2011/10/28 by Benjamin Schwarz, Schwarz, Benjamin, Henrik Seppänen +1
Mathematics · #17C50 #22E46 #32L05 #32M15 #53D20 #Complex Variables (math.CV) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.CV #math.RT #math.SG #msc:17C50 #msc:22E46 #msc:32L05 #msc:32M15 #msc:53D20
paper · pdf · doi:10.48550/arxiv.1110.6324
arxiv created 2011/10/28 · arxiv updated 2011/10/31
Let G be a complex simple Lie group, and let U ⊆ 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 \mathscrL → 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 (H0(X, \mathscrLk))k ∈ \N of irreducible G-representations. We describe explicitly the moment polytopes for the moment maps X → \fk^* 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^\C-orbits on X, to find the decompositions of the spaces H0(X,\mathscrLk). We also construct a natural Okounkov body for \mathscrL 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.