On the cells and associated varieties of highest weight Harish-Chandra modules

Let $G$ be a Hermitian type Lie group with the complexified Lie algebra $\mathfrak{g}$. We use $L(\lambda)$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $\lambda$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-w\rho-\rho$. In this paper we prove that there is only one Kazhdan--Lusztig right cell such that the corresponding highest weight Harish-Chandra modules $L_w$ have the same associated variety. Then we give a characterization for those $w$ such that $L_w$ is a highest weight Harish-Chandra module and the associated variety of $L(\lambda)$ will be characterized by the information of the Kazhdan--Lusztig right cell containing some special $w_{\lambda}$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.