The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements

The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\mathrm{\ell }$-dimensional Euclidean space was introduced and studied by K. Saito. The filtration is equivalent data to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.