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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)

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