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

AbstractLet (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 KC-orbits in p+ and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Yk the closure of the KC-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 Yk 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