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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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