Expression of an epsilon factor of pairs by an integral formula
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.22020
DOI: 10.4153/CJM-2013-042-4
author: Raphaël Beuzart-Plessis
published in: Canadian Journal of Mathematics
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1082
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 6355666
zbMATH Keywords: epsilon factors
zbMATH Keywords: twisted groups
zbMATH Keywords: p-adic reductive groups
zbMATH Keywords: representation theory
zbMATH Keywords: local Gan-Gross-Prasad conjecture
MaRDI profile type: Publication
arXiv ID: 1212.1082
OpenAlex ID: W2313977610
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title: Expression of an epsilon factor of pairs by an integral formula
review text: Let \(E/F\) be a quadratic extension of non-archimedean local fields of characteristic \(0\) and let \(\pi\) and \(\sigma\) be two tempered irreducible smooth representations of GL\((d,E)\) and GL\((m,E)\), where \(d\) and \(m\) are nonnegative integers of distinct parities. Assume moreover that \(\pi\) and \(\sigma\) are conjugate-dual, that is \(\pi^c\simeq \check{\pi}\), \(\sigma^c\simeq \check{\sigma}\), where \(c\) denotes the action of the non-trivial \(F\)-automorphism of \(E\), and a symbol \(\check{}\) denotes a contragredient representation. In particular \(\pi\) and \(\sigma\) may be extended to unitary representations \(\tilde{\pi}\) and \(\tilde{\sigma}\) of GL\((d,E)\ltimes \{ 1,c\}\) and GL\((m,E)\ltimes \{ 1,c\}\) respectively. Finally let \(\varepsilon (1/2 ,\pi\times\sigma ,\psi_E )\) be the epsilon factor of pair, for a fixed non-trivial additive character \(\psi_E\) of \(E\), as defined in [\textit{H. Jacquet} et al., Am. J. Math. 105, 367--464 (1983; Zbl 0525.22018)].NEWLINENEWLINEThe author proves an integral formula for \(\varepsilon (1/2 ,\pi\times\sigma ,\psi_E )\) involving the Harish-Chandra characters of \(\check{\pi}\) and \(\check{\sigma}\). He hopes that one should be able to apply this formula to obtain a proof of the local Gan-Gross-Prasad conjecture for unitary groups, cf. conjecture 17.3 of [\textit{W. T. Gan} et al., Sur les conjectures de Gross et Prasad. I. Paris: Société Mathématique de France (SMF) (2012; Zbl 1257.22001)]. This formula is the exact analogue of that obtained by Waldspurger for epsilon factor of pairs of auto-dual representations of general linear groups [\textit{C. Mœglin} and \textit{J.-L. Waldspurger}, Sur les conjectures de Gross et Prasad. II. Paris: Société Mathématique de France (SMF) (2012; Zbl 1257.22002)].
reviewed by: Paul Broussous
description: scientific article; zbMATH DE number 6355666