Distribution of lattice orbits on homogeneous varieties

Alex Gorodnik, Barak Weiss

Research output: Contribution to journalArticlepeer-review

Abstract

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 || • ||.

Original languageEnglish
Pages (from-to)58-115
Number of pages58
JournalGeometric and Functional Analysis
Volume17
Issue number1
DOIs
StatePublished - Apr 2007
Externally publishedYes

Keywords

  • Equidistribution
  • Lattices in Lie groups
  • Values of quadratic forms
  • Volume asymptotics

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