Hyperplane arrangements and Vinberg's $\hat{I}¸$-groups

Abstract

Let $\mathfrak{g}={⨁}_{i\in \mathbb{Z}/m\mathbb{Z}}{\mathfrak{g}}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c}\subset {\mathfrak{g}}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g}={⨁}_{i\in \mathbb{Z}/m\mathbb{Z}}{\mathfrak{g}}_i$.

17 pages, v2: minor modifications, acknowledgments updated. Final version to appear on Journal of Algebra