Finite group actions on Higgs bundle moduli spaces and twisted equivariant structures

We consider the moduli space ${\cal M}(G)$ of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group, and study the action of a finite group $\Gamma$ on ${\cal M}(G)$ induced by a holomorphic action of $\Gamma$ on $X$ and $G$, and a character of $\Gamma$. The fixed-point subvariety for this action is given by a union of moduli spaces of $G$-Higgs bundles equipped with a certain twisted $\Gamma$-equivariant structure involving a $2$-cocycle of $\Gamma$ with values in the centre of $G$. This union is paremeterized by the non-abelian first cohomology set of $\Gamma$ in the adjoint group of $G$. We also describe the fixed points in the moduli space of representations of the fundamental group of $X$ in $G$, via a twisted equivariant version of the non-abelian Hodge correspondence, which involves the $\Gamma$-equivariant fundamental group of $X$.