A Pieri rule for Hermitian symmetric pairs I

Thomas J. Enright, Markus Hunziker and Nolan R. Wallach

Vol. 214 (2004), No. 1, 23–30

DOI: 10.2140/pjm.2004.214.23

Abstract

Let (G,K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g = k ⊕ p⁺ ⊕ p⁻ be the usual decomposition of g as a k-module. K acts on the symmetric algebra S(p⁻). We determine the K-structure of all K-stable ideals of the algebra. Our results resemble the Pieri rule for Young diagrams. The result implies a branching rule for a class of finite dimensional representations that appear in the work of Enright and Willenbring (preprint, 2001) and Enright and Hunziker (preprint, 2002) on Hilbert series for unitarizable highest weight modules.

Milestones

Received: 11 April 2003

Published: 1 March 2004