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

TL;DR

This paper develops a combinatorial algorithm to compute the Gelfand-Kirillov dimension of highest weight Harish-Chandra modules for $SU(p,q)$, revealing how the dimension varies with the weight and describing the associated variety.

Contribution

It introduces a novel combinatorial method for calculating Gelfand-Kirillov dimensions of these modules and analyzes their behavior relative to weight parameters.

Findings

Gelfand-Kirillov dimension decreases as $raket{lambda+ ho,eta^} $ increases.

Provides a description of the associated variety of the modules.

Establishes a monotonic relationship between weight parameters and Gelfand-Kirillov dimension.

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) $.