A Pieri rule for Hermitian symmetric pairs I

Abstract

Let X be an irreducible Hermitian symmetric space of non-compact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m 1 ...,m r ) with m 1 >... > m r > 0. That is, the same parameters as the irreducible polynomial representations of GL(r). Let S[m 1 ,...,m r ] be the corresponding isotypic component. In this article we show that the product in S, S[m 1 ,...,m r ]S[k,0,0,...,0] is a direct sum of constituents following precisely the classical Pieri rule.