Date:	Fri, November 14, 2014
Time:	11:15
Place:	Seminar Room (120), Research I

Abstract: Let \(G\) be a connected reductive group, \(\mathfrak{g}\) its Lie algebra, and \(X = G/P\) a generalized flag variety. It is known that for the natural action of \(G\) on the cotangent bundle \(T^*X\) the image of the corresponding moment map \(T^*X \to \mathfrak{g}\) coincides with the closure of a nilpotent orbit in \(\mathfrak{g}\). Thus there is a map from the set \(F(G)\) of all generalized flag varieties of \(G\) to the set of nilpotent orbits in \(\mathfrak{g}\), and the inclusion relation between closures of nilpotent orbits induces a partial order on \(F(G)\). Thanks to a result of I. V. Losev, this partial order possesses the following remarkable property.
Theorem. Let \(X_1, X_2 \in F(G)\) be such that \(X_1 < X_2\) and let \(K \subset G\) be a connected reductive subgroup. If \(K\) acts spherically on \(X_2\), then \(K\) acts spherically on \(X_1\).
The goal of the talk is to explain how this theorem is applied for classifying all spherical actions on generalized flag varieties, with emphasis on the case \(G = GL_n\).
The talk is based on joint works with A. V. Petukhov.