On existence of double coset varieties

Abstract

Let $G$ be a complex affine algebraic group and $H, F \subset G$ be closed subgroups. The homogeneous space $G / H$ can be equipped with structure of a smooth quasiprojective variety. The situation is different for double coset varieties $\dcosets{F}{G}{H}$. In this paper we give examples showing that the variety $\dcosets{F}{G}{H}$ does not necessarily exist. We also address the question of existence of $\dcosets{F}{G}{H}$ in the category of constructible spaces and show that under sufficiently general assumptions $\dcosets{F}{G}{H}$ does exist as a constructible space.