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

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 $(G_0, G)$.