Abstract:
Let $G$ be a connected reductive group and let $B$ a Borel subgroup of $G$. It is well known that a subgroup $H \subset G$ is spherical if and only if the homogeneous space $G/H$ contains finitely many $B$-orbits (or equivalently, the flag variety $G/B$ contains finitely many $H$-orbits). In this situation, a problem of interest is to classify all $B$-orbits in $G/H$ in combinatorial terms. Until recently, such a classification existed only for two classes of spherical subgroups: parabolic and symmetric. In the talk we shall present a combinatorial description of all $B$-orbits in $G/H$ for one more wide class of spherical subgroups, namely, solvable. The main ingredients for this description are:
1) a theory of actions of one-dimensional unipotent groups on toric varieties, actively developed in the recent years;
2) a structure theory of connected solvable spherical subgroups developed by the speaker several years ago.
As an application, we shall describe the action of the Weyl group on the set of all $B$-orbits in $G/H$ that was defined by F. Knop. Besides, we plan to discuss a combinatorial model for this action in terms of weight polytopes.
The above-mentioned results naturally generalize those of D. A. Timashev obtained in 1994 for the particular case $H = TU'$, where $U$ is the unipotent radical of $B$, $U'$ is the derived subgroup of $U$, and $T$ is a maximal torus of $B$.
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