A Prym–Narasimhan–Ramanan construction of principal bundle fixed points

Let X be a compact Riemann surface and G be a connected reductive complex Lie group with center Z. Consider the moduli space M(X,G) of polystable principal holomorphic G-bundles on X. There is an action of the group H1(X,Z) of isomorphism classes of Z-bundles over X on M(X,G) induced by the multiplication Z×G→G. Let Γ be a finite subgroup of H1(X,Z). In this paper, we give a Prym–Narasimhan–Ramanan-type description of the fixed points of M(X,G) under the action of Γ. A main ingredient in this construction is the theory of twisted equivariant bundles on an étale cover of X developed in our joint paper with Gothen and Mundet i Riera.