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 ${\mathbb{P}}_{\mathbb{C}}^1$, where $\mathcal{G}\to {\mathbb{P}}_{\mathbb{C}}^1$ is a a Bruhat-Tits group scheme.