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Sur le comportement, par torsion, des facteurs epsilon de paires

Published online by Cambridge University Press:  20 November 2018

Colin J. Bushnell
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS, United Kingdom. e-mail: bushnell@mth.kcl.ac.uk
Guy Henniart
Affiliation:
Département de Mathématiques, UMR 8628 du CNRS, Bâtiment 425, Université de Paris-Sud, 91405 Orsay cedex, France. e-mail: Guy.Henniart@math.u-psud.fr
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Résumé

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Soient $F$ un corps commutatif localement compact non archimédien et $\psi$ un caractère additif non trivial de $F$. Soient $n$ et ${n}'$ deux entiers distincts, supérieurs à 1. Soient $\pi$ et ${\pi }'$ des représentations irréductibles supercuspidales de $\text{G}{{\text{L}}_{n}}\left( F \right)$, $\text{G}{{\text{L}}_{{{n}'}}}\left( F \right)$ respectivement. Nous prouvons qu’il existe un élément $c=c\left( \pi ,{\pi }',\psi \right)$ de ${{F}^{\times }}$ tel que pour tout quasicaractère modéré $\mathcal{X}$ de ${{F}^{\times }}$ on ait $\mathcal{E}\left( \chi \pi \times {\pi }',s,\psi \right)=\chi {{\left( c \right)}^{-1}}\mathcal{E}\left( \pi \times {\pi }',s,\psi \right)$. Nous examinons aussi certains cas où $n={n}',{\pi }'={{\pi }^{\text{v}}}$. Les résultats obtenus forment une étape vers une démonstration de la conjecture de Langlands pour $F$, qui ne fasse pas appel à la géométrie des variétés modulaires, de Shimura ou de Drinfeld.

Abstract

Abstract

Let $F$ be a non-Archimedean local field, and $\psi $ a non-trivial additive character of $F$. Let $n$ and ${n}'$ be distinct positive integers. Let $\pi $, ${\pi }'$ be irreducible supercuspidal representations of $\text{G}{{\text{L}}_{n}}\left( F \right)$, $\text{G}{{\text{L}}_{{{n}'}}}\left( F \right)$ respectively. We prove that there is $c=c\left( \pi ,{\pi }',\psi \right)$$\in $${{F}^{\times }}$ such that for every tame quasicharacter $\mathcal{X}$ of ${{F}^{\times }}$ we have $\mathcal{E}\left( \chi \pi \times {\pi }',s,\psi \right)=\chi {{\left( c \right)}^{-1}}\mathcal{E}\left( \pi \times {\pi }',s,\psi \right)$. We also treat some cases where $n={n}'$ and ${\pi }'={{\pi }^{\text{V}}}$. These results are steps towards a proof of the Langlands conjecture for $F$, which would not use the geometry of modular—Shimura or Drinfeld—varieties.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2001

References

Références

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