I will report on the recent proof of a conjecture in Langlands’ program called the fundametal lemma. This conjecture of number theoretic flavor, is about certain equality of p-adic orbital integral. The proof turns out to be of geometric nature and rely on the geometry of the Hitchin fibration. I will explain the connection between p-adic orbital integral and the Hitchin fibration, and more precisely the connection between Langlands’ endoscopy theory and the natural symmetries of the Hitchin fibers. Another important ingredient is the study of the support of simple perverse constituents which occur in the decomposition of the direct image of the constant sheaf by the Hitchin fibration. This support theorem provide a good control on the cohomology of singular Hitchin fibers which are the geometric incarnation of orbital integrals.