Infiniteness of Double Coset Collections in Algebraic Groups

Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show that if $X\backslash G/P$ is finite then $X$ is spherical or the Levi factor of $P$ is spherical. This implies that it is rare for $X\backslash G/P$ to be finite. The primary method of proof is to descend to calculations at the finite group level and then to use elementary character theory.