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

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