On $$A_1^2$$A12 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 and Terao (Tohoku Math J 52: 369–383, 1993), and later by a uniform argument by Douglass (Represent Theory 3: 444–456, 1999). Prior to this, Orlik and Solomon (Proc Symp Pure Math Amer Math Soc 40(2): 269–292, 1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik et al. (Adv Stud Pure Math 8: 461–77, 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 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 (Proc Nat Acad Sci USA 41:967–970, 1955).