Distribution of lattice orbits on homogeneous varieties

Abstract

Given a lattice Γ in a locally compact group G and a closed subgroup of G, one has a natural action of Γ on the homogeneous space V = H G. For an increasing family of finite subsets {Γ T : T > 0}, a dense orbit υ· Γ, υ∈V and compactly supported function φ on V, we consider the sums $$ S\varphi ,\upsilon{\left( T \right)} = {\sum\nolimits_{\gamma \in \Gamma _{T} } {\varphi {\left( {\upsilon\gamma } \right)}} } $$ . 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 or G is semisimple. When G is a group of matrices equipped with a norm, we have $$S_{\varphi,\upsilon}(T)\sim \int_{G_{T}}\varphi(\upsilon g) dg,$$ 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 $$\int_{V}\varphi d \nu,$$ where ν is an explicit limiting density depending on the choice of υ and || · ||.