Let $\pi$ be a Fuchsian group acting on the upper half plane with elliptic fixed points. Let $G$ be a semisimple simply connected algebraic group and $K_{G}$ be a maximal compact subgroup of $G$. In this talk I will show that the space of (irreducible) representations of $\pi \to K_{G}$ (with fixed conjugacy classes of the finite order elements) modulo conjugacy can be given a modular description in terms of (stable) polystable torsors of a canonically defined Bruhat-Tits group scheme associated to a parahoric datum coming from the datum of the conjugacy classes. I will indicate several developments which have arisen from this perspective.