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Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Published online by Cambridge University Press:  02 May 2017

RODOLPHE RICHARD  and
NIMISH A. SHAH Open the ORCID record for NIMISH A. SHAH [Opens in a new window]
RODOLPHE RICHARD
Affiliation:
University College London, London WC1E 6BT, UK email Rodolphe.RICHARD@Normalesup.org
NIMISH A. SHAH
Affiliation:
The Ohio State University, Columbus, OH 43210, USA email shah@math.ohio-state.edu
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Abstract

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 7 , October 2018 , pp. 2780 - 2800
Copyright
© Cambridge University Press, 2017 

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References

Borel, A. and Harish-Chandra. Arithmetic subgroups of Algebraic Groups. Ann. of Math. (2) 75(3) (1962), 485–535.Google Scholar
Borel, A.. Linear Algebraic Groups (Graduate Texts in Mathematics, 126) , 2nd edn. Springer, New York, 1991.Google Scholar
Bourbaki, N.. Groupes et algèbres de Lie. Chapitre 1: Algèbres de Lie (Éléments de mathématique, XXVI) (Actualités Scientifiques et Industrielles, 1285) . Hermann, Paris, 1960, pp. 7–136.Google Scholar
Bourbaki, N.. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie (Éléments de mathématique, Fasc. XXXVII) . Hermann, Paris, 1972, Actualités Scientifiques et Industrielles, No. 1349.Google Scholar
Chambert-Loir, A. and Tschinkel, Y.. Igusa integrals and volume asymptotics in analytic and adelic geometry. Confluentes Math. 2(3) (2010), 351–429.Google Scholar
Dani, S. G.. On invariant measures, minimal sets and a lemma of Margulis. Invent. Math. 51(3) (1979), 239–260.Google Scholar
Dani, S. G.. On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 4(1) (1984), 25–34.Google Scholar
Dani, S. G. and Margulis, G. A.. Orbit closures of generic unipotent flows on homogeneous spaces of SL(3, R). Math. Ann. 286(1–3) (1990), 101–128.Google Scholar
Dani, S. G. and Margulis, G. A.. Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 101(1) (1991), 1–17.Google Scholar
Dani, S. G. and Margulis, G. A.. Limit distributions of orbits of unipotent flows and values of quadratic forms. I. M. Gel’fand Seminar (Advances in Soviet Mathematics, 16, Part 1) . American Mathematical Society, Providence, RI, 1993, pp. 91–137.Google Scholar
Duke, W., Rudnick, Z. and Sarnak, P.. Density of integer points on affine homogeneous varieties. Duke Math. J. 71(1) (1993), 143–179.Google Scholar
Dani, S. G. and Smillie, J.. Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51(1) (1984), 185–194.Google Scholar
Eskin, A. and McMullen, C.. Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71(1) (1993), 181–209.Google Scholar
Eskin, A., Mozes, S. and Shah, N.. Unipotent flows and counting lattice points on homogeneous varieties. Ann. of Math. (2) 143(2) (1996), 253–299.Google Scholar
Eskin, A., Mozes, S. and Shah, N.. Non-divergence of translates of certain algebraic measures. Geom. Funct. Anal. 7(1) (1997), 48–80.Google Scholar
Gorodnik, A., Maucourant, F. and Oh, H.. Manin’s and Peyre’s conjectures on rational points and adelic mixing. Ann. Sci. Éc. Norm. Supér. (4) 41(3) (2008), 383–435.Google Scholar
Gorodnik, A. and Oh, H.. Rational points on homogeneous varieties and equidistribution of adelic periods; with an appendix by Mikhail Borovoi. Geom. Funct. Anal. 21(2) (2011), 319–392.Google Scholar
Helgason, S.. Differential Geometry, Lie Groups, and Symmetric Spaces (Pure and Applied Mathematics, 80) . Academic Press, New York, 1978.Google Scholar
Kempf, G. R.. Instability in invariant theory. Ann. of Math. (2) 108(2) (1978), 299–316.Google Scholar
Kleinbock, D. Y. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148(1) (1998), 339–360.Google Scholar
Kleinbock, D. and Tomanov, G.. Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comment. Math. Helv. 82(3) (2007), 519–581.Google Scholar
Margulis, G. A.. On the action of unipotent groups in the space of lattices. Proc. Summer School of the Bolyai János Mathematical Society on Lie Groups and Their Representations (Budapest, 1971). Halsted, New York, 1975, pp. 365–370.Google Scholar
Mostow, G. D.. Self-adjoint groups. Ann. of Math. (2) 62 (1955), 44–55.Google Scholar
Mostow, G. D.. Some new decomposition theorems for semi-simple groups. Mem. Amer. Math. Soc. 1955(14) (1955), 31–54.Google Scholar
Platonov, V. and Rapinchuk, A.. Algebraic Groups and Number Theory (Pure and Applied Mathematics, 139) . Academic Press, Boston, MA, 1994.Google Scholar
Ratner, M.. On Raghunathan’s measure conjecture. Ann. of Math. (2) 134(3) (1991), 545–607.Google Scholar
Richard, R.. Sur quelques questions d’équidistribution en géométrie arithmétique. PhD Thesis, Université Rennes 1, 2009, http://tel.archives-ouvertes.fr/docs/00/43/85/15/PDF/Rodolphe-Richard-These-fois12-v2.pdf.Google Scholar
Richard, R. and Zamojski, T.. Limit distribution of translated pieces of possibly irrational leaves in S-arithmetic homogeneous spaces. Preprint arXiv:1604.08494, 52pp.Google Scholar
Shah, N. A.. Uniformly distributed orbits of certain flows on homogeneous spaces. Math. Ann. 289(2) (1991), 315–334.Google Scholar
Shah, N. A.. Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci. 106(2) (1996), 105–125.Google Scholar
Tomanov, G.. Orbits on homogeneous spaces of arithmetic origin and approximations. RIMS Symposium: Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama–Kyoto, Japan, 4–8 August 1997 (Advanced Studies in Pure Mathematics, 26) . Mathematical Society of Japan, Tokyo, 2000, pp. 265–297.Google Scholar