Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules

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. There is a natural correspondence between K C -orbits in p + and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Y k the closure of the K C -orbit in p + that is associated to the kth Wallach representation. In this article we give explicit formulas for the numerator polynomials of the Hilbert series of the varieties Y k by using BGG resolutions of unitarizable highest weight modules. A preliminary result gives a new branching formula for a certain two-parameter family of finite dimensional representations of the even orthogonal groups. Our work is an extension of previous work by Enright and Willenbring.