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

Let (G,K) be an irreducible Hermitian symmetric pair of non-compact type with G=SU(p,q), and let λ be an integral weight such that the simple highest weight module L(λ) is a Harish-Chandra (g,K-module. We give a combinatorial algorithm for the Gelfand-Kirillov (GK) dimension of L(λ). This enables us to prove that the GK dimension of L(λ) decreases as the integer {λ+ρ,βvee increases, where ρ is the half sum of positive roots and β is the maximal non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of L(λ).