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Let $X$ be a Hilbert modular variety and $\mathbb{V}$ a non-trivial localsystem over $X$ with infinite monodromy. In this paper we study Saito's mixedHodge structure (MHS) on the cohomology group $H^k(X,\mathbb{V})$ using themethod of Higgs bundles. Among other results we prove the Eichler-Shimuraisomorphism, give a dimension formula for the Hodge numbers and show that themixed Hodge structure is split over $\mathbb{R}$. These results are analogousto Matsushima-Shimura [Annals of Mathematics 78, 1963] in the cocompact caseand complement the results in Freitag [Book: Hilbert modular forms,Springer-Verlag, Berlin, 1990] for constant coefficients.

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