Let $A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $A$. A multiplicity $bfm : Arightarrow Z$ 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.
Let $A$ be an irreducible Coxeter arrangement and $bfk$ be a multiplicity of $A$. We study the derivation module $D(A, bfk)$. Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action
of the Coxeter group. In this paper, we will {explicitly} construct a basis for $D(A, bfk)$ assuming $bfk$ is constant on each orbit. Consequently we will determine the exponents of $(A, bfk)$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.
