Moduli spaces of Λ-modules on abelian varieties

We give a description of MX(Λ,n) in terms of a symmetric product of a certain fibre bundle over the dual abelian variety Xˆ. We also give a moduli interpretation to the associated Hilbert scheme as the classifying space of Λ-modules with extra structure. Finally, we study the non-abelian Hodge theory associated to these new moduli spaces.