Moduli spaces of twisted equivariant G-bundles over a curve

Let $X$ be a compact Riemann surface, $\Gamma$ a finite group of automorphisms of $X$ and $G$ a connected reductive complex Lie group with center $Z$. If we equip this data with a homomorphism $\theta:\Gamma\to\text{Aut}(G)$ and a 2-cocycle $c:\Gamma\times\Gamma\to Z$, there is a notion of $(\theta,c)$-twisted $\Gamma$-equivariant $G$-bundle over $X$. The aim of this paper is to construct a coarse moduli space of isomorphism classes of polystable $(\theta,c)$-twisted equivariant $G$-bundles over $X$, according to the definition of polystability given by Garc\'ia-Prada--Gothen--Mundet i Riera. This generalizes the well-known construction of the moduli space of $G$-bundles given by Ramanathan. It also gives, in particular, a GIT construction of the moduli space of $\Gamma$-equivariant $G$-bundles, and the moduli space of $\hat G$-bundles for $\hat G$ non-connected by our joint work with Garc\'ia-Prada, Gothen and Mundet i Riera -- complementing the construction of a projective good moduli space for the moduli stack of $\hat G$-bundles given by Olsson--Reppen--Tajakka.