If $\,\Delta_{\scriptsize\mathbb{R}^n}u = \lambda u\,$ and $\,u(x)=u(x+y)$, then $\,u_{y}(x):=e^{2 \pi i \left\langle x,y\right\rangle}$

Question

Related to Analysis on Manifolds via the Laplacian page $52$, I would like someone explain to me why if we have a function $u$ such that $\,\Delta_{\mathbb{R}^{n\,}}u = \lambda u\,$ and $u(x)=u(x+y)$ with $y \in \Gamma^*$, then this function have to be on the form $\,u_{y}(x):=e^{2 \pi i \left\langle x,y\right\rangle}$? I have some ideas, but it is confusing actually. Thanks!