Abstract
We consider the natural action of maximal tori of real algebraic groups on homogeneous spaces defined as quotients of the real algebraic groups by their arithmetic subgroups. The study of the orbit closures in this context is closely related to important problems from number theory, in particular, with the notable Littlewood conjecture. We mainly concentrate on two classes of orbits: 1) relatively compact and, 2) divergent orbits. In the former case we generalize recent results of Lindenstrauss and Weiss. In the latter one we announce new results (due to Margulis, Weiss and the author) which show that the divergent and, more generally, the closed orbits are always standard.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Borel, Linear Algebraic Groups. Second Enlarged Edition, Springer, 1991.
A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, 1969.
A. Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math., vol. 224 (1966), 78–89.
A. Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math., vol. 75 (1962), 485–535.
A. Borel, G. Prasad, Values of isotropic quadratic forms at S-integer points, Compositio Math., vol. 83 (1992), 347–372.
A. Borel and J. Tits, Groupes reductifs, Inst. Hautes Études Sci. Publ. Math., vol. 27 (1965), 55–150.
C. Chevalley, Theorie des groupes de Lie, Hermann, Paris, 1951.
S. G. Dani, Invariant measures, minimal sets, and a lemma of Margulis, Invent. Math., vol.64 (1981), 357–385.
S. G. Dani and G. A. Margulis, Limit distributions of unipotent flows and values of quadratic forms. In I.M. Gelfand Seminar pp.91-137. Amer. Math. Soc, Providence, RI, 1993.
A. Katok, R.J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, vol.16 (1996), 751–778, and Corrections to “ Invariant measures for higher-rank hyperbolic abelian actions”, Ergodic Theory Dynam. Systems, vol.18 (1998), no.2, 503-507.
M. Kneser, Lectures on Galois Cohomology, Bombay, Tata Inst. of Fund. Research, 1969.
E. Lindenstrauss and B. Weiss, On Sets Invariant under the Action of the Diagonal Group, to appear in Ergodic Theory Dynam. Systems.
G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, 1991
G. A. Margulis, Formes quadratiques indéfinies et flots unipotent sur les espaces homogenes, C. R. Acad. Sci. Paris Ser.1, vol.304 (1987), 247–253.
G. A. Margulis, Discrete Subgroups and Ergodic Theory. Proc. of conference “Number Theory, trace formulas and discrete groups” in honor of A. Selberg (Oslo, 1987), 377–398, Academic Press, Boston, 1989.
G. A. Margulis, Oppenheim Conjecture. Fields Medalists’ lectures, 272-327, World Sci. Ser. 20th Century Math.5 World Sci. Publishing River Edge, NJ, 1997.
G. A. Margulis, Problems and Conjectures in Rigidity Theory. Mathematics: frontiers and perspectives, 161-174, Amer. Math. Soc, Providence, RI, 2000.
G. A. Margulis, G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., vol. 116 (1994), 347–392.
G. A. Margulis, G. M. Tomanov, Measure rigidity for almost, linear groups and its applications, J. Anal. Math., vol. 69 (1996), 25–54.
G. D. Mostow, Intersections of discrete subgroups with Cartan Subgroups, J. Indian Math. Soc, vol. 34 (1970), 203–214.
A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. USA, vol.15 (1929), 724–727.
A. Oppenheim, The minima of indefinite quaternary quadratic forms, Ann. of Math., vol. 32 (1931), 271–298.
R. S. Pierce, Associative Algebras Springer, 1982.
G. Prasad and M. S. Raghunathan, Cartan subgroups and lattices in semisimple groups, Ann. of Math., vol. 96 (1972), 296–317.
M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, 1972.
M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math., vol. 134 (1992), 545–607.
M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J., vol. 63 (1991), 235–280.
M. Ratner, Raghunathan’s conjectures for cartesian products of real and p-adic Lie groups, Duke Math. J., vol. 77 (1995), 275–382.
M. Rees, Some ℝ2-Anosov flows, University of Minnesota, Mathematics Report 82-123.
N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., vol. 289 (1991), 315–334.
W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, 1985.
J. Tits, Classification of algebraic semisimple groups, algebraic groups discontinuous subgroups, Symposium Colorado, Boulder 1965, Proc. Symp. Pure Math. No. 9, Amer. Math. Soc, Providence,1966
G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, Adv. Stud. Pure Math., vol. 27 (2000), 265–297.
G. Tomanov, B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, preprint, 2001.
A. Weil, Basic Number Theory, Springer, 1967.
A. Weil, Adels and Algebraic Groups, Inst. for Advanced Study, Princeton, NJ, 1961.
R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Tomanov, G. (2002). Actions of Maximal Tori on Homogeneous Spaces. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-662-04743-9_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07751-7
Online ISBN: 978-3-662-04743-9
eBook Packages: Springer Book Archive