The residual spectrum of the group of type \(G_ 2\)
type: Item
instance of: scholarly article
title: The residual spectrum of the group of type \(G_ 2\)
zbMATH Open document ID: 0886.11030
DOI: 10.1016/S0021-7824(97)89971-0
author: Siniša Žampera
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1997-09-15T00:00:00Z
review text: Let \(G\) be the split group of type \(G_2\) defined over a number field \(k\) and \(G({\mathcal A})\) its adele group. Let \(L_d^2\) be the discrete part of the space of square-integrable complex valued functions on \(G(k)\setminus G({\mathcal A})\), i.e. the direct sum of irreducible \(G({\mathcal A})\)-submodules. The author describes this space in terms of cuspidal representations of the standard Levi factors following Langlands' method. The most surprising result concerns the part of the discrete spectrum coming from a Borel subgroup; in addition to the trivial representation (the constants) and the representations attached to the grössencharacters of order 2 and 3, it contains a series of representations arising from the trivial character of the maximal torus. This series consists of irreducible quotients of an induced representation satisfying a strange condition on the cardinality of the ramified components \((|S|\neq 1)\). A part of this result is proved by C. Mœglin and J.-L. Waldspurger.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 1060802
zbMATH Keywords: residual spectrum
zbMATH Keywords: cuspidal representations
zbMATH Keywords: standard Levi factors
MaRDI profile type: Publication
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Wikidata QID: Q127792875
description: scientific article; zbMATH DE number 1060802