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 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.