Characterization of the local Langlands conjecture by \(\varepsilon\)-factors for pairs
type: Item
instance of: scholarly article
title: Characterization of the local Langlands conjecture by \(\varepsilon\)-factors for pairs
zbMATH Open document ID: 0810.11069
DOI: 10.1007/BF01244309
author: Guy Henniart
published in: Inventiones Mathematicae
publication date: 1995-04-11T00:00:00Z
full work available at URL: https://eudml.org/doc/144129
review text: Let \(F\) be a non-archimedean local field with finite residue field. In the equal characteristic case \textit{G. Laumon}, \textit{M. Rapoport} and \textit{U. Stuhler} [\(D\)-elliptic sheaves and the Langlands correspondence, Invent. Math. 113, 217-338 (1993; Zbl 0809.11032), Theorem 15.7] construct bijections \({\mathcal A}_ F^ 0(n) \leftrightarrow {\mathcal G}_ F^ 0 (n)\), \(\pi\mapsto \sigma_ \pi\) between irreducible supercuspidal representations of \(\text{GL}_ n(F)\) with finite central character and \(n\)-dimensional irreducible representations of the Galois group \(G_ F= \text{Gal} (\overline{F}/ F)\) resp. for all \(n\geq 1\). The main properties of these bijections are \[ L(\pi\times \pi',s)= L(\sigma_ \pi\otimes \sigma_{\pi'},s), \qquad \varepsilon(\pi\times \pi',s,\psi)= \varepsilon(\sigma_ \pi\otimes \sigma_{\pi'}, s,\psi), \] means that \(L\)- and \(\varepsilon\)-factors of pairs are preserved. In the paper under review the author proves the following uniqueness statement, which is stated in terms of the map \(\sigma\mapsto \pi_ \sigma\) in opposite direction: Theorem 1.2: Let \(n\geq 2\) be an integer and let \(\sigma\in {\mathcal G}_ F^ 0 (n)\), \(\pi\in {\mathcal A}_ F^ 0 (n)\) have the property \[ \varepsilon(\pi\times \pi_ \tau, s,\psi)= \varepsilon(\sigma\otimes \tau,s,\psi) \] for all \(\tau\in {\mathcal G}_ F^ 0(r)\), \(r=1,\dots, n-1\). Then \(\pi\cong \pi_ \sigma\). The proof is by reduction to Theorem 1.1: Let \(n\geq 2\) be an integer and let \(\pi\), \(\pi'\) be irreducible generic representations of \(\text{GL}_ n(F)\). Then the equality of ``Gamma factors'' \(\gamma(\pi\times \tau,s,\psi)= \gamma(\pi'\times \tau, s,\psi)\) for all irreducible generic representations \(\tau\) of \(\text{GL}_{n-1}(F)\) implies \(\pi\cong \pi'\). The last theorem does not depend on \(\text{char}(F)\) but uses the fact that irreducible generic representations \(\pi\) have a Whittaker model \({\mathcal W} (\pi,\psi)\) and that certain complex valued smooth fu…
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 549657
zbMATH Keywords: local Langlands conjecture
zbMATH Keywords: non-archimedean local field
zbMATH Keywords: irreducible representations
zbMATH Keywords: \(\varepsilon\)-factors
zbMATH Keywords: Whittaker model
reviewed by: Ernst-Wilhelm Zink
MaRDI profile type: Publication
cites work: Q5676301
cites work: Automorphic forms on \(GL(3)\). I, II
cites work: Q3894004
cites work: Rankin-Selberg Convolutions
cites work: \({\mathcal D}\)-elliptic sheaves and the Langlands correspondence
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
OpenAlex ID: W2059876925
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description: scientific article; zbMATH DE number 549657