{"title":"论平滑同构形式和平滑同构表示的抛物面和尖顶支持的概念","authors":"Harald Grobner, Sonja Žunar","doi":"10.1007/s00605-024-01965-z","DOIUrl":null,"url":null,"abstract":"<p>In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily <span>\\(K_\\infty \\)</span>-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.\n</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations\",\"authors\":\"Harald Grobner, Sonja Žunar\",\"doi\":\"10.1007/s00605-024-01965-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily <span>\\\\(K_\\\\infty \\\\)</span>-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.\\n</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-01965-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01965-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在本文中,我们描述了数域上一般连通还原群的光滑自形式空间(即,不一定是 \(K_\infty \)-无限自形式)的表征理论基础的几个新方面。我们对这种光滑-自变形式空间的角色模型是自变形式空间的 "光滑版本",其内部结构是弗朗克著名论文(Ann Sci de l'ENS 2:181-279, 1998)的主题。我们证明,沿着抛物线支撑的重要分解,以及沿着自形的尖顶支撑的更精细--结构上更重要--的分解,都以拓扑版本转移到了更大的光滑自形空间中。这样,我们就建立了 Franke 和 Schwermer (Math Ann 311:765-790, 1998) 以及 Mœglin 和 Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995) 第 III.2.6 节主要结果的平滑自变形式版本。
On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations
In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily \(K_\infty \)-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a “smooth version” of the space of automorphic forms, whose internal structure was the topic of Franke’s famous paper (Ann Sci de l’ENS 2:181–279, 1998). We prove that the important decomposition along the parabolic support, and the even finer—and structurally more important—decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765–790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.