Reductions for branching coefficients

TL;DR

This paper studies the cone of dominant character pairs related to the branching problem in reductive groups, showing that regular faces lead to reduction rules for multiplicities, generalizing previous results.

Contribution

It establishes that each regular face of the branching cone induces a reduction rule for multiplicities, extending prior work with new methods.

Findings

Regular faces of the cone yield reduction rules.

Multiplicities can be computed via Levi subgroup representations.

Generalizes previous results by Brion, Derksen-Weyman, Roth.

Abstract

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. We are interested in the branching problem. Fix maximal tori and Borel subgroups of $G$ and $\hat G$. Consider the cone $lr(G,\hat G)$ generated by the pairs $(\nu,\hat nu)$ of dominant characters such that $V_\nu^*$ is a submodule of $V_{\hat nu}$. It is known that $lr(G,\hat G)$ is a closed convex polyhedral cone. In this work, we show that every regular face of $lr(G,\hat G)$ gives rise to a {\it reduction rule} for multiplicities. More precisely, we prove that for $(\nu,\hat nu)$ on such a face, the multiplicity of $V_\nu^*$ in $V_{\hat nu}$ equal to a similar multiplicity for representations of Levi subgroups of $G$ and $\hat G$. This generalizes, by different methods, results obtained by Brion, Derksen-Weyman, Roth...