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 $\mathcal{A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H\in \mathcal{A}$ choose ${\alpha }_H\in V^{\ast }$ such that $H=\ker ({\alpha }_H).$ For each nonnegative integer $m$, define the derivation module $\sD^{(m)}({\cal A}) = \{θ\in {\rm Der}_S | θ(α_H) \in S α^m_H\}$. 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 $((m-1)h/2)+m_i(1\le i\le \ell )$ (when $m$ is odd). Here $m_1\le ...\le m_{\ell }$ 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.

dedication and a footnote (thanking a grant) added

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