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 {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 H 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 || · ||.
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Submitted: March 2005 Revision: April 2006 Accepted: June 2006
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Gorodnik, A., Weiss, B. Distribution of lattice orbits on homogeneous varieties. GAFA, Geom. funct. anal. 17, 58–115 (2007). https://doi.org/10.1007/s00039-006-0583-6
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DOI: https://doi.org/10.1007/s00039-006-0583-6