A note on branching of V(ρ)

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