On the cells and associated varieties of highest weight Harish-Chandra modules (1909.00705v2)

Abstract: Let $G$ be a Hermitian type Lie group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. 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 will 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.