{"title":"A note on the Loewy lengths of baby Verma modules for modular Lie algebras","authors":"Yiyang Li , Bin Shu , Yufeng Yao","doi":"10.1016/j.jalgebra.2024.09.023","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>G</em> be a connected reductive algebraic group over an algebraically closed field <strong>k</strong> of prime characteristic <em>p</em>, and <span><math><mi>g</mi><mo>=</mo><mtext>Lie</mtext><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We study the representations of the reductive Lie algebra <span><math><mi>g</mi></math></span> with <em>p</em>-character <em>χ</em> of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. <span><span>[10, II.Lem.7.20]</span></span>). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"663 ","pages":"Pages 913-932"},"PeriodicalIF":0.8000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324005295","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and . We study the representations of the reductive Lie algebra with p-character χ of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. [10, II.Lem.7.20]). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.
设 G 是在素特性 p 的代数闭域 k 上的连通还原代数群,且 g=Lie(G) 。在本注释中,我们将研究具有标准列维形式 p 字符 χ 的还原性列代数 g 的表示。我们得到了与代数群表示平行的简单模块的平移函子和壁交函子的类似结果(参见 [10, II.Lem.7.20])。此外,只要沃根猜想成立,我们就能得到韦尔马婴孩模块的洛维长度。
期刊介绍:
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.