Non-archimedean and motivic integrals on the Hitchin fibration

Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between `stringy’ Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using non-archimedean integrals on these moduli spaces, building on work of Denef-Loeser and Batyrev. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô.In my talk I will outline the main arguments of the proofs and discuss the adjustments needed, in order to replace non-archimedean integrals by motivic ones. The latter is joint work with François Loeser.