(λ), 77 ; L(σ), 46 ; L_{S,χ}(s, σ), 92 L_{S,χ}^{s}(s, σ), 92

l(g), 55 M, 18 M(t), 81 M̂, 19, 27 m_{d}(s, γ, σ), 30 m_{χ}(s, γ, σ), 30 m_{χ}^{s}(s, σ), 30 N, 18 n_{Γ}(g), 51 P_{σ}, 46 Q_{σ}, 46 q, 73 R, 77 R(K), 19 R(M), 19 R(s^{2}), 96 R_{p,χ}(s), 132 r (number of cusps), 73 S, 29 S_{χ}^{s}(s, σ), 92 S_{χ}(s, σ), 92 S(z), 71 SX, 18 SY, 18 s^{±}, 20 T, 46 T_{χ}, 130 T_{p,χ}, 136 V (t), 77 V_{χ}(σ), 89 V_{d}(γ), 27

V (γ), 27 ; V_{Y,χ}(γ), 27 ; W, 18 ; X, 17 ; X_{d}, 18 ; Y, 17 ; Z_{R,χ}(s, σ), 90 ; Z_{R}(s), 126 ; Z_{S,χ}(s, σ), 91 ; Z_{R,χ}(s), 131 ; 𝔞, 18 ; 𝔞^{+}, 18 ; 𝔤, 18 ; 𝔤_{d}, 18 ; 𝔨, 18 ; 𝔪, 18 ; 𝔫, 18 ; 𝔭, 18 ; 𝔭_{d}, 18 ; Γ, 17 ; γ, 27 ; γ′, 28 ; γ^{s}, 28 ; Δ_{χ,k}, 130 ; ϵ_{σ}, 46 ; η(D), 70 ; θ_{χ}(t, σ), 64 ; θ_{d}(t, σ), 64 ; θ_{Y} (t), 72 ; θ_{p}(t), 71 ; θ_{s}(t), 72 ; θ_{χ}^{s}(t, σ), 64 ; θ̃_{Y} (t), 75 ; θ̃_{p}(t), 73 ; θ̃_{s}(t), 75 ; μ_{χ,σ}(c), 90 ; ρ, 18 ; ρ_{𝔪}, 27 ; Σ, 71 σ, 18, 27

σ-admissible, 48 Φ^{+}(𝔤, 𝔞), 18 Φ^{+}(𝔤_{ℂ}, 𝔥), 39 χ_{a}(Y), 135 (χ, V_{χ}), 27 Ψ, 96 Ω, 28

analytic torsion, 130

Cartan-Helgason Theorem, 39 Casimir, 28 Clifford multiplication, 29 compact dual space, 18

Dirac operator, 29 dual theta function, 64 dynamical Lefschetz formula, 155

eta invariant, 70

geodesic flow, 89

Hadamard’s construction, 31 holomorphic torsion, 136 hyperbolic contribution, 50, 73

identity contribution, 50, 73 Iwasawa decomposition, 18

Mini Trace Formula, 74 multiplicities, 30

Ruelle zeta function, 90

determinant representation, 129,

131, 135 functional equation, 126 singularities, 148

scattering determinant, 71 scattering matrix, 71 Selberg zeta function, 91

determinant representation, 112 functional equation, 110 logarithmic derivative, 92 spectral singularities, 108 topological singularities, 108

super theta function, 64 super zeta function, 92

theta function, 64, 72 functional equation, 65 singularities, 65

trace formula, 49, 50, 73

wave equation, 30

zeta regularized determinants, 118