Associated varieties of integral minimal highest weight modules

TL;DR

This paper characterizes the associated varieties of integral minimal highest weight modules over complex simple Lie algebras, showing they are irreducible and correspond to specific orbital varieties.

Contribution

It establishes a precise description of the associated varieties for integral minimal modules, linking them to minimal length elements in Kazhdan--Lusztig right cells.

Findings

Associated variety of any integral minimal module is irreducible.

Associated variety equals the orbital variety of the minimal length element in a Kazhdan--Lusztig right cell.

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $L(\lambda)$ be a highest weight module of $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. A simple $\mathfrak{g}$-module $L_w:=L(-w\rho)$ is called integral minimal if the associated variety of its annihilator ideal equals the closure of the minimal special nilpotent orbit. In this paper, we find that the associated variety of any integral minimal module $L_w$ is irreducible and equal to the orbital variety corresponding to the minimal length element in the Kazhdan--Lusztig right cell containing $w$.