Multiderivations of Coxeter arrangements

Let V be an l-dimensional Euclidean space. Let G ⊂ O(V ) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D(A) = {θ ∈ DerS | θ(αH) ∈ SαmH}. 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(A). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ l) (when m is odd). Here m1 ≤ · · · ≤ ml are the exponents of G and h = ml + 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. Mathematics Subject Classification (2000): 32S22, 05E15, 20F55