  Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup
and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite.
Finiteness is proven using geometric arguments about the action of $X$ on
subspaces of the natural module for $G$. Infiniteness is proven using a
dimension criterion which involves root systems.

Group Theory