Description
The non-abelian Hodge correspondence over a compact Riemann surface X establishes a homeomorphism between the moduli space of G-Higgs bundles and the G-character variety of the fundamental group of X, for a connected semisimple complex Lie group G. Even though the group G is connected, in the study of the geometry of G-Higgs bundles, one has to deal with Higgs pairs of different sorts with non-connected structure group for which, in particular, one requires a Hitchin-Kobayashi type correspondence. This appears, for example, in the study of cyclic G-Higgs bundles, as well as in the Cayley parametrization of higher Teichmueller components. In this talk, I will explain a way of dealing with this situation, reducing the proof of the correspondence to the connected case. A particular application is an extension of the non-abelian Hodge correspondence itself to non-connected groups.