On the cells and associated varieties of highest weight Harish-Chandra modules
Zhanqiang Bai, Yixin Bao, Zhao Liang, Xun Xie
TL;DR
This work characterizes highest weight Harish-Chandra modules for Hermitian-type Lie groups through their associated varieties and Kazhdan–Lusztig right cells. It proves the existence of a unique Harish-Chandra right cell supporting modules with a given associated variety and provides explicit criteria, via special elements w_λ and minimal antidominant shifts, for when L_w is a Harish-Chandra module and what its associated variety is. The results are worked out in detail across classical types A, B, C, D and in selected exceptional types, including precise w_λ expressions, translation-functor behavior, and counts of Harish-Chandra cells; the analysis leverages Robinson–Schensted combinatorics, Wallach representations, and PyCox enumerations. Together, these findings connect the geometry of orbital varieties with representation-theoretic cell decompositions, delivering a uniform framework to compute and compare associated varieties in a broad landscape of Hermitian groups.
Abstract
Let $G$ be a Hermitian type Lie group with the complexified Lie algebra $\mathfrak{g}$. We use $L(λ)$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $λ$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-wρ-ρ$. 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 $L(λ)$ will be characterized by the information of the Kazhdan--Lusztig right cell containing some special $w_λ$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.