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Le lemme fondamental pondéré. I. Constructions géométriques

Part of: Algebraic number theory: global fields Lie groups Discontinuous groups and automorphic forms Families, fibrations

Published online by Cambridge University Press:  19 May 2010

Pierre-Henri Chaudouard  and
Gérard Laumon
Pierre-Henri Chaudouard
Affiliation:
CNRS et Université Paris-Sud, UMR 8628, Mathématique, Bâtiment 425, F-91405 Orsay Cedex, France (email: [email protected])
Gérard Laumon
Affiliation:
CNRS et Université Paris-Sud, UMR 8628, Mathématique, Bâtiment 425, F-91405 Orsay Cedex, France (email: [email protected])
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Abstract

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This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg traceformula.

Résumé

Ce travail est la partie géométrique de notre démonstration du lemme fondamental pondéré qui prolonge celle du lemme fondamental de Langlands–Shelstad due à Ngô Bao Châu. L’approche de Ngô repose sur l’étude de partie elliptique de la fibration de Hitchin. Cette fibration a pour espace total le champ des fibrés de Hitchin et pour base l’espace affine des «polynômes caractéristiques». Au-dessus de l’ouvert elliptique, elle est propre et le nombre de points de ses fibres sur un corps fini s’exprime en termes d’intégrales orbitales. Dans cet article, on étudie la fibration de Hitchin au-dessus d’un ouvert plus gros que l’ouvert elliptique, le lieu «génériquement semi-simple régulier». Les fibres ne sont en général ni de type fini ni même séparées. Par analogie avec les troncatures d’Arthur, nous introduisons le champ des fibrés de Hitchin ξ-stables. Nous montrons que celui-ci est un champ de Deligne–Mumford, lisse sur le corps de base et propre au-dessus de la base des polynômes caractéristiques. Nous exprimons le nombre de points d’une fibre ξ-stable sur un corps fini en termes d’intégrales orbitales pondérées d’Arthur qui apparaissent dans la formule des traces d’Arthur–Selberg.

Keywords

Hitchin fibrationHiggs bundlesweighted orbital integralArthur–Selberg trace formulaLanglands–Shelstad fundamental lemmaweighted fundamental lemma

MSC classification

Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11R39: Langlands-Weil conjectures, nonabelian class field theory 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 6 , November 2010 , pp. 1416 - 1506
Copyright
Copyright © Foundation Compositio Mathematica 2010

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