Equivariant multiplicities of Coxeter arrangements and invariant bases

Let $\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\A$. A multiplicity $\bfm : \A\rightarrow \Z$ is said to be equivariant when $\bfm$ is constant on each $W$-orbit of $\A$. In this article, we prove that the multi-derivation module $D(\A, \bfm)$ is a free module whenever $\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\A, \bfm)^{W}$ for any multiplicity $\bfm$ is a free module over the $W$-invariant subring.