$B'$-orbits on flag varieties and symmetry breaking

TL;DR

This paper classifies pairs of groups and parabolic subgroups where Borel subgroup orbits on flag varieties are finite and describable by invariant functions, advancing understanding of symmetry breaking in representation theory.

Contribution

It provides a complete classification of such pairs and explicit descriptions of the double coset spaces and closed orbits, based on invariant functions.

Findings

Classified all pairs (G', P) with finitely many B'-orbits on G/P.

Explicitly described the double coset space B'ackslash G/P.

Identified conditions under which B'-orbits are determined by invariant functions.

Abstract

Motivated by branching problems for principal series representations of the Lie group $G = GL(n,\mathbb R)$, we consider all pairs $(G', P)$ with $G'$ being the Levy factor of a parabolic subgroup of $G$ and $P$ a parabolic subgroup of $G$ for which a Borel subgroup $B'$ of $G'$ has finitely many orbits on $G/P$. We classify all such pairs $(G',P)$ for which $B'$-orbits on the generalized flag variety $G/P$ are determined by invariant functions inspired from the Bruhat decomposition. We also describe explicitly the double coset space $B'\backslash G/P$ as well as the closed $B'$-orbits on $G/P$ whenever $B'$-orbits are computed by these invariant functions.