Schubert varieties, inversion arrangements, and Peterson translation

Abstract : We show that an element $\mathcal{w}$ of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement $\mathcal{I} (\mathcal{w})$ associated to the inversion set of \mathcal{w} is inductively free, and the product $(d_1+1) ...(d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval [e,w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of $\mathcal{w}$.