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Some qualitative properties of branching multiplicities

Let G be a connected reductive subgroup of a complex connected reductive groupĜ. We consider the multiplicities c G,Ĝ as a function from the set of pairs of dominant weights to the set of integers. We recall that this function is piecewise quasipolynomial. Its support is a finitely generated semigroup ; we describe an irredundant list of inequalities determining the cone generated. The relation with the projection of coadjoint orbits for the Lie algebras of the compact forms of G andĜ is also recalled. We also consider the multiplicities for the fusion products for G. More precisely, we explain how the small quantum cohomology rings of homogeneous spaces G/P allow to parametrize an irredundant set of inequalities determining the multiplicative eigenvalue problem for the compact form K of G.