Containment Relations among Spherical Subgroups

TL;DR

This paper provides a combinatorial characterization of containment relations among spherical subgroups in reductive groups, extending previous work and enabling the computation of Luna data for connectedness analysis.

Contribution

It generalizes previous characterizations of spherical subgroup containment and introduces methods to compute Luna data for the identity component of spherical subgroups.

Findings

Provides a combinatorial criterion for subgroup containment.

Computes Luna data for the identity component of spherical subgroups.

Characterizes connectedness of spherical subgroups.

Abstract

A closed subgroup $H$ of a connected reductive group $G$ is called $\textit{spherical}$ if a Borel subgroup in $G$ has an open orbit on $G/H$. We give a combinatorial characterization for a spherical subgroup to be contained in another one which generalizes previous work by Knop. As an application, we compute the Luna datum of the identity component of a spherical subgroup which yields a characterization of connectedness for spherical subgroups.