Communications in Analysis and Geometry

Volume 20 (2012)

Number 5

Principal bundles over a real algebraic curve

Pages: 957 – 988



Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

Jacques Hurtubise (Department of Mathematics, McGill University, Montreal, Quebec, Canada)


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 ${\rm genus}(X) \geq 3$. 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.

Published 9 April 2013