On some modules of covariants for a reflection group

Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$andWeylgroup$W$\mathrm{.}Webuildupagradedmap$(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$of$(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules,where$\mathcal H$isthespaceof$W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.