Le lemme fondamental pond\'er\'e I : constructions g\'eom\'etriques

TL;DR

This paper extends Ngô Bao Châu's proof of the fundamental lemma by studying the Hitchin fibration over a larger set, introducing a new stable substack, and expressing fiber point counts via weighted orbital integrals.

Contribution

It introduces the $\xi$-stable Hitchin bundles as a proper Deligne-Mumford stack over a broader set, enabling new expressions for fiber point counts in terms of weighted orbital integrals.

Findings

The $\xi$-stable Hitchin fibration is proper and smooth.

Point counts of fibers relate to weighted orbital integrals.

Extension of Ngô's approach to a larger base set.

Abstract

This work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ng\^o Bao Ch\^au's proof of the Langlands-Shelstad fundamental lemma. Ng\^o'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 bigger than the elliptic set, namely the "generically regular semi-simple set". The fibers are in general neither of finite type nor separeted. By analogy with Arthur's truncation, we introduce the substack of $\xi$-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 $\xi$-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur-Selberg trace formula.