On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations

Harald Grobner, Sonja Žunar
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Abstract

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.

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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
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。
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