On the projection of functions invariant under the action of a crystallographic group

We study functions defined in (n+1)-dimensional domains that are invariant under the action of a crystallographic group. We give a complete description of the symmetries that remain after projection into an n-dimensional subspace and compare it to similar results for the restriction to a subspace. We use the Fourier expansion of invariant functions and the action of the crystallographic group on the space of Fourier coefficients. Intermediate results relate symmetry groups to the dual of the lattice of periods.