Journal of Lie Theory Volume 6 (1996) 179–190 ◯^{C} 1996 Heldermann Verlag

Torus actions on compact quotients

Anton Deitmar

Communicated by K.-H. Neeb

Abstract. A Lefschetz formula for actions of noncompact tori on compact quotients of Lie groups is given.

Introduction

Let G denote a Lie group and Γ a uniform lattice in G. We fix a maximal torus T in G and consider the action of T on the compact quotient Γ\G. Assuming T to be noncompact we will prove a Lefschetz formula relating compact orbits as local data to the action of the torus T on a global cohomology theory (tangential cohomology). Modulo homotopy, the compact orbits are parametrized by those conjugacy classes [γ] in Γ whose G-conjugacy classes meet T in points which are regular in the split component. Having a bijection between homotopy classes and conjugacy classes in the discrete group we will identify these two. For a class [γ] let X_{γ} be the union of all compact orbits in that class. Then it is known that X_{γ} is a smooth submanifold and with χ_{r}(X_{γ}) we denote its de-twisted Euler characteristic (see sect. 2.). Note that χ_{r}(X_{γ}) is local, i.e. it can be expressed as the integral over X_{γ} of a canonical differential form (generalized Euler form). On the other hand χ_{r}(X_{γ}) can be expressed as a simple linear combination of Betti numbers (see sect. 2.). Next, λ_{γ} will denote the volume of the orbit and P_{s} the stable part of the Poincaré map around the orbit. Then the number

L(γ) := \frac{λ_{γ}χ_{r}(X_{γ})}{det(1 − P_{s})}

will be called the Lefschetz number of [γ] (compare [8]). The class [γ] defines a point a_{γ} in the split part A of the torus T modulo the action of the Weyl group. In the case when the Weyl group has maximal size (for example when T is maximally split) our Lefschetz formula is an equality of distributions:

∑_{[γ]} L(γ)δ_{a_{γ}} = tr(.|H^{∗}(F)),

where H^{∗} is the tangential cohomology of the unstable/neutral foliation F induced by the torus action. In [6] a similar formula is proven to hold up to a smooth

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag function in the case of a flow. The present paper extends results of Andreas Juhl [10], [13] in the real rank one case. See also [11], [12].

1. Euler-Poincaré functions

In this section and the next we list some technical results for the convenience of the reader. Let G denote a real reductive group of inner type [14] and fix a maximal compact subgroup K. Let (τ, V_{τ}) be a finite dimensional unitary representation of K and write (τ̆, Vτ̆) for the dual representation. Assume that G has a compact Cartan subgroup T ⊂ K. Let 𝔤_{0} = 𝔨_{0} ⊕ 𝔭_{0} be the polar decomposition of the real Lie algebra 𝔤_{0} of G and write 𝔤 = 𝔨 + 𝔭 for its complexification. Choose an ordering of the roots Φ(𝔤, 𝔱) of the pair (𝔤, 𝔱). This choice induces a decomposition 𝔭 = 𝔭_{−} ⊕ 𝔭_{+}.

Proposition 1.1. For (τ, V_{τ}) a finite dimensional representation of K there is a compactly supported smooth function f_{τ} on G such that for every irreducible unitary representation (π, V_{π}) of G we have:

tr π(f_{τ}) = ∑_{p=0}^{dim(𝔭)}(−1)^{p} dim(V_{π} ⊗ ∧^{p}𝔭 ⊗ Vτ̆)^{K}.

Proof. [5]. □

Proposition 1.2. Let g be a semisimple element of the group G. If g is not elliptic, then the orbital integral O_{g}(f_{τ}) vanishes. If g is elliptic we may assume g ∈ T, where T is a Cartan in K and then we have

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

where c_{g} is Harish-Chandra’s constant, it does only depend on the centralizer G_{g} of g. Its value is given for example in [3]. Proof. [5]. □

Proposition 1.3. For the function f_{σ} we have for any π ∈ Ĝ:

tr π(f_{σ}) = ∑_{p=0}^{dim 𝔤/𝔨}(−1)^{p} dim Ext_{(𝔤,K)}^{p}(V_{σ}, V_{π}),

i.e. f_{σ} gives the Euler-Poincaré numbers of the (𝔤, K)-modules (V_{σ}, V_{π}). This justifies the name Euler-Poincaré function. Proof. [5]. □

2. De-twisted Euler characteristics

Let C^{+} denote the category of complexes of C-vector spaces which are zero in negative indices and have degreewise finite dimensional cohomology, i.e. the dimension of H^{j}(E) is finite for all j. Let K^{+} denote the weak Grothendieck group of C^{+}, i.e. K^{+} is the abelian group generated by all isomorphism classes of objects modulo the relations A = B + C, whenever any object in A is isomorphic to the direct sum of an object in B and one in C. An element E = E_{+} − E_{−} of K^{+} is called a virtual complex. Define the de-twist of an element E of K^{+} as

E′ = ∑_{k=0}^{∞} E[−k],

where E[k]_{j} = E_{k+j}. Since the sum is degreewise finite this defines a new element of K^{+}. The higher de-twists are defined inductively, so E^{(0)} = E and E^{(r+1)} = E^{(r)}′.

We need to extend the notion of an Euler characteristic to infinite virtual complexes by

χ(E) = ∑_{k=0}^{∞}(−1)^{k} dim H^{k}(E),

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

Call a virtual complex cohomologically finite if dim 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 de-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} dim H^{j}(E).

This is easily proven by induction on r. This motivates the following definition: The r-th de-twisted 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} dim H^{j}(E).

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 2.1. Let M,N be compact manifolds. We have

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

Proof. See [4]. □

To give another example of a situation in which higher Euler characteristics occur we will describe a situation in Lie algebra cohomology which will show up later.

We consider a short exact sequence

o → 𝔫 → l → 𝔞 → 0

of finite dimensional complex Lie algebras where 𝔞 is abelian. In such a situation an l-module V is called acceptable , if the 𝔞-module H^{q}(𝔫, V) is finite dimensional. Note that V itself need not 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.

Proposition 2.2. Let

o → 𝔫 → l → 𝔞 → 0

be an exact sequence of finite dimensional complex Lie algebras. Assume that the Lie algebra 𝔞 is 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. [5]. □

3. The Lefschetz formula

Let G be a connected Lie group and Γ ⊂ G a uniform lattice. Note that the existence of Γ forces G to be unimodular. Fix a Haar measure on G and consider the representation of G on the Hilbert space L^{2}(Γ\G) given by R(g)φ(x) = φ(xg). For any smooth compactly supported function f on G define R(f)φ(x) := ∫_{G} f(y)φ(xy)dy, then a calculation shows that R(f) is an integral operator with smooth kernel k(x, y) = ∑_{γ∈Γ} f(x^{−1}γy). From this it follows that R(f) is a trace class operator. Since this holds for any f we conclude that L^{2}(Γ\G) decomposes under G as a discrete sum of irreducibles with finite multiplicities:

L^{2}(Γ\G) = ⨁_{π∈Ĝ} N_{Γ}(π)π.

It follows that trR(f) = ∑_{π∈Ĝ} N_{Γ}(π)trπ(f). On the other hand, the trace of R(f) equals the integral over the diagonal of the kernel, so

trR(f) = ∫_{Γ\G} k(x, x)dx
= ∑_{[γ]} vol(Γ_{γ}\G_{γ})O_{γ}(f),

where O_{γ}(f) := ∫_{G_{γ}\G} f(x^{−1}γx)dx is the orbital integral. Note that this expression depends on the choice of a Haar measure on G_{γ}. So we state the Selberg trace formula as

∑_{π∈Ĝ} N_{Γ}(π)trπ(f) = ∑_{[γ]} vol(Γ_{γ}\G_{γ})O_{γ}(f).

From now on we will assume:

(A1) G is a semidirect product:

G ≅ H⋉R

of an abelian Lie group R and a semisimple connected Lie group H with finite center.

For the following fix a maximal torus T of H, write T = AB, where A is the split component and B is compact. Let P = MAN a parabolic then B ⊂ M. Let A^{reg} be the set of regular elements of the split torus A. Since H acts on R it acts on the unitary dual R̂. Our second assumption is

(A2) Any element of A^{reg}M acts freely on R − {0} and on R̂ − {triv}.

For any τ ∈ R̂ let H_{τ} be its stabilizer in H. For the trivial representation we clearly have H_{triv} = H. The condition (A2) says that for any nontrivial τ ∈ R̂ we have H_{τ} ∩ A^{reg}M = Ø.

Example. Clearly any semisimple connected G with finite center would give an example but there are also a lot of nonreductive examples such as the following: Let R := Mat_{2}(ℝ) with the addition, H := SL_{2}(ℝ) and let H act on R by matrix

multiplication from the left. Let G := H⋉N and T := \{( \begin{matrix}a \\ & a^{−1} \end{matrix} )\}. It is easily

seen that our assumptions are satisfied in this case.

We will only consider uniform lattices of the form Γ = Γ_{H}⋉Γ_{R}, where Γ_{H} and Γ_{R} are uniform lattices in H and R. We will further assume Γ_{H} to be weakly neat, this means, Γ_{H} is a cocompact torsion free discrete subgroup of H which is such that for any γ ∈ Γ_{H} the adjoint Ad(γ), acting on the Lie algebra of H does not have a root of unity ≠ 1 as an eigenvalue. Any arithmetic group has a weakly neat subgroup of finite index [1].

Example. Take up the above example and let D denote a quaternion division algebra over ℚ which splits over ℝ. So we have D ↪ GL_{2}(ℝ) and D^{1} ↪ SL_{2}(ℝ), where D^{1} is the set of elements of reduced norm 1. Let O denote an order in D and O^{1} := O ∩ D^{1}. Then Γ := O^{1}⋉O is a uniform lattice in SL_{2}(ℝ)⋉Mat_{2}(ℝ).

Write the real Lie algebras of G, H, M, A, N, R as 𝔤_{0}, 𝔥_{0}, 𝔪_{0}, 𝔞_{0}, 𝔫_{0}, 𝔯_{0} and their complexifications as 𝔤, 𝔥, 𝔪, 𝔞, 𝔫, 𝔯. Let Φ(𝔥, 𝔞) denote the set of roots of the pair (𝔥, 𝔞). The choice of the parabolic P amounts to the same as a choice of a set of positive roots Φ^{+}(𝔥, 𝔞). Let A^{−} ⊂ A denote the negative Weyl chamber corresponding to that ordering, i.e. A^{−} consists of all a ∈ A which act contractingly on the Lie algebra 𝔫. Further let \overline{A^{−}} be the closure of A^{−} in G, this is a manifold with boundary. Let K_{M} be a maximal compact subgroup of M. We may suppose that K_{M} contains B. Fix an irreducible unitary representation (τ, V_{τ}) of K_{M}. Let K be a maximal compact subgroup of H. We may assume K ⊃ K_{M}.

Since Γ_{H} is the fundamental group of the Riemannian manifold

X_{Γ_{H}} = Γ_{H}\X = Γ_{H}\H/K

it follows that we have a canonical bijection of the homotopy classes of loops:

[S^{1} : X_{Γ_{H}}] → Γ_{H}/conjugacy.

For a given class [γ] let X_{γ} denote the union of all closed geodesics in the corresponding class in [S^{1} : X_{Γ}]. Then X_{γ} is a smooth submanifold of X_{Γ_{H}} [7]. Let χ_{r}(X_{γ}) denote the r-fold de-twisted Euler characteristic of X_{γ}, where r = dim A.

Let E_{P} (Γ) denote the set of all conjugacy classes [γ] in Γ such that γ_{H} is in H conjugate to an element a_{γ}b_{γ} of A^{−}B.

Take a class [γ] in E_{P} (Γ). Modulo conjugation assume γ ∈ T = AB, then the centralizer Γ_{H,γ} projects to a lattice Γ_{A,γ} in the split part A. Let λ_{γ} be the covolume of this lattice. Normalize the measure on R such that vol(Γ_{R}\R) = 1.

Theorem 3.1. (Lefschetz formula, first version) Let φ be compactly supported on \overline{A^{−}}, dim G-times continuously differentiable and suppose φ vanishes on the boundary to order dim G + 1. Then we have that the expression

∑ ; \begin{matrix}π ∈ Ĝ \\ π|_{R} ≡ 1 \end{matrix} N_{Γ}(π) ∑_{p,q}(−1)^{p+q} ∫_{A^{−}} φ(a)tr(a|(H^{q}(𝔫, π) ⊗ ∧^{p}𝔭_{M} ⊗ Vτ̆)^{K_{M}})da

equals

(−1)^{dim(N)} ∑_{[γ]∈E_{P} (Γ)} λ_{γ}χ_{r}(X_{γ})\frac{φ(a_{γ})trτ(b_{γ})}{det(1 − a_{γ}b_{γ}|𝔫)det(1 − γ|𝔯)}.

Proof. Let H act on itself by conjugation, write h.x = hxh^{−1}, write H.x for the orbit, so H.x = {hxh^{−1}|h ∈ H} as well as H.S = {hsh^{−1}|s ∈ S, h ∈ H} for any subset S of H. We are going to consider functions that are supported on the closure of the set H.(MA^{−}). At first let f_{τ} be the Euler-Poincaré function defined on M attached to the representation (τ, V_{τ}) of K_{M}. Next fix a smooth function η on N which has compact support, is positive, invariant under K_{M} and satisfies ∫_{N} η(n)dn = 1. Given these data let ϕ = ϕ_{η,τ,φ} : H → C be defined by

ϕ(knma(kn)^{−1}) := η(n)f_{τ}(m)\frac{φ(a)}{det(1 − (ma)|𝔫)},

for k ∈ K, n ∈ N, m ∈ M, a ∈ \overline{A^{−}}. Further ϕ(h) = 0 if h is not in H.(M\overline{A^{−}}).

Next choose any compactly supported positive function ψ on R with ∫ ψ = 1. Let Φ(h, r) := ϕ(h)ψ(r). We will plug Φ into the trace formula. For the geometric side let γ = (γ_{H}, γ_{R}) ∈ Γ. We have to calculate the orbital integral:

O_{γ}(Φ) = ∫_{G_{γ}\G} Φ(x^{−1}γx)dx.

Now let x = (h, r) ∈ G and compute

x^{−1}γx = (h^{−1}γ_{H}h, r + h^{−1}γ_{R} − h^{−1}γ_{H}hr).

So (h,r) lies in the centralizer G_{γ} iff h ∈ H_{γ_{H}} and r ∈ R satisfies

(1 − h^{−1})γ_{R} = (1 − γ_{H}^{−1})r.

Note that by (A2) to any γ such that γ_{H} is conjugate to an element of A^{reg}M, and to any h ∈ H_{γ_{H}} such an r exists and is unique. But this condition on γ is satisfied if φ(h^{−1}γ_{H}h) ≠ 0. So suppose γ_{H} is in H.(A^{reg}M). In this case we have the integration rule

∫_{G_{γ}\G} f(g)dg = ∫_{H_{γ_{H}} \H} ∫_{R} f(h, r)drdh.

This is proven by showing that the right hand side is in fact G-invariant. We compute

∫_{R} Φ((h, r)^{−1}γ(h, r))dr = \frac{φ(h^{−1}γ_{H}h)}{det(1 − γ_{H}|𝔯)},

from which we see that the geometric side of the trace formula coincides with our claim.

Now for the spectral side let π ∈ Ĝ then the restriction of π to R is a direct integral over R̂. The irreducibility of π implies that the corresponding measure is supported on a single orbit o of the H -action on R̂. So we have

π|_{R} = ∫_{o} V_{π}(τ)dm(τ),

where m is a scalar valued measure and V_{π}(τ) is a multiple of τ. Fix τ_{0} ∈ o then the stabilizer H_{τ_{0}} acts trivially on τ_{0} and not only its class since by dim τ_{0} = 1 these two notions coincide. It follows that as H_{τ_{0}} -modules we have V_{π}(τ) ≅ η ⊗ τ for some representation η of H_{τ_{0}} . The measure m induces a measure on G_{τ_{0}}\G also denoted m which is quasi-invariant. It follows that π = ind_{H_{τ_{0}}⋉R}^{G}(η ⊗ τ) and hence η must be irreducible since π is. Let λ(x, y) denote the Radon-Nikodym derivative of the translate m_{x} with respect to m. We conclude that π(Φ) is given as an integral operator on G_{τ_{0}}\G with kernel

k(x, y) = ∫_{G_{τ_{0}}} Φ(x^{−1}zy)λ(x^{−1}zy, x)^{\frac{1}{2}} (η ⊗ τ_{o})(z)dz.

From this we get

trπ(Φ) = ∫_{G_{τ_{0}}} tr(∫_{G_{τ_{0}}} Φ(x^{−1}zx)λ(x^{−1}zx, x)^{\frac{1}{2}} (η ⊗ τ_{0})(z)dz)dx.

Consider the term Φ(x^{−1}zx) = φ(x_{H}^{−1}z_{H}x_{H})ψ(…). By (A2) this expression vanishes unless τ_{0} is the trivial character. In the case τ_{0} = triv it follows that π(R) = 1, so π may be viewed as an element of Ĥ.

To evaluate trπ(Φ) further we will employ the Hecht-Schmid character formula [9]. For this let

(MA)^{−} = interior in MA of the set

{g ∈ MA|det(1 − ga|𝔫) ≥ 0 for all 𝔞 ∈ A^{−}} .

The character Θ_{π}^{G} of π ∈ Ĝ is a locally integrable function on G. In [9] it is shown that for any π ∈ Ĥ, denoting by π^{0} the underlying Harish-Chandra module we have that all Lie algebra cohomology groups H^{p}(𝔫, π^{0}) are Harish-Chandra modules for MA. The main result of [9] is that for ma ∈ (MA)^{−} ∩ H^{reg}, the regular set, we have

Θ_{π}^{H}(ma) = \frac{∑_{p=0}^{dim 𝔫}(−1)^{p}Θ_{H_{p}(𝔫,π^{0})}^{MA}(ma)}{det(1 − ma|𝔫)}.

Let f be supported on H.(MA^{−}), then the Weyl integration formula states that

∫_{H} f(x)dx = ∫_{H/MA} ∫_{MA^{−}} f(hmah^{−1})|det(1 − ma|𝔫 ⊕ n̄)dadmdh.

So that for π ∈ Ĥ:

trπ(ϕ) = ∫_{H} Θ_{π}^{H}(x)ϕ(x)dx
= ∫_{MA^{−}} Θ_{π}^{H}(ma)f_{τ}(m)φ(a)|det(1 − ma|𝔫̄)|dadm
= (−1)^{dim N} ∫_{MA^{−}} f_{τ}(m)Θ_{H^{∗}(𝔫,π^{0})}^{MA}(ma)φ(a)dadm,

where we have used the isomorphism H_{p}(𝔫, π^{0}) ≅ H^{dim N−p}(𝔫, π^{o}) ⊗ ∧^{top}𝔫. This gives the claim. □

In the second version of the Lefschetz formula we want to substitute the character of the representation τ by an arbitrary central function on K_{M}. A smooth function f on K_{M} is called central if f(kk_{1}k^{−1}) = f(k_{1}) for all k, k_{1} ∈ K_{M}. Since B is a Cartan subgroup of the compact group K_{M}, any k ∈ K_{M} is conjugate to some element of B so the restriction gives in isomorphism from the space of smooth central functions on K_{M} to the space of smooth functions on B, invariant under the Weyl group. Hence we are led to consider Weyl group invariant functions on T.

Let A denote the convolution algebra of all W (H, T)-invariant smooth functions on T with compact support. Let S ⊂ T be the set of all ab with singular a-part.

For any t = ab in T let 𝔫_{t} be the space of all X ∈ ad(t)𝔤 on which t acts contractingly. Then 𝔫_{t} is a nilpotent Lie subalgebra of 𝔤.

Theorem 3.2. (Lefschetz formula, second version) Let φ ∈ A and suppose φ vanishes on the singular set to order dim G + 1 then the expression

∑ ; \begin{matrix}π ∈ Ĝ \\ π|_{R} ≡ 1 \end{matrix} N_{Γ}(π) ∑_{q}(−1)^{q} ∫_{T/W (H,T)} φ(t)tr(t|H^{q}(𝔫_{t}, π))dt

equals

(−1)^{dim(N)} ∑_{[γ]∈E_{P} (Γ)} λ_{γ}χ_{r}(X_{γ})\frac{φ(t_{γ})}{det(1 − t_{γ}|𝔭_{M} ⊕ 𝔫_{h_{γ}})det(1 − γ|𝔯)}.

Proof. Extend b ↦ φ(ab) to a central function on K_{M}. Then expand φ into K_{M} -types:

φ(ab) = ∑_{τ∈K̂_{M}} c_{τ} trτ(b)φ_{τ}(a),

since φ is smooth the coefficients c_{τ} are rapidly decreasing so the expressions of Theorem 3.1 when plugging in φ_{τ}|_{A^{−}} converge to

∑ ; \begin{matrix}π ∈ Ĝ \\ π|_{R} ≡ 1 \end{matrix} N_{Γ}(π) ∑_{p,q}(−1)^{p+q} ∫_{T/W (H,T)} φ(t)tr(t|H^{q}(𝔫_{t}, π) ⊗ ∧^{p}𝔭_{M})dt,

which equals

(−1)^{dim(N)} ∑_{[γ]∈E_{H}(Γ)} λ_{γ}χ_{r}(X_{γ})\frac{φ(t_{γ})}{det(1 − t_{γ}|𝔫_{h_{γ}})det(1 − γ|𝔯)}.

Now replace φ(t) by φ(t)/det(1 − t|𝔭_{M}) which gives the claim. □

At last we also mention a reformulation in terms of relative Lie algebra cohomology. Again, fix a parabolic P = MAN and now fix also a finite dimensional irreducible representation (σ, V_{σ}) of M.

Theorem 3.3. (Lefschetz formula, third version) Let φ be compactly supported on \overline{A^{−}}, dim G-times continuously differentiable and suppose φ vanishes on the boundary to order dim G + 1. Then we have that the expression

∑ ; \begin{matrix}π ∈ Ĝ \\ π|_{R} ≡ 1 \end{matrix} N_{Γ}(π) ∑_{q}(−1)^{q} ∫_{A^{−}} φ(a)tr(a|H^{q}(𝔪 ⊕ 𝔫, K_{M}, π ⊗ Vσ̆)

equals

(−1)^{dim(N)} ∑_{[γ]∈E_{P} (Γ)} λ_{γ}χ_{r}(X_{γ})\frac{φ(a_{γ})trσ(b_{γ})}{det(1 − a_{γ}b_{γ}|𝔫)det(1 − γ|𝔯)}.

Proof. Extend Vσ̆ to a 𝔪 ⊕ 𝔫-module by letting 𝔫 act trivially. We then get

H^{p}(𝔫, π^{0}) ⊗ Vσ̆ ≅ H^{p}(𝔫, π^{0} ⊗ Vσ̆).

The (𝔪, K_{M})-cohomology of the module H^{p}(𝔫, π^{0} ⊗ Vσ̆) is the cohomology of the complex (C^{∗}) with

C^{q} = Hom_{K_{M}} (∧^{q}𝔭_{M}, H^{p}(𝔫, π^{0}) ⊗ Vσ̆)
= (∧^{q}𝔭_{M} ⊗ H^{p}(𝔫, π^{0}) ⊗ Vσ̆)^{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^{0}) ⊗ ∧^{q}𝔭_{M} ⊗ Vσ̆)^{K_{M}} ≅ ∑_{q}(−1)^{q}H^{q}(𝔪, K_{M}, H^{p}(𝔫, π^{0} ⊗ Vσ̆)).

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 ⊗ Vσ̆. We have

E_{2}^{p,q} = H^{q}(𝔪, K_{M}, H^{p}(𝔫, π^{0} ⊗ Vσ̆))

and

E_{∞}^{p,q} = Gr^{q}(H^{p+q}(𝔪 ⊕ 𝔫, K_{M}, π^{0} ⊗ Vσ̆)).

Now the module in question is just

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

Since the differentials in the 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^{0}) ⊗ ∧^{q}𝔭_{M} ⊗ Vσ̆)^{K_{M}} ≅ ∑_{j}(−1)^{j}H^{j}(𝔪 ⊕ 𝔫, K_{M}, π^{0} ⊗ Vσ̆).

The claim follows. □

4. Geometric interpretation

Now consider the first version of the Lefschetz formula in the case R = 0. The representation τ defines a homogeneous vector bundle E_{τ} over G/K_{M} and by homogeneity this pushes down to a locally homogeneous bundle over Γ\G/K_{M} = MX_{Γ}. The tangent bundle T (_{M}X_{Γ}) can be described in this way as stemming from the representation of K_{M} on

𝔤/𝔨_{M} ≅ 𝔞 ⊕ 𝔭_{M} ⊕ 𝔫 ⊕ 𝔫̄.

We get a splitting into subbundles

T (_{M}X_{Γ}) = T_{c} ⊕ T_{n} ⊕ T_{u} ⊕ T_{s}.

These bundles can be characterized by dynamical properties: the action of A ≅ ℝ^{r} is furnished with a positive time direction given by the positive Weyl chamber A^{+}. Then T_{s}, the stable part is characterized by the fact that A^{+} acts contractingly on T_{s}. On the unstable part T_{u} the opposite chamber A^{−} acts contractingly. T_{c}, the central part is spanned by the “flow” A itself and T_{n} is an additive neutral part. Note that T_{n} vanishes if we choose H to be the maximal split torus. The bundle T_{n} ⊕ T_{u} is integrable, so it defines a foliation F. To this foliation we have the tangential cohomology H^{∗}(F) and also for its τ -twist: H^{∗}(F ⊗ τ). The flow A acts on the tangential cohomology whose alternating sum we will consider as a virtual A-module. For any φ ∈ C_{c}^{∞}(\overline{A^{−}}) we define L_{φ} = ∫_{A^{−}} φ(a)(a|H^{∗}(F ⊗τ))da as a virtual operator on H^{∗}(F ⊗ τ). Then we have

Proposition 4.1. Under the assumptions of Theorem 3.1 the virtual operator L_{φ} is of trace class and the RHS of Theorem 3.1 can be written as

∑_{q}(−1)^{q} tr(L_{φ}|H^{q}(F ⊗ τ)).

References

[1] Borel, A., “Introduction aux groupes arithmétiques, ” Hermann, Paris, 1969.

[2] Borel, A., and N. Wallach, “Continuous Cohomology, Discrete Groups, and Representations of Reductive Groups, ” Ann. Math. Stud. 94, Princeton 1980.

[3] Deitmar, A., Higher torsion zeta functions, Adv. Math. 110 (1995), 109– 128.

[4] Deitmar, A., Product expansions for zeta functions attached to locally homogeneous spaces, to appear in Duke Math. J.

[5] —, On some zeta functions attached to Shimura manifolds of higher rank, preprint dg-ga/9511006.

[6] Duistermaat, J., and V. Guillemin,, The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics, Invent. math. 29 (1975), 39–79.

[7] Duistermaat, J. J., J. A. C. Kolk, and V. S. Varadarajan, Spectra of locally symmetric manifolds of negative curvature, Invent. math. 52 (1979), 27– [8] Guillemin, V., Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977), 485–517.

[9] Hecht, H., and W. Schmid, Characters, asymptotics and 𝔫-homology of Harish-Chandra modules, Acta Math. 151 (1983), 49–151.

[10] Juhl, A., “Zeta-Funktionen, Index-Theorie und hyperbolische Dynamik, ” Habilitationsschrift. Humboldt-Universität zu Berlin 1993.

[11] Osborne, M. S., “Lefschetz formulas on non-elliptic complexes, ” Thesis, Yale University, 1972.

[12] —, Lie algebra cohomology of certain infinite dimensional representations, Bull. Amer. Math. Soc. 80 (1980), 852–854.

[13] Schubert, V., “Ein kohomologischer Zugang zu verallgemeinerten Selbergschen Zeta-Funktionen auf lokalsymmetrischen kompakten Rang-1-Räu-men, ” Dissertation. Göttingen 1994.

[14] Wallach, N., “Real Reductive Groups I, ” Academic Press 1988.

Mathematisches Institut Universität Heidelberg Im Neuenheimer Feld 288 69126 Heidelberg Germany

Received March 21, 1996