Spherical subgroups and double coset varieties

Let $\text{GroupG}$ be a connected reductive algebraic group, $\GroupH \subsetneq \GroupG$areductivesubgroupand$\GroupT \subset \GroupG$ a maximal torus. It is well known that if charactersitic of the ground field is zero, then the homogeneous space $\text{GroupG}\mathrm{/}\text{GroupH}$ is a smooth affine variety, but never an affine space. The situation changes when one passes to double coset varieties $\text{dcosets}\text{GroupF}\text{GroupG}\text{GroupH}$. In this paper we consider the case of $\text{GroupG}$ classical and $\text{GroupH}$ connected spherical and prove that either the double coset variety $\text{dcosets}\text{GroupT}\text{GroupG}\text{GroupH}$ is singular, or it is an affine space. We also list all pairs $\text{GroupH}\subset \text{GroupG}$ such that $\text{dcosets}\text{GroupT}\text{GroupG}\text{GroupH}$ is an affine space.