Spherical subgroups and double coset varieties

TL;DR

This paper investigates the structure of double coset varieties for classical groups with spherical subgroups, showing they are either affine spaces or singular, and classifies all cases where they are affine spaces.

Contribution

It proves that double coset varieties are either affine spaces or singular for classical groups with spherical subgroups, and provides a complete classification of affine cases.

Findings

Double coset varieties are either affine spaces or singular.

Classification of all pairs where the double coset variety is an affine space.

Results hold for classical groups with connected spherical subgroups.

Abstract

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