Orsay

Coefficients of the characteristic polynomial are generators of the ring of polynomial functions on the space of matrices which are invariant under the conjugation. This was generalized by Chevalley to general reductive groups. By looking closely on the centralisers, one is lead to a very natural 2-category attached to Chevalley characteristic morphism. This abstract, but yet elementary, construction helps one to understand the symmetries of the fibres of the Hitchin fibration, as well as those of affine Springer fibers.

We will also explain how these groups of symmetries are related to the notion of endoscopic groups, which was introduced by Langlands in his stabilisation of the trace formula. We will also briefly explain how the symmetry groups help one to acquire a rather good understanding of the cohomology of the Hitchin fibration and eventually the proof of the fundamental lemma in Langlands' program.