The subrepresentation theorem for automorphic representations
type: Item
instance of: scholarly article
title: The subrepresentation theorem for automorphic representations
zbMATH Open document ID: 1384.11071
author: Marcela Hanzer
published in: Pacific Journal of Mathematics
publication date: 2013-05-29T00:00:00Z
full work available at URL: p06.xhtml
review text: The purpose of this short note is to provide the proof of the analogue of the Casselman subrepresentation theorem in the setting of automorphic representations. The subrepresentation theorem of \textit{W. Casselman} [in: Proc. Int. Congr. Math., Helsinki 1978, Vol. 2, 557--563 (1980; Zbl 0425.22019)] is one of the most basic theorems in representation theory of reductive groups over local fields, saying that any irreducible representation (in the appropriate category) is a subrepresentation of certain parabolically induced representation from a simpler representation of an appropriate subgroup. It is a refinement of the subquotient theorem of \textit{Harish-Chandra} [Trans. Am. Math. Soc. 76, 26--65, 234--253 (1954; Zbl 0055.34002)], which claims that any irreducible representation is a subquotient of such induced representation. In the setting of automorphic representations, the subquotient theorem is due to \textit{R. P. Langlands} [Proc. Symp. Pure Math. 33, 1, 203--207 (1979; Zbl 0414.22021)]. It says that, for a connected reductive group \(G\) defined over an algebraic number field \(k\), any irreducible automorphic representation of \(G(\mathbb{A})\) in the sense of \textit{A. Borel} and \textit{H. Jacquet} [Proc. Symp. Pure Math. 33, 1, 189--202 (1979; Zbl 0414.22020)], where \(\mathbb{A}\) is the ring of adèles of \(k\), can be realized as a subquotient of a parabolically induced representation from a cuspidal automorphic representation of an appropriate subgroup. The paper under review refines this result of Langlands. It shows that any irreducible subrepresentation of the space of automorphic forms on \(G(\mathbb{A})\) may be realized as a subrepresentation of a parabolically induced representation from a cuspidal automorphic representation of an appropriate subgroup. The proof is a slight refinement of Langlands' original proof in [loc. cit.], combined with the spectral decomposition results in [\textit{C. Mœglin} and \textit{J.-L. Waldspurger}, Spectral …
reviewed by: Neven Grbac
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 6168050
zbMATH Keywords: automorphic representation
zbMATH Keywords: subrepresentation theorem
zbMATH Keywords: cuspidal representation
MaRDI profile type: Publication
OpenAlex ID: W2033880895
DOI: 10.2140/PJM.2013.261.389
description: scientific article; zbMATH DE number 6168050