Reflexive modules on normal surface singularities and representations of the local fundamental group

We consider the Riemann–Hilbert correspondence on the complement of a normal surface singularity (X,x). Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group image and the category of left image-modules that are reflexive as image-modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left image-modules. We prove that the fundamental module is an invariant module if and only if (X,x) is a quotient singularity. Finally we investigate some algebraisation aspects.