Moduli space of stable principal $G$-bundles on a compact Riemann surface.

Question

Let $C$ be a compact Riemann surface. I'm looking for some references in order to try to understand what is the moduli space of stable principal $G$-bundles on $C$, where $G$ is a simple Lie group. Could you give me some references? For example if $G=U(1)$ and $\mathbb{C}P^1$ how can I describe the moduli space of stable principal $U(1)$-bundles on $\mathbb{C}P^1$?

Answer 1

For rank $1$ Lie groups ($\mathbb{C} \mathbf{P}^1$ is not a Lie group by the way), you can look at Goldman-Xia, Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces