Principal bundles over a real algebraic curve

TL;DR

This paper studies real algebraic structures on principal G-bundles over a Riemann surface with anti-holomorphic involution, establishing a correspondence between real points of moduli spaces and certain bundle classes, with implications for stability and representations.

Contribution

It characterizes real points of moduli spaces of principal G-bundles over real algebraic curves and links stability notions with unitary representations.

Findings

Real points correspond to bundles compatible with involutions.

Stable, semistable, and polystable bundles are defined and related.

A bijection between unitary representations and polystable bundles is proven.

Abstract

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution \sigma. Let G be a connected complex reductive affine algebraic group, and let \sigma_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to \sigma and \sigma_G. We prove that the points defined over $\mathbb R$ of the smooth locus of a moduli space of principal G-bundles on X are precisely these objects, under the assumption that {\rm genus}(X) > 2. Stable, semistable and polystable bundles are defined in this context. Relationship between any of these properties and the corresponding property of the underlying holomorphic principal G-bundle is explored. A bijective correspondence between unitary representations and polystable objects is established.