Torus actions on compact quotients

For a reductive Lie group G and a uniform lattice Γ in G we consider the compact space Γ\G. We fix a maximal torus H in G and consider its action on the compact quotient Γ\G. Assuming H to be noncompact we will prove a Lefschetz formula relating compact orbits as local data to the action of the torus H on a global cohomology theory (tangential cohomology). The compact orbits are parametrized modulo homotopy by those conjugacy classes [γ] in Γ whose G-conjugacy classes meet H. On the other hand the homotopy also has to be taken into account, so, for a class [γ] let Xγ be the union of all compact orbits in that class then it is known thatXγ is a smooth submanifold and with χr(Xγ) we denote its detwisted Euler characteristic (see sect. 2). Note that χr(Xγ) is local, i.e. it can be expressed as the integral over Xγ of a canonical differential form (generalized Euler form). On the other hand it can be expressed, as in sect 2, as a simple linear