Distinguished representations of \(\mathrm{GL}(n)\) and local converse theorems
type: Item
instance of: scholarly article
title: Distinguished representations of \(\mathrm{GL}(n)\) and local converse theorems
zbMATH Open document ID: 1336.22015
author: Jeffrey Hakim
author: Omer Offen
published in: Manuscripta Mathematica
publication date: 2015-09-02T00:00:00Z
review text: Let \(E\) be a quadratic extension of a nonarchimedean local field \(F\) and \(\pi\) be an irreducible supercuspidal representation of \(\mathrm{GL}_n(E)\). Assume \(\psi\) is a nontrivial character of \(E\) whose restriction to \(F\) is trivial. The paper gives a criterion on when \(\pi\) is distinguished over \(\mathrm{GL}_n(F)\). The criterion is that the central character of \(\pi\) is trivial on \(F^\times\) and the Rankin-Selberg gamma factors \(\gamma(\frac12, \pi\times\tau,\psi)=1\) for all \(\mathrm{GL}_{n-2}(F)\) distinguished, irreducible unitary generic representation \(\tau\) of \(\mathrm{GL}_{n-2}(E)\). In the analogous situation where \(E=F\oplus F\), the above result gives a criterion on when two irreducible supercuspidal representations \(\pi_1,\pi_2\) are isomorphic, namely when the gamma factors \(\gamma(\frac12,\pi_i\times\tau,\psi)\) match. This is a (previously known) local converse theorem.
reviewed by: Zhengyu Mao
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11S37
zbMATH DE Number: 6477572
zbMATH Keywords: distinguished representation
zbMATH Keywords: converse theorem
MaRDI profile type: Publication
full work available at URL: s00229-015-0740-z
OpenAlex ID: W1991805976
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DOI: 10.1007/S00229-015-0740-Z
description: scientific article; zbMATH DE number 6477572