Hyperplane arrangements and Vinberg's $θ$-groups

TL;DR

The paper provides a uniform geometric proof that, for a periodically graded semisimple complex Lie algebra, the hyperplane arrangement obtained by restricting roots to a Cartan subspace $\mathfrak{c}\subset\mathfrak{g}_1$ coincides with the hyperplane arrangement generated by complex reflections in the little Weyl group $W$. Building on Vinberg's $\theta$-group framework and the Cartan-subspace theory, the authors give a case-free argument that the hyperplanes from $\Sigma$ match those from $W$, extending prior case-by-case work of de Graaf and Le and covering outer diagram automorphisms uniformly. The proof uses Levi-subalgebra reductions and a rank-one analysis to show each restricted-root hyperplane is stabilized by a corresponding reflection, thereby proving $\mathcal{H}_\Sigma=\mathcal{H}_W$. Beyond the abstract argument, the paper provides explicit constructions of reflection lifts for many families of automorphisms (diagram, inner, and outer) and outlines a uniform method for obtaining these lifts, offering concrete tools for representation-theoretic applications and potential links to arithmetic and geometric representation theory.

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

Theorems & Definitions (24)

• Theorem 1.1
• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Remark 3.1
• Remark 3.2
• Definition 4.1
• Lemma 4.2
• proof