Distribution of lattice orbits on homogeneous varieties

Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H G. For an increasing family of finite subsets > 0, a dense orbit υ• Γ, υ V and compactly supported function φ on V, we consider the sums S (T) = ΓT. Understanding the asymptotic behavior of S _φ,υ (T) is a delicate problem which has only been considered for certain very special choices of H,G and Γ _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 dg, where G _T = g G: ||g|| < T and Γ _T = G _T Γ. We also show that the asymptotics of S _{φ, υ} (T) is governed by where ν is an explicit limiting density depending on the choice of υ and || • ||.