## Duke Mathematical Journal

### Closed orbits for actions of maximal tori on homogeneous spaces

#### Abstract

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be any torus containing a maximal $\mathbb{R}$-split torus. We prove that the closed orbits for the action of $T$ on $G/\Gamma$ admit a simple algebraic description. In particular, we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is a product of a compact torus and a torus defined over $\mathbb{Q}$, and it is divergent if and only if the maximal $\mathbb{R}$-split subtorus of $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following:

· there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit;

· if $\rank_{\mathbb{Q}}G<\rank_{\mathbb{R}} G$, there are no divergent orbits for $T$.

#### Article information

Source
Duke Math. J., Volume 119, Number 2 (2003), 367-392.

Dates
First available in Project Euclid: 23 April 2004

https://projecteuclid.org/euclid.dmj/1082744736

Digital Object Identifier
doi:10.1215/S0012-7094-03-11926-2

Mathematical Reviews number (MathSciNet)
MR1997950

Zentralblatt MATH identifier
1040.22005

Subjects

#### Citation

Tomanov, George; Weiss, Barak. Closed orbits for actions of maximal tori on homogeneous spaces. Duke Math. J. 119 (2003), no. 2, 367--392. doi:10.1215/S0012-7094-03-11926-2. https://projecteuclid.org/euclid.dmj/1082744736

#### References

• A. Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966), 78--89.
• --------, Introduction aux groupes arithmetiques, Publ. Inst. Math. Univ. Strasbourg 15, Actualités Sci. Indust. 1341, Hermann, Paris, 1969.
• --------, Linear Algebraic Groups, 2d enlarged ed., Grad. Texts in Math. 126, New York, Springer, 1991.
• A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111--164.
• A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55--150.
• S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55--89.
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.
• D. A. Ka žhdan and G. A. Margulis, A proof of Selberg's hypothesis (in Russian), Mat. Sb. (N.S.) 75 (117) (1968), 163--168.
• D. Kleinbock, N. Shah, and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory'' in Handbook of Dynamical Systems, Vol. 1A, ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur, North-Holland, Amsterdam, 2002, 813--930. \CMP1 928 528
• G. Margulis, Problems and conjectures in rigidity theory'' in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, 2000, 161--174.
• G. Prasad and M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups, Ann. of Math. (2) 96 (1972), 296--317.
• M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb. 68, Springer, New York, 1972.
• J. Tits, Classification of algebraic semisimple groups'' in Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), ed. A. Borel and G. D. Mostow, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence, 1966, 32--62.
• G. Tomanov, Actions of maximal tori on homogeneous spaces'' in Rigidity in Dynamics and Geometry (Cambridge, U.K., 2000), ed. M. Burger and A. Iozzi, Springer, Berlin, 2002, 407--424. \CMP1 919 414
• È. B. Vinberg and A. L. Onishchik, A Seminar on Lie Groups and Algebraic Groups (in Russian), 2d ed., URSS, Moscow, 1995.