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 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. 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 implies that it is rare 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.