Finite group actions on Higgs bundle moduli spaces

Let ${\cal M}(X,G)$ be the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a semisimple complex Lie group with centre $Z$. We describe the fixed points of the action of a finite group $\Gamma$ on ${\cal M}(X,G)$, induced by holomorphic actions of $\Gamma$ on $X$ and $G$, a character of $\Gamma$ and a homomorphism from $\Gamma$ to the group of $Z$-bundles over $X$. Two important ingredients in this study are provided by the theory of twisted $\Gamma$-equivariant bundles developed by Barajas--Garc\'ia-Prada--Gothen--Mundet i Riera, and the Prym--Narasimhan--Ramanan construction given by Barajas--Garc\'ia-Prada. Via the non-abelian Hodge correspondence, our results provide a description of the fixed-point subvarieties of certain finite group actions on the $G$-character variety of the fundamental group of $X$.