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Journal of Lie Theory 22 (2012), No. 2, 505--522
Copyright Heldermann Verlag 2012

Spherical Subgroups and Double Coset Varieties

Artem Anisimov
Dept. of Higher Algebra, Faculty of Mechanics and Mathematics, Lomonosov State University, Leninskie Gory 1 / GSP-1, Moscow 119991, Russia


\def\dcosets #1#2#3 {#1 \hskip-1pt \backslash \hskip-3pt \backslash \hskip-0.8pt{#2}\hskip-1pt\slash\hskip-3pt\slash #3 \hskip1pt} Let $G$ be a connected reductive algebraic group, $H \subset G$ a reductive subgroup and $T \subset G$ a maximal torus. It is well known that if charactersitic of the ground field is zero, then the homogeneous space $G/H$ is a smooth affine variety, but never an affine space. The situation changes when one passes to double coset varieties $\dcosets{F}{G}{H}$. In this paper we consider the case of $G$ classical and $H$ connected spherical and prove that either the double coset variety $\dcosets{T}{G}{H}$ is singular, or it is an affine space. We also list all pairs $H \subset G$ such that $\dcosets{T}{G}{H}$ is an affine space.

Keywords: Double coset varieties.

MSC: 14L30,14M17

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