Geometric Zeta Functions, L^{2}-Theory, and Compact Shimura Manifolds

Anton Deitmar

Contents

1 Determinants, L^{2}-Determinants 8

1.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 Ray-Singer Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 Fredholm Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.3 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 L^{2}-determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Von Neumann Dimension and Trace . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 The Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 The L^{2}-Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Convergence of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Residually Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 The Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.3 Convergence of Characteristic Functions . . . . . . . . . . . . . . . . . . . 18 1.3.4 Convergence of the Determinant . . . . . . . . . . . . . . . . . . . . . . . 21

2 Torsion and Euler Characteristic 24

2.1 Classical Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.1 General Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.2 Combinatorial and Analytic Torsion . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 Holomorphic Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Higher Torsion and Euler Characteristics . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Determinants of Laplacians . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 Twists and Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.3 Higher Euler Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.4 Lie Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 L^{2}-Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 The Geometry of Shimura Manifolds 38

3.1 The Geometry of Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 Geodesics on the Universal Covering . . . . . . . . . . . . . . . . . . . . . 38

3.1.2 Geodesics ”Downstairs” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Euler Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 On Higher Euler Characteristics . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.2 The Fundamental Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.3 The Pseudocuspform Condition . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Haar Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Geometric Zeta Functions in the Rank One Case 46 4.1 The Zeta Functions of Selberg and Ruelle . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Rank One Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 K-separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 Selberg’s Zeta for Even Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 The Trace Formula for M-Types . . . . . . . . . . . . . . . . . . . . . . . 50 4.3.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Ruelle’s Zeta for Even Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.5 Selberg’s Zeta Function for Odd Dimensions . . . . . . . . . . . . . . . . . . . . . 61 4.6 Ruelle’s Zeta for Odd Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 The Holomorphic Torsion Zeta Function 63 5.1 The trace of the heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3 The Rank One Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.4 The Generalized Selberg Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 The generalized Selberg zeta function . . . . . . . . . . . . . . . . . . . . 85 5.4.2 The generalized Ruelle Zeta Function . . . . . . . . . . . . . . . . . . . . 88

5.5 The holomorphic torsion zeta function . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Higher Fundamental Rank 91 6.1 ”Good” Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 The Heat Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.3 Decomposition of The Lifted Heat Operator . . . . . . . . . . . . . . . . . . . . . 95 6.4 The Case of Odd Fundamental Rank . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5 The Generalized Selberg Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Dynamical Systems 102

7.1 Juhl’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.1.1 The Hecht-Schmid Character Formula . . . . . . . . . . . . . . . . . . . . 103 7.1.2 Nice Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.1.3 The Lefschetz Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 The Guillemin Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8 The Functional Equation of Theta Series 107

8.1 The Rank One Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2 Higher Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Introduction

Zeta functions encoding geometric information such as zeta functions of algebraic varieties over finite fields or zeta functions of finite graphs will loosely be called geometric zeta functions in the sequel. Sometimes the geometric situation gives one tools at hand to prove analytical continuation, functional equation and an adapted form of the Riemann hypothesis. This works as follows: The geometrical data is related to the fixed point set of an operator, this fixed point set is considered as the local data. There is a Lefschetz formula relating these to the action of this operator on a global object such as a suitable cohomology theory, thus expressing the zeta function by determinants of the operator. This is what we will call a determinant formula.

This could have been the guiding idea when Selberg in 1956 [Sel] first defined a geometric zeta function for compact Riemannian surfaces by now known as the Selberg zeta function. To play the rôle of a Lefschetz formula Selberg designed his by now famous trace formula which soon was generalized to higher dimensional Shimura manifolds, i.e. quotients of symmetric spaces. The generalization of the zeta function took somewhat more time.

In 1977 R. Gangolli [Gang] defined a zeta functions for all compact rank one Shimura manifolds. In 1985 M. Wakayama defined twisted versions of these [Wak].

Up to this point all the work was formulated in terms of the harmonic analysis not obeying the above philosophy and there was no such a thing as a determinant formula which would have given a more comprehensive way to formulate results and deeper insights into the connection between zeta functions and the topology of Shimura manifolds. Since the generalized Selberg zeta functions have infinitely many zeroes it was clear that a determinant formula would require a notion of a determinant of infinite rank operators. The latter was provided by the Ray-Singer regularized determinant [RS-AT]. In 1986 D. Fried [Fr] proved a determinant formula for the real hyperbolic spaces which are special symmetric spaces of rank one. Fried further showed that certain products of the generalized Selberg functions in these cases give the Ruelle zeta function of the geodesic flow. D. Ruelle had defined the zeta function carrying his name ten years earlier in a more general context [Rue] where he was able to show the existence of a meromorphic continuation but not very much more than that. Fried’s work however gave a determinant formula and in that way a functional equation and the generalized Riemann hypothesis. Fried showed that the special value at the center of the functional equation is given by a topological invariant: the analytical torsion.

The extension to higher rank seemed impenetrable until in 1991 H. Moscovici and R. Stanton [MS-2] found a way to circumvent the analysis of orbital integrals in that they used higher supersymmetry arguments to provoke a cancellation in the terms of higher rank. Their approach produced a kind of a Ruelle zeta function which still had analytic torsion as a special value. Unfortunately this only worked for the symmetric spaces attached to SL_{3}(ℝ) or SO(p, q) with pq odd.

In [D-Hitors] the author was able to extend their results to all Shimura manifolds of odd dimension, defining higher twisted versions of the analytic torsion which showed up as special values there. The present work will extend this to all Shimura manifolds and also give various other types of zeta functions in the higher rank situation. Apparently in even dimensions the invariants to expect are not the analytic (resp. holomorphic) torsion numbers but quotients of those and their analogues in the L^{2}-theory of Atiyah [At]. The term L^{2}-theory here means that in constructing the invariants the usual trace is replaced by a certain ”canonical trace”. The invariants thus arising are called L^{2}-invariants.

Further sources for Selberg and Ruelle zeta functions are [BuOl], [Fr2], [Fr3], [Fr4], [Ju], [MS-1], [Wak2], [Wak3], [Wak4]. For related other zeta functions read [Hash1], [Hash2], [Hash3].

The first Chapter contains an introduction to the theory of Ray-Singer determinants, their L^{2}-analogues and convergence theorems relating one to the other. In general it is an open question whether Ray-Singer determinants in a tower of coverings do converge to their L^{2} analogues, but this can be shown for determinant functions which also might be called characteristic functions.

The second chapter introduces the reader to the classical, twisted and L^{2}-torsion. In the third chapter one finds some folklore concerning Shimura manifolds.

Chapter 4 contains a treatment of the two kinds of zeta functions in the rank one case. They are expressed in a ”complete” determinant formula which also contains the ”factor at infinity”.

In chapter 5 one finds a definition of a zeta function for holomorphic Shimura manifolds that has the holomorphic torsion as special value. Of particular neatness is the case where the symmetric space is a power H^{m}, m ∈ ℕ of the hyperbolic plane H, in which case the Euler product is strikingly simple (sec. 5.2).

Holomorphic Shimura manifolds come from symmetric spaces of fundamental rank zero. Chapter 6 deals with spaces of higher fundamental rank. In section 6.4 we give the appropriate Ruelle zeta function and in 6.5 we find a generalization of the Selberg function to this setting.

In chapter 7 we present a different approach to the geometric zeta functions in the rank one case which gives rise to some interesting speculations about possible generalizations.

In the 8^{th} chapter we give the analytic continuation of theta functions which are somewhat dual objects to the zeta functions in the following way: The geometrical zeta functions considered here are defined by geometrical data (closed geodesics) and appear to have their singularities (zeroes and poles) in spectral points (Laplace eigenvalues). The theta functions in turn are defined by the very same spectral data and happen to have their singularities in the geometrical data. The duality carries on in the proof that we give in that the functional equation of the theta function is not gotten from the functional equation of the zeta function but from its Euler product.

All Shimura manifolds considered here are compact. There is, however also a Selberg zeta function in the noncompact case. The extension of the contents of this paper to the noncompact case will be the subject of future work.

I thank U. Bunke, A. Juhl, M.Olbrich and K. Takase for valuable comments. This paper was written during my stay at the MSRI at Berkeley in the special year on automorphic forms 1994/95. I thank the organizers for inviting me and all people there for generating a concentrated working atmosphere. Further I thank the staff for their continuous support and for giving me an office with a terrific view to the San Francisco Bay.

I finally thank the Deutsche Forschungsgemeinschaft for financing this research.

Notation

We will write ℕ, ℤ, ℚ, ℝ and ℂ for the natural, integer, rational, real and complex numbers.

By a virtual vector space we understand a formal difference V = V_{+} − V_{−} of vector spaces. If the components are finite dimensional, we define the dimension of V to be dim(V) = dim(V_{+}) − dim(V_{−}). An endomorphism A of virtual a virtual vector space V is a formal difference A = A^{+} − A^{−} where A^{±} is an endomorphism of V_{±}. We then define the trace and the determinant of A as tr(A) = tr(A^{+}) − tr(A^{−}) and det(A) = det(A^{+})/det(A^{−}).

When V = ⊕_{k∈ℤ}V_{k} is a ℤ-graded vector space we will automatically consider it as a virtual vector space

V_{∗} = V_{even} − V_{odd} = ⊕_{keven}V_{k} − ⊕_{kodd}V_{k}

and if both are finite dimensional we define

dim(V_{∗}) = dim(V_{even}) − dim(V_{odd}).

For any Hilbert H space we denote by B(H) the algebra of bounded linear operators on H. When we speak of an operator on a Hilbert H space this will usually mean a densely defined possibly unbounded linear operator on H.

For a smooth vector bundle E over some smooth manifold we denote by C^{∞}(E) the space of smooth sections of E and if E is defined over a Riemannian manifold and equipped with a metric we denote by L^{2}(E) the Hilbert space of measurable sqare integrable sections of E modulo the space of measurable sections vanishing outside a set of measure zero.

Let A be an abelian category. A complex over A is a sequence … → E_{−1} → E_{0} → E_{1} → …, where the E_{j} are objects of A and the differentials d_{j} : E_{j} → E_{j+1} satisfy d_{j}d_{j+1} = 0. A complex is said to be positive, if E_{−j} = 0 for j ∈ ℕ and bounded from below, if there is a j_{0} ∈ ℕ such that E_{−j} = 0 for j ≥ j_{0}. A complex E is said to be finite, if there are a, b ∈ ℤ, a ≤ b such that E_{j} = 0 for j < a or j > b. The number b − a + 1 is then called the length of the complex E.

Let A denote an abelian category. The Grothendieck group G of A is the abelian group generated by the set of isomorphism classes of objects in A, subject to the relations [A] − [B] − [C] = 0 for every exact sequence 0 → B → A → C → 0 in A. Every element x in G can be written as x = x_{+} − x_{−}, where x_{±} ∈ A.

Lie algebras of real Lie groups G, H, K, … are denoted by the corresponding small case german letters with index zero: 𝔤_{0}, 𝔥_{0}, 𝔨_{0} and their complexifications are denoted without any index, so 𝔤 = 𝔤_{0} ⊗ ℂ.

For any locally compact group G we write Ĝ for the unitary dual, i.e.: Ĝ is the set of equivalence classes of irreducible unitary representations of G.

Chapter 1

Determinants, L^{2}-Determinants

In this chapter we give an introduction to the classical theory of Ray-Singer determinants and the not so classical L^{2}-determinants. We give a comparison of the Ray-Singer determinant to the seemingly more direct Fredholm determinant which a fortiori shows that they amount to the same. We then consider the L^{2}-theoretical counterpart in the case of positive Gromov-Novikov-Shubin invariants. We finally show that in a tower of coverings the determinant functions converge to the corresponding L^{2}-determinant functions.

1.1 Determinants

1.1.1 Ray-Singer Determinants

To extend the notion of a determinant to infinite dimensions we follow the regularizing procedure by Ray and Singer [RS-RM]. So let H be a Hilbert space and consider a positive operator A, densely defined on H. The operator A is called admissible, if there is a natural number n such that A^{−n} is of trace class, and the zeta function of A:

ζ_{A}(s) =^{def} tr(A^{−s}), Re(s) > n,

admits an analytic continuation to some function holomorphic at s = 0. For an admissible operator A we define its determinant by

det(A) =^{def} exp(−\frac{d}{ds} |_{s=0} ζ_{A}(s)).

Example 1.1.1 On l^{2}(ℕ) with the canonical basis (e_{j})_{j∈ℕ} define an operator A by A(e_{j}) = je_{j}. Then A^{−2} is if trace class and ζ_{A}(s) = ζ(s), the Riemann zeta function. We have ζ′(0) = −\frac{1}{2}log(2π) and thus

det(A) = \sqrt{2π}.

CHAPTER 1 Determinants, L^{2}-Determinants 9

Since \overline{ζ_{A}(s̄)} = ζ_{A}(s) the determinant is real and positive and there are some easy rules which we list as follows:

• det(A ⊕ B) = det(A)det(B) if both are admissible.

• det(cA) = c^{ζ_{A}(0)}det(A) for c > 0.

• det(A^{s}) = det(A)^{s} for s > 0.

If the operator A has a nontrivial kernel we will write det(A) = 0, but if A′ = A |_{(ker(A))^{⊥}} is admissible we will call A admissible as well and define:

det′(A) =^{def} det(A′).

Proposition 1.1.1 A positive elliptic differential operator D defined on a hermitian vector bundle over a compact smooth Riemannian manifold is admissible and the determinant is independent of the choice of the metrics.

Proof: See ([Gilk], Lemma 1.10.1) for the admissibility, for the independence of metrics recall that det(D) only depends on the eigenvalues of D. ■

Since the determinant of a finite dimensional operator is the product of the eigenvalues, the regularized determinant may give the definition of a regularized product . So let (a_{j})_{j∈ℕ} be a sequence of real numbers tending to +∞. The sequence (a_{j})_{j∈ℕ} is called admissible, if the operator D, which has the a_{j} as eigenvalues, is admissible. Under these circumstances we define the regularized product as

∏_{j∈ℕ}a_{j} =^{def} det(D).

The above example thus gives

∏_{n∈ℕ}n = \sqrt{2π}.

1.1.2 Fredholm Determinants

There is also a more direct way to define determinants of operators on infinite dimensional spaces, namely the Fredholm determinant, which is defined as follows: Let T be a trace class operator, then we define

det_{F r}(1 + T) =^{def} ∑_{k=0}^{∞} tr ∧^{k} T.

The sum is absolutely convergent. Moreover, assume T normal with eigenvalues (λ_{n})_{n∈ℕ} then it follows that the infinite product

∏_{n∈ℕ}(1 + λ_{n})

is convergent and equals the Fredholm determinant of 1 + T. If the operator norm ∥ T ∥ of T is less than 1 we also have

det_{F r}(1 + T) = exp(− ∑_{n=1}^{∞} \frac{(−1)^{n}}{n} trT^{n}).

The connection between Fredholm determinant and zeta regularized determinant is:

Proposition 1.1.2 Let A and A + 1 be admissible with ker(A) = 0. Assume A^{ϵ−1} is of trace class for some ϵ > 0, then

det_{F r}(1 + A^{−1}) = \frac{det(A + 1)}{det(A)}.

Proof: Since the claim is certainly true for finite dimensions we can divide both sides by the factors stemming from the finitely many eigenvalues of A which are less than 1 in value. This means, we may assume ∥ A^{−1} ∥< 1 so that we have at hand the exponential series description of the Fredholm determinant. Let (λ_{n}) denote the eigenvalues of T then we have:

\frac{∂}{∂s}ζ_{A+1}(s) = − ∑_{n=1}^{∞} log(λ_{n} + 1)(λ_{n} + 1)^{−s}

= − ∑_{n=1}^{∞} log(λ_{n})(λ_{n} + 1)^{−s} − ∑_{n=1}^{∞} log(1 + \frac{1}{λ_{n}})(λ_{n} + 1)^{−s}.

The second term converges to −logdet_{F r}(1 + A^{−1}) as s → 0. The first equals

− ∑_{n=1}^{∞} log(λ_{n})λ_{n}^{−s} − ∑_{n=1}^{∞} log(λ_{n})λ_{n}^{−s}((1 + \frac{1}{λ_{n}})^{−s} − 1),

in which the first summand is \frac{∂}{∂s}ζ_{A}(s). Let f_{s}(x) = (1+x)^{−s}−1 then the Taylor series expansion of f_{s} around x = 0 is

f_{s}(x) = ∑_{n=1}^{∞} \frac{(−1)^{n}}{n!} s(s + 1) … (s + n − 1)x^{n},

so f_{s}(x) = sxh(s, x) where h is differentiable around (x, s) = (0, 0), so we have |f_{s}(x)| ≤ c_{1}sx for some c > 0 thus

|∑_{n=1}^{∞} log(λ_{n})λ_{n}^{−s}((1 + \frac{1}{λ_{n}})^{−s} − 1)| ≤ c_{2}s ∑_{n=1}^{∞} log(λ_{n})λ_{n}^{−s−1}

≤ c_{3}s ∑_{n=1}^{∞} λ_{n}^{ϵ−s−1}

so this contribution vanishes at s = 0. ■

1.1.3 Characteristic Functions

We will from now on restrict our attention to generalized Laplace operators. Much of what we are going to say will hold in a more general context but our restriction gives us some serious technical simplifications.

Let E denote a smooth vector bundle over a smooth Riemannian manifold M. A generalized Laplacian on E is a second order differential operator D such that for the principal symbol of D we have

σ_{2}(x, ξ) =∥ ξ ∥^{2} Id.

Furthermore we insist that D is semipositive, i.e. D ≥ 0.

It then follows that there is a connection ∇^{E} on E such that D equals the connection Laplacian up to an operator of order zero ([BGV], Prop. 2.5). It further follows that the heat operator e^{−tD}, t > 0 is of trace class on L^{2}(E) and that

tr(e^{−tD}) ∼ ∑_{k=0}^{∞} c_{α_{k}}t^{α_{k}},

as t → ∞, where α_{k} ∈ ℝ, α_{k} → ∞.

Proposition 1.1.3 Let D denote a generalized Laplacian over a compact smooth Riemannian manifold M. Then the characteristic function: λ ↦ det(D + λ), λ > 0 of D extends to an entire function with zeroes at the eigenvalues of −D, the order of a zero being the multiplicity of the eigenvalue. For λ → ∞ we have the following asymptotics:

\begin{aligned}−logdet(A + λ) & ∼ ∑_{α_{k}≠0,−1,−2,...} c_{α_{k}}Γ(α_{k})λ^{−α_{k}} \\ & − ∑_{k=0,−1,−2,...} c_{k} (logλ − ∑_{j=1}^{k} \frac{1}{j}) \frac{(−λ)^{k}}{k!} , \end{aligned}

Proof: Let M(s, λ) = Γ(s)ζ_{A+λ}(s). We have the differential equation:

\frac{∂}{∂λ}M(s, λ) = −M(s + 1, λ).

Now let m ∈ ℕ, we get

(\frac{∂}{∂λ})^{m+1}ζ_{A+λ}(s) = (−1)^{m+1}(s + m)(s + m − 1) … sζ_{A+λ}(s + m + 1),

so that for m large enough it follows

(\frac{∂}{∂λ})^{m+1}ζ_{A+λ}(0) = 0.

It is clear that we have log det(D + λ) = lim_{s→0}(M(s, λ) − ζ_{A+λ}(0))/s and the limit in s may be interchanged with the λ-derivation. It follows that for m large enough we have

\begin{aligned}(\frac{∂}{∂λ})^{m+1}log det(D + λ) & = (−1)^{m+1}M(m + 1, λ) \\ & = (−1)^{m+1}Γ(m + 1) ∑_{n=1}^{∞} \frac{1}{(λ + λ_{m})^{m+1}} . \end{aligned}

From this the claim follows. For the asymptotics see [Voros]. ■

1.2 L^{2}-determinants

1.2.1 Von Neumann Dimension and Trace

From now on we will fix the following situation: X_{Γ} will denote a smooth compact oriented Riemannian manifold with universal covering X and fundamental group Γ. We will assume Γ infinite. Over X_{Γ} we will have a smooth hermitian vector bundle E_{Γ} with pullback E over X. On E_{Γ} we fix a generalized Laplacian D_{Γ} and we denote its pullback to E by D.

Let L^{2}(Γ) denote the L^{2}-space over Γ with respect to the counting measure. The group Γ acts on L^{2}(Γ) by left translations. Inside the algebra B(L^{2}(Γ)) of all bounded linear operators on the Hilbert space L^{2}(Γ) we consider the von Neumann algebra of Γ, denoted V N(Γ) generated by the left translations (L_{γ})_{γ∈Γ}. There is a canonical trace on V N(Γ) defined by tr(∑_{γ} c_{γ}L_{γ}) = c_{e} which makes V N(Γ) a type II_{1} von Neumann algebra [GHJ], which is a factor if and only if every nontrivial conjugacy class in Γ is infinite.

It is easy to see that the commutant of V N(Γ) is the von Neumann algebra generated by the right translations.

Fix a fundamental domain F of the Γ-action on X. We get isomorphisms of unitary Γmodules:

L^{2}(E) ≅ L^{2}(Γ) ˆ⊗L^{2}(E|_{F}) ≅ L^{2}(Γ) ˆ⊗L^{2}(E_{Γ}).

So for the von Neumann algebra B(L^{2}(E))^{Γ} of operators commuting with the Γ-action we get

B(L^{2}(E))^{Γ} ≅ V N(Γ) ˆ⊗_{w}B(L^{2}(E_{Γ})),

where ˆ⊗_{w} means the weak closure of the algebraic tensor product, i.e. the tensor product in the category of von Neumann algebras.

Taking the canonical trace on V N(Γ) and the usual trace on B(L^{2}(E_{Γ})) we obtain a type II_{∞} trace on the algebra B(L^{2}(E))^{Γ} which we will denote by tr_{Γ}. The corresponding dimension function is denoted by dim_{Γ}. Assume for example, a Γ-invariant operator T on L^{2}(E) is given as integral operator with a smooth kernel k_{T}, then a computation shows

tr_{Γ}(T) = ∫_{F} tr(k_{T} (x, x))dx.

The reader familiar with the Selberg trace formula will immediately recognize this as the ”term of the identity”.

1.2.2 The Heat Kernel

Since the operator D is the pullback of an operator on Γ\X, it commutes with the Γ-action and so does its heat operator e^{−tD}. This heat operator e^{−tD} ∈ B(L^{2}(E))^{Γ} has a smooth kernel < x | e^{−tD} | y >. We get:

tr_{Γ}e^{−tD} = ∫_{F} tr < x | e^{−tD} | x > dx.

From this we read off that tr_{Γ}e^{−tD} satisfies the same small time asymptotics as tre^{−tD_{Γ}} [BGV].

Let D′ = D |_{(ker(D))^{⊥}}. Unfortunately very little is known about large time asymptotics of tr_{Γ}e^{−tD′} (see [LL]). Let

GNS(D) =^{def} sup{α ∈ ℝ | tr_{Γ}e^{−tD′} = O(t^{−α/2}) as t → ∞}

denote the Gromov-Novikov-Shubin invariant of D ([GrSh], [LL]). Then GNS(D) is always ≥ 0. J. Lott showed in [Lo] that the GNS-invariants of Laplacians are homotopy invariants of a manifold. J.Lott and W. Lück conjecture in [LL] that the Gromov-Novikov-Shubin invariants of Laplace operators are always positive rational or ∞.

1.2.3 The L^{2}-Zeta Function

Throughout we will assume that the Gromov-Novikov-Shubin invariant of D is positive. We consider the integral

ζ_{D_{Γ}}^{1}(s) =^{def} \frac{1}{Γ(s)} ∫_{0}^{1} t^{s−1}tr_{Γ}e^{−tD′}dt,

which converges for Re (s) >> 0 and extends to a meromorphic function on the entire plane which is holomorphic at s = 0, as is easily shown by using the small time asymptotics ([BGV],Thm 2.30).

Further the integral

ζ_{D_{Γ}(s)}^{2}(s) =^{def} \frac{1}{Γ(s)} ∫_{1}^{∞} t^{s−1}tr_{Γ}e^{−tD′}dt

converges for Re (s) < GNS(D), so in this region we define the L^{2}-zeta function of D_{Γ} as

ζ_{D_{Γ}}^{(2)}(s) =^{def} ζ_{D_{Γ}}^{1}(s) + ζ_{D_{Γ}}^{2}(s).

Assuming the Gromov-Novikov-Shubin invariant of D to be positive we define the L^{2}- determinant of D_{Γ} as

det^{(2)}(D_{Γ}) =^{def} exp(−\frac{d}{ds} |_{s=0} ζ_{D_{Γ}}^{(2)}(s)).

Proposition 1.2.1 Let D_{Γ} denote a generalized Laplacian over the manifold X_{Γ}. Then the L^{2}-characteristic function: λ ↦ det^{(2)}(D_{Γ} + λ), λ > 0, extends to a holomorphic function on ℂ\(−∞, 0].

Proof: Let λ ∈ ℂ\(−∞, 0] and s ∈ ℂ with Re (s) >> 0. Then the operator (D_{Γ} + λ)^{−s}, defined by the spectral theorem, has a continuous kernel whose diagonal, as function in S, continues to a meromorphic function in s, holomorphic at s = 0 [Sh]. So the same counts for the zeta function

ζ_{D_{Γ}+λ}^{(2)}(s) = tr_{Γ}(D_{Γ} + λ)^{−s}
= ∫_{F} tr < x | (D_{Γ} + λ)^{−s} | x > dx.

By the proof of Thm 12.1 in [Sh] it is clear that ζ_{D_{Γ}+λ}^{(2)}(s) depends holomorphically on λ. Whence the proposition. ■

1.3 Convergence of Determinants

In this section we will show that when the fundamental group is residually finite, the characteristic functions of a tower converge to the L^{2}-characteristic function.

1.3.1 Residually Finite Groups

Let Γ be an infinite group. A tower of subgroups is a sequence Γ_{1} ⊃ Γ_{2} ⊃ … of normal subgroups of Γ, each of which has finite index in Γ and the sequence satisfies ∩_{j}Γ_{j} = {1}. A group Γ that has a tower is called residually finite.

Finitely generated subgroups of GL_{n}(ℂ) are residually finite [Alp]. See [Kir] for more criteria of groups to be residually finite.

As a special example from arithmetic consider the following: Let H denote a nonsplit quaternion algebra over ℚ which splits at ∞. Consider the group schemes H^{∗} and H^{1} of units and norm one elements in H. Since via the splitting homomorphism π : H^{∗}(ℝ) → GL_{2}(ℝ) the norm becomes the determinant, H^{1}(ℝ) is mapped to SL_{2}(ℝ). Let R denote an order in H then Γ(1) = π(H^{1}(R)) is a cocompact discrete subgroup of SL_{2}(ℝ). For an ideal I in R let Γ(I) = {γ ∈ Γ(1) | γ ≡ 1 mod I} the principal congruence subgroup. Now let (I_{n})_{n∈ℕ} be a decreasing sequence of nontrivial ideals in R with ∩_{j}I_{j} = 0 then the groups Γ_{j} = Γ(I_{j}) form a tower.

1.3.2 The Convergence Theorem

Now assume Γ = π_{1}(X_{Γ}) is residually finite and fix a tower (Γ_{j})_{j∈ℕ}. For any continuous function f on ℝ we define the operator f(D) by means of the spectral theorem as follows: Let μ denote a spectral resolution of D, i.e. μ is a projection valued measure on the spectrum Spec(D) of the operator D such that on L^{2}(E) we have

D = ∫_{Spec(D)} λdμ(λ).

We then define

f(D) =^{def} ∫_{Spec(D)} f(λ)dμ(λ).

It follows

tr_{Γ}(f(D)) = ∫_{Spec(D)}(f(λ))dtr_{Γ}μ(λ),

where the integral will converge if f(D) has a sufficiently smooth kernel.

Choose n ∈ ℕ so large that (1 + D_{Γ_{j}})^{−n} is of trace class and has a Schwartz kernel which is (dim X + 1)-times continuously differentiable. Let F_{ev}^{n} denote the space of even C^{∞}-functions on ℝ which decrease more rapidly than (1 + x^{2})^{−n}, i.e. for f ∈ C^{∞}(ℝ)_{even} we have

f ∈ F_{ev}^{n} ⇔ lim_{x→∞} f(x)(1 + x^{2})^{n} = 0.

Note that the even Schwartz functions lie in F_{ev}^{n}.

Theorem 1.3.1 (Compare [Don].) For any f ∈ F_{ev}^{n} we have

\frac{trf(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} tr_{Γ}f(\sqrt{D}).

For any Schwartz function g ∈ S(ℝ) we have

\frac{trg(D_{Γ_{j}})}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} tr_{Γ}g(D).

Before proving the theorem we give some immediate applications:

Corollary 1.3.1 (Kazhdan inequality, compare [Lü]) Let h(D_{Γ_{j}}) denote the dimension of the kernel of D_{Γ_{j}} and h^{(2)}(D_{Γ}) the Γ-dimension of ker(D) then we have

lim sup_{j} \frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} ≤ h^{(2)}(D_{Γ}).

We say that D has a spectral gap at zero if there is ϵ > 0 such that SpecD∩]0, ϵ[= ∅. Let

N_{j}(x) =^{def} ∑_{λ≤x} dim Eig(D, λ).

Corollary 1.3.2 Assume there is an x > 0 such that

\frac{N_{J}(x) − N_{j}(0)}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} 0.

then D has a spectral gap at zero.

Proof of Corollary 1: Let F denote the set of all f ∈ S(ℝ) with f(0) = 1, f(x) ≥ 0 for all x ∈ ℝ, then we have

h(D_{Γ_{j}}) ≤ trf(D_{Γ_{j}})

for all f ∈ F. It follows

lim sup_{j} \frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} ≤ lim_{j} \frac{trf(D_{Γ_{j}})}{[Γ : Γ_{j}]} = tr_{Γ}f(D)

for all f ∈ F. So that

lim sup_{j} \frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} ≤ inf_{f∈F} tr_{Γ}f(D) = h^{(2)}(D_{Γ}).■

Proof of Corollary 2: Assume x > 0 as in the corollary. Let f ∈ C_{c}^{∞}(0, x) be positive, f(y) ≤ 1 for all y, then

N_{j}(x) − N_{j}(0) ≥ trf(D_{Γ_{j}}) ≥ 0.

Hence for all such f we have tr_{Γ}f(D) = 0 which gives the claim. ■

Proof of the theorem: Fix x_{0} ∈ X and v ∈ E_{x_{0}}^{∗} the dual space to E_{x_{0}}. Recall that the distribution u_{s} = cos(s\sqrt{D})v(δ_{x_{0}}) ∈ C^{∞}(E)′ satisfies the wave equation

(\frac{∂^{2}}{∂s^{2}} + D)u_{s} = 0.

By general results on hyperbolic equations ([Tayl], chap IV) it follows that cos(s\sqrt{D}) has propagation speed ≤| s |. Now for f ∈ F_{ev}^{n} the formula

f(\sqrt{D}) = \frac{1}{2π} ∫_{ℝ} f̂cos(s\sqrt{D})ds

implies that f(\sqrt{D}) has finite propagation speed if its Fourier Transform f̂ has compact support. Let P W denote the space of all f ∈ F_{ev}^{n} such that f̂ has compact support. Then P W is dense in f ∈ F_{ev}^{n} with respect to the norm

∥ f ∥= sup_{x∈ℝ} | f(x) | (1 + x^{2})^{n}.

Let f ∈ P W, we will prove the assertion of the theorem for f.

Schwartz kernels of operators on sections of the bundle E_{Γ} are distributional sections of the bundle E_{Γ} ⊗_{ext} E_{Γ}^{∗} over X_{Γ} × X_{Γ}. Where ⊗_{ext} means the exterior tensor product, i.e.: (E ⊗_{ext} F)_{(x,y)} ≅ E_{x} ⊗F_{y}. Since X_{Γ} = Γ\X, these can be identified with Γ×Γ-invariant sections of E ⊗_{ext} E^{∗}.

Lemma. 1.3.1 Modulo the above identification we have for f ∈ P W the identity of Schwartz kernels

< Γx | f(\sqrt{D_{Γ}}) | Γy >= ∑_{γ∈Γ} < x | f(\sqrt{D}) | γy > .

Proof of the lemma: Take φ ∈ C^{∞}(E_{Γ}) and identify φ with its pullback to E, which is a γ-invariant section of E. The operator cos(s\sqrt{D}) may be applied to φ in the distributional sense. Since cos(s\sqrt{D}) is Γ-invariant we obtain a Γ-invariant distribution, so cos(s\sqrt{D})φ ∈ C^{∞}(E_{Γ})′. we thus get an operator cos(s\sqrt{D}) from C^{∞}(E_{Γ}) to C^{∞}(E_{Γ})′ satisfying the same differential equation and initial value conditions as cos(s\sqrt{D_{Γ}}), hence it equals the latter. For f ∈ P W the formula f(\sqrt{D}) = \frac{1}{2π} ∫_{ℝ} f̂cos(s\sqrt{D})ds gives f(\sqrt{D}) |_{C^{∞}(E_{Γ})}= f(\sqrt{D_{Γ}}). Since f(\sqrt{D_{Γ}}) is a smoothing operator we may write

\begin{aligned}f(\sqrt{D})φ(x) & = ∫_{X} < x | f(\sqrt{D}) | y > φ(y)dy \\ & = ∑_{γ∈Γ} ∫_{F_{Γ}} < x | f(\sqrt{D}) | γy > φ(y)dy, \end{aligned}

where F_{Γ} denotes a fundamental domain of the Γ-action on X. Now the sum

∑_{γ∈Γ} < x | f(\sqrt{D}) | γy >

is locally finite and this gives the lemma. ■

To continue the proof of the theorem fix fundamental domains F_{Γ} ⊂ F_{Γ_{1}} ⊂ F_{Γ_{2}} ⊂ … in a way that there are representatives for the classes in Γ/Γ_{j} such that F_{Γ_{j}} = ∪_{σ:Γ/Γ_{j}}σF_{Γ}. We then get

\frac{trf(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} = \frac{1}{[Γ : Γ_{j}]} ∑_{γ∈Γ_{j}} ∫_{F_{Γ_{j}}} tr < x | f(\sqrt{D}) | γx > dx

= ∑_{γ∈Γ_{j}} \frac{1}{[Γ : Γ_{j}]} ∑_{σ:Γ/Γ_{j}} ∫_{F_{Γ}} tr < σx | f(\sqrt{D}) | γσx > dx

= ∑_{γ∈Γ_{j}} \frac{1}{[Γ : Γ_{j}]} ∑_{σ:Γ/Γ_{j}} ∫_{F_{Γ}} tr < x | f(\sqrt{D}) | σ^{−1}γσx > dx.

The last equality stems from the Γ-invariance of the operator f(\sqrt{D}). Since Γ_{j} is normal in Γ we end up with

\frac{trf(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} = ∑_{γ∈Γ_{j}} ∫_{F_{Γ}} tr < x | f(\sqrt{D}) | γx > dx.

As j → ∞ we get less and less summands, only the summand of γ = e remains, but this is just tr_{Γ}f(\sqrt{D}), so the claim follows for f ∈ P W.

Now let f ∈ F_{ev}^{n} be arbitrary. From Theorem 1.4 in [CGT] we get that there is a constant C > 0 such that

\frac{tr(1 + D_{Γ_{j}})^{−n}}{[Γ : Γ_{j}]} ≤ C for all j ∈ ℕ.

Let ϵ > 0. To our given f ∈ F_{ev}^{n} there is a g ∈ P W such that

sup_{x∈ℝ} | f(x) − g(x) |< \frac{ϵ}{(1 + x^{2})^{n}} .

We get

| \frac{trf(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} − tr_{Γ}f(\sqrt{D}) | ≤ | \frac{trf(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} − \frac{trg(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} |

+ | \frac{trg(\sqrt{D_{Γ_{j}}})}{[Γ : Γ_{j}]} − tr_{Γ}g(\sqrt{D}) |

+ | tr_{Γ}g(\sqrt{D}) − tr_{Γ}f(\sqrt{D}) | .

The first summand on the right hand side is less than \frac{tr((1+D_{Γ_{j}})^{−n})}{[Γ:Γ_{j}]} ≤ ϵC. The second summand tends to zero as j → ∞ and the third is less that ϵtr_{Γ}(1 + D)^{−n}. The theorem follows. ■

1.3.3 Convergence of Characteristic Functions

For later use we prove:

Lemma. 1.3.2 The function

t ↦ \frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tD_{Γ}}, t > 0,

is rapidly decreasing at zero. Moreover this holds uniformly in j, that is, there is a function f :]0, 1[→ [0, ∞[ which is rapidly decreasing at zero such that

| \frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tD_{Γ}} |≤ f(t)

for all j ∈ ℕ and all t ∈]0, 1[.

Proof: Arguing as in the proof of the preceding lemma we see that

< Γx | e^{−tD_{Γ}} | Γy >= ∑_{γ∈Γ} < x | e^{−tD} | γy >,

where convergence remains to be checked, however. To this end we write d(x, y) for the Riemann distance of the points x, y ∈ X and we use Theorem 3.1 of [CGT] to get for d(x, y) ≥ a > 0:

|< x | e^{−tD} | γy >|≤ \frac{C}{\sqrt{t}}e^{−(d(x,y)−a)^{2}/t},

where | . | here denotes the matrix L^{2}-norm, so | A |= \sqrt{trAA^{∗}}.

By Proposition 4.1 in [CGT] we conclude that the function x ↦ e^{−(d(x,y)−a)^{2}/t} is in L^{1}(X) and the L^{1}-norm is clearly bounded by a constant independent of y. From this the convergence of the sum above follows. As in the previous section we get:

\frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tD_{Γ}} = ∑_{1≠γ∈Γ_{j}} ∫_{F_{Γ}} tr < x | e^{−tD} | γx > dx.

Now fix x_{0} ∈ F_{Γ} and r > 0 such that the ball of radius r around x_{0} contains F_{Γ}, i.e. F_{Γ} ⊂ B_{r}(x_{0}). Choose R > max(r, 1) such that y ∈ F_{Γ}, x ∈ X with d(x_{0}, x) > R implies d(x, y) > r. Choose j_{0} ∈ ℕ such that for all j ≥ j_{0} and all γ ∈ Γ_{j}\{1} we have γ̄F_{Γ} ∩ B_{R}(x_{0}) = ∅. Under these circumstances we get for j ≥ j_{0}:

| \frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tD_{Γ}} | ≤ ∑_{γ∈Γ_{j}\\{1\}} \frac{C}{\sqrt{t}} ∫_{F_{Γ}} e^{−(d(x,γx)−r)^{2}/t}dx

≤ \frac{C}{\sqrt{t}} ∫_{X\B_{R}(x_{0})} e^{−(d(x,B_{r}(x_{0}))−r)^{2}/t}dx
≤ \frac{C}{\sqrt{t}} ∫_{R}^{∞} e^{−(ξ−r)^{2}/t}vol(d(x, x_{0}) = ξ)dξ
≤ \frac{C_{1}}{\sqrt{t}} ∫_{R}^{∞} e^{−(ξ−r)^{2}/t}e^{C_{2}ξ}dξ,

for some C_{1}, C_{2} > 0 by Proposition 4.1 in [CGT]. But this can be calculated as C_{1} ∫_{R/\sqrt{t}} e^{−(x−( \frac{r}{\sqrt{t}} +^{\frac{C_{2}\sqrt{T}}{2}} ))^{2}+r+ \frac{C^{2}_{2}t}{4}} dx ≤ C_{3} ∫_{R/\sqrt{t}} e^{−x^{2}}dx

≤ C_{3} ∫_{R/\sqrt{t}} xe^{−x^{2}}dx for t < 1
= \frac{C_{3}}{2} e^{−R^{2}/t}

which gives the assertion of the lemma. ■

For the following theorem we need not require the Gromov-Novikov-Shubin invariant to be positive.

Theorem 1.3.2 As j → ∞ we have

det(D_{Γ_{j}} + λ)^{\frac{1}{[Γ:Γ_{j}]}} ⟶ det^{(2)}(D_{Γ} + λ)

locally uniformly in λ ∈ ℂ\(−∞, 0].

Proof: For Re λ > 0 and s ∈ ℂ we consider

F_{j}(λ, s) = \frac{ζ_{D_{Γ_{j}} +λ}(s)}{[Γ : Γ_{j}]} − ζ_{D_{Γ}+λ}^{(2)}(s)

= \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}(\frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tDΓ})e^{−tλ}dt

which converges by the previous lemma. The formula

F_{j}(λ, s) = \frac{tr(D_{Γ_{j}} + λ)^{−s}}{Γ : Γ_{j}]} − tr_{Γ}(D + λ)^{−s} Re (s) >> 0

together with the analytic continuation of the kernel of (D_{Γ_{j}} + λ)^{−s} or (D + λ)^{−s} as in ([Sh]) shows that F_{j} extends to a holomorphic function on (ℂ\(−∞, 0]) × ℂ. As Re (λ) → ∞ we have F_{j}(λ, s) → 0 uniformly in {Re (s) < K} for all K ∈ ℝ. We want to show that this convergence is uniform in j ∈ ℕ. To this end let Re (λ) > 0 and split the integral above as ∫_{0}^{1} + ∫_{1}^{∞}. The integrand in the ∫_{0}^{1}-part obeys a uniform estimate by the previous lemma and thus the convergence is uniform in this part.

For the ∫_{1}^{∞}-part recall that there is a constant C > 0 such that \frac{tr(D_{Γ_{j}} +1)^{−n}}{[Γ:Γ_{j}]} < C, from which

it follows that \frac{tre^{−tD_{Γ_{j}}}}{[Γ:Γ_{j}]} is bounded for t ≥ 1 with a bound not depending on j. From this it follows

F_{j}(λ, s) \underrightarrow{Re (λ) → ∞} 0

locally uniformly in s and uniformly in j ∈ ℕ.

On the other hand we have a differential equation

\frac{∂}{∂λ}F_{j}(λ, s) = F_{j}(λ, s + 1).

For ϵ > 0 let G_{ϵ} denote the region

G_{ϵ} =^{def} \{z ∈ ℂ | Re (z) > −\frac{1}{ϵ}and dist(z, (−∞, 0]) > ϵ\}.

From the above it follows that there is a R > 0 such that for Re (s) > R we have

F_{j}(λ, s) \underrightarrow{j → ∞} 0 uniformly in G_{ϵ} × \{Re (s) > R\}.

We want to show that, having this convergence for Re (s) > R it already follows for Re (s) > R − 1. To achieve this let Re (s) > R − 1 and consider

F_{j}(λ, s) − F_{j}(μ, s) = ∫_{μ}^{λ} F_{j}(z, s + 1)dz,

which tends to zero as j → ∞. But on the other hand

F_{j}(λ, s) − F_{j}(μ, s) \underrightarrow{μ → ∞} F_{j}(λ, s),

as the latter convergence is uniform in j we get the claim. From this we get F_{j}(λ, s) \underrightarrow{j → ∞} 0

uniformly in a neighborhood of s = 0 and this gives the theorem. ■

1.3.4 Convergence of the Determinant

Write h(D_{Γ}) for the dimension of the kernel of D_{Γ} and h^{(2)}(D_{Γ}) for the Γ-dimension of the kernel of D. Lück showed in [Lü]:

\frac{h(△_{p,Γ_{j}})}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} h^{(2)}(△_{p,Γ})

where △_{p} is the p-th Laplacian. Unfortunately the combinatorial arguments he used do not carry over to the analytic situation. So we only can show such a convergence under additional assumptions.

Theorem 1.3.3 Suppose that there is an ϵ > 0 such that Spec(D_{Γ_{j}})∩]0, ϵ[= ∅ for all j, then

\frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} h^{(2)}(Γ)

and

det(D_{Γ_{j}})^{\frac{1}{[Γ:Γ_{j}]}} \underrightarrow{j → ∞} det^{(2)}(D_{Γ}).

Examples of this are Shimura manifolds X_{Γ}, where Γ has Kashdan property (T). This for example holds if the universal covering X of X_{Γ} is a symmetric space with every simple factor of rank > 1.

Proof: Apply the proof of the previous theorem to D − ϵ. ■

Theorem 1.3.4 Assume there exist constants C, α > 0 such that

\frac{tre^{−tD′_{Γ_{j}}}}{[Γ : Γ_{j}]} ≤ Ct^{−α}

for t > 1 and all j ∈ ℕ, then the Gromov-Novikov-Shubin invariant of D is ≥ 2α and

\frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} h^{(2)}(Γ)

as well as

det(D_{Γ_{j}})^{\frac{1}{[Γ:Γ_{j}]}} \underrightarrow{j → ∞} det^{(2)}(D_{Γ}).

Examples for this are The Laplacians of flat tori or Heisenberg manifolds.

Proof: We have that \frac{tre^{−tD_{Γ_{j}}}}{[Γ:Γ_{j}]} tends to \frac{h(D_{Γ_{j}})}{[Γ:Γ_{j}]} as t → ∞ and by the assumption this is uniform

in j. Thus from \frac{tre^{−tD_{Γ_{j}}}}{[Γ:Γ_{j}]} \underrightarrow{j → ∞} tr_{Γ}e^{−tD} and tr_{Γ}e^{−tD} \underrightarrow{t → ∞} h^{(2)}(D_{Γ}) the second assertion

follows. The first is now easy and for the third recall that for 0 < Re (s) < α we have

\frac{ζ_{D_{Γ_{j}}} (s)}{[Γ : Γ_{j}]} − ζ_{D_{Γ}}^{(2)}(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}(\frac{tre^{−tD_{Γ_{j}}}}{[Γ : Γ_{j}]} − \frac{h(D_{Γ_{j}})}{[Γ : Γ_{j}]} − tr_{Γ}e^{−tD} + h^{(2)}(D_{Γ}))dt

We split this integral into a ∫_{0}^{1}- and a ∫_{1}^{∞}-part. The ∫_{0}^{1}-part defines an entire function converging locally uniformly to zero in a neighborhood of s = 0 as j → ∞. In the ∫_{1}^{∞}-part the integrand is dominated by some constant times t^{−α} which allows us to interchange the integration with the limit as j → ∞, so this part vanishes as j → ∞. Together we get

\frac{ζ_{D_{Γ_{j}}} (s)}{[Γ : Γ_{j}]} \underrightarrow{j → ∞} ζ_{D_{Γ}}^{(2)}(s)

uniformly around s=0 which gives the last claim. ■

Question 1.3.1 Does for any generalized Laplacian the sequence \frac{dim kerD_{Γ_{j}}}{[Γ:Γ_{j}]} converge to dim_{Γ} kerD?

Question 1.3.2 Does for any generalized Laplacian the sequence det(D_{Γ_{j}})^{\frac{1}{[Γ:Γ_{j}]}} converge to det^{(2)}(D_{Γ}) if GNS(D_{Γ}) > 0 ?

Let for j ∈ ℕ and x ∈ ℝ:

N_{j}(x) =^{def} ∑_{λ≤x} dim Eig(D_{Γ_{j}}; λ)

the eigenvalue counting function of D_{Γ_{j}}. The condition of the last theorem would follow from the existence of constants a, b, C > 0 such that

N_{j}(x) − N_{j}(0) ≤ C(x^{a} + x^{b}) x ≥ 0,

for all j ∈ ℕ. If this holds, we say the the counting functions satisfy a global growth estimate.

In the following chapters we will specialize to the case where X is a symmetric space without compact factors.

Question 1.3.3 Suppose X is a symmetric space of the noncompact type and D a generalized Laplacian which is invariant under the group of orientation preserving isometries of X. Do the counting functions of D then satisfy a global growth estimate?

Torsion and Euler Characteristic

An introduction to the classical notions of analytical and combinatorial torsion in the beginning of this chapter is followed by a section dealing with the higher torsion and Euler characteristics which are the appropriate invariants to look at in the higher rank case.

2.1 Classical Torsion

2.1.1 General Torsion

A finite complex of Hilbert spaces E_{0} \underrightarrow{d} E_{1} \underrightarrow{d} … \underrightarrow{d} E_{n} is called admissible if all

Laplacians △_{p} = dd^{∗} + d^{∗}d |_{E_{p}} are admissible in the sense of 1.1.1. Examples are finite dimensional complexes or the complexes of L^{2}-sections of elliptic complexes over Riemannian manifolds. For an admissible complex E we define its torsion as

τ_{1}(E) =^{def} ∏_{p=0}^{n} det′(△_{p})^{p(−1)^{p}}.

2.1.2 Combinatorial and Analytic Torsion

Let F denote a locally constant sheaf over a connected oriented finite CW complex X of dimension n such that the stalks of F are finite dimensional unitary vector spaces, the scalar products being locally constant as well. This can be viewed as attaching to each cell e a unitary space F_{e} and an unitary isomorphism d_{e,e′} : F_{e} → F_{e′} if e′ lies in the closure of e. Let

E_{j} =^{def}  ⨁ ; \begin{matrix}e cell \\ dim e = n − j \end{matrix} F_{e}

then the boundary operator ∂ makes E = E_{0} → … → E_{n} a finite dimensional complex whose torsion was considered by Reidemeister [Reid], who showed that torsion and Betti numbers are the only numbers attached to (X, F) which are stable under subdivision. It turned out that torsion is a homeomorphy invariant, but not a homotopy invariant and thus can be used to separate homotopy equivalent spaces which are not homeomorphic such as the lens spaces.

To give a locally constant hermitian sheaf amounts to the same as giving a unitary representation φ of the fundamental group π_{1}(X). This in turn defines a flat hermitian vector bundle E_{φ} over X provided X is a smooth manifold. We then could also consider the complex of E_{φ}-valued differential forms Ω_{φ}^{∗} and its torsion τ_{1}(Ω_{φ}^{∗}). It was conjectured by Ray and Singer [RS-AT] that the numbers τ_{1}(Ω_{φ}^{∗}) and τ_{1}(F) should coincide. This was later proven independently by Cheeger [Che] and Müller [Mü]. By this identity it is justified to write τ_{1}(φ) from now on.

We now give some properties of the torsion of a flat bundle.

Proposition 2.1.1 If the dimension of X is even we have τ_{1}(φ) = 1.

Proof: [RS-AT]. ■

Proposition 2.1.2 The number τ_{1}(φ) does not depend on the metrics.

Proof: [RS-AT]. ■

Proposition 2.1.3 If X, Y are oriented compact manifolds we have

τ_{1}(X × Y, φ ⊗ ψ) = τ_{1}(X, φ)^{χ(Y)}τ_{1}(Y, ψ)^{χ(X)}

where χ denotes the Euler characteristic.

Proof: [RS-RM]. ■

2.1.3 Holomorphic Torsion

Now assume the compact Riemannian manifold X is Kähler and let Ω_{φ}^{p,.} denote the Dolbeault complex of E_{φ}-valued (p, .)-forms, where E_{φ} is the flat holomorphic hermitian bundle attached to the unitary representation φ of π_{1}(X).

The holomorphic torsion of φ is defined as

T_{hol}^{p}(φ) =^{def} τ_{1}(Ω_{φ}^{p,.}).

We give some properties of T_{hol}^{p}:

Proposition 2.1.4 We have

∏_{p=0}^{dim_{ℂ} X} T_{hol}^{p}(φ)^{(−1)^{p}} = 1.

Proof: [RS-HT]. ■

Proposition 2.1.5 Let X = ℂ/Γ, where Γ is the lattice generated by 1 and z, Im(z) > 0. Suppose φ is a nontrivial character given by

φ(mz + n) = e^{2πi(mu+nv)} 0 ≤ u, v < 1.

Then, for p = 0, 1 we have

T_{hol}^{p}(φ) =| \frac{e^{−πiv^{2}z}}{Θ(u − zv, z)} |

where Θ is the theta function

Θ(w, z) = −e^{πi(w+z/6)} ∏_{k=−∞}^{∞}(1 − e^{2πi(|k|z−ϵ_{k}w)})

ϵ_{k} = sign (k + \frac{1}{2}) = \frac{k + \frac{1}{2}}{| k + \frac{1}{2} |}.

Proof: [RS-CM]. ■

The holomorphic torsion of compact hermitian symmetric spaces is computed in [Koe]. Holomorphic torsion enters into Arakelov theory in the following way: Let X denote a

compact Kähler manifold and E a holomorphic hermitian vector bundle over X. Consider the cohomology groups H^{q}(X, E) of the E-valued Dolbeault complex and the one dimensional vector space

λ(E) = ⨂_{q≥0} ∧^{max}H^{q}(X, E)^{(−1)^{q}},

where V^{−1} denotes the dual of a vector space V. The L^{2}-scalar product induces a metric h_{L^{2}} on λ(E). Quillen [Qu] defined a new metric h_{Q} on λ(E) by

h_{Q} = h_{L^{2}}T_{hol}^{0}(E).

Now let f : X → Y be a smooth proper map of Kähler manifolds. Let E be a holomorphic hermitian vector bundle on X and let f_{∗}λ(E) = detRF_{∗}(E) be the determinant of the direct image of E as defined in [KnMf]. This is a holomorphic line bundle f_{∗}λ(E) on Y such that for every y ∈ Y

f_{∗}λ(E)_{y} = ⨂_{y≥0} ∧^{max}H^{q}(X_{y}, E)^{(−1)^{q}}.

It then follows [BGS I-III], that the Quillen metric defined fibrewise is smooth. Further its curvature was computed loc.cit.

Thus analytic torsion became part of the direct image in the arithmetic Chow ring in which setting Gillet and Soulé were able to prove an arithmetic Riemann-Roch Theorem relating the determinant of the direct image in the K-theory to the direct image in the arithmetic Chow ring [GS].

2.2 Higher Torsion and Euler Characteristics

2.2.1 Determinants of Laplacians

We extend the notion of an admissible complex to infinite complexes E = E_{0} → E_{1} → … of Hilbert spaces in the obvious manner. So E is called admissible if all Laplacians △_{p} = dd^{∗} + d^{∗}d |_{E_{p}} p = 0, 1, 2, … are admissible.

Let C_{a} denote the category whose objects are the admissible positive complexes and whose morphisms are the homomorphism of complexes φ : E → F such that the domain of φ_{j} contains all eigenvectors of the Laplacian and φ interchanges the Laplace operators, i.e.: φ△_{E} = △_{F} φ. Call a virtual complex E = E_{+} − E_{−} pseudofinite, if there is a k_{0} such that for all k ≥ k_{0} we have

det′(△ | E_{k}) = 1.

(Recall our notational convention that says det(△′ | E_{k}) = \frac{det′(△|(E_{+})_{k})}{det′(△|(E_{−})_{k})}.) For a pseudofinite virtual complex E we define the determinant of the Laplacian on E to be

det′(△ | E) = ∏_{k=0}^{∞} det′(△ | E_{k})^{(−1)^{k}}.

Lemma. 2.2.1 Assume E is a finite complex in C_{a} then

det′(△ | E) = 1.

Proof: Let v be an eigenvector of d^{∗}d for some nonvanishing eigenvalue. Then dv is an eigenvector for dd^{∗} for the same eigenvalue. So we have

det′(d^{∗}d | E_{p}) = det′(dd^{∗} | E_{p+1}).

Denoting E as E_{0} → … → E_{N} we get

det′(△ | E) = ∏_{p=0}^{N} det′(dd^{∗} | E_{p})^{(−1)^{p}}det′(d^{∗}d | E_{p})^{(−1)^{p}}

= ∏_{p=0}^{N} det′(dd^{∗} | E_{p})^{(−1)^{p}}det′(dd^{∗} | E_{p+1})^{(−1)^{p}}

= 1.■

2.2.2 Twists and Torsion

For a virtual complex E we consider the twist E′ of E defined by

E′ = ∑_{k=0}^{∞}(−1)^{k}E[−k],

where E[j] denotes the j-shifted complex, so E[j]_{k} = E_{k+j}. We also define the higher twists by iteration, so E^{(0)} = E, E^{(n+1)} = (E^{(n)})′.

If E is written as E_{+} − E_{−} we write E′ = E′_{+} − E′_{−}, where

\begin{aligned}(E′_{+})_{j} & = ∑_{k=0}^{∞}(E_{+})_{j−2k} + ∑_{k=0}^{∞}(E_{−})_{j−2k+1} \\ (E′_{−})_{j} & = ∑_{k=0}^{∞}(E_{−})_{j−2k} + ∑_{k=0}^{∞}(E_{+})_{j−2k+1}. \end{aligned}

Since the E_{±} are positive complexes the sums are finite.

Proposition 2.2.1 Let E be a pseudofinite virtual complex with det′(△ | E) = 1 then the twist E′ is a pseudofinite virtual complex and the determinant of the twist E′ equals the inverse torsion of E, i.e.

det′(△ | E′) = τ_{1}(E)^{−1}.

Proof: By our formulas for (E_{±})_{j} we have

det′(△ | (E′)_{j}) = ∏_{k=0}^{∞} \frac{det′(△ | E_{j−2k})}{det′(△ | E_{j−2k+1})}.

By det(△ | E) = 1 it follows that fir j large we have

det′(△ | (E′)_{j}) = det′(△ | E)^{(−1)^{j}} = 1,

so that E′ is a pseudofinite complex. It the follows that for some N ∈ ℕ

det′(△ | E′) = ∏_{j=0}^{N} det′(△ | E′_{j})^{(−1)^{j}}

= ∏_{j=0}^{N} ∏_{k=0}^{∞} \frac{det′(△ | E_{j−2k})^{(−1)^{j}}}{det′(△ | E_{j−2k+1})^{(−1)^{j}}}

= ∏_{n=0}^{∞} det′(△ | E_{n})^{(−1)^{n}(N−n+1)}

= (∏_{n=0}^{∞} det′(△ | E_{n})^{n(−1)^{n}})^{−1}
= τ_{1}(E)^{−1}. ■

This might lead us to the definition of higher torsion as follows: Define the zeroth torsion of a pseudofinite complex E as

τ_{0}(E) =^{def} det′(△ | E)^{−1}.

Suppose, the torsion numbers τ_{0}(E), …, τ_{r−1}(E) are defined and are all trivial, then the r-th twist E^{(r)} is pseudofinite and we define the r-th torsion as

τ_{r}(E) =^{def} τ_{0}(E^{(r)}).

Proposition 2.2.2 Let E be a pseudofinite virtual complex whose r-th torsion is defined, i.e. the twists E^{(0)}, …, E^{(r−1)} have trivial determinants, then the r-th torsion is given by the formula

τ_{r}(E) = ∏_{p=0}^{∞} det′(△ | E_{p})^{((−1)^{p+r−1}( \begin{matrix}p \\ r \end{matrix} ))},

where the product is in fact finite.

Proof: By induction on r. The cases r = 0, 1 are clear. For the step r − 1 → r assume τ_{0}(E^{(0)}) = … = τ_{0}(E^{r−1}) = 1, then

τ_{r}(E) = τ_{0}(E^{(}r)) = τ_{0}((E′)^{(r−1)})

= ∏_{p=0}^{N} det′(△|(E′)_{p})^{(−1)^{p+r}( \begin{matrix}p \\ r − 1 \end{matrix} )}

= ∏_{p=0}^{N} ∏_{k=0}^{p} det′(△|E_{p−k})^{−1)^{k+p+r}( \begin{matrix}p \\ r − 1 \end{matrix} )}

= ∏_{p=0}^{N} det′(△|E_{n})^{(−1)^{n+r}(∑_{p=n}^{N}( \begin{matrix}p \\ r − 1 \end{matrix} ))}

Now because of

∑_{p=n}^{N} ( \begin{matrix}p \\ r − 1 \end{matrix} ) = ( \begin{matrix}N + 1 \\ r \end{matrix} ) − ( \begin{matrix}n \\ r \end{matrix} )

the claim follows. ■

For the moment assume, we are in the situation of 2.1.4. again, so X_{Γ} is a compact Kähler manifold with universal covering X and fundamental group Γ. Fix a finite dimensional unitary representation φ of Γ. Then we have

Proposition 2.2.3 Denoting by T_{hol}^{p} the p-th holomorphic torsion we have

∏_{p=0}^{dim_{ℂ} X} T_{hol}^{p}(φ)^{p(−1)^{p}} = τ_{2}(φ)^{− \frac{1}{2}} .

Proof: We use the previous proposition to calculate

log(τ_{2}(φ)^{− \frac{1}{2}} ) = \frac{1}{2} ∑_{n=0}^{dim X} ( \begin{matrix}n \\ 2 \end{matrix} ) log det(△_{n,φ})

= \frac{1}{2} ∑_{n=0}^{dim X}(−1)^{n} ∑_{p+q=n} ( \begin{matrix}p + q \\ 2 \end{matrix} ) log det(△_{p,q,φ})

= \frac{1}{2} ∑_{n=0}^{dim X}(−1)^{n} ∑_{p+q=n}(p^{2} + 2pq + q^{2} − p − q)log det(△_{p,q,φ})

= ∑_{n=0}^{dim X}(−1)^{n} ∑_{p+q=n} pqlog det(△_{p,q,φ})

= ∑_{p=0}^{dim_{ℂ} X} p(−1)^{p} ∑_{q+0}^{dim_{ℂ} X} q(−1)^{q}log det(△_{p,q,φ}). ■

2.2.3 Higher Euler Characteristics

Let C^{+} denote the category of complexes of ℂ-vector spaces which are bounded from below and have degreewise finite dimensional cohomology, i.e. dim H^{j}(E) < ∞ for all j. Let K^{+} denote the Grothendieck group of C^{+}, define the twist as before, so for E ∈ K^{+} the higher twists E^{(1)}, … are defined as in the previous section. We need to extend the notion of an Euler characteristic to infinite virtual complexes by

χ(E) =^{def} ∑_{k=−∞}^{∞}(−1)^{k} dim H^{k}(E),

provided dim H^{k}(E) = dim H^{k}(E_{+}) − dim H^{k}(E_{−}) vanishes for almost all k.

As an example consider the de Rham complex over a compact oriented manifold X. Let there be a fibre bundle

𝕋 → X → B,

where 𝕋 = {z ∈ ℂ : |z| = 1} is the circle group. Let X_{T} denote the vector field on X spanned by the 𝕋-action. Assume that there is a real closed 1-form ω such that ω(X_{T}) nowhere vanishes. (Note that this assumption forces the first Betti number of X to be nonzero.) By integration we may assume ω to be 𝕋-invariant. Let F denote the subcomplex of E consisting of the 𝕋-invariant forms. We get

F = Ω(X)^{𝕋} = Ω(B) ⊕ Ω(B)∧ω,

where Ω(B) is identified with its pullback to X. Since ω is closed, this decomposition is stable under the exterior differential. Since furthermore 𝕋 is connected, it acts trivially on cohomology, so the cohomology of X is computed by F. This gives

χ(E′) = χ(F′) = χ(B).

So we get the Euler characteristic of B from the de Rham complex of X.

One is thus led to define the higher Euler characteristics as χ(E^{(r)}) for a virtual complex E but unfortunately this does not always make sense, simply because the cohomology of the virtual complex E^{(r)} needn’t vanish in high degrees.

So call a complex cohomologically finite if H^{j}(E) = 0 for large j, in other words, the total cohomology H(E) is finite dimensional.

Observation: Let the virtual complex E be cohomologically finite and assume that the Euler characteristic χ(E) vanishes. Then the twist E′ is cohomologically finite.

So start with a cohomologically finite virtual complex E. If E^{(1)}, …, E^{(r)} are cohomologically finite we have

χ(E^{(0)}) = … = χ(E^{(r−1)}) = 0

and

χ(E^{(r)}) = (−1)^{r} ∑_{j=0}^{∞} ( \begin{matrix}j \\ r \end{matrix} ) (−1)^{j}dimH^{j}(E).

This is easily proven by induction on r. This motivates the following Definition: The r-th Euler characteristic of a cohomologically finite virtual complex E is defined by

χ_{r}(E) := (−1)^{r} ∑_{j=0}^{∞} ( \begin{matrix}j \\ r \end{matrix} ) (−1)^{j}dimH^{j}(E).

The r-th Euler characteristic of a manifold is given by any complex that computes its cohomology. So in case of a fibration

𝕋 → X → B,

as above we have

χ_{1}(X) = χ(B).

To every compact manifold M we now can attach a sequence of Euler numbers

χ_{0}(M), …, χ_{n}(M),

where n is the dimension of M. The most significant of these is, as we shall see, the first nonvanishing one, so define the generic Euler number of M as

χ_{gen}(M) = χ_{r}(M), where r is the least index with χ_{r}(M) ≠ 0.

Proposition 2.2.4 Let M,N be compact Manifolds. We have

χ_{gen}(M × N) = χ_{gen}(M)χ_{gen}(N).

Proof: Let r, s ≥ 0. By the Künneth-rule we get

χ_{r+s}(M × N) = ∑_{j=0}^{∞} ( \begin{matrix}j \\ r + s \end{matrix} ) (−1)^{j} ∑_{μ+ν=j} a_{μ}b_{ν},

where a_{μ}, b_{ν} are the Betti numbers of M and N. Suppose now that we have

χ_{0}(M) = … = χ_{r−1}(M) = χ_{0}(N) = … = χ_{s−1}(N) = 0,

then we have

χ_{r}(M)χ_{s}(N) = ∑_{r′+s′=r+s} χ_{r′} (M)χ_{s′} (N)

= ∑_{r′+s′=r+s} ∑_{j=0}^{∞}(−1)^{j} ∑_{μ+ν=j} ( \begin{matrix}μ \\ r′ \end{matrix} ) ( \begin{matrix}ν \\ s′ \end{matrix} ) a_{μ}b_{ν}

= ∑_{j=0}^{∞}(−1)^{j} ∑_{μ+ν=j} a_{μ}b_{ν} ∑_{r′+s′=r+s} ( \begin{matrix}μ \\ r′ \end{matrix} ) ( \begin{matrix}ν \\ s′ \end{matrix} )

because of

∑_{r′+s′=r+s} ( \begin{matrix}μ \\ r′ \end{matrix} ) ( \begin{matrix}ν \\ s′ \end{matrix} ) = ( \begin{matrix}μ + ν \\ r + s \end{matrix} )

the claim follows. ■

2.2.4 Lie Algebra Cohomology

Just to give another example of a situation in which higher Euler characteristics occur we will describe a typical situation in Lie algebra cohomology. Further examples are given in chapter 3.

We consider a short exact sequence

o → 𝔫 → 𝔩 → 𝔞 → 0

of finite dimensional complex (just for convenience) Lie algebras where 𝔞 is abelian. In such a situation a 𝔩-module V is called acceptable , if the 𝔞-module H^{q}(𝔫, V) is a finite dimensional 𝔞-module. Note that V itself needn’t be finite dimensional.

Example 1. Any finite dimensional 𝔩-module will be acceptable.

Example 2. Let 𝔤_{0} denote the Lie algebra of a semisimple Lie group G of the Harish-Chandra class, i.e. G is connected and has a finite center. Let K be a maximal compact subgroup of G and let G = KAN be an Iwasawa decomposition of G. Write the corresponding decomposition of the complexified Lie algebra as 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫. Now let 𝔩 = 𝔞 ⊕ 𝔫 with the structure of a subalgebra of 𝔤. Consider an admissible (𝔤, K)-module V. A theorem of [HeSchm] assures us that V then is an acceptable 𝔩-module.

Theorem 2.2.1 Let

o → 𝔫 → 𝔩 → 𝔞 → 0

be an exact sequence of finite dimensional complex Lie algebras with 𝔞 abelian. Let V be an acceptable 𝔩-module then with r = dim(𝔞) we have

χ_{0}(H^{∗}(𝔩, V)) = … = χ_{r−1}(H^{∗}(𝔩, V)) = 0,

and

χ_{r}(H^{∗}(𝔩, V)) = χ_{0}(H^{∗}(𝔫, V)^{𝔞}),

where H^{∗}(𝔫, V)^{𝔞} denotes the 𝔞-invariants in H^{∗}(𝔫, V).

Proof: There is a Hochschild-Serre spectral sequence to the module V, which we denote by E. So

E_{2}^{p,q} = H^{p}(𝔞, H^{q}(𝔫, V)),

and

E_{∞}^{p,q} = Gr^{q}(H^{p+q}(𝔩, V))

for some filtration on H^{p+q}(𝔩, V). We will consider the twist of E defined as

(E′)_{s}^{p,q} =^{def} ∑_{k=0}^{∞}(−1)^{k}E_{s}^{p−k,q},

which is a virtual spectral sequence , i.e. a formal difference of two spectral sequences. One can iterate this process to obtain a sequence

E^{(0)} = E, E^{(1)} = E′, …, E^{(n)} = (E^{(n−1)})′, …

of virtual spectral sequences. Of particular interest will be

F =^{def} E^{(r)}, r = dim(𝔞).

To understand the 2-term of our spectral sequences we prove:

Lemma. 2.2.2 With r = dim(𝔞) and M any finite dimensional 𝔞-module we have

dim H^{p}(𝔞, M) = ( \begin{matrix}r \\ p \end{matrix} ) dim(M^{𝔞}).

Proof: At first we consider the case dim(M) = 1, so 𝔞 operates through a linear functional λ. We will compute the Lie algebra cohomology via the standard complex , i.e. for a Lie algebra 𝔩, an 𝔩-module V and q ≥ 0 we consider the cochain space

C^{q} =^{def} Hom_{ℂ}(∧^{q}𝔩, V)

and the differential d : C^{q} → C^{q+1} defined by

df(x_{0} ∧ … ∧ x_{q}) = ∑_{i=0}^{q−1}(−1)^{i}x_{i}.f(x_{0} ∧ … x̂_{i} … ∧ x_{q})
+ ∑_{i<j}(−1)^{i+j}f([x_{i}, x_{j}] ∧ x_{0} ∧ … x̂_{i} … x̂_{j} … x_{q}).

Since 𝔞 is abelian, the second sum vanishes in our situation and if λ = 0 the first does so too, so the differential is zero then and the claim for λ = 0 follows.

For the case λ ≠ 0 assume f ∈ C^{q} with df = 0. Let H ∈ 𝔞 with λ(H) = (−1)^{q+1} then define g ∈ C^{q−1} by

g(x_{0} ∧ … ∧ x_{q−2}) =^{def} f(x_{0} ∧ … ∧ x_{q−2} ∧ H).

We get

dg(x_{0} ∧ … ∧ x_{q−1}) = ∑_{i}(−1)^{i}x_{i}.g(x_{0} ∧ … x̂_{i} … ∧ x_{q−1})
= ∑_{i}(−1)^{i}x_{i}.f(x_{0} ∧ … x̂_{i} … ∧ x_{q−1} ∧ H)
= (−1)^{q+1}H.f(x_{0} ∧ … ∧ x_{q−1})
= f(x_{0} ∧ … ∧ x_{q−1}),

so dg = f and the claim follows.

Now that we have finished the case dim(M) = 1 we will proceed via induction on dim(𝔞). Therefore assume dim(𝔞) = 1. In this case we use a subinduction on dim(M). The case dim(M) = 1 has been dealt with. Using the Jordan normal form theorem we may assume that dim(M^{𝔞}) = 1. Consider the exact sequence

0 → M^{𝔞} → M → M/M^{𝔞} → 0,

the corresponding long exact sequence of cohomology gives a exact sequence

0 → (M/M^{𝔞})^{𝔞} → H^{1}(𝔞, M^{𝔞}) → H^{1}(𝔞, M) → H^{1}(𝔞, M/M^{𝔞}) → 0.

Using the induction assumption we get dim(M/M^{𝔞})^{𝔞} = dim H^{1}(𝔞, M^{𝔞}) = 1, so the middle arrow is zero, hence dim H^{1}(𝔞, M) = dim H^{1}(𝔞, M/M^{𝔞}) = 1 by the induction assumption.

Now assume the assertion of the lemma proven for dim(𝔞) ≤ r and assume we are given an 𝔞 of dimension r + 1. Write 𝔞 = A + B with dim(A) = 1. Consider the Hochschild-Serre spectral sequence E with

E_{2}^{p,q} = H^{p}(A, H^{q}(B, M)),
E_{∞}^{p,q} = Gr^{q}(H^{p+q}(𝔞, M)).

Since dim(A) = 1 we get E_{2}^{p,q} ≠ 0 ⇒ p = 0, 1 so that this spectral sequence degenerates. Therefore

dim H^{k}(𝔞, M) = dim Gr^{k}(H^{k}(𝔞, M)) + dim Gr^{k−1}(H^{k}(𝔞, M))
= dim H^{0}(A, H^{q}(B, M)) + dim H^{1}(A, H^{q−1}(B, M))
= dim H^{q}(B, M^{A}) + dim H^{q−1}(B, M^{A})
= dim M^{𝔞}(( \begin{matrix}r \\ q \end{matrix} ) + ( \begin{matrix}r \\ q − 1 \end{matrix} )).■

■

From this lemma we conclude that in the Grothendieck group of vector spaces we have

F_{2}^{p,q} ≅ \{ \begin{matrix}H^{q}(𝔫, V)^{𝔞} & if p = 0, \\ 0 & else. \end{matrix}

This means that we have the identity

χ_{0}(H^{∗}(𝔫, V)^{𝔞}) = χ_{0}(F_{2}^{0,∗}) = χ(F_{2}),

where χ(F_{2}) is the Euler characteristic of the spectral sequence F , which is defined by

χ(F_{2}) =^{def} ∑_{p,q}(−1)^{p+q} dim(F_{2}^{p,q}).

Note that, since the differential of a spectral sequence always maps a component to a component of the other parity (= (−1)^{p+q}) we get

χ(F_{2}) = χ(F_{3}) = … = χ(F_{∞})

and this number is going to be denoted by χ(F). We conclude

χ_{0}(H^{∗}(𝔫, V)^{𝔞}) = ∑_{p,q}(−1)^{p+q} dim(F_{∞}^{p+q})

= ∑_{q=0}^{∞}(−1)^{q} ∑_{p=0}^{∞}(−1)^{p} ( \begin{matrix}r \\ p \end{matrix} ) dim(Gr^{p}(H^{p+q}(𝔩, V)))

= ∑_{p=0}^{∞}(−1)^{p} ( \begin{matrix}r \\ p \end{matrix} ) dim(H^{p}(𝔩, V)).
= χ_{r}(H^{∗}(𝔩, V)). ■

2.3 L^{2}-Torsion

2.3.1 L^{2}-Torsion

We now return to the notation of 1.2, so X_{Γ} is an oriented compact smooth Riemannian manifold with universal covering X and fundamental group Γ. Let φ denote a finite dimensional unitary representation of Γ. Let F_{φ} denote the flat hermitian vector bundle over X_{Γ} defined by φ. We denote by △_{φ,p} the Laplace operator on F_{φ}-valued p-forms. We assume that the Gromov-Novikov-Shubin invariants of all △_{φ,p} are positive and define the L^{2}-torsion of φ by

τ^{(2)}(φ) =^{def} ∏_{p=0}^{dim X} det^{(2)}(△′_{φ,p})^{p(−1)^{p}}.

Assuming X_{Γ} Kählerian we also define the p-th holomorphic torsion as

T_{hol,p}^{(2)} =^{def} ∏_{q=0}^{dim_{ℂ} X} det^{(2)}(△_{φ,p,q})^{q(−1)^{q}},

where △_{φ,p,q} is the Hodge-Laplacian on F_{φ}-valued (p, q)-forms. We do not consider higher twists of L^{2}-torsion since they will play no rôle in the sequel.

Proposition 2.3.1 We have τ^{(2)}(φ) = 1 if dim X is even.

Proof: [Lo]. ■

Proposition 2.3.2 If X, Y are oriented compact manifolds we have

τ^{(2)}(X × Y) = τ^{(2)}(X)^{χ(Y)}τ^{(2)}(Y)^{χ(X)}

where χ denotes the Euler characteristic.

CHAPTER 2 Torsion and Euler Characteristic 37

Proof: We easily get

tr_{π_{1}(X)×π_{1}(Y)}(e^{−}t△_{X×Y,p}) = ∑_{a+b=p} tr_{π_{1}(X)}(e^{−t△_{X},a})tr_{π_{1}(Y)}(e^{−t△_{Y} ,b}).

From this the claim is derived by a purely formal computation. ■

The Geometry of Shimura Manifolds

This chapter contains some preparatory remarks on the geometry of Shimura manifolds, their sphere bundles and the geodesic homotopy classes. The higher Euler characteristics of the latter are computed in terms of root systems. This makes an important step in our computation of orbital integrals but it is to some extent a detour since a different choice of Haar measures would directly give the final result identifying the factor that goes with the orbital integral as the higher Euler characteristic of the homotopy class of the corresponding closed geodesic. But unfortunately we are forced to stick to certain standard normalizations since we pick up the results of Harish-Chandra on orbital integrals.

3.1 The Geometry of Geodesics

3.1.1 Geodesics on the Universal Covering

Let X_{Γ} denote a compact orientable locally symmetric manifold whose universal cover X is globally symmetric without compact factors. We will call such a space X_{Γ} a Shimura manifold. If the group Γ is arithmetic we say that X_{Γ} is an arithmetic Shimura manifold. Do not mix this term with the notion of a Shimura variety , which is a special arithmetic Shimura manifold together with an algebraic structure [Del].

The group G of orientation preserving isometries of X is a centerless semisimple Lie group which acts transitively on X. More precisely let K denote a maximal compact subgroup of G then X ≅ G/K. The reflection at the point eK in X defines a Cartan involution θ on G that fixes K pointwise. We get a Cartan decomposition 𝔤_{0} = 𝔨_{0} + 𝔭_{0} where 𝔤_{0} and 𝔨_{0} denote the Lie algebras of G and K and 𝔭_{0} is the −1-eigenspace of the Cartan involution θ. Let 𝔞_{0} denote a maximal abelian subspace of 𝔭_{0} and A_{0} = exp 𝔞_{0} the corresponding abelian subgroup of G. Let 𝔞_{0}^{+} denote a positive Weyl chamber in 𝔞_{0} and A_{0}^{+} = exp(𝔞_{0}^{+}). Every oriented geodesic c in X can be ”turned into A_{0}^{+} ” the following way: At first we may assume c(0) = eK since G acts transitively on X. Then, modulo K we may assume that c(t) = exp(tH_{c})K for some H_{c} ∈ \overline{𝔞_{0}^{+}} = closure of the Weyl chamber = interior points plus walls and the origin. Let W_{0}, …, W_{n} denote a complete set of nonconjugate walls of 𝔞_{0}^{+} with W_{0} = 𝔞_{0}^{+} and let

M_{j} = cent(W_{j} | K)

the centralizer of W_{j} in K. If H_{c} ∈ W_{j}, then the isotropy group of c is M_{j} and we thus get a decomposition of the tangent bundle of X as a G-set:

T X − 0 ≅ ⋃_{j=0}^{n}(G/M_{j}) × W_{j} as a G − set.

The sphere bundle thus becomes

SX ≅ ⋃_{j=0}^{n}(G/M_{j}) × SW_{j} as a G − set,

where SW_{j} denotes the set of norm 1 elements in W_{j}.

3.1.2 Geodesics ”Downstairs”

Now let Γ ⊂ G be the fundamental group of X_{Γ} then Γ is a torsion free lattice in G and

X_{Γ} = Γ\G/K.

Further, let γ ∈ Γ then, since Γ is the fundamental group of X_{Γ}, the element γ defines a free homotopy class of closed paths in X_{Γ} and this gives a bijection

{nontrivial conjugacy classes in Γ} ↔ {free homotopy classes of closed paths in X_{Γ}}.

The sphere bundle over X_{Γ} is given as:

SX_{Γ} ≅ ⋃_{j=0}^{n}(Γ\G/M_{j}) × SW_{j}

Let γ ∈ Γ and G_{γ}, Γ_{γ}, K_{γ}, denote the centralizers of γ in G, Γ, K.Let γ be in G conjugate to an element of W_{j}K then modulo conjugation we have K_{γ} ⊂ M_{γ}. So we have a map

X_{γ} = Γ_{γ}\G_{γ}/K_{γ} → Γ\G/M ↪ SX.

By Proposition 5.15 of [DKV] we get that X_{γ} actually injects in SX_{Γ} and X_{Γ} and that the image of X_{γ} is the union of all closed geodesics which map to closed paths in X_{Γ} which are in the free homotopy class of γ.

3.2 Euler Characteristics

3.2.1 On Higher Euler Characteristics

Proposition 3.2.1 Let χ_{n} denote the n-th Euler characteristic. If γ is conjugate to an element of W K, where W is a wall of dimension d then χ_{n}(X_{γ}) = 0 for n = 0, …, d − 1.

Proof: Let A ≅ ℝ^{d} the group generated by W. There is a Cartan subgroup which contains A and γ, hence we have A ⊂ G_{γ}. Furthermore the split part of γ is regular in A so we have G_{γ} ⊂ Z(A), the centralizer of A. From this we deduce that A acts on X_{γ} ≅ Γ_{γ}\G_{γ}/K_{γ} by a : Γ_{γ} g K_{γ} ↦ Γ_{γ} ga K_{γ}. Moreover A is the central split component of G_{γ}. Now by the considerations in [MS] we see that there is a torsionfree normal subgroup Γ_{γ}^{0} of finite index in Γ_{γ} such that the A-action turns X_{γ}^{0} = Γ_{γ}^{0}\G_{γ}/K_{γ} into a 𝕋^{d}-fibre bundle. Let p denote the projection X_{γ}^{0} → X_{γ} and let S denote the finite group Γ_{γ}/Γ_{γ}^{0}. The cohomology of X_{γ} is computed by the complex of S-invariant forms on X_{γ}^{0}, which we denote by Ω(X_{γ}^{0})^{S}. Since the torus 𝕋^{d} is compact and connected and the actions of 𝕋^{d} and S on X_{γ}^{0} commute, this cohomology is also computed by Ω(X_{γ}^{0})^{S𝕋^{d}}. We will show that the latter can be written as

Ω(X_{γ}^{0})^{S𝕋^{d}} ≅ ⨁_{0≤i_{1}<...<i_{n}≤d} ω_{i_{1}} ∧ … ∧ ω_{i_{n}} ∧ p^{∗}Ω(X_{γ}^{0}/𝕋^{d})^{S},

where the ω_{i} are closed 1-forms dual to the vector fields that span the 𝕋^{d}-action. Since the ω_{j} are closed this is a decomposition of the de Rham complex and gives an equivalent decomposition of the cohomology which gives the claim.

To show the existence of ω_{1}, …, ω_{d} recall that differentiable 1-forms on the homogeneous space G_{γ}/K_{γ} are given by

Ω_{1}^{∞}(G_{γ}/K_{γ}) ≅ (C^{∞}(G_{γ}) ⊗ 𝔭_{γ})^{K_{γ}}.

So the G_{γ}-invariant forms are in one to one correspondence with 𝔭_{γ}^{K_{γ}} and this space contains the wall 𝔞 with exp(𝔞) = A. Let now ω_{1}, … ω_{d} correspond to a basis of 𝔞 then these forms will be harmonic hence closed. ■

The number χ_{d}(X_{γ}) is computed in terms of root systems in [D-Hitors] and to state the result we have to introduce some notation.

Assume γ lies in a Cartan H which is supposed to have minimal splitrank among those Cartans that contain γ. We say H is fundamental relative γ.

We write 𝔤 and 𝔨 for the complexifications of the Lie algebras 𝔤_{0} and 𝔨_{0}. Let 𝔥_{0} be the Lie algebra of H and 𝔥 its complexification then 𝔥 is a Cartan subalgebra of 𝔤 and we write Φ(𝔥, 𝔤) for the root system of the pair (𝔥, 𝔤). Let x ↦ x^{c} denote the complex conjugation on 𝔤 with respect to the real form 𝔤_{0}. The complex conjugation acts on roots via α^{c}(x) = \overline{α(x^{c})}. A root is called imaginary if α^{c} = −α, real if α^{c} = α and complex if neither holds.

Let 𝔤 = 𝔥 ⊕ ⊕_{α}𝔤_{α} denote the root space decomposition. To every root space 𝔤_{α} there exists a generator X_{α} such that

[X_{α}, X_{−α}] = H_{α} with B(H_{α}, H) = α(H),
B(X_{α}, X_{−α}) = 1

and X_{α}^{c} = X_{α^{c}} if α complex and X_{α}^{c} = ±X_{−α} if α is imaginary. An imaginary root is called compact if X_{α}^{c} = −X_{−α} and noncompact otherwise. Let Φ_{n} denote the set of noncompact imaginary roots and Φ^{+} a set of positive roots as well as Φ_{n}^{+} = Φ_{n} ∩ Φ^{+}.

Let Φ_{γ} denote the set of roots of (𝔤_{γ}, 𝔥) where 𝔤_{γ} is the complexified Lie algebra of the centralizer G_{γ}. Write Φ_{γ,n}, Φ_{γ,n}^{+} accordingly.

Let ν_{γ} = dim(G_{γ}/K_{γ}) − rank(G_{γ}/K_{γ}), the latter denoting the rank as symmetric space. Denote for any compact subgroup L of G the standard volume [HC-HA1, sec.7] of L by

v_{B}(L) and let vol denote the normalized volume (loc. cit.).

For any pair of groups G ⊃ H let W (G, H) denote the group theoretical Weyl group, i.e.:

W (G, H) =^{def} N_{G}(H)/Z_{G}(H),

where N_{G}(H) is the normalizer of H in G and Z_{G}(H) is the centralizer of H in G.

For any topological group G let G^{0} denote the connected component of the unit element. Finally write ρ_{γ} for the half of the sum of the positive roots of (𝔤_{γ}, 𝔥).

Let A_{γ} denote the flat spanned by γ, i.e. A_{γ} = ∩_{γ⊂A}A, where the intersection runs over all maximal flats containing γ. Now let λ_{γ} = vol(A_{γ}).

A discrete subgroup Γ of G is called nice if for every γ ∈ Γ the set of eigenvalues of the adjoint Ad(γ) does not contain any root of unity. Every arithmetic group contains a finite index nice subgroup [Bor,17.1].

Proposition 3.2.2 Assume that Γ is nice and let γ ∈ Γ and H as above of splitrank r Assume γ is of splitrank r too. The action of the split part H_{ℝ} of H on X_{γ} makes X_{γ} the total space of a (S^{1})^{r}-fibre bundle and

χ_{r}(X_{γ}) = \frac{| W (𝔤_{γ}, 𝔥) | ∏_{α∈Φ_{γ}^{+}}(ρ_{γ}, α)}{λ_{γ}c_{γ}[G_{γ} : G_{γ}^{0}]}vol(Γ_{γ}\G_{γ}).

with

c_{γ} = (−1)^{|Φ_{γ,n}^{+}|}(2π)^{|Φ_{γ}^{+}|}2^{ν_{γ}/2} \frac{v(H_{I})}{v(K_{γ})} | W (G_{γ}, H) | .

Proof: [D-Hitors]. ■

3.2.2 The Fundamental Rank

We proved Proposition 3.2.1 with direct geometric arguments, i.e. without use of spectral theory and harmonic analysis. Introducing these tools we are now able to extend this proposition considerably.

We will first introduce the notion of the fundamental rank: The rank of G is the common dimension of all Cartan subgroups of G. The splitrank is the maximal dimension of the split part of a Cartan subgroup and the fundamental rank is minimal dimension of the split part of a Cartan subgroup. In other words the fundamental rank F R(G) of G is

F R(G) = rank(G) − rank(K).

It is by definition that G has a compact Cartan subgroup if and only if F R(G) = 0. By root space decomposition one finds that

F R(G) ≡ dim X mod (2).

As an example consider G = SL_{n}(ℝ) for n ≥ 2. Then we have K = SO(n) and so we get rank(G) = n − 1, rank(K) = [\frac{n}{2}] and so

F R(SL_{n}(ℝ)) = n − [\frac{n}{2}] − 1.

3.2.3 The Pseudocuspform Condition

Let now △_{k} denote the k-th Laplacian on X. The space of smooth k-forms on X can be described as

Ω_{k}^{∞} = (C^{∞}(G) ⊗ ∧^{k}𝔭)^{K},

where K acts via right translations on C^{∞}(G) and via the adjoint representation on ∧^{k}𝔭. The heat operator e^{−t△_{k}} has a rapidly decreasing kernel h_{t}^{k} in

(S(G) ⊗ End(∧^{k}𝔭))^{K×K},

where S(G) = ∩_{p>0}C^{p}(G) is Harish-Chandra’s Schwartz space [BM] and K × K acts on S(G) via right and left translation and on End(∧^{k}𝔭) via the adjoint representation. We put for r ≥ 0

f_{t}^{r} =^{def} ∑_{k=0}^{dim X}(−1)^{k+r} ( \begin{matrix}k \\ r \end{matrix} ) tr h_{t}^{k},

where tr means the trace in End(∧^{k}𝔭).

We want to use the inversion formula for orbital integrals as in [HC-S]. We compute the principal series contributions to the inversion formula. So let H denote a θ-stable Cartan subgroup, let H_{ℝ} denote its split component and H_{I} the compact component then we have a direct product H = H_{ℝ}H_{I}. Let P denote a cuspidal parabolic subgroup with Langlands decomposition P = MH_{ℝ}N. Let (ξ, W_{ξ}) be an irreducible unitary representation of M and e^{ν} a quasicharacter of H_{ℝ}. Set π_{ξ,ν} = Ind_{P}^{G}(ξ ⊗ e^{ν} ⊗ 1) the principal series representation. This is the right regular representation of G on the space

H_{ξ,ν} = {f : G → W_{ξ} | f(m exp(h_{ℝ})ng) = ξ(m)e^{(ρ+ν)(h_{ℝ})}f(g)}.

Recall the notion of a pseudocuspform [MS2]: A rapidly decaying function f on G is called a pseudocuspform if tr π_{ξ,ν}(f) = 0 whenever H is not fundamental.

Proposition 3.2.3 Assume the fundamental rank of G is positive.The function f_{t}^{r} with r = F R(G) is a pseudocuspform. More precisely we have:

tr π_{ξ,ν}(f_{t}) = \{ \begin{matrix}0 & if H is not fundamental \\ −e^{tπ_{ξ,ν}(C)}dim(W_{ξ} ⊗ ∧^{∗}𝔭_{𝔥})^{K∩M^{0}} & otherwise \end{matrix} .

Here C is the Casimir operator of G and for a finite dimensional vector space V we write

∧^{∗}V = ∑_{k=0}^{dim V} (−1)^{k} ∧^{k} V.

Further 𝔭_{𝔥} denotes the orthocomplement of 𝔥_{ℝ} in 𝔭, where 𝔥_{ℝ} = LieH_{ℝ}.

For r < F R(G) we have

tr π_{ξ,ν}(f_{t}) = 0.

Proof: The decomposition 𝔭 = 𝔥_{ℝ} + 𝔭_{𝔥} is K ∩ M-stable. As K ∩ M-modules we get

∧^{k}𝔭 ≅ ∑_{i+j=k} ∧^{j}𝔥_{ℝ} ⊗ ∧^{i}𝔭_{𝔥}.

Recall that 𝔥_{ℝ} is a trivial K ∩ M-module. Let r′ = dim𝔥_{ℝ}. We get by Frobenius reciprocity:

trπ_{ξ,ν}(f_{t}) = trπ_{ξ,ν}(∑_{k=0}^{d} ( \begin{matrix}k \\ r \end{matrix} ) (−1)^{k}e^{−t ˜△_{k}})

= e^{tπ_{ξ,ν}(C)} ∑_{k=0}^{d} ( \begin{matrix}k \\ r \end{matrix} ) (−1)^{k}dim(H_{ξ,ν} ⊗ ∧^{k}𝔭)^{K}

= e^{tπ_{ξ,ν}(C)} ∑_{k=0}^{d} ( \begin{matrix}k \\ r \end{matrix} ) (−1)^{k}dim(W_{ξ} ⊗ ∧^{k}𝔭)^{K∩M}

= e^{tπ_{ξ,ν}(C)} ∑_{k=0}^{d} ∑_{j=0}^{k}(−1)^{k} ( \begin{matrix}k \\ r \end{matrix} ) ( \begin{matrix}r′ \\ j \end{matrix} ) dim(W_{ξ} ⊗ ∧^{k−j}𝔭_{𝔥})^{K∩M}

= e^{tπ_{ξ,ν}(C)} ∑_{i=0}^{d} ∑_{j=0}^{d}(−1)^{i+j} ( \begin{matrix}i + j \\ r \end{matrix} ) ( \begin{matrix}r′ \\ j \end{matrix} ) dim(W_{ξ} ⊗ ∧^{i}𝔭_{𝔥})^{K∩M}.

Now let for r, r′ ∈ ℕ; i ∈ ℕ_{0}:

A(i, r, r′) = ∑_{j=0}^{r′} ( \begin{matrix}i + j \\ r \end{matrix} ) ( \begin{matrix}r′ \\ j \end{matrix} ) (−1)^{j}.

The claim now follows from the fact that r=r’ if and only if H is fundamental and the elementary

Lemma. 3.2.1 We have

A(i, r, r′) = (−1)^{r′} ( \begin{matrix}i \\ r − r′ \end{matrix} ) .

Now consider the Laplace operator △_{k,Γ} on X_{Γ} and the heat operator e^{−t△_{k,Γ}}. Plugging the function f_{t}^{r} into the Selberg Trace Formula we get

∑_{k=0}^{dim X}(−1)^{k+r} ( \begin{matrix}k \\ r \end{matrix} ) tr e^{−t△_{k,Γ}} = ∑_{[γ]⊂Γ} vol(Γ_{γ}\G_{γ})O_{γ}(f_{t}^{r}).

The sum on the right runs over all conjugacy classes in Γ and the expression O_{γ}(f) stands for the orbital integral:

O_{γ}(f) = ∫_{G/G_{γ}} f(gγg^{−1})dg.

Note that the left hand side of the trace formula tends to χ_{r}(X_{Γ}) as t → ∞.

Now by Harish-Chandra’s formula for the Fourier transform of the orbital integrals and the fact that trπ_{ξ,ν}(f_{t}^{r}) vanishes for r < F R(G) we get that for r < F R(G) all orbital integrals vanish. This gives

Theorem 3.2.1 All the Euler Characteristics χ_{0}(X_{Γ}), …, χ_{F R(G)−1}(X_{Γ}) vanish. ■

3.3 Haar Measures

The bilinear form B on 𝔤 which we always take to be a positive multiple of the Killing form induces a Haar measure on the group G the following way: Firstly B defines a positive inner product on the vector space 𝔭_{0}, which is K-invariant. By the differential of the projection G → G/K = X the space 𝔭_{0} is mapped isomorphically to the tangent space of X at eK. By translation we get a Riemannian metric on X. Let μ = μ_{B} denote the volume form of this metric. Let dk denote the Haar measure on K with total volume 1. Then the formula

∫_{G} fdμ = ∫_{X} ∫_{K} f(xk)dkdμ_{B}(x)

defines a Haar measure also denoted μ om G the so called standard measure (see [HC-HA1], sec. 7).

In the case when the fundamental rank is zero there is however a different measure which we are going to use, the Euler-Poincaré measure μ_{EP}, which is determined by the property that

μ_{EP} (Γ\G) = (−1)^{\frac{dim X}{2}} χ(X_{Γ})

for every torsionfree uniform lattice Γ. To express the relation between these two measures let T be a compact Cartan subgroup and let

c = (∏_{α>0}(α, ρ))(2π)^{−|Φ^{+}|}2^{(rankG/K−dim G/K)/2} \frac{v(K)|W (G_{ℂ}, T_{ℂ})|}{v(T)|W (K, T)|}.

Then we have

μ_{EP} = cμ.

Geometric Zeta Functions in the Rank One Case

This chapter contains a treatment of two kinds of zeta functions in the rank one case. They are expressed in a ”complete” determinant formula which also gives a determinant expression for the ”factor at infinity”.

4.1 The Zeta Functions of Selberg and Ruelle

Before preceding in a more general setting let Y denote a compact Riemannian surface of genus g ≥ 2. There is a unique hyperbolic metric on Y of curvature −1. Consider the infinite product

R(s) =^{def} ∏_{c}(1 − e^{−sl(c)})

convergent for Re (s) >> 0, where the product is taken over all primitive closed geodesics c in Y and l(c) is the length of the geodesic c.

By the uniformisation theorem Y is a Shimura manifold Y = X_{Γ} = Γ\H_{2}(ℝ), where H_{2}(ℝ) = \{z ∈ ℂ|z = x + iy, y > 0\} is the upper half plane in ℂ with the hyperbolic metric \frac{dx^{2}+dy^{2}}{y^{2}} . Then the group of orientation preserving isometries of X is G = P SL_{2}(ℝ)^{0} = SL_{2}(ℝ)/ ± 1, the connected component of P SL_{2}(ℝ). For the sphere bundle we actually have SX_{Γ} = Γ\G. The complex Lie algebra 𝔤 of G is sl_{2}(ℂ) = {M ∈ Mat_{2}(ℂ)|trM = 0}. Let

H = \frac{1}{2} ( \begin{matrix}1 \\ & −1 \end{matrix} ) ∈ 𝔤

then via the identification SX_{Γ} = Γ\G the geodesic flow becomes

Φ_{t} : SX_{Γ} = Γ\G → Γ\G

Φ_{t}(Γg) = Γg exp(tH).

Now let 𝔫 = \{( \begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix} )\} ⊂ 𝔤 and 𝔫̄ = \{( \begin{matrix}0 & 0 \\ ∗ & 0 \end{matrix} )\} ⊂ 𝔤 then with 𝔞 = ℂH the decompo-

sition

𝔤 = 𝔫̄ ⊕ 𝔞 ⊕ 𝔫

induces a decomposition of the tangent bundle of G which pushes down to a decomposition of the tangent bundle of SX_{Γ} = Γ\G as

T SX_{Γ} = T^{s} ⊕ T^{0} ⊕ T^{u}

into stable, central and unstable part, where the geodesic flow is contracting on T^{s} and expanding on T^{u}, i.e. for v ∈ T^{s} the metric norm of DΦ_{t}(v) decreases exponentially as t → ∞ and for v ∈ T^{u} the norm of DΦ_{t}(v) increases exponentially. Further the norm of DΦ_{t}(v) for v in the line bundle T^{0} remains constant. These properties are summarized in saying that Φ is an Anosov flow.

The function R(s) above is called the Ruelle zeta function since it was introduced by David Ruelle in [Rue]. Ruelle introduced it in the more general setting of arbitrary Anosov flows which satisfy a mild analyticity condition. For these flows Ruelle was able to show that R(s) admits an analytic continuation to a meromorphic function of finite order. In the general setting there is not very much more one can prove concerning R(s). For a geodesic flow of a Riemann surface however, one has more at hand.

Atle Selberg [Sel] introduced the zeta function

Z(s) =^{def} ∏_{c} ∏_{N≥0}(1 − e^{−(s+N)l(c)})

and he showed that Z(s) extends to an entire function whose zero set can be split into two parts, the spectral zeroes and the topological zeroes: The spectral zeroes all lie in the set [0, 1] ∪ (\frac{1}{2} + iℝ) and s is a spectral zero if and only if s(1 − s) is an eigenvalue of the Laplacian △_{Γ} of X_{Γ}, the multiplicity of the zero being equal to the multiplicity of the eigenvalue. The topological zeroes are at s = 0, −1, −2, … and the order of a zero at s = −k is (2k+1)(2g-2). The orders at s=0 add up.

All this information can be summarized to

Theorem 4.1.1 [CaVo] The Selberg zeta function satisfies the determinant formula:

Z(s) = det(△_{Γ} + s(1 − s))(e^{−(s− \frac{1}{2} )^{2}}det(P + s − \frac{1}{2}))^{2g−2},

where P = \sqrt{△^{d} + \frac{1}{4}} and △^{d} is the Laplacian of the 2-sphere S^{2} with the usual metric. This means that the eigenvalues of P are the numbers j + \frac{1}{2} for j = 0, 1, … and the eigenvalue j + \frac{1}{2} has multiplicity 2j + 1.

From this we also get that the function Ẑ(s) = Z(s)(e^{−(s− \frac{1}{2} )^{2}}det(P + s − \frac{1}{2}))^{2−2g} satisfies the functional equation

Ẑ(s) = Ẑ(1 − s).

Back to the Ruelle zeta function it is easy to see that

R(s) = \frac{Z(s)}{Z(s + 1)}

so that all poles and zeroes of R(s) may be read off. Even more, we find that R(s) satisfies a functional equation as s is replaced by −s and in the center of the functional equation we find the following first Taylor coefficient:

R(s) = \frac{T_{hol}(X_{Γ})}{T_{hol}^{(2)}(X_{Γ})}s^{2g−2} + higher order terms,

where T_{hol} is the holomorphic torsion, T_{hol}^{(2)} the holomorphic L^{2}-torsion.

This is going to be generalized in the following chapters.

4.2 Rank One Spaces

In this chapter we assume X to be a rank one symmetric space. By definition this means that the group of orientation preserving isometries G acts transitively on the sphere bundle SX of X. By 3.1.1 it then follows that any maximal abelian subspace 𝔞_{0} of 𝔭_{0} has dimension one and thus we conclude that G has splitrank one. There are three series and one sporadic space (see [Helg]), we have:

• the real hyperbolic space of dimension n = 2, 3, …

X = H_{n}(ℝ) = SO(n, 1)^{+}/SO(n),

• the complex hyperbolic space of dimension 2n; n = 1, 2, …

X = H_{2n}(ℂ) = SU(n, 1)/S(U(n) × U(1)),

• the quaternionic hyperbolic space of dimension 4n; n = 1, 2, …

X = H_{4n}(ℍ) = Sp(n, 1)/Sp(n),

and

• the Cayley hyperbolic space of dimension 16

X = H_{16}(Cal) = (𝔣_{4(−20)}, 𝔰𝔬(9)).

The only coincindences here are H_{2}(ℝ) ≅ H_{2}(ℂ) and H_{4}(ℝ) ≅ H_{4}(ℍ).

Since the splitrank of G is one, the fundamental rank can only be zero or one which is in this setting equivalent to the dimension being even or odd. So we only have fundamental rank equals 1 in the case of H_{n}(ℝ) for n odd, in all other cases we have F R(G) = 0. The only spaces occurring here that are hermitian symmetric spaces, i.e. that have a G-invariant holomorphic structure are the complex hyperbolic spaces.

We will fix a maximal abelian space 𝔞_{0} ⊂ 𝔭_{0} and the corresponding split torus A = exp(𝔞_{0}). Further let M denote the centralizer of A in K and fix a minimal parabolic P = MAN.

Since it doesn’t change the geometry we will not use G = Iso^{+}(X) but a finite cover of this, namely we assume G = G(ℝ), where G is a simply connected algebraic group.

4.2.1 K-separability

A finite dimensional representation (τ, V) of a maximal compact subgroup K of G defines a

vector bundle E_{τ} over X via E_{τ} =^{def} (G × V)/K where K acts on G × V from the right by

(g, v)k =^{def} (gk, τ(k^{−1})v). The projection E_{τ} → X is the obvious one: (g, v)K ↦ gK. The group

G then acts on E_{τ} via g′((g, v)K) =^{def} (g′g, v)K and this action makes E_{τ} a homogeneous

vector bundle, that is, the G-action is smooth, maps fibres to fibres, is fibrewise linear and the projection E_{τ} → X is G-equivariant.

The other way round given a homogeneous vector bundle E over X the K-action on the fibre over the point eK ∈ X = G/K defines a finite dimensional representation (τ, V) of K and it is easy to see that E ≅ E_{τ}. This defines an equivalence of categories between the category of homogeneous vector bundles over X and the category of finite dimensional representations of the compact group K.

By rank(X) = 1 we have for the sphere bundle SX:

SX ≅ G/M.

The pullback of any homogeneous vector bundle E_{τ} to G/M will be a homogeneous vector bundle over G/M corresponding to the M-representation τ|_{M}. Actually, what we are going to do is to represent homogeneous vector bundles over G/M as linear combinations of bundles over G/K, hence we have to study the restriction of representations from K to M.

For any compact group H let Rep(H) denote the Representation ring of H, as abelian group Rep(H) is the Grothendieck group of the category of finite dimensional unitary representations of H and the tensor product makes it a ring.

The restriction gives a ring homomorphism res : Rep(K) → Rep(M). A representation σ of M is called K-separable if σ lies in the image of res.

Proposition 4.2.1 If dim X is even then every finite dimensional representation of M is Kseparable. If dim X is odd and σ ∈ M̂ is not K-separable then σ + σ^{∗} is, where σ^{∗} is the dual representation. Furthermore, the constituents of the adjoint representation of M on the space ∧^{p}𝔭 are K-separable for all p.

Proof: Fix a maximal torus T in K such that T_{M} = T ∩ M is a maximal torus in M. Denote the corresponding Lie algebras by 𝔱 and 𝔱_{M}.For dim X odd we have K = Spin(2n + 1), M = Spin(2n) for some n. The claim follows easily ([BrotD]). Now assume dim X even. This means rank(g) = rank(K). Now by the minimal K-type formula ([Knapp], p.629) we see that in case of compatible orderings every highest weight of M is the restriction of a highest weight of K. To get the K-separability from this one proceeds by induction on the highest weights of M with respect to the root order we have chosen. For the trivial weight the assertion is clear. Now let λ ∈ 𝔱_{M}^{∗} be a highest weight and assume the claim proven for all highest weights λ′ < λ. Denote the M-representation with highest weight λ by R_{M}(λ). Fix a highest weight λ_{K} of K which restricts to λ and write R_{K}(λ_{K}) for the corresponding K-representation. Now let λ′ be a highest weight of M such that R_{M}(λ′) occurs in the restriction of R_{K}(λ_{K}). Since λ′ is the restriction of a highest K-weight which is smaller in the K-order than λ_{K}, it follows λ′ < λ and so we have

R_{K}(λ_{K})|_{M} = R_{M}(λ) + lower terms

and by the induction hypothesis the lower terms already lie in Rep(K). The claim now follows.

Finally to prove the assertion about the constituents of ∧^{p}𝔭 we only have to worry about the case K = SO(n) and M = SO(n − 1). Let τ_{p} denote the representation of K on ∧^{p}𝔭, then τ_{1} is just the natural representation of K = SO(n) on ℂ^{n}. Restricted to M this splits into ℂ ⊕ ℂ^{n−1}, where M acts trivially on the first summand and via the natural representation σ_{1} on the second. Therefore we get τ_{p}|_{M} = ∧^{p}σ_{1} ⊕ ∧^{p−1}σ_{1} and thus ∧^{p}σ_{1} = ⊕_{k=0}^{p}(−1)^{k+p}τ_{p}|_{M} in Rep(M). ■

4.3 Selberg’s Zeta for Even Dimensions

4.3.1 The Trace Formula for M-Types

From now on we will assume dim X even. Recall that the torus 𝔱_{M} ⊕ 𝔞 admits a real root α, unique up to sign. To fit conveniently into normalizations of Harish-Chandra we will work with the form B, a scalar multiple of the Killing form such that B(α) = 4 From [Wall] we take the following version of the trace formula. Let Γ denote a cocompact torsionfree lattice in G. Since dim X is even G admits a compact Cartan subgroup and the Euler-Poincaré measure μ_{EP} on G is nonzero. Thus μ_{EP} (Γ\G) = (−1)^{m}χ(Γ\X) = (−1)^{m}χ(Γ), the Euler-Poincaré characteristic. Let (φ, V_{φ}) denote a finite dimensional unitary representation of the group Γ. Consider the Hilbert space

L^{2}(Γ\G, φ) = {f : G → V_{φ} | f(γg) = φ(γ)f(g), ∫_{Γ\G} ∥ f(x) ∥^{2} dμ_{EP} (x) < ∞}

(modulo nullfunctions). The group G acts unitarily on L^{2}(Γ\G, φ) by right translations and it is known that we have a direct sum decomposition

L^{2}(Γ\G, φ) = ⨁_{π∈Ĝ} N_{Γ,φ}(π)π

with finite multiplicities N_{Γ,φ}(π) for all π ∈ Ĝ. Let C denote the Casimir operator of G.

Since C is central it acts on the representation space of π ∈ Ĝ as a scalar π(C) times the identity. We have the well known formula

π(C) = B(λ_{π}) − B(ρ).

Let τ ∈ K̂ and denote by χ_{τ} the character of τ. By [BM] we may insert the heat kernel e^{tC} ∗_{K} χ_{τ} into the trace formula. We introduce some notation: For an irreducible unitary representation π of G let [π : τ] = [π |_{K}: τ] denote the multiplicity of τ in the restriction π |_{K}. Let d_{π} denote the formal degree of π if π is a discrete series representation and d_{π} = 0 otherwise. This can be summarized in saying d_{π} = μ̂_{EP} ({π}), where μ̂_{EP} is the Plancherel measure corresponding to μ_{EP}.

For π ∈ Ĝ let

λ_{π} = N_{Γ,φ}(π) − (−1)^{m}χ(X_{Γ})d_{π} dim(φ).

Note that, as a consequence of the L^{2}−index theorem of Atiyah [AtSch], the number χ(X_{Γ})d_{π} = μ_{EP} (Γ\G)d_{π} is an integer. For ξ ∈ M̂ let ∧_{ξ} denote its infinitesimal character and P_{ξ} the Plancherel density on the principal series attached to ξ.

Let

f_{ξ}(t) = ∫_{−∞}^{∞} e^{−tv^{2}}P_{ξ}(v)dv.

The conjugacy classes [γ] of Γ are in bijection to the free homotopy classes of closed paths in X_{Γ}. Denote by l_{γ} the minimal length of paths in the class [γ] and by [γ_{0}] the underlying primitive class. The letter ρ denotes the half sum of positive roots and ρ_{0} = ρ|_{𝔞}. Let |ρ_{0}| denote the length of ρ_{0} under the Killing form. Since Γ is torsionfree, any γ ∈ Γ is conjugate in G to an element a_{γ}m_{γ} ∈ AM. Now let

T_{γ} = \frac{l_{γ_{0}}e^{−|ρ_{0}|l_{γ}}trφ(γ)}{det(1 − γ^{−1} | 𝔫)}.

The trace formula yields

\begin{matrix}∑_{π∈Ĝ}[π : τ]λ_{π}e^{t(π(C)+B(ρ))} \\ = ∑_{ξ∈M̂}[τ : ξ]e^{tB(∧_{ξ})} \{(−1)^{m}χ(X_{Γ}) dim(φ)f_{ξ}(t) + ∑_{[γ]≠1} T_{γ}trξ(m_{γ})\frac{e^{−l_{γ}^{2}/4t}}{\sqrt{4πt}} \} . \end{matrix}

Since we have restricted to the even dimensions there is, by Proposition 4.2.1 to every σ ∈ M̂ an element r = ∑_{τ} a_{τ} τ in Rep(K) such that

(2.3.1)  σ = r |_{M}= ∑_{τ∈K̂} a_{τ} τ |_{M} .

For π ∈ Ĝ and r ∈ Rep(K) define

N_{r}(π) = ∑_{τ∈K̂} a_{τ} [π : τ].

Note that if π = π_{ξ,ν} is a principal series representation and r ∈ Rep(K) with (2.3.1) we have

N_{r}(π) = \{ \begin{matrix}1 & if ξ = σ \\ 0 & else. \end{matrix}

Together with the trace formula this leads to the

Observation: Let r be in the kernel of the restriction map Rep(K) → Rep(M) then for every z ∈ ℂ we have

∑_{π∈Ĝ,π(C)=z} N_{r}(π)(N_{Γ}(π) − μ_{EP} (Γ\G)d_{π}) = 0.

Apply this to the case G = SL_{2}(ℝ). Here K = SO(2) and M = {±1}, so there is a lot of r as above. It follows for example for the Mock representations π_{±} (see[Lang]) that we have N_{Γ}(π_{+}) = N_{Γ}(π_{−}). Further, the discrete series representations of G = SL_{2}(ℝ) are parametrized by ℤ − {0}, so Ĝ_{d} = {π_{n} | n ≠ 0}. From the observation we get N_{Γ}(π_{n}) = d_{π_{n}}vol(Γ\G) for n ≠ ±1 and N_{Γ}(π_{±1}) = d_{π_{±1}}vol(Γ\G) + 1 (compare [Langl]).

Now for σ ∈ M̂ fix any r ∈ Rep(K) which restricts to σ. Set N_{σ}(π) = N_{r}(π) then the following expression is indeed independent of r:

∑_{π∈Ĝ} N_{σ}(π)λ_{π}e^{t(π(C)+B(ρ)−B(∧_{σ}))}

= (−1)^{m}χ(X_{Γ}) dim(φ)f_{σ}(t) + ∑_{[γ]≠1} T_{γ,σ} \frac{e^{−l_{γ}^{2}/4t}}{\sqrt{4πt}} .

4.3.2 The zeta function

Fix a minimal parabolic subgroup P of G, let P = MAN be its Langlands decomposition, so A is a one dimensional split torus, M its centralizer in K and N is the unipotent radical of P. Denote the corresponding Lie algebras by 𝔭_{0}, 𝔪_{0}, 𝔞_{0}, 𝔫_{0} and their complexifications by 𝔭, 𝔪, 𝔞, 𝔫. For N ≥ 0 let S^{N}(𝔫) denote the N-th symmetric power of 𝔫. Let Γ ⊂ G denote as before a torsion free uniform lattice in G. Since Γ is torsion free and G of real rank one, every γ ∈ Γ is in G conjugate to an element m_{γ}a_{γ} of MA.

As before, (φ, V_{φ}) denotes a finite dimensional unitary representation of Γ. We further fix a representation (σ, V_{σ}) ∈ M̂ and define the generalized Selberg Zeta Function as

Z_{σ,φ}(s) = ∏_{[γ] prime} ∏_{N≥0} det(1 − e^{−sl_{γ}}γ | V_{N}),

where γ ∈ Γ induces an operator, well defined up to conjugacy on V_{N} = V_{φ} ⊗ V_{σ} ⊗ S^{N}(𝔫) via γ ↦ φ(γ)⊗σ(m_{γ})⊗Ad^{N}((m_{γ}a_{γ})^{−1}). Recall that l_{γ} is the minimal length of a closed path freely homotopic to γ in X_{Γ}. Now in the rank one situation which we are in each free homotopy class of closed paths contains a unique geodesic. So the product above can also be interpreted as a product over all primitive closed geodesics.

For π ∈ Ĝ and z ∈ ℂ let

m_{z}^{σ} = ∑_{π∈Ĝ, π(C)=−z} N_{σ}(π)λ_{π}.

We consider the virtual operator △_{σ,φ} defined by the divisor (z, m_{z}^{σ})_{z∈I}, where I is the set of all z with m_{z}^{σ} ≠ 0. Since there are only finitely many π ∈ Ĝ with N_{σ}(π)d_{π} ≠ 0, we conclude that △_{σ,φ} is a virtual differential operator plus a finite dimensional one. More explicitly, let r = ∑_{τ} a_{τ} τ be in Rep(K) such that r restricts to σ. Every τ ∈ K̂ defines a homogeneous vector bundle E_{τ}. The Casimir acts on the sections of E_{τ} as a generalized Laplacian △_{τ}, so we define △′_{σ,φ} = ∑_{τ} a_{τ} △_{τ}, then △_{σ,φ} = △′_{σ,φ} + finite operator. It follows

Lemma. 4.3.1 The operator △_{σ,φ} is zeta admissible. Moreover, we have the asymptotics

tre^{−t△_{σ,φ}} ∼ ∑_{k=−m}^{∞} c_{k}t^{k}.■

Now set

M_{σ}(z, λ) = ∫_{0}^{∞} t^{z−1}e^{−λt}tr(e^{−t△_{σ,φ}})dt.

By taking Mellin transforms of all terms in the trace formula we get

M_{σ}(z, λ + B(ρ) − B(∧_{σ})) = M_{1}(z, λ) + M_{geo}(z, λ),

now a calculation similar to [D-Hitors, 3.12] shows

exp(M_{geo}(0, s^{2})) = Z_{σ,φ}(s + |ρ_{0}|)

and

M_{1}(z, λ) = (−1)^{m}χ(X_{Γ}) dim(φ) ∫_{0}^{∞} ∫_{−∞}^{∞} e^{−t(v^{2}+λ)}P_{σ}(v)t^{z−1}dvdt.

This will give the analytic continuation to the entire plane. Because of

exp(M_{σ}(0, λ)) = det(△_{σ,φ} + λ)

it only remains to consider M_{1}(z, λ).

In order to compute M_{1}(z, λ) we write down the Plancherel formula in our normalization of measures. For this purpose let Ĝ_{d} denote the set of discrete series representations of G and for π ∈ Ĝ_{d} we get the formal degree as

d_{π} = \frac{| W (T, K) |}{| W (T_{ℂ}, G_{ℂ}) |} ∏_{α>0} \frac{(α, λ)}{(α, ρ)},

where λ is the Harish-Chandra parameter of π [Knapp]. To derive this, we take the formula for the formal degree provided by [AtSch]. Recall the notion of the dual symmetric space X^{d} to X: Let G^{d} denote a maximal compact subgroup of the complexification G_{ℂ} of G such that K ⊂ G^{d}. Then X^{d} is by definition equal to G^{d}/K. The Haar-measure used by [AtSch] is normalized to give the dual space X^{d} of X the volume 1. By means of the Hopf-Samelson formula one gets that in our normalization the dual space has the volume

vol(X^{d}) = χ(X^{d}) = \frac{| W (G^{d}) |}{| W (K) |}.

From this the assertion follows. To σ ∈ M̂ let ∧_{σ} denote the corresponding infinitesimal character, i.e. ∧_{σ} = w_{σ} + ρ_{M} where w_{σ} is the highest weight.

There is a distinguished element γ_{0} in the center of M [Knapp,p.487] of order two. The set M̂ can be devided into a even and an odd part depending on whether the eigenvalue of γ_{0} is ±γ_{0}^{ρ}. So σ ∈ M̂_{ev} ⇔ σ(γ_{0}) = γ_{0}^{ρ}. Now, from [Knapp] we take the Plancherel formula: Suppose f ∈ C_{c}^{∞}(G), then we have

f(1) = ∑_{π∈Ĝ} d_{π}tr(π(f)) + i\frac{(−1)^{m}}{4} \frac{| W (K) |}{| W (G_{ℂ}) |}× \{ ∑_{σ∈M̂_{ev}} ∫_{−∞}^{∞} trπ_{σiν}(f) ∏_{α>0} \frac{< ∧_{σ} + iνν_{0}, α >}{< ρ, α >} coth(πν/2)dν ∑_{σ∈M̂_{odd}} ∫_{−∞}^{∞} trπ_{σiν}(f) ∏_{α>0} \frac{< ∧_{σ} + iνν_{0}, α >}{< ρ, α >} tanh(πν/2)dν\} ,

where ν_{0} ∈ 𝔞^{∗} is such that B(ν_{0}, α_{0}) = B(α_{0}, α_{0})/2 for the unique positive real root α_{0} of 𝔞+𝔱_{m}. This means that the Plancherel density P_{σ} is given by

P_{σ}(ν) = i\frac{(−1)^{m}}{4} \frac{| W (K) |}{| W (G_{ℂ}) |} ∏_{α>0} \frac{< ∧_{σ} + iνν_{0}, α >}{< ρ, α >} coth(πν/2)

if σ is even and

P_{σ}(ν) = i\frac{(−1)^{m}}{4} \frac{| W (K) |}{| W (G_{ℂ}) |} ∏_{α>0} \frac{< ∧_{σ} + iνν_{0}, α >}{< ρ, α >} tanh(πν/2)

if σ is odd. Note that P_{σ} is always a nonnegative function.

The term p_{≤k} will from now on stand for a polynomial of degree ≤ k. The expressions 4.3 and 3.4 give

logZ_{σ,φ}(s + |ρ_{0}|) = M_{geo}(0, s^{2})
= (−1)^{m+1} ∫ ^{m+1} M_{σ}(m + 1, s^{2} + B(ρ) − B(∧_{σ}))d(s^{2})
+(−1)^{m} ∫ ^{m+1} M_{1}(m + 1, s^{2})d(s^{2}) + p_{≤m}(s^{2}),

where ∫^{k} means integrating k-times (well defined up to p_{≤k−1}). Thus

log(Z_{σ,φ}(s + |ρ_{0}|)) = logdet(△_{σ,φ} + s^{2} − B(ρ) + B(∧_{σ}))
− dim φ logdet^{(2)}(△_{σ,φ} + s^{2} − B(ρ) + B(∧_{σ}))
= logdet(△_{σ,φ} + s^{2} − B(ρ) + B(∧_{σ}))
+(−1)^{m} ∫ ^{m+1} M_{1}(m + 1, s^{2})d(s^{2}) + p_{≤m}(s^{2}).

Further we have

M_{1}(m + 1, s^{2}) = m!(−1)^{m}χ(X_{Γ}) ∫_{−∞}^{∞} \frac{P_{σ}(ν)}{(ν^{2} + s^{2})^{m+1}} dν.

For λ > 0 and n > k ≥ 0 define

F_{n}^{k}(λ) = (−1)^{n+1}Γ(n) ∫_{−∞}^{∞} \frac{r^{2k−1} tanh(\frac{πr}{2} )}{(r^{2} + λ)^{n}}dr

and

F_{0}^{k}(λ) = ∫ ^{k+1} F_{k+1}^{k}(λ)dλ mod p_{≤k}(λ).

Further we set

G_{n}^{k}(t) = (−1)^{n+1}Γ(n) ∫_{−∞}^{∞} \frac{r^{2k−1} tanh(\frac{πr}{2} )}{(r^{2}t + 1)^{n}}dr.

Then

\begin{aligned}\frac{∂}{∂λ}F_{n}^{k}(λ) & = & F_{n+1}^{k}(λ) & \frac{∂}{∂t}G_{n}^{k}(t & = & G_{n+1}^{k+1}(t) \\ F_{n}^{k}(λ) & = & λ^{−n}G_{n}^{k}(\frac{1}{λ}) & G_{n}^{k}(t) & = & t^{−n}F_{n}^{k}(\frac{1}{t}) \end{aligned}

CHAPTER 4 Geometric Zeta Functions in the Rank One Case 56 The residue calculus gives

F_{2}^{1}(λ) = −\frac{2}{\sqrt{λ}} ∑_{n=0}^{∞} \frac{1}{(2n + 1 + \sqrt{λ})^{2}} .

For j ≥ 0 let D_{j} denote the operator with divisor

divD_{j} = (2n + 1, (2n + 1)^{j})_{n≥0},

then its zeta function is given by

ζ_{D_{j}+a}(s) = ∑_{n=0}^{∞} \frac{(2n + 1)^{j}}{(2n + 1 + a)^{s}} Res >> 0.

We define ζ̂_{D_{j}+a}(n) = (−1)^{n+1}Γ(n)ζ_{D_{j}+a}(n) and get

\frac{∂}{∂a}ζ̂_{D_{j}+a}(n) = ζ̂_{D_{j}+a}(n + 1).

We further let ζ̂_{D_{j}+a}(0) = ∫^{j+2} ζ̂_{D_{j}+a}(j + 2)da + p_{≤j+1}. Proposition 4.3.1

F_{0}^{k}(a^{2}) = 8(−1)^{k}ζ̂_{D_{2k−1}+a}(0).

Proof: We first derive the equation:

F_{0}^{k+1} = ∫ F_{0}^{k}(λ)dλ − λF_{0}^{k}(λ),

this is done as follows:

F_{0}^{k+1}(λ) = ∫ ^{k+2} F_{k+2}^{k+1}(λ)dλ

= ∫ ^{k+2} λ^{−k−2}G_{k+2}^{k+1}(\frac{1}{λ})dλ

= − ∫ ^{k+2} λ^{−k} \frac{∂}{∂λ}(G_{k+1}^{k}(\frac{1}{λ}))dλ

= − ∫ ^{k+2} λ^{−k} \frac{∂}{∂λ}(λ^{k+1}F_{k+1}^{k}(λ))dλ

= −(k + 1) ∫ ^{k+2} F_{k+2}^{k}(λ)dλ − ∫ ^{k+2} λF_{k+2}^{k}(λ)dλ.

Iterated partial integration gives the claim.

Now we prove the proposition by induction. The case k=1 is easy. The induction step amounts to

ζ̂_{D_{2k+1}+a}(1) = a^{2}ζ̂_{D_{2k−1}+a}(1) + p_{≤2k+1},

which is shown by differentiating 2k + 2 times. ■

Now let E_{j} denote the operator attached to the divisor

divE_{j} = (2n, (2n)^{j})_{n∈ℕ}.

Define

H_{n}^{k}(λ) = (−1)^{n+1}Γ(n) ∫_{−∞}^{∞} \frac{r^{2k−1} coth(\frac{πr}{2} )}{(r^{2} + λ)^{n}}dr

and

H_{0}^{k}(λ) = ∫ ^{k+1} H_{k+1}^{k}(λ)dλ mod p_{≤k}(λ).

We get in perfect analogy to the above

Proposition 4.3.2

H_{0}^{k}(a^{2}) = 8(−1)^{k+1}ζ̂_{E_{2k−1}+a}(0).

For the proof we only remark that residue calculus gives

H_{2}^{1}(a^{2}) = −\frac{1}{a^{3}} − \frac{2}{a} ∑_{n=1}^{∞} \frac{1}{(2n + a)^{2}} .

Let Q_{σ} denote the polynomial over ℚ

Q_{σ}(x) = ∏_{α>0} \frac{< ∧_{σ} + xν_{0}, α >}{< ρ, α >}.

Lemma. 4.3.2 We have Q_{σ}(ℤ) ⊂ ℤ and there is a k ∈ ℕ such that Q_{σ}(ℕ + k) ⊂ ℕ.

Proof: The existence of such a k follows from the Weyl dimension formula, since for n large enough Q_{σ}(n) is the dimension of a representation. For the rest let d be a natural number such that F = dQ_{σ} ∈ ℤ[x] the we have F (ℕ + k) ⊂ dℕ or F (n + k) ≡ 0(d) for n ∈ ℕ. Since n+k runs through all classes modulo d as n runs through the natural numbers it follows that F (c) ≡ 0(d) for all c ∈ ℤ. ■

As we will see, this lemma gives the analytic continuation of Z_{σ} itself whereas Wakayama [Wak] could only show the continuation of a power of Z_{σ}.

Define the operator D_{σ} by its divisor according to the cases:

• if σ is even:

divD_{σ} = (2n, Q_{σ}(2n))_{n∈ℕ}

• if σ is odd:

divD_{σ} = (2n − 1, Q_{σ}(2n − 1))_{n∈ℕ}.

Putting things together we have proven that there is a polynomial of degree ≤ 2m such that

Z_{σ,φ}(|ρ_{0}| + s) = det(△_{σ,φ} + s^{2} − B(ρ) + B(∧_{σ}))

×(det(D_{σ} + s) exp(P (s)))^{2(−1)^{m} dim(φ)χ(X_{Γ})/χ(X^{d})}.

We finally want to show that P is even. To this end recall that logZ_{σ,φ}(|ρ_{0}| + s) tends to zero as s → +∞. It is thus possible to compute P (s) by means of the asymptotics according to Proposition 1.1.3. Since s only enters in the △_{σ,φ}-term via s^{2} there is only need to consider the D_{σ}-term. Recall the operators D_{j} and E_{j}. An easy computation shows

e^{−tD_{j}} = (−1)^{j} \frac{j!}{2}t^{−(j+1)} + O(1) as t → 0

and the same formula holds for D_{j} replaced by E_{j}. Since only odd j enter in our operators D_{σ} there are only even powers in the asymptotic of D_{σ}, so we get

Theorem 4.3.1 Let c_{σ} = B(∧_{σ}) − B(ρ). The Selberg Zeta Function Z_{σ,φ} = Z can be expressed as a product of determinants

Z(|ρ_{0}| + s) = det(△_{σ,φ} + s^{2} + c_{σ}))det^{(2)}(△_{σ,φ} + s^{2} + c_{σ})^{− dim φ}

= det(△_{σ,φ} + s^{2} + c_{σ})(det(D_{σ} + s) exp(P (s^{2})))^{2(−1)^{m} dim(φ)χ(X_{Γ})/χ(X^{d})}.

where P is a polynomial of degree ≤ m. It follows that Z_{σ,φ} extends to a meromorphic function with finitely many poles and that

\frac{Z(|ρ_{0}| + s)}{Z(|ρ_{0}| − s)} = (\frac{det(D_{σ} + s)}{det(D_{σ} − s)})^{2(−1)^{m} dim(φ)χ(X_{Γ})/χ(X^{d})} .

Further Z satisfies a modified Riemann hypothesis, its singularities are devided into the spectral ones and the topological ones. A complex number s is a spectral singularity if and only if the number −c_{σ} − s^{2} is an eigenvalue of △_{σ,φ} and its order is its multiplicity then, i.e. spectral singularities are zeroes. The complex number s is a topological singularity if and only if s = −2n, n ∈ ℕ if σ is even or s = −2n + 1, n ∈ ℕ if σ is odd. Its order is then equal to the number (−1)^{m}Q_{σ}(s) dim(φ)χ(X_{Γ})/χ(X^{d}).

Finally, we compare our results with that of Juhl [Ju]. Set φ = 1. We introduce some notation. Let G^{d} denote the dual group to G. This is a compact form of G which we may assume to contain K. The compact dual A^{d} of A is a torus in G^{d}. The Lie algebra 𝔫 defines a complex structure on the compact manifold G^{d}/MA^{d} by

T^{(1,0)}(G^{d}/MA^{d}) ≅ G^{d} ×_{MA^{d}} 𝔫.

For a character χ^{d} of A with χ^{d}(γ_{0})σ(γ_{0}) = 1, where γ_{0} is defined as in 4.4 let ind(𝔫, σ, χ^{d}) denote the index of the Dolbeault complex with values in the vector bundle E_{σ⊗χ^{d}} defined by σ ⊗ χ^{d} over G^{d}/MA^{d}. Let H_{0} ∈ 𝔞 as before defined by α(H_{0}) = 1 for the short root α of A. Set

Z_{(𝔫,σ)}^{+}(s) = ∏ ; \begin{matrix}χ^{d} ∈ Â^{d} \\ χ^{d}(H_{0}) > ρ_{0} \\ χ^{d}(γ_{0})σ(γ_{0}) = 1 \end{matrix} (s + χ^{d}(H_{0}))^{ind(𝔫,σ,χ^{d})}.

where the product denotes ζ-regularized product.

Let X^{d} denote the compact dual space to X. A. Juhl has shown [Ju, p. 184]:

\frac{Z_{σ,φ}(|ρ_{0}| + s)}{Z_{σ,φ}(|ρ_{0}| − s)} ≅ (\frac{Z_{(𝔫,σ)}^{+}(|ρ_{0}| + s)}{Z_{(𝔫,σ)}^{+}(|ρ_{0}| − s)})^{−2χ(X_{Γ})/χ(X^{d})} .

where ≅ means equality up to the exponential of a polynomial. Since Z_{(n,σ)} is holomorphic and nonzero for Res > 0 we conclude

Z_{(𝔫,σ)}^{+}(s) = det(D_{σ} + s)^{(−1)^{m+1}},

it further follows that we have equality above instead of ”≅”. Let A^{d} denote the compact dual of A acting on the compact form G^{d} then we easily get

ind(𝔫, σ, χ^{d}) = (−1)^{m+1} ∏_{α>0} \frac{< ∧_{σ} + χ^{d} + ρ |_{𝔞}, α >}{< ρ, α >}.

4.4 Ruelle’s Zeta for Even Dimensions

We extend the definition of the zeta function Z_{σ,φ} to finite dimensional unitary representations σ which are not necessarily irreducible in the obvious way: If σ = ⊕_{ξ∈M̂}n_{ξ}ξ is the isotypical decomposition of σ then

Z_{σ,φ}(s) =^{def} ∏_{ξ∈M̂}(Z_{ξ,φ}(s))^{n_{ξ}}.

Let 𝔞 denote the complex Lie algebra of A, so that M is the centralizer of 𝔞 in K. Let P = MAN be a minimal parabolic. Since we have normalized the form B such that the real root α_{r} has length 2 the possible root length of (𝔤, 𝔞) are 1 and 2. Let 𝔫 = 𝔫_{1} ⊕ 𝔫_{2}, where A acts according to α_{r} on 𝔫_{2} and according to \frac{1}{2}α_{r} on 𝔫_{1}. Let σ_{1}, σ_{2} be the representations of M on 𝔫_{1}, 𝔫_{2} and write τ_{p} for the K-representation on ∧^{p}𝔭. Then we have

τ_{1}|_{M} = σ_{0} ⊕ σ_{1} ⊕ σ_{2},

where σ_{0} is the trivial one dimensional representation.

Theorem 4.4.1 For the Ruelle zeta function

R_{φ}(s) = ∏_{[γ]prime} det(1 − e^{−sl_{γ}}φ(γ))

we have

R_{φ}(s) = ∏_{l=0}^{dim 𝔫_{1}} ∏_{k=0}^{dim 𝔫_{2}} Z_{∧^{l}σ_{1}⊗∧^{k}σ_{2},φ}(s + l + 2k)^{(−1)^{l+k}}.

Proof: For an arbitrary representation σ of M we write

logZ_{σ,φ}(s) = ∑_{[γ]prime} ∑_{N≥0} trlog(1 − e^{−sl_{γ}}γ|V_{N})

= − ∑_{[γ]prime} ∑_{N≥0} ∑_{n=1}^{∞} \frac{1}{n}e^{−snl_{γ}}trφ(γ^{n})trσ(m_{γ}^{n})trAd^{N}((m_{γ}a_{γ})^{−n})

= − ∑_{[γ]} trφ(γ)\frac{e^{−sl_{γ}}}{μ_{γ}} \frac{trσ(m_{γ})}{det(1 − γ^{−1}|𝔫)}.

Now we have

det(1 − γ^{−1}|𝔫) = det(1 − γ^{−1}|𝔫_{1})det(1 − γ^{−1}|𝔫_{2})
= ∑_{l=0}^{dim 𝔫_{1}} ∑_{k=0}^{dim 𝔫_{2}}(−1)^{k+l}tr(γ^{−1}| ∧^{l} 𝔫_{1} ⊗ ∧^{k}𝔫_{2})

= ∑_{l=0}^{dim 𝔫_{1}} ∑_{k=0}^{dim 𝔫_{2}}(−1)^{k+l}e^{−(l+2k)l_{γ}}tr(m_{γ}| ∧^{l} 𝔫_{1} ⊗ ∧^{k}𝔫_{2}),

from which the proposition follows. ■

4.5 Selberg’s Zeta Function for Odd Dimensions

Here we assume n = dim X odd. It follows X = H_{n}(ℝ) the real hyperbolic space. We have G = SO(n, 1)^{+}, K = SO(n) and M = SO(n − 1). We will normalize the form B in such a way that X has curvature −1. This means

B = \frac{1}{n − 1}B_{Killing},

where B_{Killing} denotes the Killing form. The Haar measure μ = μ_{B} will be the standard measure attached to B.

We will follow the notation of 4.3. Since there is no compact Cartan subgroup there is no discrete series, so we needn’t consider the numbers λ_{π}. Indeed, let σ ∈ M̂ be K-separable, choose r ∈ Rep(K) which restricts to σ and let N_{σ}(π) = N_{r}(π). We then get

∑_{π∈Ĝ} N_{σ}(π)N_{Γ,φ}(π)e^{t(π(C)+B(ρ)−B(∧_{σ}))} = μ(Γ\G) dim(φ)f_{σ}(t) + ∑_{[γ]≠1} T_{γ,σ} \frac{e^{−l_{γ}^{2}/4t}}{\sqrt{4πt}} .

Here again f_{σ}(t) = ∫_{−∞}^{∞} e^{−tv^{2}}P_{σ}(v)dv but now the Plancherel density P_{σ} is an even polynomial of degree n − 1 ([Knapp], p.485). This makes the factor at infinity much easier to handle. Let

ζ_{a}(s) =^{def} \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}f_{σ}(t)e^{−ta^{2}}dt.

For a > 0 fixed this extends to a meromorphic function, holomorphic at s = 0 and with

Z_{1}(a) =^{def} exp(−ζ′_{a}(0)) we have

Lemma. 4.5.1

Z_{1}(a) = exp(2π ∫_{0}^{a} P_{σ}(iy)dy)

Proof: ([Fried], p.533). ■

Denote the polynomial 2π ∫_{0}^{a} P_{σ}(iy)dy by P̃_{σ}(a). Along the lines of section 4.3 we get

Theorem 4.5.1 Let X be a real hyperbolic space of odd dimension. Fix a K-separable representation σ ∈ M̂. Consider the zeta function

Z_{σ,φ}(s) = ∏_{[γ]prime} ∏_{N≥0} det(1 − e^{−sl_{γ}}γ|V_{N}),

where V_{N} = V_{φ} ⊗ V_{σ} ⊗ S^{N}(𝔫) and γ operates as φ(γ) ⊗ σ(M_{σ}) ⊗ Ad^{N}((m_{γ}a_{γ})^{−1}). Then Z_{σ,φ} satisfies the equation

Z_{σ,φ} = det(△_{σ,φ} + s^{2} + c_{σ}) exp(dim φμ(Γ\G)P̃_{σ}(s)).

It follows that Z_{σ,φ} extends to an entire function and that

Ẑ_{σ,φ} = Z_{σ,φ}(s) exp(− dim φμ(γ\G)P̃_{σ}(s))

satisfies

Ẑ_{σ,φ}(s) = Ẑ_{σ,φ}(n − 1 − s).

4.6 Ruelle’s Zeta for Odd Dimensions

We still stick to the situation of the preceding subsection, i.e. X = H_{n}(ℝ), n odd. Denote by τ_{p} the representation of K on ∧^{p}𝔭 the we have τ_{p}|_{M} = σ_{p} + σ_{p−1} for some σ_{j} ∈ M̂. Define the Ruelle zeta function as

R_{φ}(s) = ∏_{[γ]prime} det(1 − e^{−sl_{γ}}φ(γ))

Then we get

Proposition 4.6.1 ([Fried],p.532)

R_{φ}(s) = ∏_{p=0}^{n−1} Z_{σ_{p},φ}(s + p)^{(−1)^{p}}.

From this one reads off:

Theorem 4.6.1 ([Fried]) The Ruelle zeta function R_{φ}(s) extends to a meromorphic function on ℂ, its zeroes and poles are all in the set ℝ ∪ (−\frac{n−1}{2} + iℝ) ∪ (−\frac{n−1}{2} + 1 + iℝ ∪ … ∪ (\frac{n−1}{2} + iℝ) and can be expressed in terms of Laplace eigenvalues for the Laplacians △_{φ,p}, for p = 0, 1, …, n.

The function R_{φ}(s) satisfies a functional equation

R_{φ}(s) = R_{φ}(−s) exp(dim φμ(Γ\G)Q(s)),

where Q is the polynomial

Q(s) = ∑_{p=0}^{n−1} P̃_{σ_{p}}(s + p) − P̃_{σ_{p}}(n − 1 − s − p).

In the center of the functional equation we have the special value

R_{φ}(0) = τ(ϕ),

the analytic torsion of φ. ■

The Holomorphic Torsion Zeta Function

In this chapter we are going to consider a different generalization of the Selberg Zeta Function. From this we will construct a zeta function which has the holomorphic torsion as special value. Of particular neatness is the case of the m-th power H^{m}, m ∈ ℕ, of the hyperbolic plane H, where the Euler product looks very simple (sec. 5.2).

5.1 The trace of the heat kernel

Let X̄ denote a compact locally symmetric space whose universal covering X is hermitian globally symmetric of the noncompact type. Then X = G/K where G is the group of orientation preserving isometries of X and K is a maximal compact subgroup of G. Then there is a uniform lattice Γ in G such that X̄ = Γ\X = Γ\G/K. Further, Γ ≅ π_{1}(X̄), the fundamental group of X̄. We will therefore write X_{Γ} instead of X̄. It follows that G is a semisimple Lie group without center that admits a compact Cartan subgroup T ⊂ K. We denote the real Lie algebras of G, K and T by 𝔤_{0}, 𝔨_{0} and 𝔱_{0} and their complexifications by 𝔤, 𝔨 and 𝔱. We will denote the Killing form of 𝔤 by B. As well we will write B for the diagonal of the Killing form so B(X) = B(X, X). Denote by 𝔭_{0} the orthocomplement of 𝔨_{0} in 𝔤_{0} with respect to B then via the differential of exp the space 𝔭_{0} is isomorphic to the real tangent space of X = G/K at the point eK. Let Φ(𝔱, 𝔤) denote the system of roots of (𝔱, 𝔤), let Φ_{c}(𝔱, 𝔤) = Φ(𝔱, 𝔨) denote the subset of compact roots and Φ_{nc} = Φ − Φ_{c} the set of noncompact roots. To any root α let 𝔤_{α} denote the corresponding root space. Fix an ordering Φ^{+} on Φ = Φ(𝔱, 𝔤) and let 𝔭_{±} = ⨁_{α∈Φ_{nc}^{+}} 𝔤_{±α}. Then the complexification 𝔭 of 𝔭_{0} splits as 𝔭 = 𝔭_{+} ⊕ 𝔭_{−} and the ordering can be chosen such that this decomposition corresponds via exp to the decomposition of the complexified tangent space of X into holomorphic and antiholomorphic part.

Let θ denote the Cartan involution on 𝔤_{0} and on G corresponding to the choice of K. Extend θ linearly to 𝔤. Let H denote a θ-stable Cartan subgroup of G then H = AB where A is a split torus and B compact. The use of the letter B here will not cause any confusion. The dimension of A is called the split rank of H. Let 𝔞 denote the complex Lie algebra of A. Then 𝔞 is an abelian subspace of 𝔭 = 𝔭_{+} ⊕ 𝔭_{−}. Let X ↦ X^{c} denote the complex conjugation on 𝔤 according to the real form 𝔤_{0}. The next lemma shows that 𝔞 lies skew to the decomposition 𝔭 = 𝔭_{+} ⊕ 𝔭_{−}.

Lemma. 5.1.1 Let P r_{±} denote the projections from 𝔭 to 𝔭_{±} the we have dim P r_{+}(𝔞) = dim P r_{−}(𝔞) = dim 𝔥_{R}, or, what amounts to the same: 𝔞 ∩ 𝔭_{±} = 0.

Proof: 𝔞 is stable under complex conjugation which interchanges 𝔭_{+} and 𝔭_{−}. So let X ∈ 𝔭_{+} be such that X + X^{c} ∈ 𝔞. Since 𝔞 is abelian, the assumption i(X − X^{c}) ∈ 𝔞 would lead to 0 = [X + X^{c}, i(X − X^{c})] = 2i[X^{c}, X] and the latter only vanishes for X = 0. So 𝔞 does not contain X or X^{c} and the claim follows. ■

Now let 𝔞′ denote the orthocomplement of 𝔞 in P r_{+}(𝔞) ⊕ P r_{−}(𝔞). For later use we write V = 𝔞 ⊕ 𝔞′ = V_{+} ⊕ V_{−}, where V_{±} = V ∩ 𝔭_{±} = P r_{±}(𝔞).

Let n = 2m denote the real dimension of X and for 0 ≤ p, q ≤ m let Ω^{p,q}(X) denote the space of smooth (p, q)-forms on X. Fix a finite dimensional unitary representation (φ, V_{φ}) of the group Γ. Then φ defines an hermitian flat holomorphic vector bundle E_{φ} = X ×_{Γ} V_{φ} over X_{Γ}. On the space Ω^{p,q}(X_{Γ}, E_{φ}) of E_{φ}-valued forms we have a Laplacian △_{p,q,Γ}^{φ} and by construction it is clear that △_{p,q,Γ}^{φ} = △_{p,q,Γ} ⊗ 1 where △_{p,q,Γ} is the Laplacian on Ω^{p,q}(X_{Γ}) and we have used Ω^{p,q}(X_{Γ}, E_{φ}) = C^{∞}(∧^{p}T_{hol}^{∗}(X_{Γ}) ⊗ ∧^{q}T_{ahol}^{∗}(X_{Γ}) ⊗ E_{φ}). Further △_{p,q,Γ} is the pushdown of the Laplacian △_{p,q} on Ω^{p,q}(X).

By [BM] the heat operator e^{−t△_{p,q}} has a smooth kernel h_{t}^{p,q} of rapid decay in

(C^{∞}(G) ⊗ End(∧^{p}𝔭_{+} ⊗ ∧^{q}𝔭_{−}))^{K×K}.

Now fix p and set for t > 0

f_{t}^{p} = ∑_{q=0}^{m} q(−1)^{q}tr h_{t}^{p,q},

where tr means the trace in End(∧^{p}𝔭_{+} ⊗ ∧^{q}𝔭_{−}).

We want to compute the trace of f_{t}^{p} on the principal series representations. To this end let H = AB be a θ-stable Cartan subgroup with split part A and compact part B. Let P denote a parabolic subgroup of G with Langlands decomposition P = MAN. Note that M is of inner type (see [HC-HA1], Lemma 4.9). Let (ξ, W_{ξ}) denote an irreducible unitary representation of M, e^{ν} a quasicharacter of A and set π_{ξ,ν} = Ind_{P}^{G}(ξ ⊗ e^{ν+ρ_{P}} ⊗ 1), where ρ_{P} is the half of the sum of the P -positive roots.

Proposition 5.1.1 For the trace of f_{t}^{p} under π_{ξ,ν} we have

trπ_{ξ,ν}(f_{t}^{p}) = \{ \begin{matrix}0 & if dim 𝔞 > 1 \\ −e^{tπ_{ξ,ν}(C)} ∑_{q=0}^{dim(𝔭_{−})−1}(−1)^{q}dim(W_{ξ} ⊗ ∧^{p}𝔭_{+} ⊗ ∧^{q}(𝔞^{⊥} ∩ 𝔭_{−}))^{K∩M} & if dim 𝔞 = 1. \end{matrix}

Proof: As before let 𝔞′ the span of all X − X^{c}, where X + X^{c} ∈ 𝔞, X ∈ 𝔭_{+} then V = 𝔞 ⊕ 𝔞′ = V_{+} ⊕ V_{−} where V_{±} = V ∩ 𝔭_{±}. The group K_{M} = K ∩ M acts trivially on 𝔞 so for x ∈ K_{M} we have X + X^{c} = Ad(x)(X + X^{c}) = Ad(x)X + Ad(x)X^{c}. Since K respects the decomposition 𝔭 = 𝔭_{−} ⊕ 𝔭_{+}, we conclude that K_{M} acts trivially on V, hence on V_{−}. Let r = dim𝔞 = dimV_{−}. As a K_{M}-module we have

\begin{aligned}∧^{p}𝔭_{−} & = & ∑_{a+b=q} ∧^{a}V_{−} ⊗ ∧^{b}V_{−}^{⊥} \\ & = & ∑_{a+b=q} ( \begin{matrix}r \\ a \end{matrix} ) ∧^{b} V_{−}^{⊥}, \end{aligned}

where V_{−}^{⊥} = 𝔞^{⊥} ∩ 𝔭_{−}. By Frobenius reciprocity we get

trπ_{ξ,ν}(f_{t}^{p}) = trπ_{ξ,ν}(∑_{q=0}^{m} q(−1)^{q}h_{t}^{p,q})

= e^{tπ_{ξ,ν}(C)} ∑_{q=0}^{m} q(−1)^{q}dim(V_{π_{ξ,ν}} ⊗ ∧^{p}𝔭_{+} ⊗ ∧^{q}𝔭_{−})^{K}

= e^{tπ_{ξ,ν}(C)} ∑_{q=0}^{m} q(−1)^{q}dim(W_{π_{ξ}} ⊗ ∧^{p}𝔭_{+} ⊗ ∧^{q}𝔭_{−})^{K∩M}

= e^{tπ_{ξ,ν}(C)} ∑_{q=0}^{m} ∑_{a=0}^{q} q(−1)^{q} ( \begin{matrix}r \\ q − a \end{matrix} ) dim(W_{π_{ξ}} ⊗ ∧^{p}𝔭_{+} ⊗ ∧^{a}V_{−}^{⊥})^{K∩M}

= e^{tπ_{ξ,ν}(C)} ∑_{a=0}^{m} ∑_{q=a}^{m} q(−1)^{q} ( \begin{matrix}r \\ q − a \end{matrix} ) dim(W_{π_{ξ}} ⊗ ∧^{p}𝔭_{+} ⊗ ∧^{a}V_{−}^{⊥})^{K∩M}

By taking into account that we have a ≤ m−r we get ∑_{q=a}^{m} q(−1)^{q} ( \begin{matrix}r \\ q − a \end{matrix} ) = (−1)^{r} ( \begin{matrix}a \\ 1 − r \end{matrix} )

and the claim follows. ■

In view of the preceding proposition fix a Θ-stable Cartan subgroup H = AB with dimA = 1 and a parabolic P = MAN. Fix a system of positive roots compatible with P and let ρ denote the half sum of positive roots. For ξ ∈ M̂ let λ_{ξ} ∈ 𝔟^{∗} denote the infinitesimal character of ξ. Recall that we have

π_{ξ,ν}(C) = B(ν) + B(λ_{ξ}) − B(ρ).

Let c = c(H) denote the number of positive roots in ϕ(𝔞, 𝔤). Since 𝔞 is a split torus and there is a real root, it follows c = 1 or c = 2.

Lemma. 5.1.2 If c = 1 let 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} where 𝔤_{1} is the simple ideal of 𝔤 that contains 𝔞. Accordingly we have G ≅ G_{1} × G_{2}. Let M_{1} = M ∩ G_{1} then we have an isomorphism as K_{M}- modules:

𝔭_{±} ≅ ℂ ⊕ 𝔭_{G_{2},±} ⊕ 𝔭_{M_{1}}.

In the case c = 2 the space M/K_{M} inherits the complex structure from G/K, i.e. 𝔭_{M} = 𝔭_{M,+} ⊕ 𝔭_{M,−} where 𝔭_{M,±} = 𝔭_{M} ∩ 𝔭_{±} and we have a K_{M}-module isomorphism:

𝔭_{±} ≅ ℂ ⊕ 𝔭_{M,±} ⊕ 𝔫_{±},

where

n_{±} = ⨁ ; \begin{matrix}α ∈ ϕ_{nI}(𝔥, 𝔤) \\ ±α|_{𝔟} > 0 \end{matrix} 𝔤_{α}.

Proof: Let H be a generator of 𝔞_{0}. Write α_{r} for the unique positive real root in ϕ(𝔥, 𝔤). Since for any root α we have 2B(α, α_{r})/B(α_{r}) ∈ ±{0, 1, 2, 3} the only possible roots in ϕ^{+}(𝔞, 𝔤) are α_{r}/2, α_{r}, 3α_{r}/2. Consider an embedding 𝔤 ↪ gl_{n} such that the Cartan involution becomes θ(X) = −X^{t} and 𝔞 is mapped to the diagonal. Since [𝔤_{α_{r}}, θ(𝔤_{α_{r}})] = 𝔞 it is easy to see that 3α_{r}/2 does not occur.

Now write H = Y + Y^{c} with Y ∈ 𝔭_{+}. According to the root space decomposition 𝔤 = 𝔞 ⊕ 𝔨_{M} ⊕ 𝔭_{M} ⊕ 𝔫 ⊕ θ(𝔫) we write Y = Y_{a} + Y_{k} + Y_{p} + Y_{n} + Y_{θ(n)}. Because of θ(Y) = −Y it follows Y_{k} = 0 and Y_{θ(n)} = −θ(Y_{n}). For arbitrary Z ∈ 𝔟 we have [Z, Y] = 0 since [Z, H] = 0 and the projection P r_{+} is k_{M}-equivariant. Hence 0 = [Z, Y] = [Z, Y_{a}] + [Z, Y_{p}] + [Z, Y_{n} − θ(Y_{n})]. Since [Z, Y_{a}] = 0 and [Z, Y_{p}] ∈ 𝔭_{M} it follows [Z, Y_{n} − θ(Y_{n})] ∈ 𝔭_{M} ∩ (𝔫 ⊕ θ(n)) = 0. Therefore Y_{p} = 0 and Y_{n} ∈ 𝔫_{r} = 𝔤_{α_{r}}. Now k_{M} acts trivially on H and thus on Y, so on Y_{n}, i.e. [𝔨_{M}, Y_{n}] = 0. Further 𝔞 ∩ 𝔭_{+} is trivial, so Y_{n} ≠ 0, so it generates the root space 𝔤_{α_{r}}, so [𝔨_{M}, 𝔤_{α_{r}}] = 0.

Assume there is a root α ≠ α_{r} such that α|_{𝔞} = α_{r}|_{𝔞}. Then B(α, α_{r}) > 0 and hence β = α − α_{r} is a root. The root β is imaginary. Suppose β is compact, then 𝔤_{β} ⊂ 𝔨_{M} and we have [𝔤_{α_{r}}, 𝔤_{β}] = 𝔤_{α}, which contradicts [𝔨_{M}, 𝔤_{α_{r}}] = 0. It follows that β is noncompact and thus it follows

𝔤_{α_{r}|_{𝔞}} = [𝔭_{M}, 𝔤_{α_{r}}] ⊕ 𝔤_{α_{r}},

where 𝔤_{α_{r}|_{𝔞}} = {X ∈ 𝔤|[H, X] = α_{r}(H)X} is the root space for the restricted root α_{r}|_{𝔞}. Let 𝔤_{1} as above the simple ideal containing 𝔞, then 𝔤_{1} also contains 𝔤_{α_{r}} and hence we may substitute 𝔭_{M_{1}} in the above. Assume the root space 𝔤_{α_{r}/2|_{𝔞}} is nonzero, then (𝔞 ⊕ 𝔪 ⊕ 𝔫_{r} ⊕ 𝔤_{α_{r}/2|_{𝔞}} ⊕ 𝔤_{−α_{r}/2|_{𝔞}})𝔤_{1} would be a nontrivial ideal of 𝔤_{1}, therefore α_{r}/2 does not occur as root of (𝔞, 𝔤) and we may state

𝔫 = [𝔭_{M_{1}}, 𝔤_{α_{r}}] ⊕ 𝔤_{α_{r}}.

The map ϕ : X ↦ [X, Y_{n}] has no kernel in 𝔭_{M_{1}} since this would similarly allow us to construct a nontrivial ideal of 𝔤_{1}. The map

ψ : 𝔞 ⊕ 𝔭_{M} ⊕ 𝔫 → 𝔭
a + p + n ↦ a + p + n − θ(n)

is a K_{M}-isomorphism as is the map ϕ. So, as K_{M}-modules 𝔭 ≅ 𝔞 ⊕ 𝔫_{r} ⊕ 𝔭_{M} ⊕ 𝔭_{M}. To derive the assertion for the case c = 1 it remains to show 𝔭_{M_{1}} ∩𝔭_{+} = 0, because then 𝔭_{M_{1}} ≅ P r_{+}(𝔭_{M_{1}}). For this assume X ∈ 𝔭_{M_{1}} ∩ 𝔭_{+}, then 0 = [X, Y] = [x, Y_{a}] + [X, Y_{n} − θ(Y_{n})] and therefore [X, Y_{n}] = 0 which implies X = 0.

The remaining case is the case where there exists no such root α as above. This means 𝔫 = 𝔤_{α_{r}/2|_{𝔞}}⊕n_{r}. This gives [𝔭_{M}, 𝔫_{r}] = 0 and thus [𝔭_{M}, Y] = 0, which implies 𝔭_{M} = 𝔭_{M,+}⊕𝔭_{M,−} = 𝔭_{M} ∩ 𝔭_{+} ⊕ 𝔭_{M} ∩ 𝔭_{−}. So 𝔭_{M} inherits the holomorphic structure and so does 𝔫. Since the maps ϕ and ψ above are K_{M}-homomorphisms and 𝔭_{+} is given by a choice of positive roots the claim follows. ■

In this section we will prove a general lemma which we will apply to the group M later. So here we will not assume G connected, but G should be of inner type (see [HC-HA1] Lemma 4.9).

Let (τ, V_{τ}) denote an irreducible unitary representation of K. Since G is of inner type, the Casimir operator C_{K} of K will acts as a scalar τ(C_{K}) on V_{τ}.

Lemma. 5.1.3 Let (π, W_{π}) be an irreducible unitary representation of G and assume that

∑_{p=0}^{dim 𝔭_{−}}(−1)^{p} dim(W_{π} ⊗ ∧^{p}𝔭_{−} ⊗ V_{τ})^{K} ≠ 0

then we have

π(C) = τ(C_{K}) − B(ρ) + B(ρ_{K}).

Proof: Consider 𝔭 as a subspace of the Clifford algebra Cl(B, 𝔭). We will make S = ∧^{∗}𝔭_{−} a Cl(B, 𝔭)-module. For this let x ∈ 𝔭_{−} act on S via

x.z_{1} ∧ … ∧ z_{n} = \sqrt{2} x ∧ z_{1} ∧ … ∧ z_{n},

and y ∈ 𝔭_{+} via

y.z_{1} ∧ … ∧ z_{n} = \sqrt{2} ∑_{i=1}^{n}(−1)^{i+1}B(y, z_{i})z_{1} ∧ … ẑ_{i} … ∧ z_{n}.

This prescription turns S into a nontrivial Cl(B, 𝔭)-module. Since there is only one such of the dimension of S, we conclude that S is the nontrivial irreducible Cl(B, 𝔭)-module. Therefore the argumentations of [AtSch] apply to S and especially the formula of Parasarathy ([AtSch], (A13)). Note that in [AtSch] everything was done under the assumption of G being connected. Since G is of inner type, however, π(C) will be a scalar. Writing G^{0} for the connected component, the representation π|_{G^{0}} will decompose as a finite sum of irreducibles on each of which the formula of Parasarathy holds. Thus it holds globally. Let d_{±} be defined as in loc. cit. then our assumption leads to ker(d_{+}d_{−})∩π⊗S(τ) ≠ 0, and therefore 0 = τ(C_{K})−π(C)−B(ρ)+B(ρ_{K}). ■

Lemma. 5.1.4 Assume c = c(H) = 2 and let (σ, W_{σ}) be an irreducible M-subrepresentation of ∧^{l}𝔫 such that W_{σ} ∩ ∧^{l}𝔫_{−} ≠ 0. Since M is of inner type there is an infinitesimal character λ_{σ} of σ and we have

B(λ_{σ}) − B(ρ) = −(1 + 2 dim(𝔫_{−}) − l)^{2}B(\frac{α_{r}}{2} ),

where α_{r} is the unique positive real root of 𝔱_{M} ⊕ 𝔞.

Proof: The assertion follows from Corollary 5.7 of [Kost], here one uses the fact that the boundary map ∂ of the Lie algebra homology H_{∗}(𝔫) maps ⋀^{∗}(𝔫_{+} ⊕ 𝔫_{−}) to 𝔫_{r}∧ ⋀^{∗}(𝔫_{+} ⊕ 𝔫_{−}) and maps ⋀^{∗} 𝔫_{−} to zero. ■

Lemma. 5.1.5 Assume c = c(H) = 2 and let (τ, V_{τ}) be an irreducible K_{M}-subrepresentation of ∧^{l}𝔫_{−}, then the Casimir operator C_{K_{M}} of K_{M} acts on V_{τ} by the scalar

τ(C_{K_{M}}) = B(ρ) − (1 + 2 dim(𝔫_{−}) − l)^{2}B(\frac{α_{r}}{2} ) − B(ρ_{K_{M}}) − \frac{2lB(ρ_{M,n})}{dim(𝔭_{M,−})},

where ρ_{M,n} is the half of the sum of the noncompact roots of M, so ρ_{M,n} = ρ_{M} − ρ_{K}. In the case that rank(X) = 1, the last summand does not occur.

Convention. In the case of rank one the last summand is undefined. To keep the formulas unified we will agree to consider it as zero then.

Proof: Let x_{1}, …, x_{n} be a basis of 𝔭_{M,+} consisting of root vectors with respect to 𝔱_{M} and such that B(x_{j}, x_{j}^{c}) = 1, where x^{c} is the complex conjugate of x. Let L be the element of the Lie algebra of 𝔨_{M} defined by

L = ∑_{j}[x_{j}, x_{j}^{c}].

Since (x_{j}^{c}) is the dual basis to (x_{j}) it follows that L lies in the center of 𝔨_{M}. Furthermore L is imaginary so that weights will take real values on L.

Consider the K_{M}-module 𝔫_{−}. We will show now that the center of 𝔨_{M} acts on n_{−} by a character. To this end recall that 𝔱 is spanned by 𝔱_{M} and X + θ(X), where 0 ≠ X ∈ 𝔫_{r}. Let Z be a generator of the center of 𝔨 then

Z = a(X + θ(X)) + bL + R

where R lies in L^{⊥} ∩ 𝔱_{M}. Since the latter is just the dual space of the span of the compact roots in 𝔱_{M}^{∗} it follows R = 0.

Now Z acts as a scalar μ on ψ(𝔫_{−}) and it is easy to see that ab ≠ 0. Let Y ∈ 𝔤_{α} ⊂ 𝔫_{−}. A computation shows

μY = a[X, θ(Y)] + b[L, Y].

So it remains to show that Y ↦ [X, θ(Y)] acts as a scalar on 𝔫_{−}. To this end let H = [X, θ(X)] ∈ 𝔞 then

(α_{r}(H)/2)Y = [H, Y] = [X, [θ(X), Y]].

Since root spaces are one dimensional there exists a number c = c_{Y} such that [X, θ(Y)] = cY. Therefore [θ(X), Y] = cθ(Y) and (α_{r}(H)/2)Y = c^{2}Y. By this c is determined up to sign. So L acts on 𝔫_{−} by at most two eigenvalues. Suppose there are two eigenvalues and an according decomposition

𝔫_{−} = 𝔫_{−}^{0} ⊕ 𝔫_{−}^{1},

where 𝔫_{−}^{0} corresponds to the character of less absolute value. We will show that 𝔤_{1} ∩ (𝔞 ⊕ 𝔪 ⊕ 𝔫_{−}^{0} ⊕ 𝔫_{−}^{o c} ⊕ θ(𝔫_{−}^{0}) ⊕ θ(𝔫_{−}^{o c}) is an ideal of 𝔤_{1} which cannot be since 𝔤_{1} is simple, thus proving that there is only one eigenvalue of L on 𝔫_{−}. For this it suffices to show [𝔭_{M,−}, 𝔫_{−}] = 0. Let X ∈ 𝔫_{−} then X − θ(X) is in 𝔭_{−} and therefore [𝔭_{M,−}, X − θ(X)] = 0. But 𝔭_{M} also leaves stable the 𝔞-root spaces, therefore [𝔭_{M,−}, X] = 0. We have shown that the center of 𝔨_{M} acts be a character χ_{𝔫_{−}} on 𝔫_{−}. As for the value of this character recall that the center of 𝔨_{M} acts by a character χ on 𝔭_{M,−}. The assumption χ ≠ χ_{𝔫_{−}} would similarly allow us to construct a nontrivial ideal, hence it follows χ = χ_{𝔫_{−}}.

Recall that each x_{j} is a root vector, say x_{j} ∈ 𝔪_{α} then x_{j}^{c} ∈ 𝔪_{−α} and [x_{j}, x_{j}^{c}] = H_{α} where H_{α} ∈ 𝔱_{M} is defined by α(H) = B(H, H_{α}). Therefore

L = ∑_{α∈ϕ_{noncompact}^{+}(𝔱_{M} ,𝔪)}H_{α}.

So let α be a noncompact negative root in ϕ(𝔱_{M}, 𝔪). Since L acts on 𝔫_{−} by the same scalar as on 𝔭_{M,−} we get

χ_{𝔫_{−}}(L) = 2B(ρ_{M,n}, α) = −\frac{2B(ρ_{M,n})}{dim(𝔭_{M,−})}.

Now let μ be the lowest weight of τ and let σ be the M-representation generated by τ. We may assume that σ is irreducible. We claim that μ is the lowest weight vector of σ. To see this it suffices to see that σ(𝔭_{M,−})V_{τ} = 0, which in turn follows from σ(𝔭_{M,−})𝔫_{−} = [𝔭_{M,−}, 𝔫_{−}] = 0. We thus have shown that μ is the lowest weight vector of σ. So by Lemma 5.1.4 we get

B(μ − ρ_{M}) − B(ρ) = −(1 + 2 dim(𝔫_{−}) − l)^{2}B(\frac{α_{r}}{2} ).

On the other hand we know τ(C_{K_{M}}) = B(μ − ρ_{K_{M}}) − B(ρ_{K_{M}}). We have

B(μ − ρ_{M}) = B(μ − ρ_{K_{M}} − ρ_{M,n}).

Note that 2ρ_{M,n} is the dual of L. Since L is in the center of 𝔨_{M} it follows

B(ρ_{K_{M}} , ρ_{M,n}) = 0.

Therefore

B(μ − ρ_{M}) = B(μ − ρ_{K_{M}}) − 2B(μ − ρ_{M,n})
= B(μ − ρ_{K_{M}}) − μ(L)
= B(μ − ρ_{K_{M}}) + \frac{2lB(ρ_{M,n})}{dim(𝔭_{M,−})}.

From this the claim follows. ■

Define f̂_{t}^{0}_{,H} as in [HC-S]. Letting b^{∗} ∈ B^{∗} we therefore get in the case that c(H) = 2:

f̂_{t}^{0}_{,H}(ν, b^{∗}) = −e^{tB(ν)}

× ∑_{l=0}^{dim(𝔫_{−})} ∑_{p=0}^{dim 𝔭_{M,−}}(−1)^{p+l}e^{−t((1+2 dim(𝔫_{−})−l)^{2}B( \frac{αr}{2} )+ \frac{2lB(ρ_{M,n})}{dim(𝔭_{M,+})})} dim(V_{b^{∗}} ⊗ ∧^{p}𝔭_{M,−} ⊗ ∧^{l}𝔫_{−})^{K_{M}} .

In the other case, c(H) = 1 we get

Lemma. 5.1.6 Assume c(H) = 1 then for π_{ξ,ν} as in Lemma 5.1.1 we get

tr(π_{ξ,ν}(f_{t}^{0})) = −e^{t(B(ν)+B(ρ_{K∩G_{2}})+B(ρ_{M_{1}})−B(ρ))}

×( ∑_{j=0}^{dim(𝔭_{M_{1}})}(−1)^{j} dim(W_{ξ_{1}} ⊗ ∧^{j}𝔭_{M_{1}})^{K∩M_{1}})( ∑_{k+0}^{dim(𝔭_{G_{2},−})}(−1)^{k} dim(W_{ξ_{2}} ⊗ ∧^{k}𝔭_{G_{2},−})^{K∩G_{2}})

Proof: We have to show that tr(π_{ξ,ν}(f_{t}^{0})) ≠ 0 implies B(λ_{ξ}) = B(ρ_{M_{1}} + ρ_{K∩G_{2}}). We have M = M_{1} × G_{2} and thus ξ = ξ_{1} ⊗ ξ_{2}. Therefore

tr(π_{ξ,ν}(f_{t}^{0})) = −e^{tπ_{ξ,ν}(C)} ∑_{q=0}^{dim(𝔭_{−})−1}(−1)^{q} ∑_{j+k=q} dim(W_{ξ_{1}} ⊗ ∧^{j}𝔭_{M_{1}})^{K∩M_{1}} dim(W_{ξ_{2}} ⊗ ∧^{k}𝔭_{G_{2},−})^{K∩G_{2}}

= −e^{tπ_{ξ,ν}(C)}( ∑_{j=0}^{dim(𝔭_{M_{1}})}(−1)^{j} dim(W_{ξ_{1}} ⊗ ∧^{j}𝔭_{M_{1}})^{K∩M_{1}})( ∑_{k+0}^{dim(𝔭_{G_{2},−})}(−1)^{k} dim(W_{ξ_{2}} ⊗ ∧^{k}𝔭_{G_{2},−})^{K∩G_{2}})

Now assume tr(π_{ξ,ν}(f_{t}^{0})) ≠ 0 then Lemma 5.1.3 implies B(λ_{ξ_{2}}) = B(ρ_{K∩G}) whereas Lemma 2.4 in [MS-2] gives B(λ_{ξ_{1}}) = B(ρ_{M_{1}}). ■

Therefore in the case c(H) = 1 we get

f̂_{t}^{0}_{,H}(ν, b^{∗}) = f̂_{t}^{0}_{,H}(ν, b_{1}^{∗} + b_{2}^{∗}) = −e^{t(B(ν)+B(ρ_{K∩G_{2}})+B(ρ_{M_{1}})−B(ρ))}

( ∑_{j=0}^{dim(𝔭_{M_{1}})}(−1)^{j} dim(V_{b_{1}^{∗}} ⊗ ∧^{j}𝔭_{M_{1}})^{K∩M_{1}})( ∑_{k+0}^{dim(𝔭_{G_{2},−})}(−1)^{k} dim(V_{b_{2}^{∗}} ⊗ ∧^{k}𝔭_{G_{2},−})^{K∩G_{2}}).

Since Γ is the fundamental group of X_{Γ}, every conjugacy class [γ] in Γ defines a free homotopy class of closed paths in X_{Γ}. It is known [DKV] that the union X_{γ} of all closed geodesics which are homotopic to [γ] is a smooth submanifold of X_{Γ}. Let χ_{1}(X_{γ}) denote the first higher Euler number of X_{γ} (see [D-Hitors]), i.e.

χ_{1}(X_{γ}) = ∑_{p=0}^{dim(X_{γ})} p(−1)^{p}b_{p}(X_{γ}),

where b_{p}(X_{γ}) is the p-th Betti number of X_{γ}.

Now let E_{H}(Γ) denote the set of nontrivial Γ-conjugacy classes, which are in G conjugate to an element of H. For l ≥ 0 define

b_{l}(H) = (\frac{c(H)}{2} + dim(𝔫) − 1 − l)|\frac{α_{r}}{c(H)}|.

If c(h) = 2 let

d_{l}(H) = \sqrt{b_{l}(H)^{2} + \frac{2lB(ρ_{M,n})}{dim(𝔭_{M,−})}}.

In the case c(H) = 1 we finally set

d(H) = \sqrt{B(ρ) − B(ρ_{K∩G_{2}}(H)) − B(ρ_{M_{1}}(H))}.

With an argumentation as in [D-Hitors] we get

Theorem 5.1.1 Let X_{Γ} be a compact locally hermitian space with fundamental group Γ and such that the universal covering is globally symmetric of the noncompact type. Let △_{p,q,φ} be the Hodge Laplacian on (p,q)-forms with values in a the flat bundle E_{φ}, then

Θ(t) = ∑_{q=0}^{dim_{ℂ} X_{Γ}} q(−1)^{q}T re^{−t△_{0,q,φ}}

= ∑ ; \begin{matrix}H/conj. \\ c(H) = 2 \end{matrix} ∑_{[γ]∈E_{H}(Γ)} \frac{l_{γ_{0}}χ_{1}(X_{γ})trφ(γ)}{|W (𝔥, 𝔤_{γ})| ∏_{α∈ϕ_{γ}^{+}} B(ρ_{γ}, α)det(1 − γ^{−1}|𝔫)} \frac{e^{−l_{γ}^{2}/4t}}{\sqrt{4πt}}

∑_{l=0}^{dim(𝔫_{−})}(−1)^{l}e^{−td_{l}(H)^{2}} \frac{ω̃_{γ}(γ^{ρ_{M}} det(1 − γ^{−1}|(𝔨/𝔱)^{+})tr(γ_{I}| ∧^{l} 𝔫_{−}))}{γ^{ρ_{M}} ∏_{α∈ϕ_{I}^{+}−ϕ_{γ}^{+}}(1 − γ^{−α})}

+ ∑ ; \begin{matrix}H/conj. \\ c(H) = 1 \end{matrix} ∑_{[γ]∈E_{H}(Γ)} l_{γ_{0}}χ_{1}(X_{γ})\frac{e^{−l_{γ}^{2}/4t}}{\sqrt{4πt}} e^{−b_{0}(H)l_{γ}}e^{td(H)^{2}}

+f_{t}^{0}(e) dimφ vol(X_{Γ}),

where ω̃_{γ} is the usual differential operator [HC-DS] p. 33. For unexplained notation we refer to [D-Hitors].

The reader should keep in mind that by its definition we have for the term of the identity:

f_{t}^{0}(e) dimφ vol(X_{Γ}) = ∑_{q=0}^{dim_{ℂ} X} q(−1)^{q}tr_{Γ}(e^{−t△_{0,q,φ}}),

where tr_{Γ} is the Γ-trace (1.2.1). Further note that by the Plancherel theorem the Novikov-Shubin invariants of all operators △_{0,q} are positive.

Now assume we are given a unitary irreducible representation (τ, V_{τ}) of K. The representation τ defines a G-homogeneous vector bundle E_{τ} over X = G/K and we will consider the space of smooth sections of E_{τ}:

C^{∞}(E_{σ}) = (C^{∞}(G) ⊗ V_{τ})^{K}.

The Casimir operator C of G acts on this space and defines a second order elliptic differential operator C_{τ} on E_{τ}. The methods of [BM] indicate that the heat operator e^{tC_{τ}} , t > 0, acts by a smooth convolution kernel h_{t}^{τ} which is in the Harish-Chandra Schwartz space S(G). For any π ∈ Ĝ it follows

trπ(h_{t}^{τ}) = e^{tπ(C)} dim(V_{π} ⊗ V_{τ})^{K}.

By Lemma 5.1.3 there is a linear combination g_{τ} of the functions h_{t}^{τ′} such that g_{τ} ∈ S(G) and

trπ(g_{τ}) = ∑_{p=0}^{dim(𝔭_{−})} dim(V_{π} ⊗ ∧^{p}𝔭_{−} ⊗ V_{τ})^{K}.

Similar argumentations as in 5.1.1 lead to trπ_{ξ,ν}(g_{τ}) = 0 for any properly induced representation π_{ξ,ν}.

Now let (σ, W_{σ}) be a finite dimensional irreducible representation of G. Assume W_{σ} is furnished with a scalar product such that σ|_{K} is unitary. Lemma 2.4 of [MS-2] says that if

∑_{p=0}^{dim 𝔭}(−1)^{p} dim(V_{π} ⊗ ∧^{p}𝔭 ⊗ W_{σ})^{K} ≠ 0,

then

B(λ_{π}) = B(λ_{σ}).

So there is a function f_{σ} ∈ S(G) such that for any π ∈ Ĝ:

trπ(f_{σ}) = ∑_{p=0}^{dim 𝔭}(−1)^{p} dim(V_{π} ⊗ ∧^{p}𝔭 ⊗ W_{σ})^{K}.

Let f_{σ}, g_{τ} as in the previous section. We want to compute orbital integrals of these functions. Recall that an element g of G is called elliptic if it lies in a compact Cartan subgroup.

Proposition 5.1.2 Let g be a semisimple element of the group G. If g is not elliptic, the orbital integrals O_{g}(f_{σ}) and O_{g}(g_{τ}) vanish. If g is elliptic we may assume g ∈ T, where T is a Cartan in K and then we have

O_{g}(f_{σ}) = \frac{\overline{trσ(g)}|W (𝔱, 𝔤_{g})| ∏_{α∈Φ_{g}^{+}}(ρ_{g}, α)}{[G_{g} : G_{g}^{0}]c_{g}},

for all elliptic g and

O_{g}(g_{τ}) = \frac{\overline{trτ(g)}}{det(1 − g^{−1}|𝔭_{+})},

if g is regular elliptic. For general elliptic g we have

O_{g}(g_{τ}) = \frac{∑_{s∈W (T,K)} det(s)ω̃_{g}g^{sλ_{τ∗}+ρ−ρ_{K}}}{[G_{g} : G_{g}^{0}]c_{g}g^{ρ} ∏_{α∈ϕ^{+}−ϕ_{g}^{+}}(1 − g^{−α})},

where c_{g} is Harish-Chandra’s constant, it does only depend on the centralizer G_{g} of g. Its value is given in [D-Hitors], further ω̃_{γ} is the differential operator as in [HC-DS] p.33.

Proof: The vanishing of O_{g}(f_{σ}) for nonelliptic g is immediate by Harish-Chandra’s formula for the Fourier transform of orbital integrals [HC-S]. Now let g ∈ K ∩ G′, where G′ denotes the set of regular elements. We apply Lemma 4.3 of [MS-1] to get

O_{g}(f_{σ}) = \overline{trσ(g)}.

This proves the proposition in the regular case. The general case is derived from this by standard considerations (see [HC-DS], p32 ff.). A word of comment is in order here. In [MS-1] everything is formulated in terms of a certain subgroup M^{+} of M. One could either do the same here or see that Lemma 4.3 in [MS-1] actually holds for M as well since the latter still is in the Harish-Chandra class (Lemma 4.9 in [HC-HA1]). One sees by Frobenius reciprocity that the right hand side of Lemma 4.2 in [MS-1] actually gives the same for M as for M^{+}.

The case g_{τ} is treated similarly. ■

5.2 A Simple Example

Let m be a natural number and consider the group G = SL_{2}(ℝ)^{m}. A maximal compact subgroup is given by K = SO(2)^{m} and we have G/K ≅ H_{2}(ℝ)^{m}. On the latter there is a canonical metric giving it the curvature −1. To induce this metric we normalize the form B such that B(α) = 1 for every root α. For Haar measure we take the Euler-Poincaré measure.

The Cartan subgroups of splitrank one are modulo conjugation given by

A × T × … × T, T × A × T × … × T, … T × … × T × A,

where

A = \{( \begin{matrix}a \\ & a^{−1} \end{matrix} ) , a ≠ 0\} , T = SO(2).

Hence it follows that all the constants c(H) in Theorem 6.1.1 are equal to \frac{1}{2}, all spaces 𝔫_{±} are trivial and thus the summation index l only takes the value zero. As to f_{t}^{0}(e) we have

f_{t}^{0}(e) = ∑_{q=0}^{m} q(−1)^{q}tr < x|e^{−t△_{0,q,φ}}|x >

= dim(φ) ∑_{q=0}^{m} q(−1)^{q}tr < x|e^{−t△_{0,q}}|x >

Now let △^{0} denote the laplacian on H_{2}(ℝ) and let △^{1} twice the Laplacian on (0, 1)-forms on H_{2}(ℝ). Then we have

Lemma. 5.2.1

tr < x|e^{−△^{1}}|x > −tr < x|e^{−△^{0}}|x >= \frac{1}{2}

Proof: Write for the moment G = SL_{2}(ℝ) and K = SO(2). Let h_{t}^{0} denote the convolution kernel of e^{−t△^{0}} in

(C^{∞}(G) ⊗ End(𝔭_{−}))^{K×K}

and write h_{t}^{1} for the △^{1}-analogue. Let g_{t}^{0}(x) = trh_{t}^{0}(x) and g_{t}^{1}(x) = trh_{t}^{1}(x), where the trace is the trace of End(𝔭_{−}). The lemma amounts to

g_{t}^{1}(e) − g_{t}^{0}(e) = \frac{1}{2}.

We apply the Plancherel theorem to g_{t}^{0} and g_{t}^{1}. For any π ∈ Ĝ we get

trπ(G_{t}^{0}) = e^{tπ(C)} dim(V_{π})^{K},
trπ(G_{t}^{1}) = e^{tπ(C)} dim(V_{π} ⊗ 𝔭_{−})^{K}.

There is, however, only one π ∈ Ĝ for which these two dimensions differ, namely π equals the discrete series representation with Harish-Chandra parameter λ_{π} = ρ and for this representation we have with our normalizations:

π(C) = 0, dim(V_{π})^{K} = 0, dim(V_{π} ⊗ 𝔭_{−})^{K} = 1

and d_{π} = \frac{1}{2}, whence the lemma. ■

CHAPTER 5 The Holomorphic Torsion Zeta Function 75

Now since X = H_{2}(ℝ)^{m} is a product space, the heat kernels can be written as products, so

e^{−t△_{0,q}} = ∑_{I⊂{1,...;m}|I|=q} e^{−t△^{1_{I} (1)}} ⊗ … ⊗ e^{−t△^{1_{I} (m)}},

where 1_{I} is the characteristic function of I. From this we take

f_{t}^{e}(e) = dim(φ) ∑_{q=0}^{m} q(−1)^{q} ∑ ; \begin{matrix}I ⊂ \{1, …; m\} \\ |I| = q \end{matrix} ∏_{j=1}^{m} tr(, x|e^{−t△^{1_{I} (j)}}|x >
= dim(φ) ∑_{q=0}^{m} q(−1)^{q} ( \begin{matrix}m \\ q \end{matrix} ) g_{t}^{0}(e)^{m−q}g_{t}^{1}(e)^{q}
= dim(φ)m ∑_{q=1}^{m} ( \begin{matrix}m − 1 \\ q − 1 \end{matrix} ) g_{t}^{0}(e)^{m−q}(−g_{t}^{1}(e))^{q}
= dim(φ)m(−g_{t}^{1}(e))(g_{t}^{0}(e) − g_{t}^{1}(e))^{m−1}
= dim(φ)m(−1)^{m}(\frac{1}{2} + g_{t}^{0}(e))2^{1−m}

These calculations now easily lead to Theorem 5.2.1 Let m be a natural number and Γ a nice cocompact lattice in G = SL_{2}(ℝ)^{m}. Let φ be a finite dimensional unitary representation of Γ. The zeta function

Z_{φ}(s) =^{def}  ∏ ; [γ]prime ℝ − rank[γ] = 1 ∏_{k,n≥0} det(1 − e^{(s+n)l_{γ}}φ(γ) ⊗ Ad(γ_{I}|S^{k}(𝔭_{M,−})))^{χ_{1}(X_{γ})}

extends to a meromorphic function on ℂ. One has

Z_{φ}(\frac{1}{2} + s) = ∏_{q=0}^{m} det(△_{0,q,φ} + s^{2} − \frac{1}{4})^{q(−1)^{q}}(exp(s^{2})det(P + s))^{−2mdimφAr(X_{Γ})} where Ar(X_{Γ}) is the arithmetic genus of X_{Γ} and P = \sqrt{△^{d} + \frac{1}{4}} as in Theorem 4.1.1■ Corollary 5.2.1 The function Ẑ_{φ}(s) = Z_{φ}(s)det(P + s − \frac{1}{2})^{2mdimφAr(X_{Γ})} satisfies

Ẑ_{φ}(s) = Ẑ_{φ}(1 − s).■

Corollary 5.2.2 The ”factor at infinity” is for Re (s) >> 0 representable as

(exp(s^{2})det(P + s))^{−2mdimφAr(X_{Γ})} = ∏_{q=0}^{m}(det^{(2)}(△_{0,q,φ} + s^{2} − \frac{1}{4}))^{q(−1)^{q+1}}.■

Corollary 5.2.3 The function Z_{φ} vanishes at s = \frac{1}{2} up to order

n_{0} = −2mdimφAr(X_{Γ}) + ∑_{q=0}^{m} q(−1)^{q}h_{0,q}(φ),

where h_{p,q}(φ) is the (p, q)-th Hodge number. With R_{φ}(s) = Z_{φ}(s)s^{−n_{0}} we have

R_{φ}(\frac{1}{2}) = \frac{T_{hol}(X_{Γ}, φ)}{T_{hol}^{(2)}(X_{Γ})^{dimφ}} .

Further we have

T_{hol}^{(2)}(X_{Γ}) = (2^{12}exp(ζ′(1)))^{−2mdimφAr(X_{Γ})},

where ζ is the zeta function of Riemann. ■

In the theorem we assumed Γ to be nice in order to have a simple Euler product. Note that in the case G = SL_{2}(ℝ) every torsionfree Γ automatically is nice. But the theorem also extends to the case of nonnice groups as follows:

Let γ ∈ Γ then γ is the power of some prime γ_{0}, but since Γ s not nice, it may happen that G_{γ} ≠ G_{γ_{0}}. But then there is a least power γ_{0}^{n} of γ_{0} such that G_{γ_{0}^{n}} = G_{γ}. Let λ_{γ} denote the length of the geodesic attached to γ_{0}^{n}, the so called generic geodesic in X_{γ}. Then let μ_{γ} = l_{γ}/λ_{γ} the generic multiplicity of γ. For example, assume that the only root of unity of γ_{0} is −1, then we have

μ_{γ_{0}^{2}n} = n μ_{γ_{0}^{2}n+1} = 2n + 1.

Further for γ ∈ Γ let P er(γ) denote the set of orders of nontrivial roots of unity occurring as eigenvalues of the adjoint representation Ad(γ), so m is in P er(γ) if and only if a primitive m-th root of unity occurs as an eigenvalue of Ad(γ). We have: Γ is nice if and only if P er(γ) = ∅ for all γ ∈ Γ.

For any subset I ⊂ P er(γ) let n_{I} denote the least common multiple of the elements of I. Set n_{∅} = 1. Consider

Z_{φ}(s) =^{def}  ∏ ; [γ]prime ℝ − rank[γ] = 1 ∏_{I⊂P er(γ)} ∏_{k,n≥0} det(1 − e^{n_{I}(s+n)l_{γ}}φ(γ) ⊗ Ad(γ_{I}|S^{k}(𝔭_{M,−})))^{m_{I}(γ)},

where

m_{I}(γ) = ∑_{J⊂I}(−1)^{|I|+|J|+1}n_{J}χ(X̂_{γ^{n_{J}}})/μ_{γ^{n_{J}}} .

Here χ(X̂_{γ^{n_{J}}}) denotes the orbifold Euler characteristic of X̂_{γ^{n_{J}}} = X_{γ^{n_{J}}} /(geodesic flow). Note that χ(X̂_{γ^{n_{J}}}) is no longer an integer but a rational number. This cannot cause any trouble since Euler factors are all close to one and the powers are defined by the principal branch of the logarithm.

Theorem 5.2.2 With Γ ⊂ SL_{2}(ℝ) torsionfree and cocompact and Z_{φ} defined as above the preceding theorem and all corollaries hold as well.■

5.3 The Rank One Case

Now assume that the rank of X is one. From the classification of globally symmetric spaces it follows that X = SU(m, 1)/S(U(m)×U(1)). So that G = SU(m, 1)/center and K = S(U(m)× U(1))/center. Since it doesn’t change the geometry we will instead work with G = SU(m, 1) and K = S(U(m) × U(1)). The Lie algebra 𝔤_{0} can be written as

𝔤_{0} = \{( \begin{matrix}Z_{1} & Z_{2} \\ Z_{2}^{∗} & −trZ_{1} \end{matrix} ) \begin{matrix}Z_{1} ∈ Mat_{n}(ℂ), Z_{1}^{∗} = −Z_{1} \\ Z_{2} ∈ ℂ^{m} arbitrary \end{matrix} \}

where for a complex matrix A we write A^{∗} = Ā^{t}. The Lie algebra 𝔤_{0} is considered as a real Lie algebra but it is formed out of complex matrices. To avoid conflicts with the complexification we will write Z ∈ 𝔤_{0} as Z = X + IY with X, Y ∈ Mat_{m+1}(ℝ) further we write Z̄ = X − IY whereas Z^{c} will denote the complex conjugation on 𝔤 as before.

Instead of the Killing form on 𝔤 we will rather use the form

B = \frac{1}{4(m + 1)} × Killing form.

Then it follows that B(X, Y) = \frac{1}{2}Re(tr(XY)) where Re stands for the real part on 𝔤_{0}. A maximal split torus of 𝔤_{0} is given by

𝔞_{0} = ℝH_{0} where H_{0} = ( \begin{matrix}0 \\ & 0 & 1 \\ & 1 & 0 \end{matrix} ) .

The Lie algebra 𝔨_{0} is given by

𝔨_{0} = \{( \begin{matrix}Z & 0 \\ 0 & −trZ \end{matrix} ) | Z^{∗} = −Z\} ,

and the centralizer of 𝔞_{0} is 𝔞_{0} + 𝔪_{0}, where

m_{0} = \{( \begin{matrix}X & 0 & 0 \\ 0 & −\frac{1}{2}trX & 0 \\ 0 & 0 & −\frac{1}{2}trX \end{matrix} ) \begin{matrix}X ∈ Mat_{m−1}(ℂ) \\ X^{∗} = −X \end{matrix} \} .

Let 𝔱_{0} denote the space of diagonal matrices in 𝔤_{0}, then T = exp(𝔱_{0}) is a compact Cartan subgroup of G lying in K. Let 𝔱_{𝔪,o} = 𝔱_{0} ∩ 𝔪_{0}. This is a Cartan subalgebra of m_{0} and 𝔱_{𝔪} ⊕ 𝔞 is a Cartan of 𝔤. We are going to consider the root system Φ(𝔱_{𝔪} ⊕ 𝔞, 𝔤) of this torus. We have Φ(𝔱_{𝔪} ⊕ 𝔞, 𝔤) = Φ(𝔱_{𝔪}, 𝔪) ∪ Φ_{nI}(𝔱_{𝔪} ⊕ 𝔞, 𝔤), here nI means nonimaginary roots. There is up to sign a unique real root α_{r}. We have α_{r}(H_{0}) = 2.

Choose a real parabolic subalgebra 𝔞_{0} ⊕ 𝔪_{0} ⊕ 𝔫_{0} and fix an ordering on the roots compatible with the choice of the parabolic. Then the unipotent radical 𝔫 splits as n = 𝔫_{r} ⊕ 𝔫_{−} ⊕ 𝔫_{+} where n_{r} = 𝔤_{α_{r}} and

𝔫_{±} = ⨁ ; \begin{matrix}α ∈ Φ_{nI}^{+} \\ ±α | 𝔱_{𝔪} > 0 \end{matrix} 𝔤_{α}.

We will make use of the results in chapter 4. In chapter 4 we considered the generalized Selberg Zeta Function:

Z_{σ,φ}(s) = ∏ ; \begin{matrix}1 ≠ [γ] \\ prime \end{matrix} ∏_{N≥0} det(1 − e^{−sl_{γ}}γ | V).

The product is taken over all nontrivial primitive conjugacy classes in Γ and any γ ∈ Γ induces an operator, well defined up to conjugacy, on V = V_{φ} ⊗ V_{σ} ⊗ S^{N}(𝔫) via γ ↦ φ(γ) ⊗ σ(m_{γ}) ⊗ Ad^{N}((m_{γ}a_{γ})^{−1}). Let λ_{σ} denote the infinitesimal character of σ and let c_{σ} = B(λ_{σ}) − B(ρ). The main result of chapter 4 is the determinant formula

Z_{σ,φ}(m + s) = det(△_{σ,φ} + s^{2} + c_{σ})(det(D_{σ} + s)exp(P_{σ}(s^{2}))^{2(−1)^{m}dimφχ(X_{Γ})/χ(X^{d})}

where △_{σ,φ} is a virtual differential operator. P_{σ} is a polynomial which can be calculated via 1.2 and the fact that logZ_{σ,φ} tends to zero as s → ∞. This will be needed later.

Now fix any K-type τ ∈ K̂ then it is known [Zel] that τ reduces to M with multiplicity one. We define Z_{τ,φ} = ∏_{σ∈M̂ σ|τ} Z_{σ,φ}, where the symbol σ | τ means that σ occurs in hermitian homogeneous vector bundle E_{τ} over X which pushes down to a bundle E_{τ,Γ} over X_{Γ}. Let △_{τ,φ} denote the Laplacian corresponding to E_{τ,Γ} ⊗ φ. Let τ_{l} denote the representation of K on ∧^{l}𝔭_{+} then it is easy to see that τ |_{M}= σ_{l} ⊕ σ_{l−1} where σ_{l} is the representation of M on ∧^{l}𝔫_{+}. Let △_{l} = ∑_{n≥0}(−1)^{l+n}△_{τ_{n}}. Where △_{σ_{l}} is defined as in chapter 4. Note that we have △_{l} = △_{σ_{l}}+ finite operator. More precisely we get

det(△_{l} + λ) = det(△_{σ_{l}} + λ) ∏_{n=0}^{l} ∏_{π∈Ĝ_{d}}(π(C) + λ)^{[π:τ_{n}]dim_{Γ}(π)(−1)^{l+n}}

a calculation now shows that only for n = l = m we have a nontrivial contribution. Namely for the π in the discrete series with Harish-Chandra parameter

λ_{π} = ρ_{G}

and therefore π(C) = 0. Further we have dim_{γ}(π) = (−1)^{m}Ar(X_{Γ}), where Ar(X_{Γ}) denotes the arithmetic genus of X_{Γ}. From this we get

△_{l} = △_{σ_{l}} for l < m and
det(△_{m} + λ) = det(△_{σ_{l}} + λ)λ^{(−1)^{m}Ar(X_{Γ})}.

Further let D_{l} = D_{σ_{l}}, we are going to give this operator a geometric meaning relating to the dual space later.

Let P = MAN the Langlands decomposition of a minimal parabolic P, fix an irreducible unitary representation ξ of M with infinitesimal character λ_{ξ}. Recall that we have for the Casimir eigenvalue on a principal series representation π_{ξ,ν} that π_{ξ,ν}(C) = B(ν) + B(λ_{ξ}) − B(ρ). Now for l ≥ 0 let Z_{φ}^{l} denote the zeta function attached to the representation of M on ∧^{l}𝔫_{−}. Let

Z_{φ}(s) = ∏_{l=0}^{m−1} Z_{φ}^{l}(s + l)^{(−1)^{l}}.

Then we easily see that

Z_{φ}(s) = ∏ ; \begin{matrix}1 ≠ [γ] \\ prime \end{matrix} ∏_{N≥0} det(1 − e^{−sl_{γ}}φ(γ) ⊗ Ad^{N}(γ^{−1} | 𝔫_{r} ⊕ 𝔫_{−}))

Proposition 5.3.1 We have

Z_{φ}(s) = ∏_{l=0}^{m−1}(\frac{det(△_{l} + (s + l − m)^{2} − (l − m)^{2})}{det^{(2)}(△_{l} + (s + l − m)^{2} − (l − m)^{2})^{dimφ}} )^{(−1)^{l}}

= ∏_{l=0}^{m−1} det(△_{l} + (s + l − m)^{2} − (l − m)^{2})^{(−1)^{l}}

(det(D_{l} + s + l − m)exp(P_{l}((s + l − m)^{2}))^{(−1)^{m+l}dimφχ(X_{Γ})/χ(X^{d})}

By the second identity Z_{φ}(s) can be continued to a meromorphic function on the entire plane. Note that χ(X_{Γ})/χ(X^{d}) = Ar(X_{Γ}), the arithmetic genus of X_{Γ}.

We further consider

Z̃_{φ}(s) = ∏_{l=0}^{m−1} Z_{φ}^{l}(s + 2m − l)^{(−1)^{l}}.

It follows that

Z̃_{φ}(s) = ∏ ; \begin{matrix}1 ≠ [γ] \\ prime \end{matrix} ∏_{N≥0} det(1 − e^{−(s+2)l_{γ}}γ | V)

where V = V_{φ} ⊗ ∧^{m−1}𝔫_{+} ⊗ (𝔫_{r} ⊕ 𝔫_{−}) and γ acts on V as φ(γ) ⊗ Ad(γ^{−1}) ⊗ Ad^{N}(γ^{−1})

Proposition 5.3.2 We have

Z̃_{φ}(s) = ∏_{l=0}^{m−1}(\frac{det(△_{l} + (s − l + m)^{2} − (l − m)^{2})}{det^{(2)}(△_{l} + (s − l + m)^{2} − (l − m)^{2})^{dimφ}} )^{(−1)^{l}}

= ∏_{l=0}^{m−1} det(△_{l} + (s − l + m)^{2} − (l − m)^{2})^{(−1)^{l}}

(det(D_{l} + s − l + m)exp(P_{l}((s − l + m)^{2}))^{(−1)^{m+l}dimφχ(X_{Γ})/χ(X^{d})}

We want to understand the determinant of the operator D_{l} from above. It will turn out that it has a geometric interpretation as a virtual differential operator on the dual space.

Let X^{d} denote the dual space to X. The decomposition 𝔭 = 𝔭_{+} ⊕ 𝔭_{−} establishes a complex structure on X^{d} as well. Let △_{p,q}^{d} denote the Hodge Laplace operator on (p, q)-forms. Further define △_{l}^{d} by △_{l}^{d} = D_{l}^{2} − (m − l)^{2}.

Theorem 5.3.1 We have det(△_{l}^{d} + s)det(△_{l−1}^{d} + s) = det(△_{0,l}^{d} + s)R_{l}(s), where R_{l}(s) is the polynomial given by R_{0}(s) = 1 and

R_{l}(s) = ∏_{\frac{m}{2} ≤n≤ \frac{m+l}{2}}(4n^{2} − (m − l)^{2} + s)^{Q_{l}(2n)}

for l > 0 and l+m even and

R_{l}(s) = ∏_{m≤2n−1≤m+l}((2n − 1)^{2} − (m − l)^{2} + s)^{Q_{l}(2n−1)}

for l > 0, l+m odd.

Proof of the theorem: Let τ_{l} ∈ K̂ be the representation of K on ∧^{l}𝔭_{−}. Then the space of (0,l)-forms on X^{d} is as G-module equal to (C^{∞}(G^{d}) ⊗ τ_{l})^{K}. Further we have τ_{l} |_{M}= σ_{l} ⊕ σ_{l−1} where σ_{−1} = 0, and the Laplacian is given by the Casimir operator C. Now let λ ∈ (𝔞 + 𝔱_{𝔪})^{∗} be a dominant integral weight and let V (λ) ∈ Ĝ^{d} denote the corresponding irreducible representation of the compact group G^{d}. The representations τ_{l} have the following remarkable property extending Theorem1.7 of [CaWo].

Lemma. 5.3.1 For λ as above we have that τ_{l} occurs in the restriction of V (λ) if and only if λ |_{𝔱_{𝔪}} is the highest weight of σ_{l} or σ_{l−1}.

Proof: Lets show the only if part first: Assume τ_{l} occurs in V (λ) and let W ⊂ V (λ) denote the space of τ_{l}. Write U(𝔤), U(𝔨), U(𝔫̄), U(𝔞) for the universal enveloping algebras of 𝔤, 𝔨, 𝔫̄, 𝔞, where 𝔫̄ is the nilpotent Lie algebra corresponding to the negative roots. The Iwasawa decomposition 𝔤 = 𝔫̄ ⊕ 𝔞 ⊕ 𝔨 and the PBW theorem show U(𝔤) = U(𝔫̄)U(𝔞)U(𝔨). Let P r denote the projection on V (λ) to the highest weight space V (λ)_{λ}. Then U(𝔫̄) and U(𝔞) commute with P r and we get V (λ)_{λ} = P rV (λ) = P r(U(𝔤)W) = P r(U(𝔫̄)U(𝔞)W) = U(𝔫̄)U(𝔞)P rW. Which yields P rW ≠ 0 and so λ |_{T_{𝔪}} must occur as a weight of σ_{l} or σ_{l−1}. But since λ was dominant, λ |_{𝔱_{𝔪}} will be M-dominant and an inspection shows that the only dominant weight of σ_{l} is its highest weight. This gives the only if direction.

For the if part we will consider the underlying algebraic groups. We also change from the torus A to a Cayley transform of A lying in K. This doesn’t change highest weight theory. So let for the moment G = GL_{m+1}, K = GL_{m} × GL_{1} embedded in G in the obvious manner, let M = GL_{m−1} × GL_{1} embedded in G as follows: (A, a) ∈ M is mapped to diag(a, A, a), Further let A = GL_{1} embedded as a ↦ diag(a, 1, a^{−1}). Under these embedding the natural ordering on the roots of G will give our chosen ordering. Note that we also have added a center since we had to consider SL_{m+1} otherwise. This simplifies argumentation and doesn’t change the subject. Now τ_{l} and σ_{l} are given by τ_{l}(k, c) = c^{−l}λ_{l}k, the representation space is ∧^{l}ℂ^{m} and σ_{l}(a, A, a) = a^{−l} ∧^{l} A, the representation space is ∧^{l}ℂ^{m−1}. Let T denote the set of diagonal matrices in G. Any character on T will be of the form χ : diag(a_{1}, …, a_{m+1}) ↦ a_{1}^{χ_{1}} … a_{m+1}^{χ_{m+1}} with χ_{i} ∈ ℤ. We will write χ = (χ_{1}, … χ_{m+1}). The condition of dominance is χ_{1} ≥ χ_{2} ≥ … ≥ χ_{m+1}. Now let λ = (\frac{n−l}{2}, 1, …, 1, 0, …, 0, \frac{−n−l}{2}) with l-ones after the \frac{n−l}{2}. The integrality gives n ≡ l(2) and the dominance n > l if l > 0 and n ≥ 0 if l=0. We assume n > 0 and we want to show that V (λ) contains τ_{l} and τ_{l+1}. Here we want to make use of the Weyl principle which says that any irreducible representation of GL_{m+n} splits, when reduced to GL_{m} × GL_{n} as a sum ⨁_{π∈GL_{n}^{irr}} π′ ⊗ π where π′ is an irreducible representation of GL_{m}, each π occurring maximally once. Now it is easy to see that V (λ) |_{GL_{1}} contains the weight (-l) and that the corresponding highest weight of K is (1, …, 1, 0, …, 0, −l) with l-ones, i.e. the highest weight of τ_{l}. In the same manner one exhibits τ_{l+1} as counterpart of (-l-1). The proof of the lemma is finished. ■

Proof of the theorem (continued): The Peter-Weyl theorem shows C^{∞}(G^{d}) = ⨁_{π∈Ĝ^{a}} π ⊗π^{∗}, as a G^{d} × G^{d} module. It is known that any π ∈ Ĝ^{d} restricts to K with multiplicity one [Zel]. From this we see that, as a G-module

(C^{∞}(G^{d}) ⊗ τ_{l})^{K} = ⨁ ; \begin{matrix}π ∈ Ĝ^{d} \\ τ_{l} | π \end{matrix} π.

So that

det(△_{0,l}^{d} + s) = ∏ ; \begin{matrix}π ∈ Ĝ^{d} \\ τ_{l} | π \end{matrix} (π(C) + s)^{dim π},

where the product means the zeta regularized product.

Now let λ = (\frac{n−l}{2}, 1, …, 1, 0, …, 0, \frac{−n−l}{2}) dominant integral and π = π_{λ} the irreducible representation of G^{d} with highest weight λ then π(C) = B(λ+ρ)−B(ρ) = B(\frac{n}{2}α_{r} +σ_{l} +ρ)−B(ρ) = (m + n)^{2} − (m − l)^{2} and for the dimension of π we have by the Weyl character formula: dim π = Q_{l}(m + n). Note now that the element γ_{0} is, in the algebraic setting, given by the matrix diag(−1, 1, −1) and so we get: σ_{l} even ⇔ l + m even. Recall that for λ = σ_{l} + \frac{n}{2}α_{r} the dominance condition was n > l so the right hand side of the equality in the theorem does contain det(△_{0,l}^{d} + s) and all factors with λ not dominant. This gives the assertion. ■

We assemble the results of this section to

Theorem 5.3.2 Let X denote a hermitian symmetric space of real rank one and let Γ a torsionfree uniform lattice in the group G of orientation preserving isometries. Let X_{Γ} denote the quotient Γ\X. Let φ denote a finite dimensional unitary representation of Γ and define the zeta function Z_{φ} as:

Z_{φ}(s) = ∏ ; \begin{matrix}1 ≠ [γ] \\ prime \end{matrix} ∏_{N≥0} det(1 − e^{−sl_{γ}}φ(γ) ⊗ Ad^{N}(γ^{−1} | 𝔫_{r} ⊕ 𝔫_{−})),

then Z_{φ} admits analytic continuation to a meromorphic function. The function Z_{φ}(s) has vanishing order at zero:

n_{0} = ord_{s=0}Z_{φ}(s) = (−1)^{m}dimφAr(X_{Γ}) + ∑_{q=0}^{m} q(−1)^{q}h_{0,q}(φ)

where h_{p,q}(φ) is the (p,q)-Hodge number of X_{Γ} with φ-coefficients. The function R_{φ}(s) = Z_{φ}(s)s^{−n_{0}} has the value:

R_{φ}(0) = \frac{T (X_{Γ}, φ)}{T^{(2)}(X_{Γ})^{dim φ}}

where T is the Dolbeault torsion and T^{(2)} the L^{2}-Dolbeault torsion. This can also be written

R_{φ}(0) = T (X_{Γ}, φ)(T (X^{d})E(m))^{(−1)^{m}dim φAr(X_{Γ})}

where Ar(X_{Γ}) = ∑_{q=0}^{m}(−1)^{q}h_{0,q}(X_{Γ}) denotes the arithmetic genus of X_{Γ} and E(m) is an elementary factor to be computed in section 4.9.

Corollary 5.3.1 From the theorem we get a formula joining the L^{2}-torsion to the torsion of the dual space:

T^{(2)}(X_{Γ}) = (T (X^{d})E(m))^{(−1)^{m+1}Ar(X_{Γ})}.

For the proof we note that the first identities in Prop. 4.4 and 4.5 show that Z_{φ}(0) = Z̃_{φ}(0). The second identities give

Z_{φ}(s)Z̃_{φ}(−s) = ∏_{l=0}^{m−1} det(△_{l} + (s + l − m)^{2} − (l − m)^{2})^{2(−1)^{l}}

(det(D_{l}^{2} − (s + l − m)^{2})exp(2P_{l}((s + l − m)^{2}))^{(−1)^{m+l}dimφχ(X_{Γ})/χ(X^{d})}.

Now Theorem 4.6 gives the assertion. ■

We will give the recipe to compute the constant E(m). We have

E(m) = ∏_{l=1}^{m−1} R_{l}(0)^{l(−1)^{l}}exp(N_{m}),

where the R_{l} are the polynomials of Theorem 3.6 and N_{m} is given as follows: Consider the polynomial

F_{l}(x) = ( \begin{matrix}m − 1 \\ l \end{matrix} ) ∏_{k=0 k≠l}^{m−1}(x^{2} − (m − 2k − l)^{2})

we get

Q_{l}(x) = \frac{xF_{l}(x)}{mF_{0}(m)} = ∑_{j=1}^{m} b_{j}^{l}x^{2j−1}.

Let P_{l}(x) = ∑_{j=1}^{m} b_{j}^{l}c_{j}x^{j}, where c_{j} = −\frac{1}{j} ∑_{r=1}^{j} \frac{1}{2r−1}, then

N_{m} = 2 ∑_{l=0}^{m−1}(−1)^{l}P_{l}((m − l)^{2}).

The P_{l}’s are the polynomials occurring in th determinant formula in Proposition 3.5. The first values of E are:

E(1) = exp(−2),
E(2) = exp(−1/2),
E(3) = 2^{−24}3^{−12}exp(−\frac{739}{3 · 2^{5}} ),

E(4) = 2^{−895}3^{−435}exp(−\frac{11 · 277}{2 · 3^{3} · 7}).

5.4 The Generalized Selberg Zeta Function

In this section we may assume X to be an arbitrary globally symmetric space of the noncompact type. Fix a θ-stable Cartan subgroup H = AB of splitrank 1 and a parabolic subgroup P with Langlands decomposition P = MAN. Let G act on itself by conjugation, write g.x = gxg^{−1}, write G.x for the orbit, so G.x = {gxg^{−1}|g ∈ G} as well as G.S = {gsg^{−1}|s ∈ S, g ∈ G} for any subset S of G. We are going to consider functions that are supported on the set G.(MA). Let σ be a finite dimensional representation of M. Then let f_{σ} be as in 5.1 but with M taking the place of G. Now consider for s ∈ ℂ and n ∈ ℕ the function g_{s}^{n} on Ā^{+} given by

g_{s}^{n}(a) = l_{a}^{n+1}e^{−sl_{a}},

where l_{a} is the length of a, so l_{a} = |log(a)|. Choose any η : N → G which has compact support, is positive and such that

∫_{N} η(n)dn = 1.

Given these data, let Φ = Φ_{η,σ,n,s} : G → ℂ be defined by

Φ(knma(kn)^{−1}) = η(n)f_{σ}(m)\frac{g_{s}^{n}(a)}{det(1 − (ma)^{−1}|𝔫)},

for k ∈ K, n ∈ N, m ∈ M and a ∈ A. Further ϕ(g) = 0 if g is not in G.(MA).

To fix notations let (φ, V_{φ}) be a finite dimensional unitary representation of Γ and recall that the group G acts by right shifts unitarily on the Hilbert space L^{2}(Γ\G, φ) consisting of all measurable functions f : G → V_{φ} such that f(γx) = φ(γ)f(x) and f is square integrable over Γ\G (modulo null functions). It is known that as a G-representation this space splits as

L^{2}(Γ\G) = ⨁_{π∈Ĝ} N_{Γ,φ}(π)π

with finite multiplicities N_{Γ,φ} < ∞.

Proposition 5.4.1 The function Φ is (n − dim(𝔫))-times continuously differentiable. For n and Re (s) large enough it goes into the trace formula and for any finite dimensional unitary representation φ of Γ we have

∑_{π∈Ĝ} N_{Γ,φ}(π)tr(Φ) = ∑_{[γ]} vol(Γ_{γ}\G_{γ})O_{m_{γ}}^{M}(f_{σ})\frac{g_{s}^{n}(a)tr(φ(γ))}{det(1 − (ma)^{−1}|𝔫)},

where the sum runs over all classes [γ] such that γ is conjugate to an element m_{γ}a_{γ} of MA^{+}.

Proof: The function f_{σ} is rapidly decreasing [BM], replace for the moment g_{s}^{n} by g_{s}^{n}χ_{T}, where χ_{T} : Ā^{+} → [0, 1] is a C^{∞}-function which is constant 1 for l_{a} ≤ T and constant 0 for l_{a} ≥ T + 1. Then Φ is rapidly decreasing and it thus goes into the trace formula. Considering each side of the trace formula separately it is easy to see that, as T tends to infinity, both converge dominatedly. ■

5.4.1 The generalized Selberg zeta function

Besides the parabolic P = MAN we also consider the opposite parabolic P̄ = MAN̄. The Lie algebra of N̄ is written 𝔫̄. Let V denote a Harish-Chandra module of G then we consider the Lie algebra homology H_{∗}(𝔫̄, V) and cohomology H^{∗}(𝔫̄, V). It is shown in [HeSchm] that these are Harish-Chandra modules of the group MA.

We will say that a discrete subgroup Γ ⊂ G is nice if for every γ ∈ Γ the adjoint Ad(γ) has no roots of unity as eigenvalues. Every arithmetic Γ has a nice subgroup of finite index [Bor]. Let H_{1} ∈ A^{+} be the unique element with B(H_{1}) = 1.

We will denote by E_{H}(Γ) the set of nontrivial Γ-conjugacy classes [γ], which are such that γ is in G conjugate to an element of H. Such an element will then be written a_{γ}b_{γ} or a_{γ}m_{γ}.

Theorem 5.4.1 Let Γ be nice and (φ, V_{φ}) a finite dimensional unitary representation of Γ. Choose a θ-stable Cartan H of splitrank one. For Re (s) >> 0 define the zeta function

Z_{H,σ,φ}(s) = ∏_{[γ]∈E_{H}(Γ)} ∏_{N≥0} det(1 − e^{−sl_{γ}}γ|V_{φ} ⊗ W_{σ} ⊗ S^{N}(𝔫))^{χ_{1}(X_{γ})},

where γ acts on V_{φ} ⊗ W_{σ} ⊗ S^{N}(𝔫) via φ(γ) ⊗ σ(m_{γ}) ⊗ Ad^{N}((m_{γ}a_{γ})^{−1}). Then Z_{H,σ,φ} has a meromorphic continuation to the entire plane. If s_{0} is a pole or zero of Z_{H,σ,φ} then s_{0} = λ(H_{1}), where λ ∈ 𝔞^{∗} is a weight of the 𝔞 action on the finite dimensional virtual space

∑_{p,q}(−1)^{p+q}(H^{q}(𝔫̄, π) ⊗ ∧^{p}𝔭_{M} ⊗ W_{σ^{∗}})^{K_{M}} ,

for some π ∈ Ĝ with N_{Γ,φ}(π) ≠ 0. Here 𝔞 only acts on the first tensor factor. Let m_{s_{0},π,σ} ∈ ℤ denote the dimension of the generalized weight space of λ in this virtual module. For the order of Z_{σ} at s = s_{0} we have

ord_{s=s_{0}}(Z_{H,σ,φ}(s)) = − ∑_{π∈Ĝ} N_{Γ,φ}(π)m_{s_{0},π,σ}

Remark. To give the reader a better feeling of what this theorem says about the singularities of Z_{H,σ,φ}, we consider a zero or pole s_{0} of Z_{H,σ,φ}. It then follows from Harish-Chandra’s theory of global characters, that there is an integer k ∈ {0, …, 2 dim(𝔫_{+}+2)} and a Weyl group element w ∈ W (𝔥, 𝔤) such that

s_{0} = wλ_{π}(H_{1}) − 2|ρ_{0}| − k|α_{r}|/2,

where λ_{π} ∈ 𝔥^{∗} is the infinitesimal character of π.

Furthermore, there has to exist a finite subset E ⊂ Φ_{nI}^{+}(𝔥, 𝔤) of the set of nonimaginary positive roots such that

B(wλ_{π}|_{𝔟} − ∑_{α∈E} α|_{𝔟}) = B(λσ̃).

Proof: From [D-Hitors] we take for Γ nice

χ_{1}(X_{γ}) = \frac{| W (𝔤_{γ}, 𝔥) | ∏_{α∈Φ_{γ}^{+}}(ρ_{γ}, α)}{λ_{γ}c_{γ}[G_{γ} : G_{γ}^{0}]}vol(Γ_{γ}\G_{γ}),

so that the geometric side of our trace formula will be

∑_{[γ]} \frac{l_{γ_{0}}χ_{1}(X_{γ})\overline{trσ(m_{γ})}l_{γ}^{n+1}e^{−sl_{γ}tr(φ(γ))}tr(φ(γ))}{det(1 − γ^{−1}|𝔫)} = (−1)^{n+1}(\frac{∂}{∂s})^{n+2}logZ_{H,σ,φ}(s).

To understand the spectral side let

(MA)^{+} = interior in MA of the set {g ∈ MA|det(1 − ga|𝔫̄) ≥ 0 ∀a ∈ A^{+}}.

Now B is a compact Cartan in M and (M.B)A^{+} is a subset of (MA)^{+}. It was proven in [HeSchm] that for any π ∈ Ĝ, writing the character of π as Θ_{π}^{G} we have for ma ∈ (MA)^{+} ∩ G^{reg}

Θ_{π}^{G}(ma) = \frac{Θ_{H_{∗}(𝔫̄,π)}^{MA}(ma)}{det(1 − (ma)^{−1}|𝔫)}.

Let f be supported on G.(MA) then applying the Weyl integration formula twice gives

∫_{G} f(g)dg = ∫_{G/MA} ∫_{MA^{+}} f(gmag^{−1})|det(1 − ma|𝔫 ⊕ 𝔫̄)dadmdg.

So that for π ∈ Ĝ

trπ(Φ) = ∫_{G} Θ_{π}^{G}(x)Φ(x)dx
= − ∫_{MA^{+}} Θ_{π}^{G}(ma)f_{σ}(m)g_{s}^{n}(a)det(1 − ma|𝔫)dadm
= − ∫_{A^{+}} ∑_{L} \frac{1}{|W (L, M)|} ∫_{L} ∫_{M/L} f_{σ}(mlm^{−1})dmΘ_{π}^{G}(la)det(1 − la|𝔫)dmdlg_{s}^{n}(a)da,

where the sum runs over all M-conjugacy classes of Cartan subgroups L of M. The integral over M/L is just an orbital integral of f_{σ} so that in the sum only the compact Cartan B will survive. Hence

trπ(Φ) = − ∫_{MA^{+}} Φ(ma)det(1 − ma|𝔫)Θ_{H_{∗}(𝔫̄,π)}^{MA}(ma)dmda
= ∫_{MA^{+}} Φ(ma)a^{2ρ}det(1 − (ma)^{−1}|𝔫)Θ_{H_{∗}(𝔫̄,π)}^{MA}(ma)dmda
= − ∫_{MA^{+}} Φ(ma)det(1 − (ma)^{−1}|𝔫)Θ_{H^{∗}(𝔫̄,π)}^{MA}(ma)dmda
= − ∫_{MA^{+}} fσ^{∗}(m)Θ_{H^{∗}(𝔫̄,π)}^{MA}(ma)dmg_{s}^{n}(a)da,

where we have used the isomorphism

H_{p}(𝔫̄, V) ≅ H^{p}(𝔫̄, V) ⊗ ∧^{max}𝔫̄,

([HeSchm], 2.18). This gives

trπ(Φ) = − ∫_{A^{+}} tr(a|(H^{∗}(𝔫̄, π) ⊗ ∧^{∗}𝔭_{M} ⊗ W_{σ^{∗}})^{K_{M}})l_{a}^{n+1}e^{−sl_{a}}da.

Now let the A-decomposition of (H^{∗}(𝔫̄, π) ⊗ ∧^{∗}𝔭_{M} ⊗ W_{σ^{∗}})^{K_{M}} be

⨁_{λ} E_{λ},

where E_{λ} is the biggest subspace on which a − e^{λ(loga)} acts nilpotently. Let m_{λ} denote the dimension of E_{λ}. Integrating over 𝔞^{+} instead over A^{+} we get

trπ(Φ) = − ∫_{0}^{∞} ∑_{λ} m_{λ}e^{(λ(H_{1})−s)t}t^{n+1}dt
= (−1)^{n+1}(\frac{∂}{∂s})^{n+1} ∑_{λ} m_{λ} \frac{1}{λ(H_{1}) − s}
= − ∑_{λ} m_{λ} \frac{1}{(s − λ(H_{1}))^{n+2}} .

From this the theorem follows. ■

The remark following the theorem will follow form the Wigner Lemma [BorWall], Cor 4.2 together with the

Proposition 5.4.2 Let σ be a finite dimensional representation of M then the order of Z_{H,σ,φ}(s) at s = s_{0} is

(−1)^{dim(N)} ∑_{π∈Ĝ} N_{Γ,φ}(π) ∑_{q=0}^{dim(𝔪⊕𝔫)}(−1)^{q} dim(H^{q}(𝔪 ⊕ 𝔫, K_{M}, π ⊗ W_{σ^{∗}})_{λ}

with λ = s_{0}λ_{1}.

Proof: Extend W_{σ^{∗}} to a 𝔪 ⊕ 𝔫-module by letting 𝔫 act trivially. We then get

H^{p}(𝔫, π) ⊗ W_{σ^{∗}} ≅ H^{p}(𝔫, π ⊗ W_{σ^{∗}}).

The (𝔪, K_{M})-cohomology of the module H^{p}(𝔫, π ⊗ W_{σ^{∗}}) is the cohomology of the complex (C^{∗}) with

C^{q} = Hom_{K_{M}} (∧^{q}𝔭_{M}, H^{p}(𝔫, π) ⊗ W_{σ^{∗}})
= (∧^{q}𝔭_{M} ⊗ H^{p}(𝔫, π) ⊗ W_{σ^{∗}})^{K_{M}} ,

since ∧^{p}𝔭_{M} is a self-dual K_{M}-module. Therefore we have an isomorphism of virtual A-modules:

∑_{q}(−1)^{q}(H^{p}(𝔫, π) ⊗ ∧^{q}𝔭_{M} ⊗ W_{σ^{∗}})^{K_{M}} ≅ ∑_{q}(−1)^{q}H^{q}(𝔪, K_{M}, H^{p}(𝔫, π ⊗ W_{σ^{∗}})).

Now one considers the Hochschild-Serre spectral sequence in the relative case for the exact sequence of Lie algebras

0 → 𝔫 → 𝔪 ⊕ 𝔫 → 𝔪 → 0

and the (𝔪 ⊕ 𝔫, K_{M})-module π ⊗ W_{σ^{∗}}. We have

E_{2}^{p,q} = H^{q}(𝔪, K_{M}, H^{p}(𝔫, π ⊗ W_{σ^{∗}}))

and

E_{∞}^{p,q} = Gr^{q}(H^{p+q}(𝔪 ⊕ 𝔫, K_{M}, π ⊗ W_{σ^{∗}})).

Now the module in question is just

χ(E_{2}) = ∑_{p,q}(−1)^{p+q}E_{2}^{p,q}.

Since the differentials in the Hochschild-Serre spectral sequence are A-homomorphisms this equals χ(E_{∞}). So we get an A-module isomorphism of virtual A-modules

∑_{p,q}(−1)^{p+q}(H^{p}(𝔫, π) ⊗ ∧^{q}𝔭_{M} ⊗ W_{σ^{∗}})^{K_{M}} ≅ ∑_{j}(−1)^{j}H^{j}(𝔪 ⊕ 𝔫, K_{M}, π ⊗ W_{σ^{∗}}).■

5.4.2 The generalized Ruelle Zeta Function

Theorem 5.4.2 Let Γ be nice and choose a θ-stable Cartan subgroup H of splitrank one. For Re (s) >> 0 define the zeta function

Z_{H,σ,φ}^{R}(s) = ∏_{[γ]∈E_{H}(Γ)} det(1 − e^{−sl_{γ}}φ(γ)),

then Z_{H,σ,φ}^{R}(s) extends to a meromorphic function on ℂ.

Proof: We consider the logarithm of the Selberg function for φ = 1

logZ_{H,σ,φ}^{R}(s) = − ∑_{[γ]} \frac{χ_{1}(X_{γ})e^{−sl_{γ}}trσ(m_{γ})tr(φ(γ))}{μ_{γ}det(1 − γ^{−1}|𝔫)}.

Consider first the case when there is only one positive root α_{r} in the root system Φ(𝔞, 𝔤), we normalize B such that |α_{r}| = 2. Then a acts on ∧^{l}𝔫̄ by a^{−lα_{r}}. Let σ_{l} denote the representation of K_{M} on ∧^{l}𝔫̄ the we get

Z_{H,σ,φ}^{R}(s) = ∏_{l=0}^{dim N} Z_{H,σ_{l},φ}(s + 2l)^{(−1)^{l}}

and hence the claim.

If we have two positive roots, say α_{r} and \frac{α_{r}}{2} , with |α_{r}| = 2 then we have 𝔫̄ = 𝔫̄_{r} ⊕ 𝔫̄_{I} where 𝔫̄_{r} has dimension one and a acts on 𝔫̄_{r} by a^{−α_{r}} and on 𝔫̄_{I} by a^{− \frac{αr}{2}} . Now it follows

Z_{H,σ,φ}^{R}(s) = ∏_{l=0}^{dim N−1}(\frac{Z_{H,∧^{l}𝔫̄_{I},φ}(s + l)}{Z_{H,∧^{l}𝔫̄_{I}⊗𝔫̄_{r},φ}(s + l + 2)})^{(−1)^{l}}.■

5.5 The holomorphic torsion zeta function

Now let X be hermitian again and let (τ, V_{τ}) be an irreducible unitary representation of K_{M}.

Theorem 5.5.1 Let Γ be nice and (φ, V_{φ}) a finite dimensional unitary representation of Γ. Choose a θ-stable Cartan H of splitrank one with c(H) = 2. For Re (s) >> 0 define the zeta function

Z_{H,τ,φ}(s) = ∏_{[γ]∈E_{H}(Γ)} ∏_{k≥0} det(1 − e^{−sl_{γ}}γ|V_{k}^{τ})^{χ_{1}(X_{γ})},

where V_{k}^{τ} = V_{φ} ⊗ V_{τ} ⊗ S^{k}(𝔫) and γ acts via φ(γ) ⊗ τ(m_{γ}) ⊗ Ad^{k}(γ^{−1}). Then Z_{H,τ,φ} has a meromorphic continuation to the entire plane. If s_{0} is a pole or zero of Z_{H,τ,φ} then s_{0} = λ(H_{1}), where λ ∈ 𝔞^{∗} is a weight of the 𝔞 action on the finite dimensional virtual space

∑_{p,q}(−1)^{p+q}(H^{q}(𝔫̄, π) ⊗ ∧^{p}𝔭_{M,−} ⊗ W_{τ^{∗}})^{K_{M}} ,

for some π ∈ Ĝ with N_{Γ,φ}(π) ≠ 0. Here 𝔞 only acts on the first tensor factor. Let m_{s_{0},π,σ} ∈ ℤ denote the dimension of the generalized weight space of λ in this virtual module. For the order of Z_{σ} at s = s_{0} we have

ord_{s=s_{0}}(Z_{σ}(s)) = − ∑_{π∈Ĝ} N_{Γ,φ}(π)m_{s_{0},π,σ}

Proof: The proof proceeds as the proof of Theorem 5.4.1 with g_{τ^{1}} instead of f_{σ}, where τ^{1} denotes the virtual representation τ ⊗ ∧^{∗}𝔭_{M,−}. One makes use of an analogue of Lemma 4.9 in [MS-2]. ■

Extend the definition of Z_{H,τ,φ}(s) to arbitrary virtual representations in the following way. Consider a finite dimensional virtual representation ξ = ⊕_{i}a_{i}τ_{i} with a_{i} ∈ ℤ and τ_{i} ∈ K̂_{M}. Then let Z_{H,ξ,φ}(s) = ∏_{i} Z_{H,τ_{i},φ}(s)^{a_{i}}.

Now for c(H) = 2 let Z_{H,l,φ}(s) = Z_{H,∧^{l}𝔫_{−},φ}(s) and

Z_{H,φ}(s) = ∏_{l=0}^{dim(𝔫_{−})} Z_{H,l,φ}(s + d_{l}(H)),

In the case c(H) = 1 let

Z_{H,φ}(s) = Z_{H,φ}^{R}(s + d(H)),

where Z_{H,φ}^{R} denotes the Ruelle zeta function. See 5.1 for the constants.

Proposition 5.5.1 Assume Γ is nice, then for λ > 0 we have the identity

∏_{q=0}^{dim_{ℂ} X}(\frac{det(△_{0,q,φ} + λ)}{det^{(2)}(△_{0,q,φ} + λ)})^{q(−1)^{q}} =

∏ ; H/conj. ; splitrank = 1 ; c(H) = 2 ∏_{l=0}^{dim 𝔫_{+}} (Z_{H,l,φ}(b_{0}(H) + \sqrt{λ + d_{l}(H)^{2}})^{(−1)^{l}}

∏ ; H/conj. ; splitrank = 1 ; c(H) = 1 Z_{H,φ}^{R}(b_{0}(H) + \sqrt{λ + d(H)^{2}}).

Proof: The equality is gotten by taking the Mellin transform of the expressions in Theorem 5.1.1. Then one uses the fact that the identity contribution to the trace formula equals the Γ-trace. ■

Now the results in this section assemble to

Theorem 5.5.2 The zeta function Z_{H,φ} extends to a meromorphic function on the entire plane. Let

Z_{φ}(s) = ∏ ; H/conj. ; splitrank = 1 Z_{H,φ}(s + b_{0}(H)).

Let n_{0} be the order of Z_{φ} at zero then Proposition 5.5.1 shows that

n_{0} = ∑_{q=0}^{dim_{ℂ}(X)} q(−1)^{q}(h_{0,q}(X_{Γ}) − h_{0,q}^{(2)}(X_{Γ})),

where h_{p,q}(X_{Γ}) is the (p, q)-th Hodge number of X_{Γ} and h_{p,q}^{(2)}(X_{Γ}) is the (p, q)-th L^{2}-Hodge number of X_{Γ}. Let R_{φ}(s) = Z_{φ}(s)s^{−n_{0}} then

R_{φ}(0) = \frac{T (X_{Γ}, φ)}{T^{(2)}(X_{Γ})^{dim φ}} .■

Higher Fundamental Rank

In this chapter we consider Shimura manifolds X_{Γ} = Γ\G/K, where G is supposed to have a fundamental rank > 0. Examples are G = SL_{n}(ℝ), where n is > 2.

This is the situation where our definitions of higher torsion and higher Euler characteristics start to unfold their meaning. It turns out that all torsion numbers of index smaller than the fundamental rank are trivial. This is paralleled on the geometrical side by the vanishing of the corresponding Euler numbers.

To define the zeta functions we have to impose a condition on the group Γ which is automatically satisfied in the cases when the fundamental rank is either zero or one.

6.1 ”Good” Groups

In this chapter we will assume Γ to be good in a sense to be explained now: A real reductive group H is called cuspidal, if H/center has a compact Cartan subgroup. An element g of G is called cuspidal, if the centralizer G_{g} of g in G is cuspidal. Note that every semisimple regular element of G is cuspidal. Further an arbitrary semisimple element g lies in a Cartan subgroup H = AB where A is the split part and B is compact. According to this decomposition we write g = a_{g}b_{g}. Now suppose H is chosen such that the dimension of A is minimal then g is cuspidal if and only if a_{g} is regular in A, which means that the centralizer G_{a_{g}} of a_{g} equals the centralizer G_{A} of A.

An element g of G is called fundamental, if g ≠ 1 and g lies in a fundamental Cartan subgroup. Now the lattice Γ is called good if every nontrivial fundamental element of Γ is cuspidal.

Note that goodness only depends on the commensurability class of Γ, i.e. when Γ_{1} and Γ_{2} are commensurable (⇔ Γ_{1} ∩ Γ_{2} has finite index in Γ_{1} and in Γ_{1}) and torsionfree and Γ_{1} is good then so is Γ_{2}.

If G has a compact Cartan subgroup, i.e. G has fundamental rank zero then every lattice is good. If G has fundamental rank one then every torsionfree lattice is good.

If G = G_{1} × G_{2}, Γ = Γ_{1} × Γ_{2} and G_{1}, G_{2} both have no compact Cartan subgroup then Γ is not good. In this case Γ is reducible. There are however examples examples of irreducible lattices which fail to be good.

Let p denote an odd prime and G = SL_{p}(ℝ) then any lattice in G is arithmetic [Marg]. So let A denote a central ℚ-division algebra of degree p [Pierce]. The A ⊗ ℝ ≅ Mat_{p}(ℝ) and the norm-1-subgroup A^{1} of A^{∗} maps into G. Any arithmetic subgroup of A^{1} gives a uniform lattice in G. Let α denote a nontrivial element of A^{1}. The subfield ℚ(α) of A must be a strictly maximal subfield of A. The centralizer of α is a simple algebra over ℚ(α) inside A, hence it must coincide with ℚ(α). Thus the centralizer of α is abelian, so α is regular and Γ is good.

Let G = SL_{2}(ℂ)^{d_{1}} × SL_{2}(ℝ)^{d_{2}}, d_{1} + d_{2} > 1 and Γ an irreducible lattice in G. Then there is a number field K with d_{1} pairs of conjugate complex places and d_{2} real places such that Γ is commensurable to A^{1}, where A is a quaternion algebra over K. We may assume Γ nice. Since on the one hand SL_{2}(ℝ) has a compact Cartan and on the other every noncentral semisimple element of SL_{2}(ℂ) is regular we conclude that Γ is good.

An example of a lattice which has nonregular elements but is good can be constructed similarly in SL_{4}(ℝ) and an example of a nongood irreducible lattice in SL_{9}(ℝ).

6.2 The Heat Trace

Now fix a finite dimensional unitary representation φ of Γ. Then φ defines a flat hermitian vector bundle E_{φ} over X_{Γ}. Let E denote the pullback of E_{φ} to X. Let △_{φ,k,Γ} be the Laplace operator on E_{φ}-valued differential forms and △_{φ,k} the Laplace operator on E-valued k-forms on X. The heat operator e^{−t△_{φ,k}} has a smooth kernel h_{t}^{k}, which is of rapid decay [BM]. We put

f_{t} = ∑_{k=0}^{dim X}(−1)^{k} ( \begin{matrix}k \\ r \end{matrix} ) tr(h_{t}^{k}),

where r = rank(G) − rank(k) denotes the fundamental rank of G and the trace means pointwise trace. Since △_{φ,k,Γ} is the pushdown of △_{φ,k} we get by the Selberg trace formula with d = dim X:

∑_{k=0}^{d}(−1)^{k} ( \begin{matrix}k \\ r \end{matrix} ) tr(e^{−t△_{φ,k}}) = ∑_{[γ]} trφ(γ) vol(Γ_{γ}\G_{γ}) O_{γ}(f_{t}),

where the sum on the right hand side is taken over the conjugacy classes [γ] in Γ and O_{γ}(f_{t}) denotes the orbital integral:

O_{γ}(f_{t}) = ∫_{G/G_{γ}} f_{t}(gγg^{−1})dg.

We want to use the Fourier transform formula for orbital integrals as in [HC-S]. Since the fundamental rank of G is > 0 there is no compact Cartan subgroup in G and hence it suffices to compute the principal series contributions to the Fourier transform. So let H denote a Θ-stable Cartan subgroup, let A denote its split component and B the compact component then we have a product H = AB. Let P denote a cuspidal parabolic subgroup with Langlands decomposition P = MAN. Let (ξ, W_{ξ}) be an irreducible unitary representation of M and e^{ν} a quasicharacter of A. Set π_{ξ,ν} = Ind_{P}^{G}(ξ ⊗ e^{ν} ⊗ 1) the principal series representation.

Similar to Proposition 5.1.1 we get:

Proposition 6.2.1 For the trace of f_{t} under π_{ξ,ν} we have:

tr π_{ξ,ν}(f_{t}) = \{ \begin{matrix}−e^{tπξ,ν(C)} dim(W_{ξ} ⊗ ∧^{∗}𝔭_{𝔥})^{K∩M} & if H is fundamental \\ 0 & otherwise. \end{matrix}

Here C is the Casimir operator of G and for a finite dimensional vector space V we write

∧^{∗}V = ∑_{k=0}^{dim V} (−1)^{k} ∧^{k} V.

Further 𝔭_{𝔥} denotes the orthocomplement of 𝔞 in 𝔭, where 𝔞 = LieA.

From now on let H = AB be a fundamental Θ-stable Cartan subgroup of G. Since exp : 𝔞 → A is an isomorphism we will frequently identify these groups to simplify notation.

As in 5.1 it follows that for an irreducible unitary representation ξ of M with infinitesimal character ∧_{ξ} we have

(W_{ξ} ⊗ ∧^{∗}𝔭_{M} ⊗ V_{μ})^{K∩M} ≠ 0 ⇒ B(∧_{ξ}) = B(μ).

Choose a system of positive roots according to the chosen parabolic P = MAN. Write ρ for the half sum of the positive roots. Recall that we have

π_{ξ,ν}(C) = B(ν) + B(∧_{ξ}) − B(ρ)

Define ′F_{f}^{H} as in [HC-HA1,sec.17]. By Theorem 15 in [HC-S] and the preceding proposition we get with A^{∗}, B^{∗} denoting the unitary duals:

′F_{f_{t}}^{H}(h) = −|W (g | H)|^{−1} ∑_{s∈W (G|H)} ∑_{B^{∗}∈B^{∗}} ϵ_{I}(s)

∫_{A^{∗}} B^{∗}(s(B))A^{∗}(s(A))e^{t(B(B^{∗})+B(A^{∗})−B(ρ))}dim(H(B^{∗}) ⊗ ∧^{∗}𝔭_{M} ⊗ ∧^{∗}𝔫)^{K∩M^{0}}dh_{R}^{∗}.

Here H(B^{∗}) means the discrete series representation of M^{0} given by B^{∗}. The normalizations of measures follow [HC-S]. We have

∫_{A^{∗}} A^{∗}(s(A))e^{tB(A^{∗})}dA^{∗} = \frac{e^{−B(A)/4t}}{\sqrt{4πt}^{r}} .

Let C_{M} be the Casimir operator of M with respect to B |_{M}. Let ρ_{M} denote the half sum of positive imaginary roots. Further let

′△_{M} = ∏_{α∈Φ_{M}^{+}}(1 − ξ_{−α}) = ′△_{M,n} ′△_{M,c}

the c and the n denoting the product over the compact and noncompact imaginary roots and ξ_{α}(h) = e^{α(log h)} for h ∈ H. Now by Lemma 4.2 in [MS2] and similar computations as in the proof of Lemma 4.3 in [MS1] we get for h = ab regular in H, where 𝔫 = Lie N ⊗ ℂ:

′F_{f_{t}}^{H}(h) = \frac{e^{− \frac{B(a)}{4t} −t(B(ρ)−B(ρ_{M}))}}{\sqrt{4πt}^{r}} \frac{′△_{M}(b)}{|′△_{M,n}(b)|^{2}} tr(e^{tC_{M}} b | ∧^{∗}𝔫).

Now let h ∈ H and write Φ_{h} for the set of roots α such that ξ_{α}(h) = 1. Write Φ_{h,c} and Φ_{h,n} for the compact and noncompact imaginary roots in Φ_{h}. Let c_{h} have the same meaning as in 3.2. Write as an M^{0}-module:

∧^{l}𝔫 = ∑_{i∈I_{l}} a_{i,l}V_{λ_{i,l}},

where V_{λ_{i,l}} is an irreducible M^{0}-module with infinitesimal character λ_{i,l} and a_{i,l} denotes the multiplicity of V_{λ_{i,l}} in ∧^{l}𝔫. Let ω̃ denote the following element of the symmetric algebra S(𝔥) over 𝔥:

ω̃ = ∏_{α∈Φ_{h}^{+}} H_{α}.

Similar to [MS2] we get for the orbital integrals for h ∈ H, h ≠ 1:

∫_{G/G_{γ}} f_{t}(xhx^{−1})dx = ∫_{G/G_{h}^{0}} f_{t}(xhx^{−1})dx [G_{h} : G_{h}^{0}]^{−1}

= [G_{h} : G_{h}^{0}]^{−1}c_{h}^{−1} \frac{e^{−(B(A)/(4t))−tB(ρ)}}{ξ_{ρ}(h) ∏_{α∈Φ_{h,c}^{+}}(1 − ξ_{−α}(h))}

∑_{l=0}^{dimN}(−1)^{l} ∑_{i∈I_{l}} a_{i,l}e^{tB(λ_{i,l})} ∑_{w∈W_{ℂ}(M)} ϵ(w)ω̃_{h}(w(λ_{i,l}))ξ_{w(λ_{i,l})}(B).

We write E_{2}(M^{0}) for the set of discrete series representations of M^{0} and E_{f}(Γ) for the set of conjugacy classes [γ] in Γ such that γ is fundamental in G. By the fact that f_{t} is a pseudocuspform we conclude that in the trace formula only conjugacy classes in E_{f} contribute. We write l_{γ} for the common length of the geodesics in the homotopy class of γ.

We end up with Lemma. 6.2.1

Θ(t) = ∑_{k=0}^{d}(−1)^{k} ( \begin{matrix}k \\ r \end{matrix} ) tre^{−t△_{φ,k}}

= ∑_{[1]≠[γ]∈E_{f} (γ)} trφ(γ)\frac{vol(Γ_{γ}\G_{γ})[G_{h} : G_{h}^{0}]^{−1}}{ξ_{ρ}(γ) ∏_{α∈Φ^{+}−Φ_{γ,c}^{+}}(1 − ξ_{−α}(γ))}c_{γ}^{−1}

\frac{e^{−(l_{γ}^{2}/4t)−tB(ρ)}}{\sqrt{4πt}^{r}} ∑_{l=0}^{dim N}(−1)^{l} ∑_{i∈I_{l}} a_{i,l}e^{tB(λ_{i.l})} ∑_{w∈W_{ℂ}(M)} ϵ(w)ω̃_{h}(w(λ_{i,l}))ξ_{w(λ_{i,l})}(B)

+ dim F vol(X) ∑_{l=0}^{dim N}(−1)^{l} ∑_{i∈I_{l}} a_{i,l} ∑_{ξ∈E_{2}(M^{0})} dim(W_{ξ} ⊗ ∧^{∗}𝔭_{M} ⊗ V_{λ_{i,l}})^{K∩M^{0}}F_{ξ}(t).

Here we wrote

F_{ξ}(t) = ∫_{A^{∗}} e^{t(B(A^{∗})+B(λ_{i,l})+B(ρ))}

and P_{ξ}(ν) is the Plancherel density of the principal series representations π_{ξ,ν}. By [HC-HAIII] we know that P_{ξ} is a polynomial of degree dim N, more precisely

P_{ξ}(ν) = c ∏_{α∈Φ^{+}} < λ_{ξ} + ν, α >,

where c > 0 is given by Theorem 24.1 in [HC-HAIII].

Choosing an orthonormal basis of A^{∗} we are lead to consider

∫_{ℝ^{r}} r^{−t∥x∥}P_{ξ}(x)dx

for a polynomial P_{ξ}. Writing P_{ξ} as linear combination of monomials x^{β} = x_{1}^{β_{1}} … x_{r}^{β_{r}} it is clear that only those monomials contribute, for which β ∈ 2ℤ. But for β = 2α we get

∫_{ℝ^{r}} e^{−t∥x∥}x^{2α}dx = \sqrt{π}^{r}2^{−|α|} ∏_{j=1}^{r}(2α_{j} − 1)!!t^{−|α|− \frac{r}{2}} ,

where (2n − 1)!! = (2m − 1)(2m − 3) … 1 and |α| = α_{1} + … + α_{r}. We conclude, that F_{ξ}(t) is t^{− \frac{r}{2}} times a polynomial in (\frac{1}{t}).

6.3 Decomposition of The Lifted Heat Operator

Recall that for the space Ω^{k} of smooth k-forms on X with values in E we have

Ω^{k} ≅ (C^{∞}(Γ\G, φ) ⊗ ∧^{k}𝔭)^{K}.

Let 𝔭_{M} denote the positive part of the Cartan decomposition of M. Write 𝔠 = 𝔞 + 𝔭_{M}. As K ∩ M^{0}-module we have 𝔭 ≅ 𝔠 + 𝔫. Let V (λ_{i.l}) denote the V_{λ_{i,l}}-isotypic subspace of ∧^{l}𝔫. Let P r^{i,j,l} : ∧^{k}𝔭 → ∧^{j}𝔠 ⊗ V (λ_{i,l}) denote the projection and I^{i,j,l} : ∧^{j}𝔠 ⊗ V (λ_{i,l}) → ∧^{k}𝔭 the injection. Let

Ω^{i,j,l} = 1 ⊗ P r^{i,j,l}(Ω^{i+j}) ⊂ (C^{∞}(Γ\G, φ) ⊗ ∧^{j}𝔠 ⊗ V (λ_{i,l}))^{K∩M^{0}}.

The space on the left hand side is only in the K ∩ M^{0}-invariants and not in the K-invariants, but e^{−t△_{φ,k}} is a convolution operator and may be regarded as an operator from K ∩ M^{0}-to K-invariants. So we may set

h_{t}^{i,j,l} = (1 ⊗ P r^{i,j,l}) ∘ e^{−t△_{φ,k}} ∘ (1 ⊗ I^{i,j,l}).

This defines an integral operator e^{−t△_{φ}^{i,j,l}}.

Lemma. 6.3.1 As K ∩ M^{0}-modules we have

∑_{j=0}^{dim 𝔭_{M} +r}(−1)^{j+l} ( \begin{matrix}j + l \\ r \end{matrix} ) (∧^{j}𝔠 ⊗ ∧^{l}𝔫) ≅ (−1)^{l+r}(∧^{∗}𝔭_{M} ⊗ ∧^{l}𝔫).

The proof is straightforward.

Set f_{t}^{i,j,l} = tr h_{t}^{i,j,l} and

f_{t}^{i,l} = ∑_{j=0}^{dim 𝔭_{M}}(−1)^{j}f_{t}^{i,j,l}.

Similar to 3.5 we find that f_{t}^{i,l} is a pseudocuspform and for the fundamental principal series we have

tr π_{ξ,ν}(f_{t}^{i,l}) = −e^{−tπ_{ξ,ν}(C)}dim(W_{ξ} ⊗ ∧^{∗}𝔭_{M} ⊗ V (λ_{i,l}))^{K∩M^{0}}.

So put Θ_{i,l}(t) = ∑_{j} tr e^{−t△_{φ}^{i,j,l}}(−1)^{j+1}.

Proposition 6.3.1 We have

Θ(t) = ∑_{l=0}^{dim N} ∑_{i∈I_{l}}(−1)^{l}Θ_{i,l}(t).

This follows from the trace formula since the orbital integrals of both sides coincide.

Note that there is some cancellation in the preceding proposition. To describe this, consider the M^{0}-decomposition

∧^{l}𝔫 = ⨁_{i∈I_{l}} a_{i,l}V_{λ_{i,l}}

CHAPTER 6 Higher Fundamental Rank 97

Now there is a differential of Lie algebra homology ∂ : ∧^{l}𝔫 → ∧^{l−1}𝔫, making ∧^{∗}𝔫 a complex. Let H_{∗}(𝔫) denote its homology, then, since ∂ is M-equivariant, we have as virtual M-modules

∧^{∗}𝔫 ≅ H_{∗}(𝔫).

Now let J_{l} ⊂ I_{l} be the subset of isotypes which also occur in H_{l}(𝔫), so

H_{l}(𝔫) = ⨁_{i∈J_{l}} b_{i,l}V_{λ_{i,l}}

for some coefficients 0 < b_{i,l} ≤ a_{i,l}, but we actually have Lemma. 6.3.2 Suppose i ∈ J_{l}, then a_{i,l} = b_{i,l}.

Proof: This follows immediately from Corollary 5.7 in [Kost]. ■

Proposition 6.3.2

Θ(t) = ∑_{l=0}^{dim N} ∑_{i∈J_{l}}(−1)^{l}Θ_{i,l}(t).

■

Lemma. 6.3.3 Let i ∈ J_{l}, then

B(λ_{i,l}) − B(ρ) = −B(ρ_{0} + χ_{i,l}),

where ρ_{0} = ρ|𝔞 ∈ 𝔞^{∗} and χ_{i,l} ∈ 𝔞^{∗} is the character of 𝔞 on V_{λ_{i,l}} ⊂ ∧^{l}𝔫.

Analogous to the above we get Lemma. 6.3.4

Θ_{i,l}(t) = ∑_{[1]≠[γ]∈E_{f} (γ)} trφ(γ)\frac{vol(Γ_{γ}\G_{γ})[G_{h} : G_{h}^{0}]^{−1}}{ξ_{ρ}(γ) ∏_{α∈Φ^{+}−Φ_{γ,c}^{+}}(1 − ξ_{−α}(γ))}c_{γ}^{−1}

\frac{e^{−l_{γ}^{2}/4t−tB(ρ)}}{\sqrt{4πt}^{r}} a_{i,l}e^{tB(λ_{i.l})} ∑_{w∈W_{ℂ}(M)} ϵ(w)ω̃_{h}(w(λ_{i,l}))ξ_{w(λ_{i,l})}(B)

+dim F vol(X)a_{i,l} ∑_{ξ∈E_{2}(M^{0})} dim(W_{ξ} ⊗ ∧^{∗}𝔭_{M} ⊗ V_{λ_{i,l}})^{K∩M^{0}}F_{ξ}(t).

We want to compute the Mellin transform of Θ_{i,l}. So we define

M_{λ}^{i,l} = ∫_{0}^{∞} t^{z−1}e^{−λt}Θ_{i,l}(t)dt.

To compute the geodesic terms we need to consider

g_{b}^{a}(z) = ∫_{0}^{∞} t^{z−1}e^{−( \frac{a}{t} +bt)}dt, P a, b > 0.

We define g_{a}(z) = g_{a}^{a}(z) and find

g_{b}^{a}(z) = \sqrt{a/b}^{z}g\sqrt{ab}(z).

We further see

\begin{aligned}g_{a}(−z) & = g_{a}(z), \\ zg_{a}(z) & = a(g_{a}(z + 1) − g_{a}(z − 1) \\ \frac{∂}{∂a}g_{a}(z) & = −g_{a}(z − 1) − g_{a}(z + 1), \\ g_{a}(−\frac{1}{2}) & = π^{\frac{1}{2}} e^{−2a}a^{− \frac{1}{2}} . \end{aligned}

For n ≥ 0 define a polynomial P_{n}(x) by

P_{0}(x) = 1, P_{1}(x) = 1,
P_{n+1}(x) = P_{n−1}(x) + (2n − 1)xP_{n}(x).

Note that deg P_{n} = n − 1, the constant term is 1 and the highest coefficient equals (2n − 3)!! = (2n − 3)(2n − 5) … 1. By the above we get

g_{a}(−\frac{2n + 1}{2}) = \sqrt{π} \frac{e^{−2a}}{\sqrt{a}} P_{n+1}(\frac{1}{2a}).

6.4 The Case of Odd Fundamental Rank

In this section we assume the fundamental rank r of G to be odd. We write

∑_{ξ∈E_{2}(M^{0})} F_{ξ}(t) dim(W_{ξ} ⊗ ∧^{∗}𝔭_{M} ⊗ V_{λ_{i,l}})^{K∩M^{0}} = ∑_{n=0}^{dim(N)} c_{n}^{i,l}t^{−n− \frac{r}{2}} .

We further write μ_{γ} for the quotient l_{γ}/λ_{γ}. If we denote by △_{φ}^{i.l} the operator △_{φ}^{∗,i,l} on Ω^{∗,i,l} then by 3.11 we get

log det(△_{φ}^{i,l} + s^{2} − B(ρ) + B(λ_{i,l}))

= ∑_{1≠[γ]∈E_{f} (γ)} tr φ(γ) χ_{r}(X_{γ}) D_{i,l}(γ)

(\frac{2s}{l_{γ}})^{\frac{1}{2} (r−1)}P_{[ \frac{1}{2} r]}(\frac{1}{sl_{γ}})\frac{\sqrt{π}}{s} \frac{e^{−sl_{γ}}}{μ_{γ}}

+ dim F vol(X_{Γ}) ∑_{n=0}^{dim(N)} c_{n}^{i,l}Γ(−n − \frac{r}{2})s^{2n+r}

with

D_{i,l} = \frac{|W_{ℂ}(G_{γ})|^{−1} ∏_{α∈Φ_{γ}^{+}}(ρ_{γ}, α)^{−1}}{ξ_{ρ_{m}} ∏_{Φ^{+}−Φ_{γ}^{+}}(1 − ξ_{−α}(γ))} ∑_{w∈W_{ℂ}(M)} ϵ(w)ω̃_{γ}(w(λ_{i,l}))ξ_{w(λ_{i,l})}(B).

Now let

Z_{φ}^{i,l} = exp( ∑_{1≠[γ]∈E_{f} (γ)} tr φ(γ) χ_{r}(X_{γ}) D_{i,l}(γ)(\frac{2s}{l_{γ}})^{\frac{1}{2} (r−1)}P_{[ \frac{1}{2} r]}(\frac{1}{sl_{γ}})\frac{\sqrt{π}}{s} \frac{e^{−sl_{γ}}}{μ_{γ}}).

Let \sqrt{.} denote the standard branch of the square root and let

Z_{φ}^{±} = ∏_{i,l} Z_{φ}^{i,l}(s ± \sqrt{B(ρ) − B(λ_{i,l}}))^{a_{i,l}(−1)^{l}}.

Let R_{r} denote the rational function

R_{r}(x, y) = (\frac{2x}{y})^{\frac{1}{2} (r−1)}P_{[ \frac{1}{2} r]}(\frac{1}{xy})x^{−1}.

Proposition 6.4.1 We get

Z_{φ}^{±}(s) = exp( ∑_{1≠[γ]∈E_{f} (γ)} tr φ(γ) χ_{r}(X_{γ}) L^{±}(s, γ)\frac{e^{−l_{γ}s}}{μ_{γ}}),

where

L^{±}(s, γ) = \frac{tr(R_{r}(s ± |χ + ρ|, l_{γ})a_{γ}^{±|χ+ρ|}γ_{I} | H^{∗}(𝔫))}{tr(γ | H^{∗}(𝔫))},

here χ stands for the weight of the A-action and |.| means absolute value on 𝔞^{∗}. As usual H^{∗}(𝔫) means Lie algebra cohomology.

For the proof let T_{s,γ}^{±} denote the operator on ∧^{∗}𝔫 acting on V (λ_{i,l}) by the scalar

R_{r}(s ± \sqrt{B(ρ) − B(λ_{i,l})}, l_{γ})e^{∓l_{γ} \sqrt{B(ρ)−B(λ_{i,l}})}.

Analogous to [MS2] we have

∑ a_{i,l}(−1)^{l}R_{r}(s ± \sqrt{B(ρ) − B(λ_{i,l})}, l_{γ})D_{i,l}(γ) = \frac{tr(T_{s,γ}^{±}γ_{I} | ∧^{∗}𝔫^{∗})}{det(1 − γ | 𝔫)}

Since T interchanges with the differential on ∧𝔫^{∗} we have

tr(T_{s,γ}^{±}γ_{I} | ∧^{∗}𝔫^{∗}) = tr(T_{s,γ}^{±}γ_{I} | H^{∗}(𝔫)).

By corollary 5.7 in [Kost] we get the claim.

Putting things together we get Theorem 6.4.1 Let the fundamental rank r of G be odd. The zeta functions

Z_{φ}^{±}(s) = exp( ∑_{1≠[γ]∈E_{f} (γ)} tr φ(γ) χ_{r}(X_{γ}) L^{±}(s, γ)\frac{e^{−l_{γ}s}}{μ_{γ}})

admit analytic continuation to meromorphic functions on the complex plane. They satisfy a functional equation

Z_{φ}^{−}(−s) = Z_{φ}^{+}(s)e^{2P (s)}

where P denotes a polynomial of degree 2 dim(N). Around zero we find

Z_{φ}^{±}(s) = τ_{r}(E_{φ})s^{χ_{r} (X_{Γ},φ)} + higher order terms.

Furthermore we have τ_{1}(E_{φ}) = … = τ_{r−1}(E_{φ}) = 1.

6.5 The Generalized Selberg Zeta Function

Let r denote the fundamental rank of G which we assume to be > 0. For γ ∈ Γ let λ_{γ} denote the volume of a maximal compact flat containing the geodesic [γ] and let μ_{γ} = \frac{l_{γ}}{λ_{γ}} denote the generic multiplicity .

Theorem 6.5.1 Let the lattice Γ be good and nice and consider the zeta function

Z_{φ}(s) = ∏ ; [γ] prime, ; fundamental ∏_{N≥0} det(1 − ξ_{−ρ_{0}}(γ)e^{−sl_{γ}}|V_{N})^{\frac{χ_{r} (Xγ)}{μγ}} ,

where V_{N} = V_{φ} ⊗ H_{∗}(𝔫) ⊗ S^{N}(𝔫) and γ acts as φ(γ) ⊗ Ad(γ_{I}) ⊗ Ad^{N}(γ^{−1}).

a) If r = 1 then Z_{φ}(s) extends to a meromorphic function. If the complex number s_{0} is a singularity (pole or zero) then s_{0}^{2} + B(ρ_{0} + χ_{i,l}) is an eigenvalue of △_{i,l} for some l and some i ∈ J_{l}. The (vanishing-) order of Z_{φ} at s_{0} is

∑_{l=0}^{dim(N)}(−1)^{l} ∑_{i∈J_{l}} dim(Eig(△_{i,l}, s_{0}^{2} + B(ρ_{0} + χ_{i,l})).

b) If r > 1 is odd, the logarithm log(Z_{φ}(s)) extends to a meromorphic function. The poles of log(Z_{φ}(s)) have order \frac{r−1}{2} and if s_{0} is a pole then s_{0}^{2} + B(ρ_{0} + χ_{i,l}) is an eigenvalue of △_{i,l} for some l and some i ∈ J_{l}. The principal of the Laurent series around s_{0} then is

(\frac{1}{s^{2} − s_{0}^{2}} )^{\frac{r−1}{2}} \sqrt{4π}^{r−1} ∑_{l=0}^{dim(N)}(−1)^{l} ∑_{i∈J_{l}} dim(Eig△_{i,l}, s_{0}^{2} + B(ρ_{0} + χ_{i,l})).

c) If r > 0 is even, the square of the logarithm, (log(Z_{φ}(s)))^{2} extends to a meromorphic function. The poles of (log(Z_{φ}(s)))^{2} have order r−1 and if s_{0} is a pole then s_{0}^{2}+B(ρ_{0}+χ_{i,l}) is an eigenvalue of △_{i,l} for some l and some i ∈ J_{l}. The principal part of the Laurent series around s_{0} then is

(\frac{1}{s^{2} − s_{0}^{2}} )^{r−1}(4π)^{r−1}( ∑_{l=0}^{dim(N)}(−1)^{l} ∑_{i∈J_{l}} dim(Eig△_{i,l}, s_{0}^{2} + B(ρ_{0} + χ_{i,l})))^{2}.

Proof: We take the formula of Lemma 6.3.4 and multiply both sides with \sqrt{4πt}^{r−1}. Taking the Mellin transform now gives on the spectral side

\frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}Θ_{i,l}(t)\sqrt{4πt}^{r−1}e^{_{t}λ}dt = \sqrt{4π}^{r−1} \frac{Γ(s + \frac{r−1}{2})}{Γ(s)} ∑_{n=1}^{∞}(λ_{n} + λ)^{−s− \frac{r−1}{2}}

where (λ_{n}) runs through the eigenvalues of △_{i,l}. From this the theorem follows easily. ■

Dynamical Systems

In this chapter we are going to present a different approach to the zeta function which gives rise to some interesting speculations concerning possible generalizations.

Let P = MAN be a minimal parabolic of G with M ⊂ K and Lie(A) ⊥ Lie(K). Then M is the centralizer of A in K, in fact, the group AM will be the total centralizer of A. Note that the dimension of A is just the ℝ-rank of G or the rank of the symmentric space X = G/K. In the rank one case the sphere bundle writes

SX = G/M → G/K.

The geodesic flow then is

Φ_{t}(gM) = g exp(tH)M,

where H ∈ Lie(A) is an element of norm 1. Now let Γ be a torsionfree uniform lattice in G and let X_{Γ} = Γ\X. Among other things we considered the Ruelle Zeta Function of the geodesic flow on SX_{Γ} = Γ\G/M:

R(s) = ∏_{c}(1 − e^{−sl_{γ}}),

and we have seen that R extends to a meromorphic function which satisfies a determinant formula, involving Laplace operators on X_{Γ}.

On the other hand we know that the geodesic flow belongs to the very general class of Anosov flows and that the Ruelle zeta function of an Anosov flow already extends to a meromorphic function under much more general conditions then ours [Rue]. However, there’s no determinant formula in general and only very little is known about the analysis of a Ruelle zeta function.

To generalize to arbitrary Anosov flows we would have to get rid of the space X_{Γ} and have an approach purely in terms of the sphere bundle SX_{Γ} = Γ\G/M. Such an approach indeed exists, it was discovered by A. Juhl [Ju]. The reason, why it works is the following: In our approach we had to resort to X_{Γ} because the irreducible representations of G have finite K-type multiplicities, but not finite M-type multiplicities. Viewing the G-representation space L^{2}(Γ\G) as a sheaf over Ĝ this means that the restriction to any K-type τ (for example to p-forms on X_{Γ}):

(L^{2}(Γ\G) ⊗ τ)^{K}

gives a sheaf with finite dimensional stalks over Ĝ.

On the other hand the group M is the maximal compact subgroup of the Levi group AM so the same would count for appropriate sheaves on ÂM. There are natural functors joining Ĝ to ÂM. Of special interest to us is one that preserves characters of representations.

7.1 Juhl’s Approach

7.1.1 The Hecht-Schmid Character Formula

For any Lie algebra L and any L-module V we write H_{q}(L, V) for the q-th Lie algebra homology of L with coefficients in V and H^{q}(L, V) for the q-the Lie algebra cohomology.

Let G be a real reductive group with maximal compact K and V be a Harish-Chandra module of G, that is, V is an admissible (𝔤, K)-module, where 𝔤 is the Lie algebra of G. Recall that V has a character Θ_{G}(.|V) defined by Harish-Chandra. Θ_{G}(.|V) is a distribution on G given by a locally integrable and conjugation invariant function also denoted by Θ_{G}(.|V), which s real analytic on the regular set of G.

Now let P = MAN be a parabolic subgroup of G. Let Φ^{+}(𝔤, 𝔞) denote the set of positive roots according to the choice of P and let A^{+} denote the positive Weyl chamber in A. Let 𝔫 be the Lie algebra of N and let 𝔫^{−} = θ(𝔫) where θ is the Cartan involution. Set

(MA)^{+} =^{def} interior in MA of the set
{g ∈ MA|det(1 − ga|𝔫^{−}) ≥ 0 for all a ∈ A^{+}}.

Note that for dim(A) maximal we have (MA)^{+} = MA^{+}. Hecht and Schmid proved in [HeSchm]:

Theorem 7.1.1 For every Harish-Chandra module V we have that

• H_{p}(𝔫^{−}, V) is a Harish-Chandra module for the reductive group MA.

• For all x ∈ (MA)^{+} ∩ G^{reg}, the set of regular elements in MA we have

Θ_{G}(x|V) = \frac{∑_{p=0}^{dim(𝔫^{−})}(−1)^{p}Θ_{MA}(x|H_{p}(𝔫^{−}, V))}{det(1 − x|𝔫^{−})}.■

7.1.2 Nice Test Functions

Now assume G semisimple with finite center and K a maximal compact subgroup. Let P = MAN be the Langlands decomposition of a minimal parabolic subgroup. Then A is a maximal split torus and we may assume M ⊂ K and Lie(A) ⊥ Lie(K). Let 𝔞_{0} denote the Lie algebra of A, 𝔞_{0}^{+} a positive Weyl chamber according to the choice of P, A^{+} = exp(𝔞_{0}^{+}). Let 𝔫_{0}^{+} denote the Lie algebra of N and 𝔫_{0}^{−} its opposite, i.e.: N_{0}^{−} = θ(𝔫_{0}^{+}), where θ is the Cartan involution. Let

D_{±}(ma) =^{def} det(1 − ma|𝔫_{0}^{±})

for ma ∈ MA. We have the integration formula

∫_{G} f(g)dg = ∫_{MA\G} ∫_{M} ∫_{A^{+}} f(g^{−1}mag)|D_{+}(ma)D_{−}(ma)|dadmdg

for f ∈ C_{c}^{∞}(U), where U = {gxg^{−1}|g ∈ G, x ∈ (MA) ∩ G^{reg}}. All Haar measures in this formula are supposed to be normalized in the sense of Harish-Chandra ([HC-HAI],sec. 7).

To create our test functions let φ ∈ C_{c}^{∞}(A^{+}) and choose η ∈ C_{c}^{∞}(MA\G) such that η ≥ 0 and ∫_{MA\G} η(g)dg = 1. Define

Φ(g^{−1}mag) = η(g)φ(a)|D_{+}(ma)|^{−1}.

Then Φ is well defined, smooth and compactly supported. Further the support of Φ lies completely in the set of conjugates of elements of MA with regular A-part.

Let π be an irreducible unitary representation of G. By the Hecht-Schmid character formula we get

trπ(Φ) = ∫_{G} Θ_{G}(x|π)Φ(x)dx
= ∫_{MA\G} ∫_{M} ∫_{A^{+}} η(g)|D_{−}(ma)|Θ_{G}(ma|π)dadmdg

= ∫_{A^{+}} φ(a) ∫_{M} ∑_{p=0}^{dim(N)}(−1)^{p}Θ_{MA}(ma|H_{p}(𝔫^{−}, V))dmda

= ∫_{A^{+}} φ(a)tr(a|H_{∗}(𝔫^{−}, V)^{M})da.

Note that the space H_{p}(𝔫^{−}, V)^{M} is finite dimensional.

7.1.3 The Lefschetz Formula

For any γ ∈ Γ such that γ is conjugate to an element of MA write m_{γ}a_{γ} for such an element. Whenever this notation is used, one has to make clear that it is independent from the particular m_{γ}a_{γ} ∈ MA chosen.

Theorem 7.1.2 (Juhl) Let φ ∈ C_{c}^{∞}(A^{+}) then we have the identity

∑_{π∈Ĝ} N_{Γ}(π) ∫_{A^{+}} φ(a)tr(a|H_{∗}(𝔫^{−}, V)^{M})da = ∑_{[γ]⊂Γ} λ_{γ} \frac{φ(a_{γ})}{|D_{+}(m_{γ}a_{γ})|},

where the sum runs over all γ ∈ Γ that are in G conjugate to an element of MA and λ_{γ} = vol(Γ_{γ}\MA) is the volume of a maximal flat containing the geodesic [γ].

Proof: One simply plugs the function Φ into the trace formula. It is clear by the above calculation that the spectral side gets the above shape. The expression for the geometric side follows directly from the definition of Φ. ■

This Theorem can easily be generalized allowing test functions φ which are defined on A^{+}, vanish on the boundary to a sufficiently high order and are decaying at infinity fast enough.

Corollary 7.1.1 Let φ ∈ C_{c}^{∞}(A^{+}) then we have the identity

∑_{π∈Ĝ} N_{Γ}(π) ∫_{A^{+}} φ(a)tr(a|H^{∗}(𝔫^{−}, V_{π})^{M})da = (−1)^{dim N} ∑_{[γ]⊂Γ} λ_{γ} \frac{φ(a_{γ})}{det(1 − m_{γ}a_{γ}|𝔫^{−})},

where the sum runs over all γ ∈ Γthat are in G conjugate to an element of MA.

Proof: We use the natural isomorphism:

H_{p}(𝔫^{−}, V) ≅ H^{d−p}(𝔫^{−}, V) ⊗ ∧^{d}𝔫^{−},

([HeSchm] (2.18)) where d denotes the dimension on 𝔫. Since M acts trivially on ∧^{d}𝔫^{−} we get

tr(a|H_{∗}(𝔫^{−}, V)^{M}) = (−1)^{d}tr(a|H^{∗}(𝔫^{−}, V)^{M}ξ_{−2ρ}(a).

So one simply replaces φ(a) by φ(a)ξ_{2ρ}(a) to achieve the assertion of the corollary out of the theorem.■

By plugging in suitable test functions φ, Juhl arrives at expressions involving the Selberg zeta function. This works well for the rank one situation. For higher rank we may put in modified test functions as in 5.4

Then we get

Theorem 7.1.3 (Higher rank Lefschetz Theorem) Suppose Γ is nice. Let H = AB any Θ-stable Cartan subgroup of G with r = dim(A) > 0. Choose a parabolic P = MAN. Then we have an equality of distributions on the positive Weyl chamber A^{+}:

∑_{π∈Ĝ} N_{Γ}(π)tr(a|H^{∗}(𝔫̄, π) ⊗ ∧^{∗}𝔭_{M})^{K_{M}}) = (−1)^{dim(N)} ∑_{[γ]∈E_{H}(Γ)} λ_{γ}χ_{r}(X_{γ})\frac{δ_{a_{γ}}}{det(1 − m_{γ}a_{γ}|𝔫̄)}.■

7.2 The Guillemin Conjecture

The spectral side of the corollary in the last section can be viewed as the A-trace on the complex

0 → (L^{2}(Γ\G) ⊗ (∧^{0}𝔫^{−})^{∗})^{M} → (L^{2}(Γ\G) ⊗ (∧^{1}𝔫^{−})^{∗})^{M} → …
… → (L^{2}(Γ\G) ⊗ (∧^{top}𝔫^{−})^{∗})^{M} → 0,

with the Lie algebra differential. But this complex also has a different interpretation. Recall that the stable bundle on SX_{Γ} = Γ\G/M could be described as

T^{s} = Γ\G ×_{M} 𝔫^{−}.

so the above complex becomes the stable complex :

o → L^{2}(∧^{0}T^{s}) → L^{2}(∧^{1}T^{s}) → … → L^{2}(∧^{top}T^{s}) → 0,

and a short computation shows that the differential is just the exterior differential of SX_{Γ} followed by a projection ∧^{p}T → ∧^{p}T^{s}, where T denotes the tangent bundle of SX_{Γ}. Note that the stable complex is not elliptic, a fact which is remedied in our situation by the cohomology having finite multiplicities as a G-representation. Now one might present the following

Conjecture 7.2.1 Let Φ_{t} denote a smooth Anosov flow on a compact manifold then the flow has a distributional trace for t > 0 on the cohomology of the stable complex such that for t > 0

tr(Φ_{t}|H^{∗}(L^{2}(∧^{.}T^{s}))) = (−1)^{dim T^{s}} ∑_{c} \frac{l(c_{0})δ_{l(c)}}{det(1 − P_{c}^{s})}.

Here the sum runs over all closed orbits c of Φ and l(c) denotes the length of the orbit c. To an orbit c let c_{0} denote the underlying primitive orbit and P_{c}^{s} the stable part of the Poincaré map around c and δ_{x} denotes the delta distribution at the point x.

Compare this to ([Gu], p.504).

Perhaps one should weaken this conjecture in insisting that there should exist, as in our case, a smooth perfect pairing

B_{x} : T_{x}^{s} × T_{x}^{u} → ℂ

which is invariant under the flow, i.e. B_{Φ_{t}x}(DΦ_{t}v, DΦ_{t}w) = B_{x}(v, w).

Once having established this conjecture in a particular setting one can proceed as in [Ju] to finally get an analysis of the Ruelle zeta function in terms of the spectrum of the flow on the stable cohomology. This would possibly clarify the relation of the Ruelle zeta function to topological invariants as torsion and Euler characteristics. Further it would show whether the Ruelle zeta function in general satisfies an analogue of the Riemann hypothesis.

The Functional Equation of Theta Series

Geometric zeta functions are defined by geometrical data (closed geodesics) and turn out to have singularities (zeroes and poles) which are related to spectral data (Laplace eigenvalues). The theta series to be discussed in this section are dual to the zeta functions in that they are defined by the very same spectral data and have their singularities at the lengths of the closed orbits.

8.1 The Rank One Case

For the contents of this subsection compare [CaVo] and [BuOl]. We consider the situation of chapter 4. So G is the group of orientation preserving isometries of the even-dimensional rank one symmetric space X, which we identify with G/K where K is a maximal compact subgroup of G. We fix a torsionfree uniform lattice Γ in G and consider the compact Shimura manifold X_{Γ} = Γ\X.

In chapter 4 we considered the zeta function

Z_{σ,φ}(s) = ∏_{[γ] prime} ∏_{N≥0} det(1 − e^{−sl_{γ}}γ | V_{N}),

where V_{N} = V_{φ} ⊗ V_{σ} ⊗ S^{N}(𝔫) and γ acts on V_{N} as γ ↦ φ(γ) ⊗ σ(m_{γ}) ⊗ Ad^{N}((m_{γ}a_{γ})^{−1}). We showed (4.3.1) that with c_{σ} = B(λ_{σ}) − B(ρ) we have

Z(ρ_{0} + s) = det(△_{σ,φ} + s^{2} + c_{σ})(det(D_{σ} + s) exp(P (s^{2})))^{2(−1)^{m} dim(φ)χ(X_{Γ})/χ(X^{d})}.

Now we are going to consider the theta series :

Θ_{σ,φ}(τ) = tr(e^{−τ\sqrt{△_{σ,φ}+c_{σ}}}),

where the root is positive on ℝ_{+} and negative imaginary on ℝ_{−}. Let

F (s) = det(△_{σ,φ} + s^{2} + c_{σ})
= ∏_{j≥0}(a_{j}^{2} + s^{2})

where a_{j} ≥ 0 or ia_{j} ≥ 0 and the product means regularized product. So, if (λ_{j})_{j∈ℕ} is the sequence of eigenvalues of the operator △_{σ,φ}, then a_{j}^{2} = λ_{j} + c_{σ}. Assume that the a_{j} are numerated in such a way that the sequence a_{j}^{2} is ascending. There are only finitely many a_{j} on the imaginary axis and there is a p > 0 such that ∑′_{j} |a_{j}|^{−p} < ∞. Let a > |a_{0}| be arbitrary. Then the theta series is

Θ_{σ,φ}(τ) = ∑_{j=0}^{∞} e^{−τa_{j}} Re (τ) > 0.

We are further going to consider the dual space theta function :

Θ_{σ,d}(τ) = tr(e^{−τD_{σ}}), Re (τ) > 0.

Since we fix the M-type σ and the Γ-representation φ we are going to leave these out in our notation so we will write Θ = Θ_{σ,φ} and Θ_{d} = Θ_{σ,d}. Our first result is

Proposition 8.1.1 The function Θ_{d} = Θ_{σ,d} extends to a meromorphic function on the entire plane. It is even, i.e.:

Θ_{d}(−τ) = Θ_{d}(τ),

its poles are all of order dim(X) and are located at τ = πik, k ∈ ℤ.

Proof: We first consider the case when σ is even in the sense of 4.3.2:

Θ_{d}(τ) = ∑_{n=1}^{∞} Q_{σ}(2n)e^{−2τn}

= Q_{σ}(−\frac{∂}{∂τ}) ∑_{n=1}^{∞} e^{−2τn}
= Q_{σ}(−\frac{∂}{∂τ})(\frac{1}{e^{2τ} − 1})

This gives the meromorphic continuation and the location and order of the poles (recall that the polynomial Q_{σ} has order dim(X) − 1. To see that Θ_{d} is even, recall that

Q_{σ}(x) = xQ̃_{σ}(x^{2})

for a polynomial Q̃_{σ}. It follows

Θ_{d}(τ) = 2Q̃_{σ}((\frac{∂}{∂τ})^{2})(\frac{1}{(e^{t} − e^{−t})^{2}} ),

which gives the claim. For σ odd we similarly get

Θ_{d}(τ) = Q_{σ}(−\frac{∂}{∂τ})\frac{1}{e^{t} − e^{−t}} ,

which gives the claim in that case. ■

Theorem 8.1.1 The theta series Θ = Θ_{σ,φ} admits a continuation to a meromorphic function on ℂ. It satisfies a functional equation:

Θ(τ) + Θ(−τ) = 2(−1)^{m+1} dim(φ)\frac{χ(X_{Γ})}{χ(X^{d})}Θ_{d}(iτ).

The function Θ(τ) has simple poles at τ = ±il_{γ}, γ ≠ 1 and the residue at τ = ±il_{γ} is

−\frac{1}{2π} \frac{l_{γ}}{μ_{γ}} \frac{trφ(γ)trσ(m_{γ})}{det(1 − γ^{−1}|𝔫)}.

The function Θ(τ) further has poles of order dim(X) at the nonpositive poles of Θ_{d}(iτ). This concludes the list of poles of Θ.

Proof: The argument principle shows that

Θ(τ) = n_{0} + \frac{1}{2πi} ∫_{C} e^{iτz}(\frac{F′}{F} (z) − \frac{2n_{0}}{z})dz,

where n_{0} = ord_{z=0}F (z) and C is the contour shown at the following picture:

[figure: C; ia_{0}; ia_{1}; ia_{2}; ia_{3}; ia_{4}; ia_{5}; −ia_{0}; −ia_{1}; 0; −a; a]

At first the contour goes from −a + i∞ to −a, then from −a to a circumventing the ±ia_{j} which are nonzero as shown and then it goes from a to a + i∞. If one of the a_{j} happens to be zero, which is equivalent to the fact that n_{0} ≠ 0, the contour goes right trough it.

Let C^{+} = C ∩ {Re (z) > 0} and C^{−} = C ∩ {Re (z) < 0}. Since F is even it follows that the logarithmic derivative \frac{F′}{F} is odd and hence

Θ(τ) = n_{0} + \frac{1}{2πi} ∫_{C^{+}} e^{iτz}(\frac{F′}{F} (z) − \frac{2n_{0}}{z})dz.

Now by the determinant formula we get

\frac{F′}{F} (z) − \frac{2n_{0}}{z} = \frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z} + 2αzP′(z^{2}) + \frac{d}{dz} logdet(D_{σ} + z),

where α = 2(−1)^{m+1} dim(φ)\frac{χ(X_{Γ})}{χ(X^{d})}.

Therefore Θ(τ) splits as follows:

Θ(τ) = n_{0} + Θ_{Z}(τ) + Θ_{pol}(τ) + Θ_{dual}(τ).

Let’s start with the simplest part

1. The polynomial contribution This is

Θ_{pol} = \frac{α}{πi} ∫_{C^{+}} e^{iτz}zP′(z^{2})dz
= \frac{α}{π} \frac{∂}{∂τ} P′(−(\frac{∂}{∂τ})^{2})(∫_{C^{+}} e^{iτz}dz − ∫_{−C^{−}} e^{−iτz}dz)
= \frac{α}{π} \frac{∂}{∂τ} P′(−(\frac{∂}{∂τ})^{2})(\frac{i}{τ} − \frac{i}{τ})
= 0.

So the polynomial contribution vanishes.

2. The zeta contribution Here we consider

Θ_{Z}(τ) = \frac{1}{2πi} ∫_{C^{+}} e^{iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz + \frac{1}{2πi} ∫_{−C^{−}} e^{−iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz
= \frac{1}{π} ∫_{0}^{a} sin(τz)(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz

+\frac{1}{2πi} ∫_{a}^{a+i∞} e^{−iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz

+\frac{1}{2πi} ∫_{a−i∞}^{a} e^{−iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz.

The first integral should be read as a path integral over a path circumventing the singularities in the lower half plane. The first summand thus defines an odd entire function in τ. Therefore it does not contribute to the functional equation.

We may assume a to be large enough so that the Euler product for the zeta function Z converges at Re (z) = a. Then we get

logZ(ρ_{0} + z) = − ∑_{[γ]prime} ∑_{N≥0} ∑_{k=1}^{∞} \frac{1}{k} e^{−zkl_{γ}}trφ(γ^{k})trσ(m_{γ^{k}})trAd^{N}((m_{γ^{k}}a_{γ^{k}})^{−1}|𝔫)

= − ∑_{[γ]≠1} ∑_{N≥0} \frac{e^{−zl_{γ}}}{μ_{γ}} trφ(γ)trσ(m_{γ})trAd^{N}((m_{γ}a_{γ})^{−1}|𝔫)

= − ∑_{[γ]≠1} \frac{e^{−zl_{γ}}}{μ_{γ}} \frac{trφ(γ)trσ(m_{γ})}{det(1 − γ^{−1}|𝔫)},

and so

\frac{d}{ds}logZ(ρ_{0} + z) = ∑_{[γ]≠1} b_{γ}e^{−zl_{γ}},

where

b_{g}a = \frac{l_{γ}}{μ_{γ}} \frac{trφ(γ)trσ(m_{γ})}{det(1 − γ^{−1}|𝔫)}.

From this we get

\frac{1}{2πi} ∫_{a}^{a+i∞} e^{−iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz + \frac{1}{2πi} ∫_{a−i∞}^{a} e^{−iτz}(\frac{Z′}{Z} (ρ_{0} + z) − \frac{2n_{0}}{z})dz

= \frac{1}{2πi} ∑_{[γ]≠1} b_{γ}(\frac{e^{(−iτ−l_{γ})a}}{−iτ − l_{γ}} − \frac{e^{(iτ−l_{γ})a}}{iτ − l_{γ}}).

which is an odd meromorphic function.

To finish the zeta contribution it remains to consider

f(τ) =^{def} − \frac{1}{2πi} ∫_{a−i∞}^{a} e^{−iτz} \frac{2n_{0}}{z} dz − \frac{1}{2πi} ∫_{a}^{a+i∞} e^{iτz} \frac{2n_{0}}{z} dz.

We see that f is a primitive of the function \frac{2n_{0}}{π} \frac{sin(aτ)}{τ} and that lim_{τ→+∞} f(τ) = 0. From this it follows that f(0) = \frac{2n_{0}}{π} ∫_{0}^{∞} \frac{sin(x)}{x}dx = −n_{0}. Since f is the primitive of an even entire function we conclude that f is entire and

f(τ) + f(−τ) = −2n_{0},

which makes the 2n_{0} drop out of the functional equation.

CHAPTER 8 The Functional Equation of Theta Functions 112 3. The dual space contribution We consider

Θ_{dual}(τ) = \frac{α}{2πi} ∫_{C^{+}} e^{iτz}dlogdet(D_{σ} + z) + \frac{α}{2πi} ∫_{−C^{−}} e^{−iτz}dlogdet(D_{σ} + z).

We replace −C^{−} by C^{−} again and move the contour on to the imaginary axis to get

Θ_{dual}(τ) = \frac{α}{2πi} ∫_{0}^{i∞} e^{iτz}(dlogdet(D_{σ} + z) − dlogdet(D_{σ} − z))
= \frac{α}{2πi} ∫_{0}^{infty} e^{−τt}(\frac{dlogdet(D_{σ} + z)}{dz}|_{z=it} − \frac{dlogdet(D_{σ} − z))}{dz}|_{z=−it})dt

Now we have

\frac{d}{dz} logdet(D_{σ} + z) = −\frac{d}{dz} ζ′_{D_{σ}+z}(0)
= −\frac{d}{dz} \frac{d}{ds}|_{s=0}ζ_{D_{σ}+z}(s)
= −\frac{d}{ds}|_{s=0} \frac{d}{dz} ζ_{D_{σ}+z}(s)
= \frac{d}{ds}|_{s=0}sζ_{D_{σ}+z}(s + 1)
= ζ_{D_{σ}+z}(1),

since ζ_{D_{σ}+z}(s) is holomorphic at s = 1. Therefore we conclude

Θ_{dual}(τ) = \frac{α}{2πi} ∫_{0}^{∞} e^{−τt}(ζ_{D_{σ}+it}(1) − ζ_{D_{σ}−it}(1))dt.

Now assume σ odd and recall that

ζ_{D_{σ}+z}(1) = ∫_{0}^{∞} t^{s−1}e^{−tz}Θ_{d}(t)dt|_{s=1}
= ∫_{0}^{∞} t^{s−1}e^{−tz}Q_{σ}(−\frac{∂}{∂t})\frac{1}{e^{t} − e^{−t}} dt|_{s=1}
= ∫_{0}^{∞} t^{s−1} \frac{1}{e^{t} − e^{−t}} Q_{σ}(\frac{∂}{∂t})e^{−tz}dt|_{s=1}
= Q_{σ}(−z) ∫_{0}^{∞} \frac{t^{s−1}e^{−tz}}{e^{t} − e^{−t}} dt|_{s=1}
= Q_{σ}(−z) ∑_{n=0}^{∞} ∫_{0}^{∞} t^{s−1}e^{−t(z+1)}e^{−2nt}dt|_{s=1}
= Q_{σ}(−z) ∑_{n=0}^{∞} \frac{1}{(z + 2n + 1)^{s}} |_{s=1}.

And thus

ζ_{D_{σ}+it}(1) − ζ_{D_{σ}−it}(1) = Q_{σ}(−it)(∑_{n=0}^{∞} \frac{1}{(2n + 1 + it)^{s}} − ∑_{n=0}^{∞} \frac{1}{(2n + 1 − it)^{s}})|_{s=1}

= Q_{σ}(−it) ∑_{n=0}^{∞} \frac{2it}{(2n + 1)^{2} + t^{2}}

= \frac{π}{2} Q_{σ}(−it)i tanh(\frac{πt}{2}).

So for σ odd we get

Θ_{dual} = \frac{α}{4} ∫_{0}^{∞} e^{−τt}Q_{σ}(−it) tanh(\frac{πt}{2})dt
= \frac{α}{4}Q_{σ}(i\frac{∂}{∂τ}) ∫_{o}^{∞} e^{−τt} tanh(\frac{πt}{2})dt.

Since

tanh(\frac{πt}{2}) = 1 + 2 ∑_{n=1}^{∞}(−1)^{n}e^{−πnt},

we get

Θ_{dual}(τ) = \frac{α}{4i}Q̃_{σ}(−(\frac{∂}{∂τ})^{2})(\frac{1}{τ^{2}} + 2 ∑_{n=1}^{∞}(−1)^{n} \frac{1}{(τ + πn)^{2}} ),

which completes the meromorphic continuation. As for the functional equation one gets

Θ_{dual}(τ) + Θ_{dual}(−τ) = \frac{α}{2i}Q_{σ}(i\frac{∂}{∂τ}) ∑_{n∈ℤ}(−1)^{n} \frac{1}{τ + πn}

= \frac{α}{2i}Q_{σ}(i\frac{∂}{∂τ})\frac{1}{e^{t} + e^{−t}} ,

which gives the claim.

For even σ one gets analogously

Θ_{dual}(τ) = \frac{α}{4} ∫_{0}^{∞} e^{−τt}Q_{σ}(−it)(coth(\frac{πt}{2}) − \frac{2}{πt})dt.

The summand with \frac{2}{πt} gives an odd rational function thus does not contribute to the functional equation. The rest is analogous to the odd case. ■

8.2 Higher Rank

The main ingredients of the proof of the functional equation of the theta series in the rank one case were the Euler product and the determinant formula for the zeta function. It is easy to see that the theorem generalizes to all the various situations in the preceding chapters where we had established these two ingredients. We are not going to bore the reader in listing all the results. We will just restrict to the case SL_{2}(ℝ)^{m}.

We take up the situation of 5.2 again, so G = SL_{2}(ℝ)^{m} for some natural number m, Γ ⊂ G is a nice cocompact discrete subgroup and (φ, V_{φ}) is a finite dimensional unitary representation of Γ. The representation φ defines a holomorphic flat hermitian vector bundle E_{φ} over X_{Γ} = Γ\X = Γ\G/K, We denote by △_{p,q,φ} the Hodge-Laplace operator on E_{φ}-valued (p, q)-forms. We consider the theta function :

Θ_{φ}(τ) = ∑_{q=0}^{m} tre^{−τ\sqrt{△_{0,q,φ}− \frac{1}{4}}} , Re (τ) > 0.

To establish a functional equation of this theta series we also need a dual theta series:

Θ_{d}(τ) = tre^{−τP}
= ∑_{j=0}^{∞}(2j + 1)e^{−τ(j+ \frac{1}{2} )}.

Then we get

Theorem 8.2.1 The dual theta function Θ_{d}(τ)extends to an even meromorphic function. The poles of Θ_{d} are of order 2 and are located at τ = πik, k ∈ ℤ. The theta function Θ_{φ}(τ) extends to a meromorphic function on the entire plane which satisfies the functional equation

Θ_{φ}(τ) + θ_{φ}(τ) = 2m dim(φ)Ar(X_{Γ})Θ_{d}(iτ).

The poles of Θ_{φ}(τ) are the nonnegative poles of Θ_{d}(iτ) and simple poles at τ = ±il_{γ} γ ∈ Γ−{1}, ℝ − rank(γ) = 1 and the residue at τ = ±il_{γ} is

−\frac{1}{2π} \frac{l_{γ}}{μ_{γ}} \frac{tr(φ(γ))}{det(1 − γ^{−1}|𝔫)}χ_{1}(X_{γ}).

■

The remaining cases of higher rank are left as an exercise to the interested reader.

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Index

B, 44, 50 F R(G), 42 K-separable, 49 L^{2}-characteristic function, 14 L^{2}-determinant, 14 L^{2}-torsion, 36 L^{2}-zeta function, 13 L^{2}(Γ\G, φ), 51 M_{σ}(z, λ), 53 N_{Γ,φ}(π), 51 W (G, H), 41 X_{α}, 41 [π : τ], 51 Φ(𝔥, 𝔤), 40 Φ_{γ}, 41 Φ_{n}, 41 dim_{Γ}, 12 Ĝ_{d}, 54 μ_{γ}, 98, 100 ν_{γ}, 41 π(C), 51 ρ_{γ}, 41 tr_{Γ}, 12 c_{γ}, 41 d_{π}, 51 m_{z}^{σ}, 53

acceptable Ł-module, 33 admissible complex, 24, 27 admissible operator, 8 Anosov flow, 47 Arakelov theory, 26

Cartan involution, 38 characteristic function, 11 cohomologically finite complex, 31 compact root, 41 complex, 6 complex root, 40 cuspidal group, 91

determinant, 8, 27

dual space theta function, 108 dual symmetric space, 54 dual theta series, 114

eigenvalue counting function, 23

Euler characteristic of a spectral sequence,

35 Euler-Poincaré measure, 45 exterior tensor product, 17

fundamental rank, 42

generalized Laplacian, 11 generic Euler number, 32 generic geodesic, 76 generic multiplicity, 76, 100 geodesic flow, 46 good group, 91 Gromov-Novikov-Shubin invariant, 13 Grothendieck group, 6

Harish-Chandra, 103

hermitian symmetric spaces, 49 higher Euler characteristic, 31 higher holomorphic torsion, 36 higher torsion, 29 holomorphic torsion, 25 homogeneous vector bundle, 49

imaginary root, 40

Laplacian, 24

nice subgroup, 41

orbital integral, 44

pseudocuspform, 43 pseudofinite, 27

Quillen metric, 26

real root, 40 regularized product, 9 Representation ring, 49 residually finite, 14 root space decomposition, 41 root system, 40 Ruelle Zeta Function, 102 Ruelle zeta function, 47

Shimura manifold, 38 Shimura variety, 38 spectral gap at zero, 16 spectral zeroes, 47 stable complex, 106 standard complex of Lie algebra cohomol-

ogy, 34 standard measure, 44 standard volume, 41

theta function, 114 theta series, 107 topological zeroes, 47 torsion, 24 tower, 14 twist of a virtual complex, 28

virtual spectral sequence, 33 virtual vector space, 6 von Neumann algebra of a group, 12 Weyl group, 41

zeta function of an operator, 8