Geodesic
Spherical Subgroups and Double Coset Varieties
Journal of Lie theory, Tome 22 (2012) no. 2, pp. 505-522.

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\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.
Classification : 14L30,14M17
Mots-clés : Double coset varieties 
@article{JLT_2012_22_2_JLT_2012_22_2_a8,
     author = {A. Anisimov },
     title = {Spherical {Subgroups} and {Double} {Coset} {Varieties}},
     journal = {Journal of Lie theory},
     pages = {505--522},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JLT_2012_22_2_JLT_2012_22_2_a8/}
}
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A. Anisimov . Spherical Subgroups and Double Coset Varieties. Journal of Lie theory, Tome 22 (2012) no. 2, pp. 505-522. http://geodesic.mathdoc.fr/item/JLT_2012_22_2_JLT_2012_22_2_a8/