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 $⟨\lambda +\rho ,{\beta }^{\vee }⟩$ 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 $L(\lambda )$.

We add a footnote on page 9 in this version

Cited by in corpus (2)

• Gelfand-Kirillov dimensions and associated varieties of highest weight modules
• Associated varieties of minimal highest weight modules