Symplectic branching laws and Hermitian symmetric spaces

Abstract:

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 $X=G/Q=U/K$ which is a compact Hermitian symmetric space. Let $\mathscr {L} \rightarrow X$ be the ample line bundle which generates the Picard group of $X$. In this paper we study the restrictions to $K$ of the family $(H^0(X, \mathscr {L}^k))_{k \in \mathbb {N}}$ of irreducible $G$-representations. We explicitly describe the moment polytopes for the moment maps $X \rightarrow \mathfrak {k}^*$ 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 $K^{\mathbb {C}}$-orbits on $X$, to find the decompositions of the spaces $H^0(X,\mathscr {L}^k)$. We also construct a natural Okounkov body for $\mathscr {L}$ and the $K$-action, and we 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.