Spherical actions on flag varieties of the classical groups (based on joint works with A. Petukhov)

Let $G$ be a connected reductive group, $\mathfrak g$ the Lie algebra of $G$, 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 moment map coincides with the closure of a nilpotent orbit in $\mathfrak g$. This yields a map from the set $\mathscr F(G)$ 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 the set $\mathscr F(G)$. Thanks to a result of I. V. Losev, this partial order possesses the following remarkable property.
Theorem. Let $K \subset G$ be a connected reductive subgroup, $X_1, X_2 \in \mathscr F(G)$, and $X_1 \prec X_2$. If $K$ acts spherically on $X_2$ then $K$ acts spherically on $X_1$.
In the talk we shall explain how this theorem applies to classifying all spherical actions on flag varieties of classical groups. Here an important role is played by the known description of nilpotent orbits of classical Lie algebras in terms of Young diagrams.