Distribution of lattice orbits on homogenous varieties

A Gorodnik, B Weiss

Research output: Contribution to journalArticle (Academic Journal)peer-review

37 Citations (Scopus)


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 is an element of V and compactly supported function phi on V, we consider the sums S-phi, v( T) = Sigma(gamma is an element of Gamma T) phi(v gamma). Understanding the asymptotic behavior of S-phi,S-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,S-v(T) similar to integral(GT)phi(vg) dg, where G(T) = {g is an element of G : parallel to g parallel to <T} and Gamma(T) = G(T) boolean AND Gamma. We also show that the asymptotics of S-rho,S-v(T) is governed by integral(V)phi d nu, where nu is an explicit limiting density depending on the choice of v and parallel to.parallel to.
Translated title of the contributionDistribution of lattice orbits on homogenous varieties
Original languageEnglish
Pages (from-to)58 - 115
Number of pages58
JournalGeometric and Functional Analysis
Volume17 (1)
Publication statusPublished - Apr 2007

Bibliographical note

Publisher: Birkhauser Verlag AG


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