Symplectic branching laws and Hermitian symmetric spaces

Let $G$ be a complex simple Lie group, and let $U\subseteq G$ be a maximal compact subgroup. Assume that $G$ admits a homogenous space $X=G\mathrm{/}Q=U\mathrm{/}K$ which is a compact Hermitian symmetric space. Let $\mathcal{L}\to X$ be the ample line bundle which generates the Picard group of $X$. In this paper we study the restrictions to $K$ of the family $(H^0(X, \mathscr{L}^k))_{k \in \N}$ofirreducible$G$-representations. We describe explicitly the moment polytopes for the moment maps $X\to {\text{fk}}^{\ast }$ associated to positive integer multiples of the Kostant-Kirillov symplectic form on $X$, and we use these, together with an explicit characterization of the closed $K^{\text{C}}$-orbits on $X$, to find the decompositions of the spaces $H^0(X,{\mathcal{L}}^k)$. We also construct a natural Okounkov body for $\mathcal{L}$ and the $K$-action, and identify it with the smallest of the moment polytopes above. In particular, the Okounkov body is a convex polytope. In fact, we even prove the stronger property that the semigroup defining the Okounkov body is finitely generated.