Torus actions on compact quotients

We consider the action of a noncompact torus H on the compact quotient G/L, where G is a Lie group containing H and L is a uniform lattice in G. Using harmonic analysis on G we prove a formula relating the compact orbits of H to the action of H on the (infinite dimensional) tangential cohomology. The formula may be viewed as a Lefschetz formula or an equivariant index theorem for noncompact group actions.