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We consider the action of a finite subgroup of the mapping class group$Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the modulispace $\mathcal{R}(S,G)$ of representations of $\pi_1(S)$ in a connectedsemisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realizationproblem ensures the existence of an element $J$ in the Teichm\"uller space of$S$ for which $\Gamma$ can be realised as a subgroup of the group ofautomorphisms of $X=(S,J)$ which are holomorphic or antiholomorphic. Weidentify the fixed points of the action of $\Gamma$ on $\mathcal{R}(S,G)$ interms of $G$-Higgs bundles on $X$ equipped with a certain twisted$\Gamma$-equivariant structure, where the twisting involves abelian andnon-abelian group cohomology simultaneously. When the kernel of the isotropyrepresentation of the maximal compact subgroup of $G$ is trivial, the fixedpoints can be described in terms of familiar objects on $Y=X/\Gamma^+$, where$\Gamma^+ \subset \Gamma$ is the maximal subgroup of $\Gamma$ consisting ofholomorphic automorphisms of $X$. If $\Gamma=\Gamma^+$ one obtains actual$\Gamma$-equivariant $G$-Higgs bundles on $X$, which in turn correspond withparabolic Higgs bundles on $Y=X/\Gamma$ (this generalizes work of Nasatyr \&Steer for $G=SL(2,\mathbb{R})$ and Boden, Andersen & Grove and Furuta & Steerfor $G=SU(n)$). If on the other hand $\Gamma$ has antiholomorphicautomorphisms, the objects on $Y=X/\Gamma^+$ correspond with pseudorealparabolic Higgs bundles. This is a generalization in the parabolic setup of thepseudoreal Higgs bundles studied by the first author in collaboration withBiswas & Hurtubise.

Action of the mapping class group on character varieties and Higgs bundles