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Unformatted text preview: 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 com- pact 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 Poincar´ e map around the orbit. Then the number L ( γ ) := λ γ χ r ( X γ ) det(1- P s ) will be called the Lefschetz number of [ γ ] (compare ). 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: X [ 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  a similar formula is proven to hold up to a smooth ISSN 0949–5932 / $2.50 C Heldermann Verlag 180 Deitmar function in the case of a flow. The present paper extends results of Andreas Juhl ,  in the real rank one case. See also , . 1. Euler-Poincar´ 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  and fix a maximal compact subgroup K . Let ( τ, V τ ) be a finite dimensional unitary representation of K and write (˘ τ, V ˘ τ ) 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 Φ( g , t ) of the pair ( g , t ). This choice induces a decomposition p = p- ⊕ p + ....
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- Winter '11
- Manifold, Representation theory, Lie group, Lie algebra, lie groups, Lefschetz