## 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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