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Let $W_a$ be an affine Weyl group with corresponding finite root system$\Phi$. In \cite{JYS1} Jian-Yi Shi characterized each element $w \in W_a$ by a$ \Phi^+$-tuple of integers $(k(w,\alpha))_{\alpha \in \Phi^+}$ subject tocertain conditions. In \cite{NC1} a new interpretation of the coefficients$k(w,\alpha)$ is given. This description led us to define an affine variety$\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points arein bijection with $W_a$. It turns out that this variety has more than oneirreducible component, and the set of these components, denoted$H^0(\widehat{X}_{W_a})$, admits many interesting properties. In particular thegroup $W_a$ acts on it. In this article we show that the set of irreduciblecomponents of $\widehat{X}_{W(\widetilde{A}_n)}$ is in bijection with theconjugacy class of $(1~2~\cdots~n+1) \in W(A_n) = S_{n+1}$. We also compute theaction of $W(A_n)$ on $H^0(\widehat{X}_{W(\widetilde{A}_n)})$.

A symmetric group action on the irreducible components of the Shi variety associated to $W(\widetilde{A}_n)$