Limit distributions of expanding translates of certain orbits on homogeneous spaces

  • Nimish A. Shah
Article

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 gan} →e asn → ∞. Let Ω be a non-empty open subset ofU+ andni ∞ 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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [D1]
    Dani S G, Invariant measures of horospherical flows on non-compact homogeneous spaces,Invent. Math. 47 (1978) 101–138MATHCrossRefMathSciNetGoogle Scholar
  2. [D2]
    Dani S G, On orbits of unipotent flows on homogeneous spaces,Ergod. Th. Dynam. Sys. 4 (1980) 25–34MathSciNetGoogle Scholar
  3. [D3]
    Dani S G, Continuous equivariant images of lattice-actions on boundaries,Ann. Math. 119 (1984) 111–119CrossRefMathSciNetGoogle Scholar
  4. [DM1]
    Dani S G and Margulis G A, Orbit closures of generic unipotent flows on homogeneous spaces of SL(3, ℝ),Math. Ann. 286 (1990) 101–128MATHCrossRefMathSciNetGoogle Scholar
  5. [DM2]
    Dani S G and Margulis G A, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces,Proc. Indian Acad. Sci. 101 (1991) 1–17MATHMathSciNetCrossRefGoogle Scholar
  6. [DM3]
    Dani S G and Margulis G A, Limit distributions of orbits of unipotent flows and values of quadratic forms,Advances in Soviet Math. 16 (1993) 91–137MathSciNetGoogle Scholar
  7. [DR]
    Dani S G and Raghavan S, Orbits of Euclidean frames under discrete linear groups,Isr. J. Math. 36 (1980) 300–320MATHCrossRefMathSciNetGoogle Scholar
  8. [DRS]
    Duke W, Rudnick Z and Sarnak P, Density of integer points on affine symmetric varieties,Duke Math J. 71(1993) 143–180MATHCrossRefMathSciNetGoogle Scholar
  9. [EM]
    Eskin A and McMullen C, Mixing, counting and equidistribution in Lie groups,Duke Math. J. 71 (1993)181–209MATHCrossRefMathSciNetGoogle Scholar
  10. [EMS1]
    Eskin A, Mozes S and Shah N A, Non-divergence of translates of certain algebraic measures. PreprintGoogle Scholar
  11. [EMS2]
    Eskin A, Mozes S and Shah N A, Unipotent flows and counting lattice points on homogeneous varieties.Ann. Math. (to appear)Google Scholar
  12. [GR]
    Garland H and Raghunathan M S, Fundamental domains for lattices in ℝ-rank 1 semisimple Lie groups,Ann. Math. 92 (1970) 279–326CrossRefMathSciNetGoogle Scholar
  13. [Ml]
    Margulis G A, Quotient groups of discrete subgroups and measure theory, Funct.Anal. Appl. 12 (1978) 295–305MATHMathSciNetGoogle Scholar
  14. [M2]
    Margulis G A, Arithmeticity of irreducible lattices in semisimple groups of rank greater than 1,Invent Math. 76(1984) 93–120MATHCrossRefMathSciNetGoogle Scholar
  15. [MS]
    Mozes S and Shah N A, On the space of ergodic invariant measures of unipotent flows,Ergod. Th. Dynam. Sys. 15 (1978) 149–159MathSciNetGoogle Scholar
  16. [R]
    Raghunathan M S,Discrete subgroups of Lie groups. (1972) (Springer: Berlin Heidelberg New York)MATHGoogle Scholar
  17. [Ra1]
    Ratner M, On Raghunathan’s measure conjecture,Ann. Math. 134 (1991) 545–607CrossRefMathSciNetGoogle Scholar
  18. [Ra2]
    Ratner M, Raghunathan’s topological conjecture and distributions of unipotent flows.Duke Math. J. 63 (1991) 235–280MATHCrossRefMathSciNetGoogle Scholar
  19. [S]
    Schlichtkrull H,Hyperfunctions and Harmonic Analysis on Symmetric Spaces (1984) (Birkhauser: Boston)MATHGoogle Scholar
  20. [Sh1]
    Shah N A, Uniformly distributed orbits of certain flows on homogeneous spaces.Math. Ann. 289 (1991) 315–334MATHCrossRefMathSciNetGoogle Scholar
  21. [Sh2]
    Shah N A, Limit distributions of polynomial trajectories on homogeneous spaces,Duke Math. J. 75 (1994) 711–732MATHCrossRefMathSciNetGoogle Scholar
  22. [St]
    Stuck G, Minimal actions of semisimple groups: InWorkshop on Lie groups, Ergodic theory, and Geometry and problems and Geometric rigidity, April 13–17, 1992, Berkeley. MSRI preprint (August 1992).Google Scholar
  23. [SZ]
    Stuck G and Zimmer R J, Stabilizers of ergodic actions of higher rank semisimple groups,Ann. Math. 139 (1994) 723–747MATHCrossRefMathSciNetGoogle Scholar
  24. [W]
    Witte D, Measurable quotients of unipotent translations on homogeneous spaces,Trans. Am. Math. Soc. 345 (1994) 577–594MATHCrossRefMathSciNetGoogle Scholar
  25. [Z1]
    Zimmer R J, Ergodic theory, semisimple groups, and foliations by manifolds of negative curvature.I.H.E.S. Publ. Math. 55 (1982) 37–62MATHMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 1996

Authors and Affiliations

  • Nimish A. Shah
    • 1
  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia

Personalised recommendations