On descent and the generic packet conjecture
type: Item
instance of: scholarly article
title: On descent and the generic packet conjecture
zbMATH Open document ID: 1430.22013
author: Sandeep Varma
published in: Forum Mathematicum
publication date: 2019-02-01T00:00:00Z
review text: For a quasi-split connected reductive group \( G \) be over a \( p \)-adic field \( F \), the local Langlands conjectures postulate that the set of isomorphism classes of irreducible tempered representations of \( G(F) \) can be partitioned into finite subsets, called tempered \( L \)-packets, in a certain natural way. The generic packet conjecture (also called tempered packet conjecture in the literature), due to \textit{F. Shahidi} [Ann. Math. (2) 132, No. 2, 273--330 (1990; Zbl 0780.22005)], says that every such tempered \( L \)-packet should contain a generic representation. In other words, suppose \(\Pi \) is a tempered \(L\)-packet of representations of \( G(F) \), \( B=TU \) a Levi decomposition of a Borel subgroup \( B \) of \( G \), and \(\lambda \) a character of \( U(F) \) which is nondegenerate, i.e., whose stabilizer in \( T(F) \) is the center of \( G(F) \). Then there should exist at least one representation \((\pi,V) \) in \(\Pi \) which is \( \lambda \)-generic, i.e., has a nonzero space of \( (U(F),\lambda) \)-coinvariants. A stronger form of this conjecture says that (for any choice of \( (U(F), \lambda) \) as above) every tempered \( L \)-packet contains precisely one \(\lambda\)-generic representation. This stronger form is known to be true for the general linear group \( \mathrm{GL}_n \) over \( F \) (and hence also for \( \mathrm{SL}_n \)). The weaker form of this conjecture for quasi-split symplectic, special orthogonal and unitary groups over \( F \) is addressed by \textit{T. Konno} [Isr. J. Math. 129, 253--289 (2002; Zbl 1004.22008)], assuming the existence of endoscopic character identities, recently established by \textit{J. Arthur} [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)] and \textit{C. P. Mok} [Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AM…
reviewed by: Ramdin Mawia
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 7010461
zbMATH Keywords: twisted endoscopy
zbMATH Keywords: Whittaker models
zbMATH Keywords: generic packet conjecture
MaRDI profile type: Publication
full work available at URL: forum-2015-0113
OpenAlex ID: W1536069978
Wikidata QID: Q123114439
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DOI: 10.1515/FORUM-2015-0113
description: scientific article; zbMATH DE number 7010461