## Classification of smooth affine spherical varieties.(English)Zbl 1120.14042

A spherical variety is a normal variety $$X$$ acted on by a reductive group $$G$$ in such a way that a Borel subgroup $$B\subseteq G$$ has an open orbit in $$X$$. The paper deals with the classification of affine smooth spherical varieties. The latter serve as local models for multiplicity free Hamiltonian actions of a maximal compact subgroup $$K\subset G$$ [M. Brion, in: Sémin. d’algébre P. Dubreil et. M.-P. Malliavin, Proc., Paris 1986, Lect. Notes Math. 1296, 177–192 (1987; Zbl 0667.58012)], which justifies the importance of the problem.
The first step is to apply Luna’s slice theorem and deduce that $$X\simeq G\times^HV$$ is an equivariant vector bundle over a spherical homogeneous space $$G/H$$ with reductive stabilizer $$H$$ whose fiber $$V$$ is a spherical module for a certain reductive subgroup $$L\subset H$$. The latter subgroup is the image in $$H$$ of the generic isotropy subgroup for the $$G$$-action on the cotangent bundle of $$G/\widetilde{H}$$, where $$\widetilde{H}$$ is the maximal central extension of $$H$$ in $$G$$. Making use of that, the classification is derived from the classification of affine spherical homogeneous spaces [M. Krämer, Compos. Math. 38, 129–153 (1979; Zbl 0402.22006), M. Brion, Compos. Math. 63, 189–208 (1987; Zbl 0642.14011), I. V. Mikityuk, Math. USSR, Sb. 57, 527–546 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 4, 514–534 (1986; Zbl 0652.70012)] and spherical modules [V. G. Kac, J. Algebra 64, 190–213 (1980; Zbl 0431.17007), M. Brion, Ann. Sci. Éc. Norm. Supér. (4) 18, 345–387 (1985; Zbl 0588.22010), C. Benson and G. Ratcliff, J. Algebra 181, No. 1, 152–186 (1996; Zbl 0869.14021), A. S. Leahy, J. Lie Theory 8, No. 2, 367–391 (1998; Zbl 0910.22015)]. In fact, the classification is obtained only up to coverings, central tori, and $$\mathbb C^{\times}$$-fibrations, due to some problems in the case of disconnected $$H$$ and of $$G$$ or $$H$$ containing torus factors.

### MSC:

 14M17 Homogeneous spaces and generalizations 14L30 Group actions on varieties or schemes (quotients)
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