Differential operators on Hilbert modular forms

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We investigate differential operators and their compatibility with subgroups of SL 2 ( R ) n . In particular, we construct Rankin–Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin–Cohen bracket of a Hilbert–Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.