  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) $.
