L^{2}-Invariants of Locally Symmetric Spaces

Martin Olbrich

Received: August 9, 2001

Communicated by Ursula Hamenstädt

Abstract. Let X = G/K be a Riemannian symmetric space of the noncompact type, Γ ⊂ G a discrete, torsion-free, cocompact subgroup, and let Y = Γ\X be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for L^{2}(G) and results on (𝔤, K)-cohomology can be used in order to compute the L^{2}-Betti numbers, the Novikov-Shubin invariants, and the L^{2}-torsion of Y in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank m = rk_{ℂ}G−rk_{ℂ}K of G. In particular, we show the nonvanishing of the L^{2}-torsion of Y whenever m = 1.

2000 Mathematics Subject Classification: 58J35, 57R19, 22E46 Keywords and Phrases: locally symmetric spaces, L^{2}-cohomology, Novikov-Shubin invariants, L^{2}-torsion, relative Lie algebra cohomology

1 Introduction

During the last two decades L^{2}-invariants have proved to be a powerful tool in the topology of compact manifolds (see [15] for an overview). Although they can be defined in purely combinatorial terms we are interested here in their equivalent analytic versions: They are spectral invariants of the p-form Laplacians of the universal cover of the manifold. For particular nice manifolds these might be computable. Indeed, the aim of the present paper is to extract from the representation theoretic work of Harish-Chandra [9] and Borel-Wallach [2] information on the spectral decomposition of the form Laplacians on Riemannian symmetric spaces of the non-compact type which is sufficiently explicit in order to compute the spectral invariants of interest. We try to do this in a rather detailed way which, we hope, keeps the paper readable for nonspecialists in harmonic analysis. ^{1} Let X → Y be the universal cover of a compact Riemannian manifold. Set Γ := π_{1}(Y). The form Laplacian Δ_{p} = d^{∗}d + dd^{∗} defines a non-negative, elliptic, self-adjoint operator acting on L^{2}(X, Λ^{p}T^{∗}X), the square integrable p-forms on X. By Δ′_{p} and Δ_{p}^{c} = (d^{∗}d)′ we denote the restriction of Δ_{p} to the orthogonal complement of its kernel and the coclosed forms in this orthogonal complement, respectively. We consider the corresponding heat kernels e^{−tΔ_{p}^{∗}}(x, x′) := (P_{∗}e^{−tΔ_{p}})(x, x′), x, x′ ∈ X, for ∗ = ∅, ′ or c. Here P_{∗} denotes the orthogonal projection to the corresponding subspace. The local traces tr e^{−tΔ_{p}^{∗}}(x, x) are Γ-invariant functions on X. For a thorough discussion of the following definitions we refer to [14], [17], [16], and [15]. We set

Tr_{Γ}e^{−tΔ_{p}^{∗}} := ∫_{F} tr e^{−tΔ_{p}^{∗}}(x, x) dx ,

where F ⊂ X is a fundamental domain of the action of Γ on X and dx is the Riemannian volume element of X. Then the L^{2}-Betti numbers are given by

b_{p}^{(2)}(Y) := lim_{t→∞} Tr_{Γ}e^{−tΔ_{p}} ∈ [0, ∞) .

They are equal to the von Neumann dimension of ker Δ_{p} viewed as Hilbert N (Γ)-module, where N (Γ) is the group von Neumann algebra of Γ. If the spectrum of Δ_{p} has no gap around 0 the Novikov-Shubin invariants of Y are defined by

α̃_{p}(Y) := sup\{β | Tr_{Γ}e^{−tΔ′_{p}} = O(t^{− \frac{β}{2}} ) as t → ∞\} ∈ [0, ∞] .

It measures the asymptotic behaviour of the spectral density function of Δ_{p} at 0. In case of a gap around 0 we set α̃_{p}(Y) := ∞^{+}. Replacing Δ′_{p} by Δ_{p}^{c} we obtain the analogously defined Novikov-Shubin invariants α_{p}(Y) of d^{∗}d. Using the action of the exterior differential d on the Hodge decomposition of L^{2}-forms we obtain

α̃_{p}(Y) = min{α_{p}(Y), α_{p−1}(Y)} .  (1)

Finally, if α_{p}(Y) > 0 for all p (or, more generally, if X is of determinant class (see [15])), then the L^{2}-torsion of Y is defined by

ρ^{(2)}(Y) := \frac{1}{2} ∑(−1)^{p+1}p log det_{Γ}(Δ′_{p}) = \frac{1}{2} ∑(−1)^{p} log det_{Γ}(Δ_{p}^{c}) ,

where for ∗ = ′, c

− log det_{Γ}(Δ_{p}^{∗}) := \frac{d}{ds}_{|s=0} (\frac{1}{Γ(s)} ∫_{0}^{ε} Tr_{Γ}e^{−tΔ_{p}^{∗}}t^{s−1}dt)
+ ∫_{ε}^{∞} Tr_{Γ}e^{−tΔ_{p}^{∗}}t^{−1}dt

1Note added in proof: In the meanwhile the interested reader can find a discussion without proofs of the results of the present paper in the recent monograph [16] which gives a comprehensive treatment of the theory of L^{2}-invariants.

Documenta Mathematica 7 (2002) 219–237

L^{2}-Invariants of Locally Symmetric Spaces 221

for any ε > 0, where the first integral is considered as a meromorphic function in s. By Poincaré duality ρ^{(2)}(Y) = 0 for even dimensional manifolds Y. From now on let X = G/K be a Riemannian symmetric space of the noncompact type. Here G is a real, connected, linear, semisimple Lie group without compact factors, and K ⊂ G is a maximal compact subgroup. It is the universal cover of compact locally symmetric spaces of the form Y = Γ\X, where Γ ≅ π_{1}(Y) can be identified with a discrete, torsion-free, cocompact subgroup of G. It will be convenient to consider also the compact dual X^{d} of X. X^{d} is defined as follows: Let 𝔤, 𝔨 be the Lie algebras of G, K. Then we have the Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭, and 𝔤^{d} := 𝔨 ⊕ i𝔭 is another subalgebra of the complexification of 𝔤. Let G^{d} be the corresponding analytic subgroup of the complexification G_{ℂ} of G. Then G^{d} is a compact group, and X^{d} = G^{d}/K. We normalize the Riemannian metric on X^{d} such that multiplication by i becomes an isometry T_{eK}X ≅ 𝔭 → i𝔭 ≅ T_{eK}X^{d}. In this paper we are going to prove the following theorem:

Theorem 1.1 Let n = dim Y and m = m(X) := rk_{ℂ}G − rk_{ℂ}K be the fundamental rank of G. Let χ(Y) be the Euler characteristic of Y. Then

(a) b_{p}^{(2)}(Y) ≠ 0 ⇔ m = 0 and p = \frac{n}{2}.

In particular, b(2)_{\frac{n}{2}}(Y) = (−1)^{\frac{n}{2}} χ(Y) = \frac{vol(Y)}{vol(X^{d})}χ(X^{d}).

(b) α_{p}(Y) ≠ ∞^{+} ⇔ m > 0 and p ∈ [\frac{n−m}{2}, \frac{n+m}{2} − 1]. In this range α_{p}(Y) = m.

(c) ρ^{(2)}(Y) ≠ 0 ⇔ m = 1.

Note that n − m is always even and positive. Part (a) of the theorem was known for a long time, at least since Borel’s paper [1]. For special cases see also [4], [5]. For the convenience of the reader we include a proof here. In fact, we prove a stronger statement which should have been known to the experts although we were not able to find it in the literature:

Proposition 1.2 The discrete spectrum of Δ_{p} on L^{2}(X, Λ^{p}T^{∗}X) is empty unless m(X) = 0 and p = \frac{n}{2}. In this case 0 is the only eigenvalue of Δ_{p}.

The strategy of the proof of (b) can already be found in Lott’s paper [14], Section VII. To be more precise, Equation (1) implies the slightly weaker result

α̃_{p}(Y) = \{ \begin{matrix}∞^{+} & p ∉ [\frac{n−m}{2}, \frac{n+m}{2}] or m = 0 \\ m & p ∈ [\frac{n−m}{2}, \frac{n+m}{2}] and m ≠ 0 \end{matrix} .  (2)

Lott proved the first line of (2) and that α̃_{p}(Y) is finite and independent of p for the remaining values of p. He indicated how one should be able to compute the precise value of α̃_{p}(Y). But he finished the computation in the real rank one case, only. In addition, already Borel [1] showed that the range of the differential d_{p} of the L^{2}-de-Rham-complex is not closed for p ∈ [\frac{n−m}{2}, \frac{n+m}{2} −1]. Theorem 1.1 (b) can be considered as a quantitative refinement of this result. After the present paper was written I was informed by S. Mehdi that there is a joint paper of him with N. Lohoue [13] which has recently appeared in print and which contains a proof of (2). The interested reader will also find there more information concerning the material presented here in Sections 2 and 3. But he should be aware that in that paper the range of finiteness of α̃_{p}(Y) is constantly misprinted and that the proof of the second line of (2) as written down there is not quite complete (it is not mentioned that it is important to know that p_{ξ}(0) > 0, see Equation (14) below). The main motivation to do the present work was to obtain part (c) of the theorem. In our locally homogeneous situation we have for any x ∈ X

Tr_{Γ}e^{−tΔ_{p}^{∗}} = vol(Y) · tr e^{−tΔ_{p}^{∗}}(x, x)

and thus

ρ^{(2)}(Y) = vol(Y) · T^{(2)}(X)

for a certain real number T^{(2)}(X). Note that in contrast to ρ^{(2)}(Y) the number T^{(2)}(X) depends on the normalization of the invariant metric on X (of course only via the volume form). A well-known symmetry argument ([18], Proposition 2.1) yields that T^{(2)}(X) = 0 whenever m ≠ 1. Lott [14] (see also [17]) showed that T^{(2)}(H^{n}) ≠ 0 for n = 3, 5, 7, where H^{n} is the real hyperbolic space (his values for T^{(2)}(H^{n}) for n = 5, 7 were not correct). This led to the conjecture that T^{(2)}(H^{n}) ≠ 0 for all odd n which was open until the work of Hess-Schick [11] who found a trick in order to control the sign of log det_{Γ}(Δ_{p}^{c}) in terms of p and n. So they were able to show that there is a positive rational number q_{n} such that

T^{(2)}(H^{n}) = (−\frac{1}{π} )^{\frac{n−1}{2}} q_{n} .  (3)

Here the metric on H^{n} is normalized to have sectional curvature −1. Along the same lines Hess [10] obtained

T^{(2)}(SL(3, ℝ)/SO(3)) ≠ 0 .  (4)

Let us introduce

Q_{n} = \frac{2q_{n}}{(\frac{n−1}{2})!}

and rewrite (3) as

T^{(2)}(H^{n}) = (−1)^{\frac{n−1}{2}} \frac{πQ_{n}}{vol(S^{n})} .  (5)

Recall that S^{n} is the compact dual of H^{n}. The rational number Q_{n} which does not depend on the normalization of the metric has a nice interpretation in terms of Weyl’s dimension polynomial for finite-dimensional representations, see Proposition 5.3. But its significance remains to be clarified further, and it seems to be difficult to write down a practical formula valid for all odd n. One has Q_{3} = \frac{1}{3}, Q_{5} = \frac{31}{45}, Q_{7} = \frac{221}{210}. For further information see [11]. We will reduce Theorem 1.1 (c) to (4) and the positivity of Q_{n}. Let X be an arbitrary symmetric space satisfying m(X) = 1. By the classification of simple Lie groups X = X_{1} × X_{0}, where m(X_{0}) = 0 and X_{1} = SL(3, ℝ)/SO(3) or X_{1} = X_{p,q} := SO(p, q)^{0}/SO(p) × SO(q) for p, q odd. (Here as throughout the paper a superscript 0 denotes the connected component of the identity.) Note that a corresponding decomposition of Y does not necessarily exist. We show

Proposition 1.3

(a) T^{(2)}(X_{p,q}) = (−1)^{\frac{pq−1}{2}} χ(X_{p−1,q−1}^{d})\frac{πQ_{p+q−1}}{vol(X_{p,q}^{d})}.

(b) If m(X) = 1, then T^{(2)}(X) = \frac{(−1)^{\frac{n_{0}}{2}} χ(X_{0}^{d})}{vol(X_{0}^{d})}T^{(2)}(X_{1}).

Here n_{0} = dim X_{0}.

Note that X_{p−1,q−1}^{d} = SO(p + q − 2)/SO(p − 1) × SO(q − 1) and for p, q > 1

χ(X_{p−1,q−1}^{d}) = 2(\frac{p+q−2}{2} ; \frac{p−1}{2}) .

In fact, it is a classical result that χ(X^{d}) > 0 whenever m(X) = 0 (compare Theorem 1.1 (a)). It is equal to the quotient of the orders of certain Weyl groups (see Section 5). Now Theorem 1.1 (c) follows from Proposition 1.3, (4) and the positivity of Q_{n}. We are also able to identify the missing constant in (4).

Proposition 1.4 If X = SL(3, ℝ)/SO(3), then

T^{(2)}(X) = \frac{2π}{3vol(X^{d})} .

If the invariant metric on X is induced from twice the trace form of the standard representation of 𝔰𝔩(3, ℝ), then vol(X^{d}) = 4π^{3}, and we have

T^{(2)}(X) = \frac{1}{6π^{2}} .

In particular, we see that (−1)^{\frac{n−1}{2}} T^{(2)}(X) is positive for all X with m(X) = 1. Proposition 5.3 provides a uniform formula for the L^{2}-torsion of all these spaces. Acknowledgements: I am grateful to Wolfgang Lück and Thomas Schick for inspiring discussions which have provided me with a sufficient amount of motivation and of knowledge on L^{2}-invariants in order to perform the computations which led to the results of the present paper. I am also indebted to Wolfgang Lück for giving me the opportunity to report on them at the Oberwolfach conference “L^{2}-methods and K-theory”, September 1999. In addition, I benefited from discussions with J. Lott, P. Pansu, E. Hess and U. Bunke.

2 The Harish-Chandra Plancherel Theorem

We want to understand the action of the Laplacian and of the corresponding heat kernels on L^{2}(X, Λ^{p}T^{∗}X). Since the Laplacian coincides (up to the sign) with the action of the Casimir operator Ω of G (Kuga’s Lemma [2], Thm. 2.5.) it is certainly enough to understand the “decomposition” of L^{2}(X, Λ^{p}T^{∗}X) into irreducible unitary representations of G. There is an isomorphism of homogeneous vector bundles Λ^{p}T^{∗}X ≅ G ×_{K} Λ^{p}𝔭^{∗}, and, hence, of G-representations

L^{2}(X, Λ^{p}T^{∗}X) ≅ [L^{2}(G) ⊗ Λ^{p}𝔭^{∗}]^{K} .

Thus our task consists of two steps: First to understand L^{2}(G) as a representation of G × G which is accomplished by the Harish-Chandra Plancherel Theorem recalled in the present section and, second, to understand spaces of the form [V_{π} ⊗Λ^{p}𝔭^{∗}]^{K}, where (π, V_{π}) is an irreducible unitary representation of G occurring in the Plancherel decomposition. For general G, the second step will resist a naive approach. However, if π(Ω) = 0, then the space [V_{π} ⊗Λ^{p}𝔭^{∗}]^{K} has cohomological meaning, and the theory of relative (𝔤, K)-cohomology as recalled in the next section will provide a sufficient amount of information. An irreducible unitary representation (π, V_{π}) of G is called a representation of the discrete series if there is a G-invariant embedding V_{π} ↪ L^{2}(G). Let Ĝ_{d} denote the set of equivalence classes of discrete series representations of G. Then we have

Theorem 2.1 (Harish-Chandra [7]) Ĝ_{d} is non-empty if and only if m(X) = 0.

Note that m(X) = 0 means that G has a compact Cartan subgroup. The Plancherel Theorem provides a decomposition of L^{2}(G) which is indexed by discrete series representations of certain subgroups M ⊂ G which we are going to define now. Let 𝔞_{0} ⊂ 𝔭 be a maximal abelian subspace. It induces a root space decomposition

𝔤 = 𝔤_{0} ⊕ ⨁_{α∈Δ(𝔤,𝔞_{0})} 𝔤_{α} .

Choose a decomposition Δ(𝔤, 𝔞_{0}) = Δ^{+} ∪^{·} −Δ^{+} into positive and negative roots, and let Π ⊂ Δ^{+} be the subset of simple roots. For any subset F ⊂ Π we define

𝔞_{F} = {H ∈ 𝔞_{0} | α(H) = 0 for all α ∈ Π} ,  A_{F} = exp(𝔞_{F}) ,
𝔫_{F} = ⨁_{{α∈Δ^{+}|α_{|𝔞_{F}} ≠0}} 𝔤_{α} ,  N_{F} = exp(𝔫_{F}) .

Furthermore, there is a unique (possibly disconnected) subgroup M_{F} ⊂ G with Lie algebra 𝔪_{F} such that M_{F} A_{F} is the centralizer of 𝔞_{F} in G and 𝔪_{F} is orthogonal to 𝔞_{F} with respect to any invariant bilinear form on 𝔤. M_{F} is a reductive subgroup with compact center. The corresponding parabolic subgroup P_{F} := M_{F} A_{F} N_{F} is called a standard parabolic. P_{F} is called cuspidal if M_{F} has a compact Cartan subgroup. If for two subsets F, I ⊂ Π the spaces 𝔞_{F} and 𝔞_{I} are conjugated by an element of K (thus by an element of the Weyl group W (𝔤, 𝔞_{0})) we call P_{F} and P_{I} associate. (In many cases this already implies F = I.) The assignment P_{F} ↦ A_{F} T, where T is a compact Cartan subgroup of M_{F}, gives a one-to-one correspondence between association classes of cuspidal parabolic subgroups and conjugacy classes of Cartan subgroups of G. For two subsets F ⊂ I ⊂ Π we have P_{F} ⊂ P_{I}, M_{F} ⊂ M_{I}, A_{F} ⊃ A_{I}, N_{F} ⊃ N_{I}. If F is understood we will often suppress the subscript F. For illustration let us consider the two extreme cases. The minimal parabolic arises for F = ∅. Since M_{∅} is compact P_{∅} is always cuspidal. For F = Π we have P = M = G, and G is cuspidal iff m(X) = 0. For any cuspidal parabolic subgroup we have dim A ≥ m(X), and there is exactly one association class of cuspidal parabolic subgroups, called fundamental, with dim A = m(X). Theorem 2.1 also holds in the context of such reductive groups like M. Thus a parabolic P = MAN is cuspidal iff M has a non-empty discrete series M̂_{d}. Let 𝔞_{ℂ}^{∗} be the complexified dual of the Lie algebra 𝔞 of A. For a discrete series representation (ξ, W_{ξ}) of M and ν ∈ 𝔞_{ℂ}^{∗} we form the induced representation (π_{ξ,ν}, H^{ξ,ν}) by

H^{ξ,ν} = \{f : G → W_{ξ} | \begin{matrix}f(gman) = a^{−(ν+ρ_{𝔞})}ξ(m)^{−1}f(g) for all \\ g ∈ G, man ∈ MAN, f_{|K} ∈ L^{2}(K, W_{ξ}) \end{matrix} \} ,
(π_{ξ,ν}(g)f)(x) = f(g^{−1}x) .

Here ρ_{𝔞} = \frac{1}{2} ∑_{α∈Δ^{+}} α_{|𝔞}. If ν ∈ i𝔞^{∗}, then π_{ξ,ν} is unitary. An invariant bilinear

form on 𝔤 induces corresponding forms on 𝔪 and 𝔞_{ℂ}^{∗} and determines Casimir operators Ω and Ω_{M} of G and M, respectively. Then we have

π_{ξ,ν}(Ω) = ⟨ν, ν⟩ − ⟨ρ_{𝔞}, ρ_{𝔞}⟩ + ξ(Ω_{M}) .  (6)

Note that ξ(Ω_{M}) is a non-negative real scalar. Let C(G) ⊂ L^{2}(G) be the Harish-Chandra Schwartz space (for a definition see e.g. [19], 7.1.2). It is stable under the left and right regular representations l and r of G. Let C(G)_{K×K} = {f ∈ C(G) | dim span{l_{k_{1}}r_{k_{2}}f | k_{1}, k_{2} ∈ K} < ∞} be the subspace of Schwartz functions which are K-finite from the left and the right. Note that C(G)_{K×K} is dense in L^{2}(G). Suppose that ν ∈ i𝔞^{∗}. Then for f ∈ C(G)

π_{ξ,ν}(f) := ∫_{G} f(g)π_{ξ,ν}(g) dg

is a well-defined trace class operator on H^{ξ,ν} which has finite rank if f ∈ C(G)_{K×K}. Note that the map f ↦ π_{ξ,ν}(f) intertwines the G-actions in the following way: π_{ξ,ν}(l_{x}r_{y}f) = π_{ξ,ν}(x)π_{ξ,ν}(f)π_{ξ,ν}(y^{−1}) for x, y ∈ G.

The Harish-Chandra Plancherel Theorem can now be formulated as follows:

Theorem 2.2 (Harish-Chandra [9]) For each cuspidal parabolic subgroup as constructed above and any discrete series representation ξ of the corresponding group M there exists an explicitly computable analytic function p_{ξ} : i𝔞^{∗} → [0, ∞) of polynomial growth (the Plancherel density) such that for any f ∈ C(G)_{K×K} and g ∈ G

f(g) = ∑_{P} ∑_{ξ∈M̂_{d}} ∫_{𝔞^{∗}} Tr(π_{ξ,iν}(f)π_{ξ,iν}(g^{−1})) p_{ξ}(iν)dν .

Here the first sum runs over a set of representatives P = P_{F} of association classes of cuspidal parabolic subgroups of G.

For more details on the Plancherel Theorem and the structure theory behind it the interested reader may consult the textbooks [12], [19], [20]. Note that the Plancherel measures p_{ξ}(iν)dν depend on the normalization of the Haar measure dg. In the remainder of the paper we use the following one. Let dx be the Riemannian volume form of X = G/K and dk be the Haar

measure of K with total mass one. Then ∫_{G} f(g) dg = ∫_{X} f̃(x) dx, where

f̃(gK) = ∫_{K} f(gk) dk. We normalize the invariant bilinear form on 𝔤 such that

its restriction to 𝔭 ≅ T_{eK}X coincides with the Riemannian metric of X. Let dν be the Lebesgue measure corresponding to the induced form on 𝔞^{∗}. By these choices p_{ξ} is uniquely determined. We are now able to give a kind of spectral expansion of Tr_{Γ}e^{−tΔ_{p}^{∗}}.

Corollary 2.3

Tr_{Γ}e^{−tΔ_{p}} = vol(Y) ∑_{P} ∑_{ξ∈M̂_{d}} ∫_{𝔞^{∗}} e^{−t(∥ν∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))}
dim[H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K}p_{ξ}(iν)dν  (7)
= vol(Y) ∑_{P} ∑_{ξ∈M̂_{d}} ∫_{𝔞^{∗}} e^{−t(∥ν∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))}
dim[W_{ξ} ⊗ Λ^{p}𝔭^{∗}]^{K_{M}} p_{ξ}(iν)dν . (8)

Here K_{M} := K ∩ M denotes the maximal compact subgroup of M. There are only finitely many pairs (P, ξ) with [H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K} ≅ [W_{ξ} ⊗ Λ^{p}𝔭^{∗}]^{K_{M}} ≠ {0}.

Proof. We define k_{t} ∈ [C(G) ⊗ End(Λ^{p}𝔭^{∗})]^{K×K} by k_{t}(g) := e^{−tΔ_{p}}(eK, gK) ∘ g. For f ∈ L^{2}(X, Λ^{p}T^{∗}X) ≅ [L^{2}(G) ⊗ Λ^{p}𝔭^{∗}]^{K} we have e^{−tΔ_{p}}f(g_{0}) =

∫_{G} k_{t}(g)f(g_{0}g) dg. In addition, tr e^{−tΔ_{p}}(x, x) = tr k_{t}(e) for any x ∈ X.

We consider π_{ξ,iν}(k_{t}) as an operator acting on H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}. Using Kuga’s Lemma, Equation (6), and the K × K-invariance of k_{t} one derives that π_{ξ,iν}(k_{t}) = e^{−t(∥v∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))}P, where P is the orthogonal projection onto the subspace of K-invariants in H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}. The Plancherel formula now yields

tr k_{t}(e) = ∑_{P} ∑_{ξ∈M̂_{d}} ∫_{𝔞^{∗}} e^{−t(∥v∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))} dim[H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K} p_{ξ}(iν)dν .

This proves (7). Since H^{ξ,iν} ≅ L^{2}(K ×_{K_{M}} W_{ξ}) as a representation of K Equation (8) follows by Frobenius reciprocity. The last assertion is a consequence of the Blattner formula (see e.g. [19], 6.5.4) for the K_{M}-types of discrete series representations of M. □

3 (𝔤, K)-cohomology

If (π, V_{π}) is a representation of G on a complete locally convex Hausdorff topological vector space, then we can form its subspace V_{π,K} consisting of all Kfinite smooth vectors of V_{π}. V_{π,K} becomes a simultaneous module under 𝔤 and K, where both actions satisfy the obvious compatibility conditions. Such a module is called a (𝔤, K)-module (see [2], 0.2). We are interested in the functor of (𝔤, K)-cohomology V ↦ H^{∗}(𝔤, K, V) which goes from the category of (𝔤, K)-modules to the category of vector spaces. It is the right derived functor of the left exact functor taking (𝔤, K)-invariants. H^{∗}(𝔤, K, V) can be computed using the standard relative Lie algebra cohomology complex ([V ⊗ Λ^{∗}𝔭^{∗}]^{K}, d), where

dω(X_{0}, …, X_{p}) = ∑_{i=0}^{p}(−1)^{i}π(X_{i})ω(X_{0}, …, X̂_{i}, …, X_{p}) ,
ω ∈ [V ⊗ Λ^{p}𝔭^{∗}]^{K}, X_{i} ∈ 𝔭 .

Note that for V = C^{∞}(G)_{K} this complex is isomorphic to the de Rham complex of the symmetric space X. Let Z(𝔤) be the center of the universal enveloping algebra of 𝔤. If (π, V) is an irreducible (𝔤, K)-module, then any z ∈ Z(𝔤) acts by a scalar χ_{π}(z) on V. The homomorphism χ_{π} : Z(𝔤) → ℂ is called the infinitesimal character of V. The following basic result can be considered as an algebraic version of Hodge theory.

Proposition 3.1 ([2], II.3.1. and I.5.3.) Let (π, V_{π}) be an irreducible unitary representation of G, and let (τ, F) be an irreducible finite-dimensional representation of G. Then

H^{p}(𝔤, K, V_{π,K} ⊗ F) = \{ \begin{matrix}[V_{π} ⊗ F ⊗ Λ^{p}𝔭^{∗}]^{K} & π(Ω) = τ(Ω) \\ \{0\} & π(Ω) ≠ τ(Ω) \end{matrix} .

If H^{p}(𝔤, K, V_{π,K} ⊗ F) ≠ {0}, then χ_{π} = χτ̃, where τ̃ is the dual representation of τ.

The cohomology groups H^{p}(𝔤, K, V_{π,K} ⊗ F) for the representations π = π_{ξ,iν} occurring in the Plancherel Theorem have been computed in [2]. We shall need the following information.

Proposition 3.2 ([2],II.5.3., III.5.1., and III.3.3.)

(a) Let (τ, F) be an irreducible finite-dimensional representation of G and π ∈ Ĝ_{d} with χ_{π} = χτ̃. Then

dim H^{p}(𝔤, K, V_{π,K} ⊗ F) = \{ \begin{matrix}1 & p = \frac{n}{2} \\ 0 & otherwise \end{matrix} .

(b) Let (π_{ξ,ν}, H^{ξ,ν}) be a representation occurring in the Plancherel Theorem.

Then

H^{∗}(𝔤, K, H_{K}^{ξ,iν}) = {0}

unless P is fundamental, ν = 0 and ξ belongs to a certain non-empty finite subset Ξ ⊂ M̂_{d}. If P is fundamental and ξ ∈ Ξ, then

dim H^{p}(𝔤, K, H_{K}^{ξ,0}) = \{ \begin{matrix}\binom{m}{p− \frac{n−m}{2}} & p ∈ [\frac{n−m}{2}, \frac{n+m}{2}] \\ 0 & otherwise \end{matrix} .

Choose a Cartan subalgebra 𝔥 ⊂ 𝔤 and a system of positive roots. Via the Harish-Chandra isomorphism any infinitesimal character χ_{π} : Z(𝔤) → ℂ is given by an element Λ_{π} ∈ 𝔥_{ℂ}^{∗}, which is uniquely determined up to the action of the Weyl group W (𝔤, 𝔥) ([19], 3.2.4.). The set of infinitesimal characters of discrete series representations coincides (in case m = 0) with the set of infinitesimal characters of finite-dimensional representations, which are of the form μ_{τ} + ρ_{𝔤}, μ_{τ} and ρ_{𝔤} being the highest weight of τ and the half-sum of positive roots, respectively. In the following we will represent infinitesimal characters of discrete series representations by elements of this form. We introduce a partial order on 𝔥_{ℂ}^{∗} by saying that μ > ν, if μ − ν is a sum of (not necessarily distinct) positive roots. Then a careful examination of the proof of Proposition II.5.3. in [2], which only rests on some basic knowledge of the possible K-types occurring in discrete series representations, shows that slightly more than Proposition 3.2 (a) is true.

Proposition 3.3 Let (τ, F) be an irreducible finite-dimensional representation of G and π ∈ Ĝ_{d} with Λ_{π} ≮ Λτ̃. Then

dim[V_{π,K} ⊗ F ⊗ Λ^{p}𝔭^{∗}]^{K} = \{ \begin{matrix}1 & p = \frac{n}{2}, χ_{π} = χτ̃ \\ 0 & otherwise \end{matrix} .

4 L^{2}-Betti numbers and Novikov-Shubin invariants

In this section we shall prove parts (a) and (b) of Theorem 1.1 as well as Proposition 1.2. Let L^{2}(X, Λ^{p}T^{∗}X)_{d} be the discrete subspace of L^{2}(X, Λ^{p}T^{∗}X), i.e., the direct sum of the L^{2}-eigenspaces of the Laplacian. The Plancherel Theorem in particular says that as a representation of G

L^{2}(X, Λ^{p}T^{∗}X)_{d} ≅ ⨁_{π∈Ĝ_{d}} Vπ̃ ⊗ [V_{π} ⊗ Λ^{p}𝔭^{∗}]^{K} .

Let (τ_{0}, ℂ) be the trivial representation of G. Let π ∈ Ĝ_{d}. Then Λ_{π} ≮ Λ_{τ_{0}} = ρ_{𝔤}. Proposition 3.3 now yields

dim[V_{π} ⊗ Λ^{p}𝔭^{∗}]^{K} = \{ \begin{matrix}1 & p = \frac{n}{2}, χ_{π} = χ_{τ_{0}} \\ 0 & otherwise \end{matrix} .  (9)

Since in case m = 0 discrete series representations with infinitesimal character χ_{τ_{0}} always exist (see e.g. [12], Thm. 9.20 or [19], Thm. 6.8.2) and τ_{0}(Ω) = 0 this implies Proposition 1.2. In particular, b_{p}^{(2)}(Y) is non-zero exactly when m = 0 and p = \frac{n}{2}. Using that the L^{2}-Euler characteristic coincides with the usual Euler characteristic and applying Hirzebruch proportionality we obtain

b(2)_{\frac{n}{2}}(Y) = (−1)^{\frac{n}{2}} χ(Y) = \frac{vol(Y)}{vol(X^{d})}χ(X^{d}) .  (10)

We will give an alternative, purely analytic proof of that formula in the sequel of Corollary 5.2. This finishes the proof of part (a) of Theorem 1.1. We now turn to part (b). In order to compute α_{p}(Y) we need an expression for Tr_{Γ}e^{−tΔ_{p}^{c}}.

Proposition 4.1 For any triple (P, ξ, ν) appearing in (7) let B^{p}(ξ, ν) = d([H^{ξ,iν} ⊗ Λ^{p−1}𝔭^{∗}]^{K}) be the space of coboundaries in the relative Lie algebra cohomology complex and b^{p}(ξ, ν) be its dimension. Then

Tr_{Γ}e^{−tΔ_{p}^{c}} = vol(Y) ∑_{P ≠G} ∑_{ξ∈M̂_{d}} ∫_{𝔞^{∗}} e^{−t(∥ν∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))}
b^{p+1}(ξ, ν) p_{ξ}(iν)dν .  (11)

Here the first sum runs over a set of representatives P = P_{F} of association classes of proper cuspidal parabolic subgroups of G. If m > 0, P is fundamental, ξ ∈ Ξ (see Proposition 3.2 (b)), and ν ≠ 0, then

b^{p+1}(ξ, ν) = \{ \begin{matrix}\binom{m−1}{p− \frac{n−m}{2}} & p ∈ [\frac{n−m}{2}, \frac{n+m}{2} − 1] \\ 0 & otherwise \end{matrix} .  (12)

Proof. We proceed exactly as in the proof of Corollary 2.3. The kernel e^{−tΔ_{p}^{c}} determines a function k_{t}^{c} ∈ [C(G) ⊗ End(Λ^{p}𝔭^{∗})]^{K×K}. Then one computes that π_{ξ,iν}(k_{t}^{c}) = e^{−t(∥v∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))}P^{c}, where P^{c} is the projection to the orthogonal complement in [H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K} of the space of cocycles in the relative Lie algebra cohomology complex. The dimension of that complement is equal to b^{p+1}(ξ, ν). If m = 0, P = G, and ξ ∈ Ĝ_{d}, then b^{p+1}(ξ) = 0 for all p by (9). This proves (11). Let now m > 0, P be fundamental, ξ ∈ Ξ, and ν ∈ 𝔞^{∗}. In this case H^{p}(𝔤, K, H_{K}^{ξ,0}) ≠ {0} for some p. Hence Proposition 3.1 implies that π_{ξ,0}(Ω) = 0 and

h^{p}(ξ) := dim[H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K} = dim[H^{ξ,0} ⊗ Λ^{p}𝔭^{∗}]^{K} = dim H^{p}(𝔤, K, H_{K}^{ξ,0}) .

By Proposition 3.2 (b) we have h^{p}(ξ) = \binom{m}{p− \frac{n−m}{2}}. On the other hand,

dim H^{p}(𝔤, K, H_{K}^{ξ,iν}) = 0 for ν ≠ 0 implies that h^{p}(ξ) = b^{p}(ξ, ν) + b^{p+1}(ξ, ν).

(12) now follows inductively. □

If H^{p}(𝔤, K, H_{K}^{ξ,iν}) = {0} for all ν ∈ 𝔞^{∗}, then by Proposition 3.1 dim[H^{ξ,iν} ⊗ Λ^{p}𝔭^{∗}]^{K} = 0 (which is independent of ν) or inf_{ν∈𝔞^{∗}}(∥ν∥^{2} + ∥ρ_{𝔞}∥^{2} − ξ(Ω_{M})) > 0. Now Proposition 4.1 implies that the spectrum of Δ_{p}^{c} has a gap around zero, which means α_{p}(Y) = ∞^{+}, unless m > 0 and p ∈ [\frac{n−m}{2}, \frac{n+m}{2} −1]. In the latter case we obtain for some c > 0

Tr_{Γ}e^{−tΔ_{p}^{c}} = vol(Y)\binom{m − 1}{p − \frac{n−m}{2}} ∑_{ξ∈Ξ} ∫_{𝔞^{∗}} e^{−t∥ν∥^{2}} p_{ξ}(iν)dν + O(e^{−ct})

as t → ∞, where 𝔞 corresponds to a fundamental parabolic subgroup and thus has dimension m. Let P = MAN be fundamental and ξ ∈ M̂_{d}. Then dim 𝔫 =: 2u is even. Choose a compact Cartan subgroup T ⊂ M with Lie algebra 𝔱 and a system Δ^{+}(𝔪, 𝔱) of positive roots. Considering the pair (𝔪, 𝔱) instead of (𝔤, 𝔥) we can define Λ_{ξ} ∈ 𝔱_{ℂ}^{∗} in the same way as at the end of Section 3. Then 𝔥 := 𝔱 ⊕ 𝔞 is a Cartan subalgebra of 𝔤. Let Φ^{+} be a system of positive roots for (𝔤, 𝔥) containing Δ^{+}(𝔪, 𝔱). Then there exists a positive constant c_{X} depending only on the normalization of the volume form dx such that

p_{ξ}(ν) = c_{X}(−1)^{u} ∏_{α∈Φ^{+}} \frac{⟨α, Λ_{ξ} + ν⟩}{⟨α, ρ_{𝔤}⟩}  (13)

(see [9], Thm. 24.1, [20], Thm. 13.5.1 or [12], Thm. 13.11). In particular, p_{ξ} is an even polynomial of degree dim 𝔫. The factor (−1)^{u} makes it nonnegative on i𝔞^{∗}. The element Λ_{ξ} + ν gives the infinitesimal character of π_{ξ,ν}: Λ_{π_{ξ,ν}} = Λ_{ξ} + ν ([12], Prop. 8.22). Let now ξ ∈ Ξ. Then Propositions 3.2 (b) and 3.1 imply that π_{ξ,0} has the same infinitesimal character as the trivial representation. It follows that Λ_{ξ} is conjugated in 𝔥_{ℂ}^{∗} by an element of the Weyl group W (𝔤, 𝔥) to ρ_{𝔤}. By (13) we obtain p_{ξ}(0) = ±c_{X} ≠ 0. On the other hand p_{ξ}(0) ≥ 0, hence

p_{ξ}(0) > 0 .  (14)

We decompose p_{ξ}(ν) = ∑_{k=0}^{u} p_{ξ,2k}(ν) into homogeneous polynomials. Set q_{ξ,k} =

∫_{∥ν∥=1} p_{ξ,2k}(ν) dν. Then q_{ξ,0} > 0 and

∫_{𝔞^{∗}} e^{−t∥ν∥^{2}} p_{ξ}(iν)dν = ∑ q_{ξ,k} ∫_{0}^{∞} e^{−tr^{2}}r^{m−1+2k}dr
= ∑ t^{−( \frac{m}{2} +k)}q_{ξ,k} ∫_{0}^{∞} e^{−y^{2}}y^{m−1+2k}dy .

Thus for p ∈ [\frac{n−m}{2}, \frac{n+m}{2} − 1] the leading term of Tr_{Γ}e^{−tΔ_{p}^{c}} as t → ∞ is a non-zero multiple of t^{− \frac{m}{2}} . This completes the proof of Theorem 1.1 (b).

5 L^{2}-torsion

For even dimensional manifolds the L^{2}-torsion vanishes. Thus we may assume that m is odd, in particular m ≥ 1. Then Δ′_{p} = Δ_{p}. We first want to compute

k_{X}(t) := \frac{1}{2vol(Y)} ∑_{p=0}^{n}(−1)^{p}p Tr_{Γ}e^{−tΔ_{p}} .

Then

T^{(2)}(X) = \frac{d}{ds}_{|s=0} (\frac{1}{Γ(s)} ∫_{0}^{ε} k_{X}(t)t^{s−1}dt) + ∫_{ε}^{∞} k_{X}(t)t^{−1}dt .

Let P = MAN be a parabolic subgroup appearing in (8). Set K_{M} = K ∩ M and 𝔭_{𝔪} = 𝔭 ∩ 𝔪. Then an elementary calculation in the representation ring R(K_{M}) of K_{M} yields

∑_{p=0}^{n}(−1)^{p}p Λ^{p}𝔭^{∗} = 0 , if dim 𝔞 ≥ 2 ,

while for dim 𝔞 = 1 we have

∑_{p=0}^{n}(−1)^{p}p Λ^{p}𝔭^{∗} = ∑_{p=0}^{n−1}(−1)^{p+1}Λ^{p}(𝔭_{𝔪}^{∗} ⊕ 𝔫^{∗})

= ∑_{l=0}^{dim 𝔫}(−1)^{l+1}(Λ^{ev}𝔭_{𝔪}^{∗} − Λ^{odd}𝔭_{𝔪}^{∗}) ⊗ Λ^{l}𝔫

(see [18], Prop. 2.1 and Lemma 2.3). It follows from (8) that k_{X}(t) ≡ 0 for m > 1, hence ρ^{(2)}(Y) = 0. From now on let m = 1. Let P = MAN be a fundamental parabolic subgroup of G. Then (8) gives

k_{X}(t) = \frac{1}{2} ∑_{l=0}^{dim 𝔫}(−1)^{l+1} ∑_{ξ∈M̂_{d}} dim [W_{ξ} ⊗ (Λ^{ev}𝔭_{𝔪}^{∗} − Λ^{odd}𝔭_{𝔪}^{∗}) ⊗ Λ^{l}𝔫^{∗}]^{K_{M}}

∫_{𝔞^{∗}} e^{−t(∥ν∥^{2}+∥ρ_{𝔞}∥^{2}−ξ(Ω_{M}))} p_{ξ}(iν)dν . (15)

Now X = X_{1} × X_{0}, X_{1} = G_{1}/K_{1}, X_{0} = G_{0}/K_{0}, m(X_{0}) = 0 as explained in the introduction. Although (15) can be evaluated directly for general X with m(X) = 1 we prefer to reduce the computation to the irreducible case X = X_{1}. In order to compute T^{(2)}(X) it is sufficient to compare ρ^{(2)}(Y) with vol(Y) for one particular Y = Γ\X. If we choose Γ of the form Γ_{1} × Γ_{0}, where Γ_{0} ⊂ G_{0} and Γ_{1} ⊂ G_{1}, then

ρ^{(2)}(Y) = χ(Y_{0})ρ^{(2)}(Y_{1}) ,

where Y_{1} = Γ_{1}\X_{1}, Y_{0} = Γ_{0}\X_{0}. Applying Hirzebruch proportionality we obtain the assertion of Proposition 1.3 (b)

T^{(2)}(X) = \frac{ρ^{(2)}(Y)}{vol(Y_{0})vol(Y_{1})} = \frac{χ(Y_{0})}{vol(Y_{0})}T^{(2)}(X_{1})
= \frac{(−1)^{\frac{n_{0}}{2}} χ(X_{0}^{d})}{vol(X_{0}^{d})}T^{(2)}(X_{1}) .  (16)

It remains to deal with the case X = X_{1}. We can assume that G = SO(p, q)^{0}, p ≤ q odd, or G = SL(3, ℝ). Then M ≅ SO(p − 1, q − 1), 𝔫 ≅ ℝ^{p+q−2} or M ≅ ^{0}GL(2, ℝ) := {A ∈ GL(2, ℝ) | | det A| = 1}, 𝔫 ≅ ℝ^{2}, respectively, and M acts on 𝔫 via the standard representation. Note that M is not connected unless G = SO(1, q)^{0}. The M^{0}-representations Λ^{l}𝔫^{∗} ⊗ ℂ are irreducible unless G = SO(p, q) and l = u = \frac{1}{2} dim 𝔫. In the latter case Λ^{u}𝔫^{∗} ⊗ ℂ decomposes into two irreducible components Λ^{+}𝔫 ⊕ Λ^{−}𝔫. Since compact Cartan subgroups of M are connected the discrete series representations of M are induced from discrete series representations of M^{0}: W_{ξ} = Ind_{M^{0}}^{M}(W_{ξ^{0}}), ξ_{0} ∈ (M̂^{0})_{d} (see [19], 6.9 and 8.7.1). As representations of K_{M} we have W_{ξ} ≅ Ind_{K_{M}^{0}}^{K_{M}} (W_{ξ^{0}}). By Frobenius reciprocity we obtain

dim[W_{ξ} ⊗ (Λ^{ev}𝔭_{𝔪}^{∗} − Λ^{odd}𝔭_{𝔪}^{∗}) ⊗ Λ^{l}𝔫^{∗}]^{K_{M}}
= dim[W_{ξ_{0}} ⊗ (Λ^{ev}𝔭_{𝔪}^{∗} − Λ^{odd}𝔭_{𝔪}^{∗}) ⊗ Λ^{l}𝔫^{∗}]^{K_{M}^{0}} .

Note that the infinitesimal characters χ_{ξ} and χ_{ξ_{0}} coincide. By χ(𝔪, K_{M}^{0}, .) we denote the Euler characteristic of relative Lie algebra cohomology. Set v = \frac{1}{2} dim 𝔭_{𝔪}. Applying Propositions 3.1 and 3.2 (a) to M^{0} instead of G we obtain

dim[W_{ξ_{0}} ⊗ (Λ^{ev}𝔭_{𝔪}^{∗} − Λ^{odd}𝔭_{𝔪}^{∗}) ⊗ Λ^{∗}𝔫^{∗}]^{K_{M}^{0}}
= χ(𝔪, K_{M}^{0}, W_{ξ_{0},K_{M}^{0}} ⊗ Λ^{∗}𝔫^{∗})
= \{ \begin{matrix}(−1)^{v} & χ_{ξ} = χ_{Λ^{∗}𝔫} \\ 0 & otherwise \end{matrix} .

Here Λ^{∗}𝔫^{∗}, ∗ = l, +, −, denotes an irreducible component of Λ^{l}𝔫^{∗} ⊗ ℂ. In all cases under consideration the set {α_{|𝔞} | α ∈ Δ^{+}, α_{|𝔞} ≠ 0} consists of a single element α_{0} ∈ 𝔞^{∗}. It follows that ρ_{𝔞} = uα_{0}. Moreover, Ω_{M} acts on Λ^{l}𝔫 as l(2u − l)∥α_{0}∥^{2}id (compare [18], Lemma 2.5). In order to evaluate (15) further we have to determine the constant c_{X} in formula (13). This can be done in complete generality. So for a moment we drop the assumptions m = 1, X = X_{1}.

Lemma 5.1 Let P = MAN ⊂ G be fundamental. We retain the notation introduced before (13). Set W_{A} = {k ∈ K|Ad(k)𝔞 ⊂ 𝔞}/K_{M}, S_{A}^{d} = exp(i𝔞)K ⊂ X^{d}, and let Φ_{𝔨}^{+} be a positive root system for (𝔨, 𝔱) with corresponding half sum ρ_{𝔨}. Then

c_{X} = \frac{1}{|W_{A}|(2π)^{\frac{n+m}{2}}} \frac{∏_{α∈Φ^{+}}⟨α, ρ_{𝔤}⟩}{∏_{α∈Φ_{𝔨}^{+}}⟨α, ρ_{𝔨}⟩}  (17)

= \frac{1}{|W_{A}|} \frac{vol(S_{A}^{d})}{(2π)^{m}} \frac{1}{vol(X^{d})} .  (18)

Proof. Formula (17) is a combination of [8], Thm. 37.1, with [9], Cor. 23.1, Thm. 24.1 and Thm. 27.3. In order to apply these results correctly one has to take into account that Harish-Chandra’s and our normalizations of the measures dg and dν all of them starting from a fixed invariant bilinear form on 𝔤 differ by the factors 2^{\frac{n−dim 𝔞_{0}}{2}} ([8], Section 7 and Lemma 37.2) and (2π)^{m}, respectively. On the other hand we have

∏_{α∈Φ_{𝔨}^{+}}⟨α, ρ_{𝔨}⟩ = (2π)^{\frac{dim K/T}{2}} \frac{vol(T)}{vol(K)}  (19)

(see e.g. [8], Lemma 37.4). Here the volumes are the Riemannian ones corresponding to the invariant bilinear form ⟨., .⟩. Formula (19) holds for any pair (K, T) of a connected compact Lie group and a maximal torus. Applying it also to the pair (G^{d}, H^{d}), where H^{d} is the maximal torus of G^{d} with Lie algebra 𝔱 ⊕ i𝔞, we obtain

\frac{∏_{α∈Φ^{+}}⟨α, ρ_{𝔤}⟩}{∏_{α∈Φ_{𝔨}^{+}}⟨α, ρ_{𝔨}⟩} = (2π)^{\frac{n−m}{2}} \frac{vol(K)vol(H^{d})}{vol(G^{d})vol(T)} = (2π)^{\frac{n−m}{2}} \frac{vol(S_{A}^{d})}{vol(X^{d})} .  (20)

Documenta Mathematica 7 (2002) 219–237

The second equality follows from the fact that the map H^{d}/T → X^{d}, hT ↦ hK, is an isometric embedding with image S_{A}^{d}. This proves (18). □

In particular, specializing (13) and (18) to the case m = 0 we obtain as a consequence of Weyl’s dimension formula

Corollary 5.2 Let π be a discrete series representation of G having the same infinitesimal character as the finite-dimensional representation τ, then

p_{π} = \frac{dim τ}{vol(X^{d})} .

Let us now give the promised analytic proof of (10). For a fixed finitedimensional representation τ of G (which still is assumed to be connected) there are exactly |W (𝔤, 𝔱)|/|W (𝔨, 𝔱)| equivalence classes of discrete series representations with infinitesimal character χ_{τ} (see [19], Thm. 8.7.1, or [12], Thm. 12.21). But this quotient of orders of Weyl groups is equal to χ(X^{d}) (see e.g. [3]). By (7), (9) and Corollary 5.2 we obtain

b(2)_{\frac{n}{2}} = vol(Y) ∑_{π∈Ĝ_{d},χ_{π}=χ_{τ_{0}}} p_{π} = vol(Y)χ(X^{d})\frac{1}{vol(X^{d})} .

We return to the evaluation of k_{X}(t) for m = 1, X = X_{1}. The polynomial p_{ξ} only depends on the infinitesimal character of ξ. By Λ_{∗} ∈ i𝔱^{∗}, ∗ = l, +, −, we denote the infinitesimal character of the irreducible M-representation Λ^{∗}𝔫. We set

p_{l}(ν) := \{ \begin{matrix}∏α∈Φ_{+} \frac{⟨α,Λ_{l}+ν⟩}{⟨α,ρ_{𝔤}⟩} & G = SL(3, ℝ) \\ & or l ≠ u \\ ∏_{α∈Φ^{+}} \frac{⟨α,Λ_{+}+ν⟩}{⟨α,ρ_{𝔤}⟩} + ∏_{α∈Φ^{+}} \frac{⟨α,Λ_{−}+ν⟩}{⟨α,ρ_{𝔤}⟩} \\ = 2 ∏_{α∈Φ^{+}} \frac{⟨α,Λ_{+}+ν⟩}{⟨α,ρ_{𝔤}⟩} & otherwise \end{matrix} .

For fixed infinitesimal character there are |W (𝔪, 𝔱)|/|W_{K_{M}} | equivalence classes of discrete series representations, where W_{K_{M}} = {k ∈ K_{M} | Ad(k)𝔱 ⊂ 𝔱}/T. Note that there is an embedding W (𝔨_{𝔪}, 𝔱) ↪ W_{K_{M}} which becomes an isomorphism if K_{M} is connected. Furthermore, |W_{A}| = 1 except for G = SO(1, q)^{0}, where |W_{A}| = 2. In any case |W_{K_{M}} ||W_{A}| = 2|W (𝔨_{𝔪}, 𝔱)|. Let X_{M}^{d} be the compact dual of X_{M} = M/K_{M} = M^{0}/K_{M}^{0}. Then |W (𝔪, 𝔱)|/|W_{K_{M}} ||W_{A}| = \frac{1}{2}χ(X_{M}^{d}). In addition, u + v = \frac{n−1}{2} and

\frac{vol(S_{A}^{d})}{2π} = \frac{1}{∥α_{0}∥} .

Summarizing the above discussion we obtain

k_{X}(t) = (−1)^{\frac{n−1}{2}} \frac{χ(X_{M}^{d})}{4∥α_{0}∥vol(X^{d})} ∑_{l=0}^{2u}(−1)^{l+1}k_{l}(t) ,

Documenta Mathematica 7 (2002) 219–237

where

k_{l}(t) = ∫_{𝔞^{∗}} e^{−t(∥ν∥^{2}+(u−l)^{2}∥α_{0}∥^{2})}p_{l}(iν)dν .

For an even polynomial P and c ≥ 0 set

k_{P,c}(t) = ∫_{−∞}^{∞} e^{−t(y^{2}+c^{2})}P (iy)dy .

Then (compare [6], Lemma 2 and Lemma 3, [17], Lemma 6.4, [11], p.332)

\frac{d}{ds}_{|s=0} (\frac{1}{Γ(s)} ∫_{0}^{ε} k_{P,c}(t)t^{s−1}dt) + ∫_{ε}^{∞} k_{P,c}(t)t^{−1}dt

= (∫_{0}^{ε} k_{P,c}(t)t^{s−1}dt)_{|s=0} + ∫_{ε}^{∞} k_{P,c}(t)t^{−1}dt
= −2π ∫_{0}^{c} P (y) dy .

Since p_{l} = p_{2u−l} we obtain

Proposition 5.3 Let X = G/K with m(X) = 1, P = MAN ⊂ G a fundamental parabolic subgroup, u = \frac{1}{2} dim 𝔫, and X_{M} = M/K_{M}. Then

T^{(2)}(X) = (−1)^{\frac{n−1}{2}} χ(X_{M}^{d})\frac{πQ_{X}}{vol(X^{d})} ,

where

Q_{X} = ∑_{l=0}^{u−1}(−1)^{l} ∫_{0}^{u−l} p_{l}(y · α_{0}) dy .

In fact, we have proved the proposition for irreducible X, but (16) shows that it holds in the general case, too. Let us first discuss the case G = SO(p, q)^{0}, X = X_{p,q}, p ≤ q odd. Then n = pq and X_{M} = X_{p−1,q−1}. Furthermore, the polynomials p_{l} depend only on the complexification of G, i.e., on p + q. Thus Q_{X_{p,q}} = Q_{H^{p+q−1}}. This proves Proposition 1.3 (a). We emphasize again that Q_{H^{p+q−1}} is a positive rational number [11]. In fact, Hess-Schick showed that

(−1)^{l} ∫_{0}^{u−l} p_{l}(y · α_{0}) dy > 0 , l = 0, …, u − 1 .

Let X = SL(3, ℝ)/SO(3). Then n = 5 and u = 1. We find that

Q_{X} = ∫_{0}^{1} y^{2}dy = \frac{1}{3} , X_{M}^{d} = S^{2}, χ(X_{M}^{d}) = 2 .

Using for instance (20) also vol(X^{d}) can be easily computed. This proves Proposition 1.4 and finishes the proof of Theorem 1.1.

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Martin Olbrich Mathematisches Institut Universität Göttingen Bunsenstr. 3-5 37073 Göttingen Germany olbrich@uni-math.gwdg.de 238

Documenta Mathematica 7 (2002)