{"title":"The dimension formula for certain twisted Jacquet modules of a cuspidal representation of GL(n,Fq)","authors":"Kumar Balasubramanian , Himanshi Khurana","doi":"10.1016/j.laa.2025.01.027","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> be a positive integer. Let <em>F</em> be the finite field of order <em>q</em> and <span><math><mi>G</mi><mo>=</mo><mi>GL</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>. Let <span><math><mi>P</mi><mo>=</mo><mi>M</mi><mi>N</mi></math></span> be the standard parabolic subgroup of <em>G</em> corresponding to the partition <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></math></span>. Let <span><math><mi>A</mi><mo>∈</mo><mi>M</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo><mo>×</mo><mi>k</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> be a rank <em>t</em> matrix. In this paper, we compute the dimension formula for the twisted Jacquet module <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></msub></math></span> that depends on <span><math><mi>n</mi><mo>,</mo><mi>k</mi></math></span> and <em>t</em>, when <em>π</em> is an irreducible cuspidal representation of <em>G</em> and <span><math><msub><mrow><mi>ψ</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is a character of <em>N</em> associated with <em>A</em>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"710 ","pages":"Pages 151-164"},"PeriodicalIF":1.0000,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525000278","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let be a positive integer. Let F be the finite field of order q and . Let be the standard parabolic subgroup of G corresponding to the partition . Let be a rank t matrix. In this paper, we compute the dimension formula for the twisted Jacquet module that depends on and t, when π is an irreducible cuspidal representation of G and is a character of N associated with A.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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