For affine Coxeter/Weyl groups, there is a standard representation into $GL_n(\mathbb{R})$ (where $n$ is the rank of the Coxeter group) where the reflecting hyperplanes form a nice hyperplane arrangement $\mathcal{H}$ in $\mathbb{R}^n$ and the group acts on the connected components of the $\mathbb{R}^n\setminus \mathcal{H}$. Additionally, this action preserves a family of planes. It is standard to take one of these planes and intersect it with $\mathcal{H}$ to obtain a collection of hyperplanes usually denoted $H_{\alpha,k}$, for $k\in \mathbb{Z}$ and $\alpha$ a root in the underlying finite root system. Then the affine Coxeter group acts on the connected components of hyperplane complement of $\{H_{\alpha,k}\}$.
I was wondering if there is an analogous way of understanding the hyperplane arrangement and its complement for extended affine Weyl groups. Specifically, for the Weyl group corresponding to the $A_1^{(1,1)}$ root system is there a nice system of alcoves that the Weyl group acts simply transitively?
@Moishe Cohen mentioned in Conceptual question about extended affine Weyl group $\hat{W}_a$ that the extended affine Weyl group had its own system of alcoves refining the one of $\tilde A_1$.