Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α H ∈V * such that H=ker(α H ). For each nonnegative integer m, define the derivation module D (m) (?)={θ∈Der S |θ(α H )∈Sα m H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D (m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m i (1≤i≤ℓ) (when m is odd). Here m 1≤···≤m ℓ are the exponents of G and h=m ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.