Multiderivations of Coxeter arrangements

Abstract

Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $\alpha_H \in V^*$ such that $H = {\rm ker}(\alpha_H).$ For each nonnegative integer $m$, define the derivation module $\sD^{(m)}({\cal A}) = \{\theta \in {\rm Der}_S | \theta(\alpha_H) \in S \alpha^m_H\}$. The module is known to be a free $S$-module of rank $\ell$ 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 $\sD^{(m)} (\cal A)$. Their degrees are all equal to $mh/2$ (when $m$ is even) or are equal to $((m-1)h/2) + m_i (1 \leq i \leq \ell)$ (when $m$ is odd). Here $m_1 \leq ... \leq m_{\ell}$ are the exponents of $G$ and $h= m_{\ell} + 1$ 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.