Twisted endoscopy and the generic packet conjecture
type: Item
instance of: scholarly article
title: Twisted endoscopy and the generic packet conjecture
zbMATH Open document ID: 1004.22008
DOI: 10.1007/BF02773167
author: Takuya Konno
published in: Israel Journal of Mathematics
publication date: 2003-02-03T00:00:00Z
review text: Let \(F\) be a local field and let \(G\) be a connected reductive quasi-split group over \(F\). Conjecturally the set of the isomorphism classes of irreducible tempered representations of \(G(F)\) should be partitioned into a disjoint union of finite subsets, the \(L\)-packets, parametrized by the Langlands parameters for \(G\). In this situation one has the (local) generic packet conjecture: every tempered \(L\)-packet contains a generic element. A second conjecture of great importance concerns the existence and properties of the twisted endoscopic lifting for \(GL(n)\) with respect to an outer automorphism. This should give liftings of tempered \(L\)-packets for many classical groups to irreducible tempered representations of \(GL(n,F)\). In this paper the author supposes expected properties of the twisted endoscopic lifting (in particular, the matching of orbital integrals to stable orbital integrals up to a transfer factor), and then deduces the generic packet conjecture for the classical groups which lift to \(GL(n)\). In order to accomplish this, the author proves a twisted analogue of the results by \textit{F. Rodier} [in: Non-commut. harmon. Anal., Actes Colloq. Marseille-Luminy 1974, Lect. Notes Math. 466, 151-171 (1975; Zbl 0339.22014)] and \textit{C. Mœglin} and \textit{J. L. Waldspurger} [Math. Z. 196, 427-452 (1987; Zbl 0612.22008)] on the dimension of the space of degenerate Whittaker vectors. In an appendix, the author gives a classification of the isomorphism classes of elliptic twisted endoscopic data for \(GL(n)\) over a local or global field of characteristic zero. Because our knowledge of twisted endoscopy is far from complete, the hypotheses allow one to deduce the generic packet conjecture only in the case of \(U(3)\) at the moment. As it happens, a different method based on the relative trace formula gives both the local and global generic packet conjecture for that case [\textit{S. Friedberg, S. Gelbart, H. Jacquet} and \textit{J. Rogawski}, …
reviewed by: Solomon Friedberg
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 1760719
zbMATH Keywords: \(L\)-packet
zbMATH Keywords: generic packet conjecture
zbMATH Keywords: twisted endoscopic lifting
Wikidata QID: Q123200039
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1760719