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Spherical actions on flag varieties. (English. Russian original) Zbl 1327.14217
Sb. Math. 205, No. 9, 1223-1263 (2014); translation from Mat. Sb. 205, No. 9, 3-48 (2014).
Let \(G\) be a connected reductive linear algebraic group over an algebraically closed field of characteristic 0, and let \(X\) be a generalized flag variety of \(G\).
The authors address the problem of classifying the connected reductive subgroups \(K\) of \(G\) acting spherically on \(X\), that is, such that \(X\) admits an open orbit for a Borel subgroup of \(K\).
This classification is already known in some special cases. Here they complete the classification when \(G\) is the general linear group.
The key idea of the paper is the following. Let \(X=G/P\). The action of \(P\) on its unipotent radical has an open orbit, the Richardson orbit. Let \(O_X\) be the corresponding adjoint nilpotent \(G\)-orbit, with its symplectic structure. Then \(X\) is \(K\)-spherical if and only if the action of \(K\) on \(O_X\) is coisotropic. Furthermore, if \(X\) and \(Y\) are two generalized flag varieties of \(G\) with \(O_X\subset\overline{O_Y}\) and \(X\) is \(K\)-spherical, then \(Y\) is \(K\)-spherical as well.
Reviewer: Paolo Bravi (Roma)

14M15 Grassmannians, Schubert varieties, flag manifolds
14M27 Compactifications; symmetric and spherical varieties
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