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 $(G_0, G)$.
Categories: math.RT math.AG