Associated varieties of minimal highest weight modules

Bai, Zhanqiang; Ma, Jia-Jun; Xiao, Wei; Xie, Xun

doi:10.1090/ert/681

Associated varieties of minimal highest weight modules
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by Zhanqiang Bai, Jia-Jun Ma, Wei Xiao and Xun Xie;

Represent. Theory 28 (2024), 498-513

DOI: https://doi.org/10.1090/ert/681

Published electronically: November 12, 2024

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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 [Ann. Sci. École Norm. Sup. (4) 31 (1998), 17–45], 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.

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