On the set of orbits of a spherical subgroup on the flag variety

Let G be a complex reductive group and B a Borel subgroup of G, a subgroup H of G is called spherical if it acts with finitely many orbits on the flag variety G/B. Even though spherical subgroups are classified, the corresponding decompositions of G/B are not yet well understood in general. In this seminar I will consider two special cases, namely that of a solvable spherical subgroup and that of the Levi factor of a parabolic subgroup with abelian unipotent radical. In these cases, I will explain how to attach a root system to every H-orbit in G/B, and how these root systems allow to parametrize the H-orbits in G/B. These parametrizations are compatible with a general action of the Weyl group of G that Knop defined on the set of H-orbits in G/B, and I will explain how to recover the Weyl group action from the parametrization of the orbits. The talk is based on two works in collaboration, with Andrea Maffei and with Guido Pezzini."