Symmetric pairs and branching laws

TL;DR

This paper introduces a new formula for branching coefficients in symmetric pairs of compact Lie groups, linking the orbit structure of a subgroup on a flag variety to representation theory.

Contribution

It provides a novel formula for branching coefficients of symmetric pairs, parameterized by the orbit space of the complexified subgroup on the flag variety.

Findings

The action of $H_C$ on the flag variety has finitely many orbits.

A formula for branching coefficients is derived based on orbit parametrization.

The approach connects geometric orbit data with algebraic branching laws.

Abstract

Let $G$ be a compact connected Lie group and let H be a subgroup fixed by an involution. A classical result assures that the action of the complex reductive group $H_C$ on the flag variety $F$ of $G$ admits a finite number of orbits. In this article we propose a formula for the branching coefficients of the symmetric pair $(G,H)$ that is parametrized by $H_C\backslash F$.