Distribution of lattice orbits on homogenous varieties

Given a lattice Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of Gamma on the homogeneous space V = H\ G. For an increasing family of finite subsets {Gamma(T) : T > 0}, a dense orbit v . Gamma, v is an element of V and compactly supported function phi on V, we consider the sums S-phi, v( T) = Sigma(gamma is an element of Gamma T) phi(v gamma). Understanding the asymptotic behavior of S-phi,S-v(T) is a delicate problem which has only been considered for certain very special choices of H, G and {Gamma(T)}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S-phi,S-v(T) similar to integral(GT)phi(vg) dg, where G(T) = {g is an element of G : parallel to g parallel to <T} and Gamma(T) = G(T) boolean AND Gamma. We also show that the asymptotics of S-rho,S-v(T) is governed by integral(V)phi d nu, where nu is an explicit limiting density depending on the choice of v and parallel to.parallel to.