The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic $L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{ n}\times\GL_{ 1}$. We show that our results are compatible with Deligne's conjecture.
Categories: math.NT