Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL2n(R). 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.