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)(X,x). Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group π1loc(X,x) and the category of left DX,xDX,x-modules that are reflexive as OX,xOX,x-modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left DX,xDX,x-modules. We prove that the fundamental module is an invariant module if and only if (X,x)(X,x) is a quotient singularity. Finally we investigate some algebraisation aspects.