Limit distributions of expanding translates of certain orbits on homogeneous spaces Article

Received: 21 May 1995 Revised: 09 November 1995 DOI :
10.1007/BF02837164

Cite this article as: Shah, N.A. Proc. Indian Acad. Sci. (Math. Sci.) (1996) 106: 105. doi:10.1007/BF02837164 Abstract LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and\(\overline {GA} = L\) . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G )). Consider the expanding horospherical subgroup ofG associated toa defined as U^{+} ={gεG:a ^{−n} ga^{n} } →e as n → ∞ . Let Ω be a non-empty open subset ofU ^{+} andn _{i} → ∞ be any sequence. It is showed that\(\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L\) . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG -equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

Keywords Limit distributions unipotent flow horospherical patches symmetric subgroups continuous equivariant factors

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Authors and Affiliations 1. School of Mathematics Tata Institute of Fundamental Research Bombay India