This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Journal of Lie Theory Volume 6 (1996) 179190 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 com pact quotients of Lie groups is given. Introduction Let G denote a Lie group and a uniform lattice in G . We fix a maximal torus T in G and consider the action of T on the compact quotient \ 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 [ ] in whose Gconjugacy 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 [ ] let X be the union of all compact orbits in that class. Then it is known that X is a smooth submanifold and with r ( X ) we denote its detwisted Euler characteristic (see sect. 2.). Note that r ( X ) is local, i.e. it can be expressed as the integral over X of a canonical differential form (generalized Euler form). On the other hand r ( X ) can be expressed as a simple linear combination of Betti numbers (see sect. 2.). Next, will denote the volume of the orbit and P s the stable part of the Poincar e map around the orbit. Then the number L ( ) := r ( X ) det(1 P s ) will be called the Lefschetz number of [ ] (compare [8]). The class [ ] defines a point a 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: X [ ] L ( ) a = 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 09495932 / $2.50 C Heldermann Verlag 180 Deitmar 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. EulerPoincar e 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 ( , V ) be a finite dimensional unitary representation of K and write ( , V ) for the dual representation. Assume that G has a compact Cartan subgroup T K . Let g = k p be the polar decomposition of the real Lie algebra g of G and write g = k + p for its complexification. Choose an ordering of the roots ( g , t ) of the pair ( g , t ). This choice induces a decomposition p = p p + ....
View Full
Document
 Winter '11
 PhanThuongCang

Click to edit the document details