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

Abstract

Let $W_a$ be an affine Weyl group with corresponding finite root system $\Phi$. In Shi (J Lond Math Soc (2) 35(1):42–55, 1987) characterized each element $w \in W_a$ by a $\Phi ^+$-tuple of integers $(k(w,\alpha ))_{\alpha \in \Phi ^+}$ 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 $k(w,\alpha )$ 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 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 $H^0({\widehat{X}}_{W_a})$, 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 ${\widehat{X}}_{W({\widetilde{A}}_n)}$ is in bijection with the conjugacy class of $(1~2~\cdots ~n+1) \in W(A_n) = S_{n+1}$. We also compute the action of $W(A_n)$ on $H^0({\widehat{X}}_{W({\widetilde{A}}_n)})$.