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Mémoires de la SMF - Parutions - 124 (2011)

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Changement de base et induction automorphe pour en caractéristique non nulle
Guy Henniart, Bertrand Lemaire
Mémoires de la SMF 124 (2011), ii+194 pages
Acheter l'ouvrage

Résumé :
Soit E/F une extension cyclique de corps (commutatifs) locaux ou globaux, de degré fini . La théorie du changement de base de GL_n(F) à GL_n(E) et celle de l'induction automorphe de GL_m(E) à GL_md(F) sont deux illustrations du principe de fonctorialité de Langlands: pour local, elles correspondent côté galoisien à la restriction des représentations de W'_F à W'_E et à l'induction des représentations de W'_E à W'_F , où W'_F désigne le groupe Weil-Deligne de , W'_E celui de . Si est une extension finie d'un corps -adique Q_p , ces deux théories existent depuis longtemps (Arthur-Clozel, Henniart-Herb). On les étend dans ce mémoire au cas où est un corps localement compact non archimédien de caractéristique non nulle. On montre aussi, pour un corps global de fonctions , que ces deux théories locales sont compatibles aux applications globales de changement de base et d'induction automorphe déduites, via la correspondance de Langlands établie par Lafforgue, de la restriction et de l'induction des représentations galoisiennes globales.

Mots-clefs : corps local non archimédien, caractéristique non nulle, groupe linéaire, représentation admissible, correspondance de Langlands locale, changement de base local, induction automorphe locale; algèbre de Hecke sphérique, isomorphisme de Satake, extension non ramifiée, lemme fondamental pour le changement de base (resp. pour l'induction automorphe); -intégrale orbitale, caractère -tordu, représentation -discrète, fonction élémentaire, application norme, identité à la Shintani; -intégrale orbitale, caractère -tordu, représentation -discrète, facteur de transfert, identité de caractères, modèle de Whittaker; corps de fonctions, représentation automorphe, groupe de Weil, correspondance de Langlands globale, changement de base global, induction automorphe globale.

Abstract:
Base change and automorphic induction for in positive characteristic
Let E/F be a finite cyclic extension of local or global fields, of degree . The theory of base change from GL_n(F) to GL_n(E) and the theory of automorphic induction from GL_m(E) to GL_md(F) are two instances of Langlands' functoriality principle: when is local, they correspond respectively to restriction to of representations of the Weil-Deligne group of , and induction to of representations of the Weil-Deligne group of . If is a finite extension of a -adic field Q_p , these theories were established long ago (Arthur-Clozel, Henniart-Herb). In this memoir we extend them to the case where is a non-Archimedean locally compact field of positive characteristic. We also prove, for a global functions field , that these two local theories are compatible with the global maps of base change and automorphic induction deduced, via the Langlands correspondence proved by Lafforgue, from restriction and induction of global Galois representations.

Keywords: non-Archimedean local field, positive characteristic, linear group, admissible representation, local Langlands correspondence, local base change, local automorphic induction; spherical Hecke algebra, Satake isomorphism, unramified extension, fundamental lemma for base change (resp. for automorphic induction); -orbital integral, -twisted character, -discrete series, elementary function, norm map, à la Shintaniidentity; -orbital integral, -twisted character, -discrete series, transfer factor, characters identity, Whittaker model; functions field, automorphic representation, Weil group, global Langlands correspondence, global base change, global automorphic induction.

Class. math. : 22E50

ISBN : 978-85629-311-9
ISSN : 0249-633-X
Publié avec le concours de : Centre National de la Recherche Scientifique

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