La fibration de Hitchin-Frenkel-Ngo et son complexe d'intersection

TL;DR

This paper constructs the Hitchin fibration for SL_{2} using Vinberg's semigroup, proving a transversality result that aids in establishing the fundamental lemma for spherical Hecke algebras.

Contribution

It extends the Hitchin fibration construction to groups following Frenkel-Ngo's scheme, utilizing Vinberg's semigroup and proving key transversality results.

Findings

Construction of Hitchin fibration for SL_{2} using Vinberg's semigroup

Proof of transversality between intersection complex and diagonal in the stack of G-bundles

Application to the fundamental lemma for spherical Hecke algebra

Abstract

In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngo in the case of SL_{2}. This construction uses as a decisive tool the Vinberg's semigroup. The total space of Hitchin is obtained by taking the fiber product of the Hecke stack with the diagonal of the stack of G-bundles $Bun_{G}$; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $Bun_{G}$, over a sufficiently big open subset, in order to get local applications, such that the fundamental lemma for the spherical Hecke algebra. Along the proof of this theorem, we establish a result concerning the integral conjugacy classes of the points of a simply connected group in a local field.