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 (Tôhoku 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).

令\（\ mathcal {A} \）是\（\ ell \）维欧几里得空间中的Weyl排列。\（\ mathcal {A} \）的限制自由度首先由Orlik和Terao逐案解决（Tokuhoku Math J 52：369-383，1993），然后由Douglass通过统一论证解决。 （代表理论3：444-456，1999）。在此之前，Orlik和Solomon（Proc Symp Pure Math Amer Math Soc 40（2）：269-292，1983）已经通过穷竭完全确定了这些安排的指数。一个经典的结果归功于Orlik等人。（Adv Stud Pure Math 8：461–77，1986）断言，任何\（A_1 \）限制的指数，即\（\ mathcal {A} \）对超平面的限制，由下式给出：\（\ {m_1，\ ldots，m _ {\ ell -1} \} \），其中\（\ exp（\ mathcal {A}）= \ {m_1，\ ldots，m _ {\ ell} \} \）与\（m_1 \ le \ cdots \ le m _ {\ ell} \）。作为对限制指数进行概念理解的下一步，我们将研究\（A_1 ^ 2 \）限制，即\（\ mathcal {A} \）对类型\（A_1 ^ 2 \ ）。在本文中，我们对指数进行了组合描述，并根据Kostant（Proc Nat Acad Sci USA 41：967）的相关根的经典概念，描述了\（A_1 ^ 2 \）限制的推导模块的基础。–970，1955年）。