Distribution of lattice orbits on homogeneous varieties

Abstract

Given a lattice Γin a locally compact group G and a closed subgroup H of G, one has a natural action of Γon the homogeneous space V=H\G. For an increasing family of finite subsets {Γ_T: T>0}, a dense orbit vΓ, v\in V, and compactly supported function ϕon V, we consider the sums S_{ϕ,v}(T)=\sum_{γ\in Γ_T} ϕ(v γ). Understanding the asymptotic behavior of S_{ϕ,v}(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 S_{ϕ,v}(T) \sim \int_{G_T} ϕ(vg) dg, where G_T={g\in G:||g||<T} and Γ_T = G_T \cap Γ. We also show that the asymptotics of S_{ϕ,v}(T) is governed by \int_V ϕdν, where νis an explicit limiting density depending on the choice of v and the norm.