On computing the extended weight semigroups for spherical homogeneous spaces

A closed subgroup $H$ of a connected reductive algebraic group $G$ is said to be spherical if the homogeneous space $G/H$ contains an open orbit for the induced action of a Borel subgroup $B \subset G$. According to a known result of Vinberg and Kimelfeld, $H$ is spherical if and only if for every irreducible finite-dimensional representation $V$ of $G$ and every character $\chi$ of $H$ the subspace of all $H$-semiinvariant vectors of weight $\chi$ in $V$ is at most one-dimensional. All pairs $(V, \chi)$ for which the above-mentioned subspace is nontrivial (and hence precisely one-dimensional) are described by the so-called extended weight semigroup of $G/H$.
The talk is devoted to the problem of computing the extended weight semigroup for a given spherical subgroup $H \subset G$ in terms of a regular embedding of $H$ in a parabolic subgroup of $G$. The main result is an explicit procedure for computing generators of the semigroup in question provided that we know one more invariant of the spherical subgroup $H$, namely, the set of simple spherical roots. We also plan to discuss known results and open questions related to the problem of computing the simple spherical roots.