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

Let (G,K) be a Hermitian symmetric pair and let and denote the corresponding complexified Lie algebras. Let be the usual decomposition of as a -module. There is a natural correspondence between -orbits in and a distinguished family of unitarizable highest weight modules for called the Wallach representations. We denote by Yk the closure of the -orbit in 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.