Gelfand-Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $SU(p,q)$

Abstract

Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ \lambda $ be an integral weight such that the simple highest weight module $ L(\lambda) $ is a Harish-Chandra $ (\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(\lambda) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(\lambda) $ decreases as the integer $ \langle\lambda+\rho,\beta^\vee\rangle $ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety of $ L(\lambda) $.