{"title":"论非阿基米德局部域上一般线性群内形式的乘积赋形剂","authors":"Kei Yuen Chan","doi":"10.1007/s00031-024-09861-4","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(G_n\\)</span> be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation <span>\\(\\sigma \\)</span> of <span>\\(G_n\\)</span>. Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some <span>\\(G_m\\)</span>, on which the parabolic induction functor <span>\\(\\tau \\mapsto \\tau \\times \\sigma \\)</span> is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Product Functor on Inner forms of the General Linear Group Over A Non-Archimedean Local Field\",\"authors\":\"Kei Yuen Chan\",\"doi\":\"10.1007/s00031-024-09861-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(G_n\\\\)</span> be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation <span>\\\\(\\\\sigma \\\\)</span> of <span>\\\\(G_n\\\\)</span>. Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some <span>\\\\(G_m\\\\)</span>, on which the parabolic induction functor <span>\\\\(\\\\tau \\\\mapsto \\\\tau \\\\times \\\\sigma \\\\)</span> is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00031-024-09861-4\",\"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/10.1007/s00031-024-09861-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the Product Functor on Inner forms of the General Linear Group Over A Non-Archimedean Local Field
Let \(G_n\) be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation \(\sigma \) of \(G_n\). Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some \(G_m\), on which the parabolic induction functor \(\tau \mapsto \tau \times \sigma \) is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.