Fixed points in Higgs bundle moduli spaces and the Prym--Narasimhan--Ramanan construction

Abstract

Let $X$ be a compact Riemann surface and $G$ a connected reductive complex Lie group with centre $Z$. Consider the moduli space $M(X,G)$ of polystable $G$-Higgs bundles on $X$. The group of isomorphism classes of $Z$-bundles on $X$, which is isomorphic to $H^1(X,Z)$, acts on $M(X,G)$ via extension of structure group by the multiplication homomorphism $Z\times G\to G$. The group $\text{Aut}(G)$ also acts on $M(X,G)$ by extension of structure group, and so does the group $\text{Aut}(X)$ of holomorphic automorphisms via pullback. Finally, $\mathbb{C}^*$ acts by multiplying the Higgs field. Combining these provides an action of the semidirect product of $H^1(X,Z)$ and $(\text{Aut}(G)\times\text{Aut}(X))\times\mathbb{C}^*$ on $M(X,G)$, where $\text{Aut}(G)$ and $\text{Aut}(X)$ act on $H^1(X,Z)$ by extension of structure group and pullback, respectively. Let $H$ be such semidirect product. Let $\Gamma$ be a finite subgroup of $H$. The goal of this thesis is to find a Prym--Narasimhan--Ramanan-type construction to describe the fixed points of the action of $\Gamma$ on $M(X,G)$. More precisely, we show that fixed points correspond to twisted equivariant Higgs pairs over certain \'etale covers of $X$. Our results generalize Garc\'ia-Prada--Ramanan, where $\Gamma$ was considered to be cyclic, and Narasimhan--Ramanan, who only consider actions of cyclic subgroups of $H^1(X,\mathbb{C}^*)$ for $G=\text{GL}(n,\mathbb{C})$.