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 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 subgroups. Finally, excluding a case in F4, we show that if X G/P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X G/P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.