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 $\mathfrak{ g} = \bigoplus_{ i \in \mathbb{ Z} /m \mathbb{ Z}} \mathfrak{ g}_i$.
Categories: math.RT math.GR