Let G be a connected reductive group over an algebraically closed field. A challenging task is the computation of the (intersection) cohomology of the moduli space of G-Higgs bundles over a curve. One way to tackle the problem is to study the derived direct image of the intersection complex of this moduli space along the G-Hitchin fibration. In the first part of the talk, we provide an overview of the G-Hitchin fibration for the moduli space of G-Higgs bundles. In the second part, we show that the derived direct image of the intersection complex for the meromorphic G-Hitchin fibration is determined by its restriction to the so-called elliptic locus, for any connected reductive group G. The cases GLn and SLn were already known thanks to the works of Chaudouard-Laumon, de Cataldo and Maulik-Shen. This is a work in progress jointly with Mark Andrea de Cataldo, Andres Fernandez Herrero and Mirko Mauri.