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

✍ Scribed by Thomas J. Enright; Markus Hunziker

• Book ID: 104140661
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 340 KB
• Volume: 273
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00159-5

✦ Synopsis

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.