Hyperplane arrangements and Vinberg's $\theta$-groups

Abstract

Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ 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. L\^e 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 $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$.