On some modules of covariants for a reflection group

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$. We build up a graded isomorphism $\smash{\bigl(\bigwedge\mathfrak{h}\otimes\mathcal H\otimes \mathfrak{h}\big)\vphantom)^W}\to \bigl(\bigwedge \mathfrak{g}\otimes \mathfrak{g}\big)^\mathfrak{g}$ of $\bigl(\bigwedge \mathfrak{g}\big)^\mathfrak{g}\cong S(\mathfrak{h})^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation.