Analysis on Symmetric and Locally Symmetric Spaces

(Multiplicities, Cohomology and Zeta functions)

Ulrich Bunke and Robert Waldmüller

1 Multiplicities of principal series

Let us first fix some notation. Let G be a semisimple Liegroup with finite center and G ≅ KAN be the Iwasawa decomposition of G, where K is a maximal compact subgroup, A a maximal ℝ-split torus of G, and N is a nilpotent subgroup. Correspondingly, we have a decomposition of the Lie algebra: 𝔤 ≅ 𝔨 ⊕ 𝔞 ⊕ 𝔫. We denote by P := MAN a minimal parabolic subgroup, where M = Z_{K}(A) is the centraliser of K in A. For λ ∈ 𝔞_{ℂ}^{∗}, we define the character a ↦ a^{λ} := exp(λ(log a)). Furthermore, let ρ ∈ 𝔞^{∗} denote H ↦ \frac{1}{2}T r(ad(H)|_{𝔫}). For a finite dimensional representation (σ, V_{σ}) of M, we define σ_{λ} : P → GL(V_{σ}) by man ↦ a^{ρ−λ}σ(m).

Definition 1 The induced representation π^{σ_{λ}} := Ind_{P}^{G}(σ_{λ}) is called the principal series representation with parameter (σ, λ).

π^{σ_{λ}} acts on the space of sections of regularity ? of the G-equivariant bundle V (σ, λ) := G ×_{P} V_{σ_{λ}}. The space of sections will be denoted by V_{π_{σ_{λ}}}^{?} := C^{?}(G/P, V (σ, λ)), where ? belongs to K − fin, ω, ∞, L^{2}, −∞ or −ω. Here, K − fin means K-finite, ω real-analytic, ∞ smooth and L^{2} square integrables. To be honest,C^{K−fin}(G/P, V (σ, λ)) is only a (𝔤, K)-module. Denote the ℂ-linear dual of σ by σ^{∗}. To define the hyperfunction and distribution valued sections, observe that V (1, −ρ) → G/P is the bundle of densities, and we have a G-equivariant pairing V (σ, λ) ⊗ V (σ^{∗}, −λ) → V (1, −ρ). Combining this pairing with integration, we have a pairing of spaces of sections.

Definition 2

C^{−ω}(G/P, V (σ, λ)) := C^{ω}(G/P, V (σ^{∗}, −λ))^{∗}
C^{−∞}(G/P, V (σ, λ)) := C^{∞}(G/P, V (σ^{∗}, −λ))^{∗}.

C^{L^{2}}(G/P, V (σ, λ)) is a G-Banach representation. If ℜ(λ) = 0, it is unitary in a natural way.

Example We consider G = SL(2, ℝ). The Iwasawa components are

K = \{(\begin{matrix}cos ϕ & − sin ϕ \\ sin ϕ & cos ϕ \end{matrix}) |ϕ ∈ [0, 2π)\} A = \{(\begin{matrix}t^{\frac{1}{2}} & 0 \\ 0 & t^{− \frac{1}{2}} \end{matrix} ) |t ∈ (0, ∞)\}

N = \{(\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) |x ∈ ℝ\}

In this case, we have 𝔞 ≅ ℝH, where H = (\begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix}). Calculating ρ(H) = \frac{1}{2},

we obtain that ρ ≅ \frac{1}{2}. Since M ≅ \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix})\} ≅ ℤ/2ℤ, M̂ = \{1, θ\}

consists of two elements. Furthermore, G/P = K/M ≅ S^{1}. Choosing σ = 1, the principal series representations are parametrized by λ ∈ ℂ. For generic λ, the representation is irreducible.

− \frac{1}{2} − n \frac{1}{2} \frac{1}{2} + n

( )

unitary principal series

complementary series

\begin{matrix}D{+ \atop 2n+1} & D{− \atop 2n+1} \\ F2n+1 \end{matrix}

\begin{matrix}F2n+1 \\ D{+ \atop 2n+1} & D{− \atop 2n+1} \end{matrix}

D^{−^{+}} is called the holomorphic/antiholomorphic discrete series, and F_{2n+1} is the 2n+1-dimensional irreducible representation of SL(2, ℝ). The unitary principal series representations, the complementary series representations, D_{2n+1}^{±} and F_{1} are unitary. Note that π^{1_{±} \frac{1}{2}} is a non-trivial extension and therefore not unitary.

Let Γ ⊂ G be a cocompact, torsion-free discrete subgroup, X := G/K, Y := Γ\X and ^{Γ}V_{π^{σ_{λ}}}^{−ω} := {ϕ ∈ V_{π^{σ_{λ}}}^{−ω}|π^{σ_{λ}}(γ)ϕ = ϕ ∀ γ ∈ Γ}, the space of Γ-invariant hyperfunction vectors.

Lemma 1 (Frobenius reciprocity)

ΓV_{π^{σ_{λ}}}^{−ω} ≅ Hom_{G}(V_{π^{σ{∗ \atop −λ}}}^{−ω} , C^{∞}(Γ\G))
ϕ ↦ (f ↦^{i_{ϕ}} (g ↦ ϕ(π^{σ^{∗}^{−λ}}(g)f)))
(f ↦ (ψ(f)(Γ1))) ← ψ

Since Γ is cocompact, we have a decomposition

C^{∞}(Γ\G) ≅ ˆ⨁_{(π,V_{π})∈Ĝ_{u}}m_{Γ}(π)V_{π}^{∞},

where Ĝ_{u} is the unitary dual of G, i.e. the set of equivalence classes of irreducible unitary representations of G. As a consequence, if π^{σ_{λ}} is unitary and λ ≠ 0, then it is irreducible and therefore m_{Γ}(π^{σ_{λ}}) = dim(^{Γ}V_{π^{σ_{λ}}}^{−ω}). Moreover, if π^{σ_{λ}} is irreducible and not unitary, then ^{Γ}V_{π^{σ_{λ}}}^{−ω} = {0}. We now return to our example and connect ^{Γ}V_{π^{σ_{λ}}}^{−ω} with the eigenvalues of the Laplacian on Y. As K-homogenous bundles, V (σ, λ) = G ×_{P} V_{σ_{λ}} = K ×_{M} V_{σ}, and for σ = 1, K ×_{M} V_{1} ≅ S^{1} × ℂ. Denote by f ∈ C^{∞}(K ×_{M} V_{1}) the unique K-invariant section with f(1) = 1. For any choice of λ, this corresponds to a real-analytic section f_{λ} of V (1, λ). Using the reciprocity homomorphism i : V_{π^{1_{λ}}}^{−ω} → Hom_{G}(V_{π^{1_{−λ}}}^{ω} , C^{∞}(G)) for trivial Γ , we define the Poisson transformation

i_{·}(f_{−λ}) : V_{π^{1_{λ}}}^{−ω} → C^{∞}(G).

Now, any ϕ ∈ V_{π^{1_{λ}}}^{−ω} defines i_{ϕ}(f_{−λ}) ∈ C^{∞}(X). The Casimir-operator Ω of 𝔤 acts on π^{1_{λ}} by a scalar, π^{1_{λ}}(Ω) = μ(λ) := \frac{1}{4} − λ^{2}, while Ω|_{C^{∞}(X)} = Δ_{X}, the Laplace operator. Therefore, i_{ϕ}(f_{−λ}) ∈ Ker(Δ_{X} − μ(λ)).

Theorem 1 (Helgasson [9]) For λ ∉ \{−\frac{1}{2}, −\frac{3}{2}, −\frac{5}{2}, ...\}, the Poisson transformation

i_{·}(f_{−λ}) : V_{π^{1_{λ}}}^{−ω} → Ker(Δ_{X} − μ(λ))

is an isomorphism.

To calculate H^{∗}(Γ, V_{π^{1_{λ}}}^{−ω}), we need the following classical theorems.

Theorem 2 Let M be a C^{ω}-manifold, E, F → M C^{ω}-vector bundles and A : C^{∞}(M, E) → C^{∞}(M, F) an elliptic operator with C^{ω}-coefficients. If M is non-compact, then A is surjective.

Theorem 3 ( [1],Lemma 2.4, 2.6) Let a discrete group U act properly on a space M and let F be a soft or flabby U-equivariant sheaf on M. Then H^{i}(U, F(M)) = 0 for all i ≥ 1.

Combining Theorem 1 and Theorem 2, we get a short exact sequence for λ ∉ \{−\frac{1}{2}, −\frac{3}{2}, −\frac{5}{2}, ...\}

0 → V_{π^{1_{λ}}}^{−ω} →^{i_{·}(f_{−λ})} C^{∞}(X) →^{Δ_{X}−μ(λ)} C^{∞}(X) → 0.

By Theorem 3, this is a Γ-acyclic resolution of V_{π^{1_{λ}}}^{−ω}, so for λ ∉ −\frac{1}{2} − ℕ_{0}, we get

\begin{matrix}i & H^{i}(Γ, V_{π^{1_{λ}}}^{−ω}) \\ 0 & ker(Δ_{Y} − μ(λ)) \\ 1 & coker(Δ_{Y} − μ(λ)) \end{matrix}

Since (Δ_{Y} − μ(λ)) is an elliptic operator, both spaces are finite dimensional. To compute ^{Γ}V_{π^{1_{λ}}}^{−ω} for λ ∈ −\frac{1}{2} − ℕ_{0}, we need H^{i}(Γ, V_{π^{1_{λ}}}^{−ω}) for i ≥ 1, λ ∈ \frac{1}{2} + ℕ as well. In the case of λ ∈ \frac{1}{2} + ℕ, we have the exact sequence

0 → F_{2n+1} → V_{π^{1_{n+ \frac{1}{2}}}}^{−ω} → D_{2n+1}^{−ω} → 0

Since in this case μ(λ) = \frac{1}{4} − λ^{2} is negative, Δ_{Y} − μ(λ) is a positive operator on a compact manifold and therefore injective,so ^{Γ}V_{π^{1_{λ}}}^{−ω} is trivial. Using the long exact sequence

... → H^{i}(Γ, F_{2n+1}) → H^{i}(Γ, V_{π^{1_{n+ \frac{1}{2}}}}^{−ω} ) → H^{i}(Γ, D_{2n+1}^{ω}) → ...

and the topological result

\begin{matrix}i & dim(H^{i}(Γ, F_{2n+1})) & dim(H^{i}(Γ, F_{1})) \\ 0 & 0 & 1 \\ 1 & (2g-2)(2n+1) & 2g \\ 2 & 0 & 1 \end{matrix} ,

where g denotes the genus of Y, we get

i dim(H^{i}(Γ, F_{2n+1})) dim(H^{i}(Γ, V_{π^{1_{λ}}}^{−ω})) dim(H^{i}(Γ, D_{2n+1}^{−ω})) 0  0  0  (2g-2)(2n+1) 1  (2g-2)(2n+1)  0  0 2  0  0  0 .

For λ = 1/2, we have that dim(ker(Δ_{Y})) = dim(coker(Δ_{Y})) = 1, so we get

\begin{matrix}i & dim(H^{i}(Γ, F_{1})) & dim(H^{i}(Γ, V_{π^{1_{1/2}}}^{−ω})) & dim(H^{i}(Γ, D_{1}^{−ω})) \\ 0 & 1 & 1 & 2g \\ 1 & 2g & 1 & 2 \\ 2 & 1 & 0 & 0 \end{matrix} .

Here, the long exact sequence gave us that dim(H^{1}(Γ, D_{1}^{−ω})) is either 1 or 2 and since D_{1}^{−ω} is the sum of two conjugate isomorphic G submodules, it has to be even. For λ ∈ −1/2 − ℕ_{0}, we have a long exact sequence

... → H^{i}(Γ, D_{2n+1}^{−ω}) → H^{i}(Γ, V_{π^{1_{λ}}}^{−ω}) → H^{i}(Γ, F_{2n+1}) → ...

Using the result for positive λ, we obtain immediately for λ ∈ −\frac{1}{2} − ℕ

i dim(H^{i}(Γ, D_{2n+1}^{−ω})) dim(H^{i}(Γ, V_{π^{1_{λ}}}^{−ω})) dim(H^{i}(Γ, F_{2n+1})) 0  (2n+1)(2g-2)  (2n+1)(2g-2)  0 1  0  (2n+1)(2g-2)  (2n+1)(2g-2) 2  0  0  0 .

In the case of λ = −1/2, it is

i dim(H^{i}(Γ, D_{1}^{−ω})) dim(H^{i}(Γ, V_{π^{1_{− \frac{1}{2}}}}^{−ω} )) dim(H^{i}(Γ, F_{1})) 0  2g  2g  1 1  2  2g+1  2g 2  0  1  1 .

Everything except dim(H^{0}(Γ, V_{π^{1_{− \frac{1}{2}}}}^{−ω} )) is determined by the long exact sequence.

For dim(H^{0}(Γ, V_{π^{1_{− \frac{1}{2}}}}^{−ω} )), we could have 2g or 2g + 1, but all invariants are in

the submodule D_{1}^{−ω} because otherwise we would get an embedding of V_{π^{1 \frac{1}{2}}}^{ω} into

L^{2}(Γ\G) which is impossible, since V_{π^{1 \frac{1}{2}}}^{ω} is not unitary.

In all these cases, the cohomology groups are finite dimensional and X (Γ, V_{π^{1_{λ}}}^{−ω}) = 0. Let us now consider the general case, so let G_{ℂ} be a connected reductive group over ℂ, G_{ℝ} a real form, K ⊂ G_{ℝ} maximal compact, and Γ ⊂ G_{ℝ} be cocompact, torsion-free and discrete. Denote by Mod(𝔤, K) the category of (𝔤, K) modules and by HC(𝔤, K) ⊂ Mod(𝔤, K) the subcategory of Harish-Chandra modules. Recall that a module is called Harish-Chandra if it is finitely generated over the universal enveloping algebra U(𝔤) and admissible, i.e. ∀γ ∈ K̂, dim(V (γ)) < ∞, where V (γ) is the γ-isotypical component of V. For a G module V denote the K-finite submodule by V_{K−fin} = {v ∈ V |dim(span(Kv)) < ∞}. For V ∈ HC(𝔤, K) denote by Ṽ := Hom(V, ℂ)_{K−fin} the dual in HC(𝔤, K).

Definition 3 For a Harish-Chandra module V, the maximal globalisation of V is MG(V) := Hom_{(𝔤,K)}(Ṽ , C^{∞}(G)_{K−fin}).

Thus, the maximal globalisation MG(V) is a particular continuous representation of G whose K-finite part is isomorphic to V. For a Harish-Chandra module V, define the hyperfunction vectors of V to be V^{−ω} := (Ṽ_{B}^{ω})^{∗}, the topological dual of the analytic vectors in any Banach globalisation of the dual representation.

Theorem 4 (Schmid [15]) MG(V) ≅ V^{−ω}.

Theorem 5 (Bunke/Olbrich [2], 1.4) For V ∈ HC(𝔤, K),

H^{∗}(Γ, MG(V)) ≅ ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)Ext_{(𝔤,K)}^{∗}(Ṽ , V_{π,K−fin}).

For the proof of this theorem, we need the generalisation of Theorem 3

H^{i}(Γ, C^{∞}(G)_{K−fin}) = 0 ∀i ≥ 1

and the following generalisation of Theorem 1.

Theorem 6 (Kashiwara-Schmidt [12])

Ext_{(𝔤,K)}^{i}(Ṽ , C^{∞}(G)_{K−fin}) = \{ \begin{matrix}MG(V), i = 0 \\ 0, i > 0 \end{matrix}

Now, let us chose a projective resolution P_{·} → Ṽ → 0 of Ṽ in Mod(𝔤, K). By Theorem 6, Hom_{(𝔤,K)}(P_{·}, C^{∞}(G)_{K−fin}) resolves MG(V) Γ-acyclically , therefore Hom_{𝔤,K}(P_{·}, C^{∞}(Γ\G)_{K−fin}) calculates H^{∗}(Γ, MG(V)). Finally, an analytic argument ( [2], Lemma 3.1) shows that only a finite part of the decomposition

C^{∞}(Γ\G)_{K−fin} = ˆ⨁_{(π,V_{π})∈Ĝ_{u}}m_{Γ}(π)V_{π,K−fin}^{∞}

contributes to Hom_{(𝔤,K)}(P_{·}, C^{∞}(Γ\G)_{K−fin}).

2 The Selberg Zeta function

In this section we will additionally assume that the real rank of G is equal to one. A group element g ∈ G is called hyperbolic if there exist m_{g} ∈ M, a_{g} ∈ A such that g is conjugate in G to m_{g}a_{g} with a_{g}^{ρ} > 1. If g is hyperbolic, a_{g} is unique, and m_{g} is unique up to conjugation in M. Note that if γ ∈ Γ, γ ≠ 1, then γ is hyperbolic. For a hyperbolic g ∈ G, we define

Z(g, σ, λ) := ∏_{k=0}^{∞} det(1 − σ_{λ−2ρ}(m_{g}a_{g}) ⊗ S^{k}(Ad(m_{g}a_{g})|\overline{𝔫})),

where σ_{λ−2ρ} denotes the representation of P as in section 1 and S^{k} is the k′th symmetric power.

In the example of G = SL(2, ℝ), a hyperbolic g is conjugate to m_{g} ( \begin{matrix}t^{1/2} & 0 \\ 0 & t^{−1/2} \end{matrix} ) ,

t > 1. Therefore, we get

Z(g, 1, λ) = ∏_{k=0}^{∞}(1 − t^{−λ−k−1/2}).

Definition 4 The conjugacy class [γ] ∈ CΓ is called primitive, if [γ] is not of the form [γ] = [γ^{′n}] for some γ′ ∈ Γ, n > 1.

Definition 5 The Selberg Zeta function is defined for ℜ(λ) > ρ by the converging product

Z(Γ, σ, λ) = ∏ ; \begin{matrix}[γ] ∈ CΓ \\ [γ]primitive \end{matrix} Z(γ, σ, λ).

Theorem 7 (Fried [8]) Z(Γ, σ, λ) has a meromorphic continuation to 𝔞_{ℂ}^{∗}.

The proof uses Ruelles thermodynamic formalism. The disadvantage of this method of proof is that is doesn’t give any information about the singularities of the continuation or about a functional equation. We use the Selberg trace formuala to calculate the singularities of the Zeta function and to determine the functional equation. To this end, we introduce the logarithmic derivative

L(Γ, σ, λ) := \frac{Z′(Γ, σ, λ)}{Z(Γ, σ, λ)},

considered as a 1-form on 𝔞_{ℂ}^{∗}. Denote by X^{d} the dual of the symmetric space X, i.e.

\begin{matrix}X & H^{m} & ℂH^{m} & ℍH^{m} & CaH^{2} \\ X^{d} & S^{n} & ℂP^{m} & ℍP^{m} & CaP^{2} \end{matrix} .

In the case of n := dimX odd, we assume that either σ is irreducible and the isomorphism class of σ is fixed under the action of the Weyl group, or σ = σ′ ⊕ σ^{′w} with w the non-trivial element of the Weyl group W (𝔤, 𝔞) ≅ ℤ/2ℤ and σ′ irreducible. The positive root system of (𝔤, 𝔞) is either of the form \{α\} or \{\frac{α}{2}, α\}. We call α the long root and identify 𝔞_{ℂ}^{∗} ≅ ℂ sending α to 1. Define the root vector H_{α} ∈ 𝔞 corresponding to a postitive root α by the condition that

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

where ⟨., .⟩ denotes an Ad(G) -invariant inner product on 𝔤. For σ ∈ M̂, we define ϵ_{α}(σ) ∈ \{0, \frac{1}{2}\} by the condition e^{2πiϵ_{α}(σ)} = σ(exp(2πiH_{α}) and ϵ_{σ} ∈ \{0, \frac{1}{2}\} by requiring ϵ_{σ} ≡ |ρ| + ϵ_{α}(σ) mod ℤ.

Theorem 8 (Bunke/Olbrich [6]) There exists a virtual elliptic operator A_{X}(σ) and a corresponding operator A_{X^{d}}(σ), such that in the case of n even

\frac{L(Γ, σ, λ)}{2λ} = ′T r′(λ^{2} + A_{Y} (σ))^{−1}  (1)
− \frac{(−1)^{\frac{n}{2}} vol(Y)π}{vol(X^{d})} \frac{P_{σ}(λ)}{2λ} \{ \begin{matrix}tan(πλ) if ϵ_{σ} = \frac{1}{2} \\ −cot(πλ) if ϵ_{σ} = 0 \end{matrix}  (2)
− \frac{(−1)^{\frac{n}{2}} vol(Y)}{vol(X^{d})}′T r′(λ^{2} − A_{X^{d}}(σ))^{−1}  (3)

and in the case of n odd,

\frac{L(Γ, σ, λ)}{2λ} = ′T r′(λ^{2} + A_{Y} (σ))^{−1}
+ (−1)^{\frac{n+1}{2}} \frac{vol(Y)π}{vol(X^{d})} \frac{P_{σ}(λ)}{2λ}

In the theorem, P_{σ} is some polynomial depending on σ. Since (λ^{2} + A_{Y}^{2} (σ))^{−1} is not trace-class, we have to use a regularised trace ′T r′. We refer to [6], 3.2. for a discussion of the regularisation. ¿From this formula, it follows that L(Γ, σ, λ) is meromorphic. In the case of n even, (2) is odd and (3) is even. The poles of the two terms cancel for λ > 0, add up for λ < 0 and I := 2λ((2) + (3)) is regular at zero ( [6], 3.2.3). Hence, the set of poles of I is ϵ_{σ} − ℕ. For A an operator on some Hilbert space, ′T r′(λ − A)^{−1} has first order poles at the eigenvalues of A with res (2λ′T r′(λ^{2} − A)^{−1}) equal to the dimension of the corresponding eigenspace, hence

res_{−ϵ_{σ}−k}I = −\frac{(−1)^{\frac{n}{2}} 2vol(Y)}{vol(X^{d})}dimE_{A_{X^{d}} (σ)}(−ϵ_{σ} − k)^{2}.

It follows from Hirzebruch proportionality that \frac{vol(Y)}{vol(X^{d})} ∈ ℤ ( [6],Prop. 3.14), so the residues of I are integral.

(1) has first order poles at ±is with residue dimE_{A_{Y} (σ)}(s^{2}), where s^{2} is a nonzero eigenvalue of A_{Y} (σ) and if zero is an eigenvalue of A_{Y} (σ), there is an additional pole at zero with residue 2dimkerA_{Y} (σ). In the case of n odd, the only poles are those coming from the first term. Note that all the residues of L are integral.

[figure: ); ρ; domain of convergence; \begin{matrix}contribution of \\ spec(A_{Y} (σ)) \end{matrix}; −ρ; (; \begin{matrix}contribution of \\ spec(A_{X^{d}} (σ)) \end{matrix}]

We’ll now come back to the example of G = SL(2, ℝ), σ = 1. The above theorem specializes to

\frac{L(Γ, σ, λ)}{2λ} :=′ T r′(Δ_{Y} −\frac{1}{4}+λ^{2})^{−1}+\frac{2g − 2}{2}′T r′(Δ_{S^{2}}+\frac{1}{4}−λ^{2})^{−1}+\frac{2g − 2}{2} \frac{π tan(λπ)}{λ},

where g is the genus of Γ\X. The eigenvalues of Δ_{S^{2}} are \frac{n^{2}}{4} , n ∈ ℕ with mult(n^{2}/4) = 2n+1. Using this, we get the following picture for ord(Z(Γ, 1, λ)).

[figure: − \frac{2n+1}{2}; (2g − 2)(2n + 1); − \frac{1}{2}; (2g − 2) + 1; 1/2; 1; dimKer(Δ_{Y} − \frac{1}{4} + λ^{2}); 2dimKer(Δ_{Y} − \frac{1}{4} )]

Theorem 9 (Bunke/Olbrich, conjectured by Patterson [1]) . For λ ≠ 0 ^{1},

ord_{λ}Z(Γ, σ, λ) = −X′(Γ, V_{π^{σ_{λ}}}^{−ω})

where X′(Γ, V_{π^{σ_{λ}}}^{−ω}) = Σ_{p=0}^{∞}(−1)^{p}p dimH^{p}(Γ, V_{π^{σ_{λ}}}^{−ω}).

Proof: In the example, the proof is just by inspection. In the general case, use Theorem 5 to obtain

H^{p}(Γ, V_{π^{σ_{λ}}}^{−ω})
= ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)Ext_{(𝔤,K)}^{p}(V_{π^{σ{∗ \atop −λ}} ,K−fin}, V_{π,K−fin})

= ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)Ext_{(𝔤,K)}^{p}(V_{π^{∗},K−fin}, V_{π^{σ_{λ}} ,K−fin})

= ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)Hom_{MA}(H_{p}(𝔫, V_{π^{∗}}) ⊕ H_{p−1}(𝔫, V_{π^{∗}}), V_{σ_{λ}})

= ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)Hom_{MA}([H^{n−1−p}(𝔫, V_{π^{∗},K−fin})
⊕H^{n−p}(𝔫, V_{π^{∗},K−fin})] ⊗ Λ^{n−1}𝔫, V_{σ_{λ}})
= ⨁_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π) [((H^{p}(𝔫, V_{π,K−fin}) ⊕ H^{p−1}(𝔫, V_{π,K−fin})) ⊗ V_{σ_{−λ}^{∗}})^{MA}]^{∗}

Setting

X (π, σ, λ) := ∑_{p=0}^{∞}(−1)^{p}dim(H^{p}(𝔫, V_{π,K−fin}) ⊗ V_{σ_{−λ}^{∗}})^{MA},

the following theorem finishes the proof.

Theorem 10 (Juhl, [10], [11])

ord_{μ}Z(Γ, σ, λ) = (−1)^{n−1} ∑_{(π,V_{π})∈Ĝ_{u}} m_{Γ}(π)X (π, σ, μ).

3 Generalisations

There are two obvious directions in which the theory could be generalised. First, one could try to generalise the group G. Unfortunately, we do not know of a satisfactory general definition of a Zeta function for groups of higher rank, but see [13], [14], [7] for special cases. The other direction is to weaken the conditions on Γ. In [3] and [4], the theory is generalised to the case of subgroups Γ ⊂ G of finite covolume. Examples of infinite covolume (convex coccompact) are considered in [5]

1For λ = 0 see [1]

References

[1] U.Bunke and M. Olbrich, Gamma-cohomology and the Selberg zeta function J. reine angew. Math. 467 (1995)

[2] U.Bunke and M. Olbrich, Cohomological Properties of the Canonical Globalizations of Harish-Chandra Modules Annals of Global Analysis and Geometry 15 (1997)

[3] U. Bunke and M. Olbrich, Resolutions of distribution globalizations of Harish-Chandra modules and cohomology, J. reine angew. Math. 497 (1998)

[4] U. Bunke and M. Olbrich, Fuchsian groups of the second kind and representations carried by the limit set, Invent. math. 127 (1997)

[5] U. Bunke and M. Olbrich, Group cohomology and the singularities of the Selberg Zeta function associated to a Kleinian group, Annals of mathematics 149 (1999)

[6] U. Bunke and M. Olbrich, Selberg Zeta and Theta Functions, Math. Res. 83, Akademie Verlag, Berlin 1995

[7] A. Deitmar, Geometric zeta functions of locally symmetric spaces, Amer. J. Math. 122 (2000)

[8] D. Fried, The zeta functions of Ruelle and Selberg I, Ann. Sci. Ec. Norm. Sup. (4) 19 (1986)

[9] S. Helgason, Groups and Geometric Analysis, Academic Press, 1984

[10] A. Juhl, Zeta-Funktionen, Index-Theorie und hyperbolische Dynamik, Habilitationsschrift, Humboldt-Universität zu berlin, 1995

[11] A. Juhl, Cohomological Theory of Dynamical Zeta Functions, Birkhäuser, Basel 2001

[12] M. Kashiwara and W. Schmid Quasi-equivariant D-modules, equivariant derived category and representations of reductive Lie groups, in J.L. Brylinski, R. Brylinski, V. Guillemin and V. Kac (eds.), Lie Theory and Geometry. In Honor of Bertram Kostant, Progr. in Math. 123, 1994

[13] H. Moscovici and R. Stanton, Eta invariants of Dirac operators on lcally symmetric manifolds, Invent. Math. 95 (1989)

[14] H. Moscovici and R. Stanton, R-torsion and zeta functions for locally symmetric manifolds, Invent. Math. 105 (1991)

[15] W. Schmid, Boundary value problems for group invariant differential equations, Soc. Math. de France, Asterisque, hors serie(1985)