On the annihilator variety of a highest weight Harish-Chandra module

Let $G$ be a Hermitian type Lie group with maximal compact subgroup $K$. Let $L(\lambda )$ be a highest weight Harish-Chandra module of $G$ with the infinitesimal character $\lambda$. By using some combinatorial algorithm, we obtain a description of the annihilator variety of $L(\lambda )$. As an application, when $L(\lambda )$ is unitarizable, we prove that the Gelfand-Kirillov dimension of $L(\lambda )$ only depends on the value of $z=(\lambda ,{\beta }^{\vee })$, where $\beta$ is the highest root.