Let $W_a$ be an affine Weyl group with corresponding finite root system ϕ. In Shi (J Lond Math Soc (2) 35(1):42-55, 1987) characterized each element w ∈ $W_a$ by a ϕ+-tuple of integers (k(w, α))$_{α∈ϕ}+$ subject to certain conditions. In Chapelier-Laget (Shi variety corresponding to an affine Weyl group. arXiv:2010.04310, 2020) a new interpretation of the coefficients k(w, α) is given. This description led us to define an affine variety X W a , called the Shi variety of W a , whose integral points are in bijection with W a . It turns out that this variety has more than one irreducible component, and the set of these components, denoted H$^0$(X$_{Wa}$), admits many interesting properties. In particular the group $W_a$ acts on it. In this article we show that the set of irreducible components of $X_{W({\widetilde{A}_n)}}$ is in bijection with the conjugacy class of (1 2 ... n + 1) ∈ $W(A_n$) = S$_{n+1}$.