{"title":"The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits","authors":"Alexander Premet, David I. Stewart","doi":"10.4310/pamq.2024.v20.n1.a12","DOIUrl":null,"url":null,"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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n1.a12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.