Branching rules related to spherical actions on flag varieties (joint work with A. Petukhov)

Let $G$ be a simply connected semisimple algebraic group, $H$ a connected subgroup of it, and $X = G/P$ a (generalized) flag variety. According to a result of E. B. Vinberg and B. N. Kimelfeld from 1978, the following conditions are equivalent:
(1) the natural action of $H$ on $X$ is spherical, that is, a Borel subgroup of $H$ has an open orbit in $X$;
(2) for every irreducible representation of $G$ realized in the space of sections of a homogeneous line bundle on $X$, the restriction to $H$ is multiplicity free.
Under conditions (1) and (2), the restrictions to $H$ of all possible irreducible representations of $G$ realized in spaces of sections of homogeneous line bundles on $X$ are described by a certain free semigroup of finite rank, called the branching semigroup. In this connection, a natural problem is to compute this semigroup for all spherical actions on flag varieties.
In the talk, we shall discuss general methods for solving the above problem and demonstrate how these methods apply to computing the branching semigroups for all known spherical actions on flag varieties (for which so far there is no complete classification).