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

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 ${\mathrm{res}}_{{\mathfrak{g}}_0}V(d\rho )$, for any saturation factor $d$ of the pair $(G_0,G)$.

6 Pages