On the cohomology groups of local systems over Hilbert modular varieties via Higgs bundles

Let X be a Hilbert modular variety and V a non-trivial local system over X. In this paper we study Deligne-Saito's mixed Hodge structure (MHS) on the cohomology groups H k (X,V) using the method of Higgs bundles. Among other results we give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over R. These results are analogous to (18) in the cocompact case and complement the results in (8) for constant coefficients.