On a conjecture of Jacquet about distinguished representations of \(\text{GL}(n)\)
type: Item
instance of: scholarly article
title: On a conjecture of Jacquet about distinguished representations of \(\text{GL}(n)\)
zbMATH Open document ID: 1019.22008
DOI: 10.1215/S0012-7094-01-10912-5
author: Dipendra Prasad
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1091737221
review text: Let \(k\) be a \(p\)-adic field, \(K\) a quadratic unramified extension of \(k\), \(\pi\) a cuspidal irreducible admissible representation of \(\text{GL}(n,K)\) whose central character is trivial on \(k^\times\), \(\check\pi\) its contragredient. Let \(\sigma\) denote the non trivial element of \(\text{Gal}(K/k)\), its componentwise action on \(\text{GL}(n,K)\), and \(^\sigma\pi:g\mapsto\pi(\sigma(g))\). Let \(\omega\) be the non trivial character on \(k^\times/N_{K/k}K^\times\). \(\pi\) is called distinguished if it admits a nonzero \(\text{GL}(n,K)\)-invariant linear form, and \(\omega\)-distinguished if there is a linear form \(L:\pi\to\mathbb C\) with \(L(\pi(g)\xi)=\omega(\text{det}(g))L(\xi)\) for all \(g\) in \(\text{GL}(n,K)\) and \(\xi\in\pi\). The main theorem (Theorem 4) asserts: \({}^\sigma\pi\simeq\check\pi\) iff \(\pi\) is distinguished, or also -- when \(n\) is even -- \(\omega\)-distinguished. The proof is based on a result of Bushnell and Kutzko which realizes any cuspidal representation of \(\text{GL}(n,K)\) by compact induction from a finite dimensional representation of an open compact-mod-center subgroup. It has the advantage of being purely local, although to extend the result to ramified quadratic extensions and to non cuspidal representations one should use the global trace formula. The result is the exact statement which according to the author was conjectured by Jacquet in [\textit{H. Jacquet} and \textit{Y. Ye}, C. R. Acad. Sci. Paris 311, 671--676 (1990; Zbl 0715.11026)]. This conjecture follows from the conjecture submitted earlier (but appeared later) in \textit{Y. Flicker} [J. Reine Angew. Math. 418, 139-172 (1991; Zbl 0725.11026)] characterizing the \(\text{GL}(n,K)\)-distinguished \(\pi\) on \(\text{GL}(n,K)\) as stable or unstable base change lifts from the quasi-split unitary group \(\text{U}(n,K/k)\), and the conjectural theory of base change for \(\text{U}(n,K/k)\). The author also refers to \textit{Y. Flicker} and \textit{J. Hak…
reviewed by: Yuval Z. Flicker
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1820833
zbMATH Keywords: distinguished representations
zbMATH Keywords: GL(n)
zbMATH Keywords: unramified quadratic extension
zbMATH Keywords: cuspidal representations
zbMATH Keywords: global trace formula
zbMATH Keywords: Steinberg representation
MaRDI profile type: Publication
Wikidata QID: Q122906468
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description: scientific article; zbMATH DE number 1820833