Institute for Mathematical Sciences

Preprint ims01-11

G. Tomanov and B. Weiss
Closed orbits for actions of maximal tori on homogeneous spaces.

Abstract: Let $G$ be a real algebraic group defined over $Q$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be a maximal $\R$-split torus. We classify the closed orbits for the action of $T$ on $G/\Gamma,$ and show that they all 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 defined over $\Q$, and is (totally) divergent if and only if $x^{-1}Tx$ is defined over $\Q$ and $\Q$-split. Our analysis also yields the following: there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit. \item if $\Q {\rm -rank}(G)<\R{\rm -rank}(G)$, there are no divergent orbits for $T$.
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