On $A_1^2$ restrictions of Weyl arrangements

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