Construction of a Galois action on modular forms for an arbitrary unitary group (Q965077)

The author extends and improves his earlier results [J. Math. Kyoto Univ. 41, No. 1, 183--231 (2001; Zbl 0988.11024)] by constructing an \(\Aut(\mathbb{C})\) action on modular forms for unitary groups over CM fields (an imaginary quadratic extension \(K\) of a totally real field \(F\)). A technical restriction is lifted, there are extra results about the Hecke action, and the proofs no longer rely on Fourier-Jacobi expansions. As the author points out, the existence of such an action is known: what is new here is that the action is described explicitly. The action is on the space of all such modular forms: various choices have to be made, namely a CM-type \(\Psi\) for \(K\), i.e.\ a choice of complex embedding over each real place of \(F\), and a suitable skew-hermitian matrix \(T\), in order to define the unitary group concretely, and the image of a form may require different choices. The precise statement of the theorem is too long to give here. It divides into two cases depending on whether the rank \(q\) of a maximal isotropic subspace under \(T\) is zero or, more interestingly, positive. In the latter case the unitary group \(U(T,\Psi)\) contains a natural copy of \(\text{Sp}(q,F)\) and the restriction of the action to the corresponding Siegel modular forms reduces to the well-known action given by Shimura. Moreover, the action is compatible with Hecke operators.