Subspaces of $L^2({\mathbb{R}}^n)$ Invariant Under Shifts by a Crystal Group

For a crystal group $\mathrm{\Gamma }$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2({\mathbb{R}}^n)$ is called $\mathrm{\Gamma }$–shift invariant if, for every $f\in \mathcal{V}$, the shifts of $f$ by every element of $\mathrm{\Gamma }$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $\mathrm{\Gamma }$–shift invariant closed subspaces of $L^2({\mathbb{R}}^n)$.