In this article, we construct the Hitchin fibration for groups following the scheme outlined by Frenkel-Ngô [?] in the case of $SL_{2}$. This construction uses as a decisive tool the Vinberg semigroup and follows the study accomplished in [?]. 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 $\mathrm {Bun}_{\mathrm {G}}$ ; we prove a transversality statement between the intersection complex of the Hecke stack and the diagonal of $\mathrm {Bun}_{\mathrm {G}}$, over a sufficiently big open subset, in order to get local applications, such as the fundamental lemma for the spherical Hecke algebra. Along the proof of this theorem, we establish a result concerning the integral conjugacy es of the points of a simply connected group in a local field.