On some modules of covariants for a reflection group

This is joint work with Paolo Papi. Let $W$ be an irreducible finite reflection group, $\mathfrak{h}$ its (complexified) reflection module. $\mathcal{H} = C[\mathfrak{h}]/I$, where $I$ is the ideal generated by polynomial invariants of positive degree. $A = (\Lambda (\mathfrak{h})\otimes \mathcal{H})^W$ is an exterior algebra and we completely determine the $A$-module structure of $N := hom_W (\mathfrak{h},\Lambda (\mathfrak{h})\otimes \mathcal{H})$.
When $\mathfrak{h}$ is the Cartan subalgebra of a simple Lie algebra $\mathfrak{g}$, it is well known and easy that $A$ is canonically isomorphic to $(\Lambda(\mathfrak{g}))^{\mathfrak{g}}$ and we verify that $N = hom_{\mathfrak{g}}(\mathfrak{g}, \Lambda(\mathfrak{g})$ as an $A$-module.
Finally if $V$ is an irreducible $g$-module whose zero weight space we denote by $V_0$, we construct a degree preserving map
$$hom_{\mathfrak{g}}(V,\Lambda(\mathfrak{g}))\to hom_W (V_0,\Lambda (\mathfrak{h})\otimes \mathcal{H})$$
which we conjecture to be injective. This conjecture implies a well known conjecture by Reeder.