  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.
