On homomorphisms of $\pi_{1}(\mathbb P^1-\mathcal R)$ into compact semisimple groups

Abstract

The aim of this paper is to give verifiable criteria for the existence of {\em irreducible} homomorphisms of $\pi_{1}(\mathbb P^1 - \mathcal R)$ into compact semisimple groups, for a finite subset $\mathcal R$ such that the conjugacy classes of the images of lassos around the marked points are fixed. By a theorem in \cite{bs}, this question reduces into one of giving verifiable criteria for the existence of stable $\mathcal G$-torsors on $\mathbb{P}^1_{\mathbb{C}}$, where $\mathcal{G} \to \mathbb{P}^1_\mathbb{C}$ is a a Bruhat-Tits group scheme.