Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 1

Special Issue dedicated to Corrado De Concini

Guest Editors: Alberto De Sole, Nicoletta Cantarini, and Andrea Maffei

The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits

Pages: 537 – 563

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n1.a12

Authors

Alexander Premet (Department of Mathematics, University of Manchester, United Kingdom)

David I. Stewart (Department of Mathematics, University of Manchester, United Kingdom)

Abstract

Let $G$ be a simple algebraic group defined over $\mathbb{C}$ and let $e$ be a rigid nilpotent element in $g = \operatorname{Lie} (G)$. In this paper we prove that the finite $W$-algebra $U(\mathfrak{g}, e)$ admits either one or two $1$-dimensional representations. Thanks to the results obtained earlier this boils down to showing that the finite $W$-algebras associated with the rigid nilpotent orbits of dimension 202 in the Lie algebras of type $E_8$ admit exactly two 1‑dimensional representations. As a corollary, we complete the description of the multiplicity-free primitive ideals of $U(\mathfrak{g})$ associated with the rigid nilpotent $G$-orbits of $\mathfrak{g}$. At the end of the paper, we apply our results to enumerate the small irreducible representations of the related reduced enveloping algebras.

Received 23 November 2022

Accepted 1 August 2023

Published 26 March 2024