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

Abstract

In this paper we will construct a certain action of Aut$(\mathbb{C})$ on holomorphic modular forms (of any weights) with respect to an arbitrary unitary group, which is compatible with Hecke operators. We can write this action explicitly and simply. The image of the action in general is a holomorphic modular form for another unitary group.