Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients

Abstract

Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity free. If V_λ is an irreducible representation of K, let S_λ denote the corresponding isotypic component of S. Write S_λ S_μ for the subspace of S spanned by products of S_λ and S_μ. If V_ν occurs as an irreducible constituent of the tensor product of V_λ and V_μ, is it true that S_ν is contained in S_λ S_μ? We investigate this question for representations arising in the context of Hermitian symmetric pairs. We show that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. We also show how the conjecture connects multiplication in the ring S to the usual Littlewood-Richardson rule.