Modules of differential operators of order 2 on Coxeter arrangements

We prove that the modules of differential operators of order 2 on the classical Coxeter arrangements are free by exhibiting bases. For this purpose, we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's criterion. In the case type $A$, we apply Cauchy-Sylvester's theorem on compound determinants to Vandermond determinant. By using the Schur polynomials, we define operators which form a part of a basis of modules of differential operators on the classical Coxeter arrangements of type $A$. In the cases of type $B$ and type $D$, the proofs go similarly to the case of type $A$ with some adjustments of operators and determinants.