Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface

We introduce real structures on L-twisted Higgs pairs over a compact connected Riemann surface X equipped with an anti-holomorphic involution, where L is a holomorphic line bundle on X with a real structure, and prove a Hitchin-Kobayashi correspon-dence for the L-twisted Higgs pairs. Real GR-Higgs bundles, where GR is a real form of a connected semisimple complex affine alge-braic group G, constitute a particular class of examples of these pairs. In this case, the real structure of the moduli space of G-Higgs pairs is defined using a conjugation of G that commutes with the one defining the real form GR and a compact conjugation of G pre-serving GR. We establish a homeomorphism between the moduli space of real GR-Higgs bundles and the moduli space of representa-tions of the fundamental group of X in GR that can be extended to a representation of the orbifold fundamental group of X into a cer-tain enlargement of GR with quotient Z/2Z. Finally, we show how real GR-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of GR-Higgs bun-dles, constructed using the real structures on G and X. A similar result is proved for the representations of the orbifold fundamental group.