Spherical subgroups and double coset varieties

Details Spherical subgroups and double coset varieties Spherical subgroups and double coset varieties

2011-08-10

arxiv.org/abs/1108.2148v2

Journal of Lie Theory 22 (2012), no. 2, 505--522

Mathematics

Algebraic Geometry

16 pages v2: improved readability of the text based on feedback from a referee of Journal of Lie Theory

Scientific paper

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.

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