Branching rules related to spherical actions on flag varieties

TL;DR

This paper reviews known cases and computes new restricted branching monoids for spherical actions of reductive subgroups on flag varieties, focusing on simple groups with symmetric subgroups and special linear groups.

Contribution

It extends the understanding of restricted branching monoids by computing them for new classes of spherical actions on flag varieties.

Findings

Restricted branching monoids are computed for all spherical actions with G simple and H symmetric.

Restricted branching monoids are computed for all spherical actions with G = SL_n.

The paper consolidates known cases and provides new explicit computations.

Abstract

Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where $H$ is a Levi subgroup of $G$) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples $(G,H,X)$ satisfying one of the following two conditions: (1) $G$ is simple and $H$ is a symmetric subgroup of $G$; (2) $G = \mathrm{SL}_n$.