{"title":"与刚性零势轨道相关的无多重性原始理想数","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":"{\"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}","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}
The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits
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.