Duke Mathematical Journal

Closed orbits for actions of maximal tori on homogeneous spaces

George Tomanov and Barak Weiss

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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

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

First available in Project Euclid: 23 April 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]


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

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