Equivariant multiplicities of Coxeter arrangements and invariant bases

Let $\text{A}$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\text{A}$. A multiplicity $\text{bfm}:\text{A}\to \mathbb{Z}$ is said to be equivariant when $\text{bfm}$ is constant on each $W$-orbit of $\text{A}$. In this article, we prove that the multi-derivation module $D(\text{A},\text{bfm})$ is a free module whenever $\text{bfm}$ 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 $D(\text{A},\text{bfm}{)}^W$ for any multiplicity $\text{bfm}$ is a free module over the $W$-invariant subring.