Let Lambda be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles are examples of Lambda-modules for particular choices of Lambda.
Simpson studied the moduli problem for the classification of Lambda-modules over Kahler varieties, proving the existence of a moduli space Lambda-modules. We give an explicit description of the moduli spaces of Lambda modules over abelian varieties in terms of the symmetric product Sym(Y) of a
certain smooth variety. In the way, we give a interpretation of Hilbert scheme Hilb(Y) as the moduli space of a certain rigidification of Simpson’s moduli problem.