Orbits of strongly solvable spherical subgroups on the flag variety

TL;DR

This paper classifies the orbits of certain strongly solvable subgroups on the flag variety of a complex reductive group, using combinatorial invariants and models related to Weyl group actions.

Contribution

It provides a classification of H-orbits on G/B for strongly solvable subgroups H and describes the Weyl group action using combinatorial models.

Findings

Classification of H-orbits on G/B using combinatorial invariants

Description of Weyl group action on H-orbits via weight polytopes

Development of a combinatorial model for the Weyl group action

Abstract

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H of B acting with finitely many orbits on the flag variety G/B, and we classify the H-orbits in G/B in terms of suitable combinatorial invariants. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G/B, and we give a combinatorial model for this action in terms of weight polytopes.