On $A_1^2$ restrictions of Weyl arrangements

Let $\mathcal{A}$ be a Weyl arrangement in an $\ell$-dimensional Euclidean space. The freeness of restrictions of $\mathcal{A}$ was first settled by a case-by-case method by Orlik-Terao (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik-Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik-Solomon-Terao (1986), asserts that the exponents of any $A_1$ restriction, i.e., the restriction of $\mathcal{A}$ to a hyperplane, are given by $\{m_1,\ldots, m_{\ell-1}\}$, where $\exp(\mathcal{A})=\{m_1,\ldots, m_{\ell}\}$ with $m_1 \le \cdots\le m_{\ell}$. As a next step after Orlik-Solomon-Terao towards understanding the exponents of the restrictions, we will investigate the $A_1^2$ restrictions, i.e., the restrictions of $\mathcal{A}$ to subspaces of the type $A_1^2$. In this paper, we give a combinatorial description of the exponents of the $A_1^2$ restrictions and describe bases for the modules of derivations in terms of the classical notion of related roots by Kostant (1955).