Equivariant multiplicities of Coxeter arrangements and invariant bases

Let AA be an irreducible Coxeter arrangement and WW be its Coxeter group. Then WW naturally acts on AA. A multiplicity m:A→Z is said to be equivariant when m is constant on each WW-orbit of AA. In this article, we prove that the multi-derivation module D(A,m) is a free module whenever m is equivariant by explicitly constructing a basis, which generalizes the main theorem of Terao (2002) [12]. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the WW-invariant part D(A,m)W for any multiplicity m is a free module over the WW-invariant subring.