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

Abstract

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 to certain conditions. In \cite{NC1} a new interpretation of the coefficients $k(w,\alpha )$ is given. This description led us to define an affine variety ${\hat{X}}_{W_a}$, 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 $H^0({\hat{X}}_{W_a})$, 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 ${\hat{X}}_{W({\tilde{A}}_n)}$ is in bijection with the conjugacy class of $(12\cdots n+1)\in W(A_n)=S_{n+1}$. We also compute the action of $W(A_n)$ on $H^0({\hat{X}}_{W({\tilde{A}}_n)})$.

16 pages, 5figures, 1 table