A Pieri rule for Hermitian symmetric pairs II

Thomas J. Enright and Nolan R. Wallach

Vol. 216 (2004), No. 1, 51–61

DOI: 10.2140/pjm.2004.216.51

Abstract

Let X be an irreducible Hermitian symmetric space of noncompact 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, (m₁,…,m_r) with m₁ ≥… ≥ m_r ≥ 0. That is, the same parameters as the irreducible polynomial representations of GL(r). Let S[m₁,…,m_r] be the corresponding isotypic component. In this article we show that the product in S, S[m₁,…,m_r]S[k,0,0,…,0] is a direct sum of constituents following precisely the classical Pieri rule.

Milestones

Received: 11 April 2003

Revised: 5 November 2003

Published: 1 September 2004