Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

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 combinatorial algorithm for the Gelfand–Kirillov (GK) dimension of $L(\lambda)$. This enables us to prove that the GK 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 non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of $L(\lambda)$.