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 $T$.