Distribution of lattice orbits on homogeneous varieties

Abstract

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\in V, and compactly supported function \phi on V, we consider the sums S_{\phi,v}(T)=\sum_{\gamma\in \Gamma_T} \phi(v \gamma). Understanding the asymptotic behavior of S_{\phi,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,v}(T) \sim \int_{G_T} \phi(vg) dg, where G_T={g\in G:||g||<T} and \Gamma_T = G_T \cap \Gamma. We also show that the asymptotics of S_{\phi,v}(T) is governed by \int_V \phi d\nu, where \nu is an explicit limiting density depending on the choice of v and the norm.