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

TL;DR

This paper develops a Prym-Narasimhan-Ramanan-type construction to describe fixed points in the moduli space of principal G-bundles on a Riemann surface under finite group actions, using twisted equivariant bundles.

Contribution

It introduces a novel construction for fixed points in moduli spaces of principal bundles using twisted equivariant bundles and Prym-type methods.

Findings

Provides a new description of fixed points under finite group actions.

Utilizes twisted equivariant bundle theory on étale covers.

Connects Prym-Narasimhan-Ramanan techniques with moduli space analysis.

Abstract

Let $X$ be a compact Riemann surface and $G$ be a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable principal holomorphic $G$-bundles on $X$. There is an action of the group $H^1(X,Z)$ of isomorphism classes of $Z$-bundles over $X$ on $M(X,G)$ induced by the multiplication $Z\times G\to G.$ Let $\Gamma$ be a finite subgroup of $H^1(X,Z)$. Our goal is to find a Prym--Narasimhan--Ramanan-type construction to describe the fixed points of $M(X,G)$ under the action of $\Gamma$. A main ingredient in this construction is the theory of twisted equivariant bundles on an \'etale cover of $X$ developed in arXiv:2208.0902(2).