The notion of norm and the representation theory of orthogonal groups
type: Item
instance of: scholarly article
title: The notion of norm and the representation theory of orthogonal groups
zbMATH Open document ID: 0852.22016
DOI: 10.1007/BF01245173
author: Freydoon Shahidi
published in: Inventiones Mathematicae
publication date: 1996-12-02T00:00:00Z
full work available at URL: https://eudml.org/doc/144247
review text: Nondiscrete tempered spectra of classical groups over local fields are studied by using the theory of endoscopy that relates them to self-dual representations of \(GL(n)\). Let \(\tau\) and \(\tau'\) be two discrete series representations of \(SO_{2m} (F)\) and \(GL_n (F)\) over a local field \(F\), respectively, and \(\tau \otimes \tau'\) the representation of the Levi factor \(SO_{2m} (F) \times GL_n (F)\) of \(SO_{2r} (F)\) with \(r = m + n\). \(I (\tau \otimes \tau')\) is then the representation of \(SO_{2r} (F)\) unitarily induced from \(\tau \otimes \tau'\). The determination of the reducibility of \(I(\tau \otimes \tau')\) is used to study the residue of the standard intertwining operator. The latter, in turn, requires the introduction of norm correspondence from a certain subset of \(\theta\)-conjugacy classes in \(GL_n (F)\) into the set of conjugacy classes in \(SO_{2m} (F)\) where \(\theta (g) = w_n^{-1} \cdot^t g^{-1} \cdot w_n\) is the automorphism of \(GL_n(F)\) defined by means of the \(n \times n\) matrix \(w_n\) whose unique nonzero elements lie on its second diagonal.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 763280
zbMATH Keywords: third order orthogonal groups
zbMATH Keywords: classical groups
zbMATH Keywords: local fields
zbMATH Keywords: theory of endoscopy
zbMATH Keywords: representations
zbMATH Keywords: Levi factor
zbMATH Keywords: intertwining operator
reviewed by: Eugene Kryachko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763280