On irreducible representations of Fuchsian groups

Let ${\mathcal {R}} \subset \mathbb {P}^1_{\mathbb {C}}$ be a finite subset of markings. Let G be an almost simple simply-connected algebraic group over $\mathbb {C}$ . Let $K_G$ denote the compact real form of G. Suppose for each lasso l around the marked point, a conjugacy class $C_l$ in $K_G$ is prescribed. The aim of this paper is to give verifiable criteria for the existence of an irreducible homomorphism of $\pi _{1}(\mathbb P^1_{\mathbb {C}} \,{\backslash}\, {\mathcal {R}})$ into $K_G$ such that the image of l lies in $C_l$ .