A note on branching of $V(\rho)$

TL;DR

This paper establishes a precise criterion for when certain highest weight representations of a subalgebra appear in the restriction of a representation of a larger Lie algebra, extending understanding of branching rules in representation theory.

Contribution

It provides a necessary and sufficient condition for the occurrence of specific subalgebra representations within restricted representations for any saturation factor.

Findings

Characterization of branching of $V( ho)$ in terms of highest weights.

Applicable to all saturation factors $d$ of the pair $(G_0, G)$.

Enhances understanding of representation restrictions in Lie theory.

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and let $\mathfrak{g}_0$ be the sub-algebra fixed by a diagram automorphism of $\mathfrak{g}$. Let $G$ be the complex, simply-connected, simple algebraic group with Lie algebra $\mathfrak{g}$, and let $G_0$ be the connected subgroup of $G$ with Lie algebra $\mathfrak{g}_0$. Let $\rho$ be the half sum of positive roots of $\mathfrak{g}$. In this article, we give a necessary and sufficient condition for a highest weight $\mathfrak{g}_0$-representation $V_0(d\mu)$ to occur in the representation ${\rm res}_{\mathfrak{g}_0}V(d\rho)$, for any saturation factor $d$ of the pair $(G_0, G)$.