Associated varieties of minimal highest weight modules

TL;DR

This paper classifies minimal highest weight modules for complex simple Lie algebras and determines their associated varieties, extending previous work and identifying all weak quantizations of minimal orbital varieties.

Contribution

It extends Joseph's classification to include all minimal highest weight modules and explicitly determines their associated varieties.

Findings

Classified all minimal highest weight modules for simple Lie algebras.

Determined the associated varieties of these modules.

Identified all weak quantizations of minimal orbital varieties.

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra. A simple $\mathfrak{g}$-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of this paper is a classification of minimal highest weight modules for $\\mathfrak{g}$. This classification extends the work of Joseph, which focused on categorizing minimal highest weight modules annihilated by completely prime ideals. Furthermore, we have determined the associated varieties of these modules. In other words, we have identified all possible weak quantizations of minimal orbital varieties.