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

Let $X$ be a Hilbert modular variety and $\Bbb{V}$ a non-trivial local system over $X$ with infinite monodromy. In this paper we study Saito's mixed Hodge structure (MHS) on the cohomology group $H^k(X,\Bbb{V})$ using the method of Higgs bundles. Among other results we prove the Eichler-Shimura isomorphism, give a dimension formula for the Hodge numbers and show that the mixed Hodge structure is split over $\Bbb{R}$. These results are analogous to the work of Y. Matsushima and G. Shimura in the cocompact case and complement the results of E. Frietag for constant coefficients.