On some modules of covariants for a reflection group

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$. We build up a graded map $(\mathcal{H}\otimes \bigwedge \mathfrak{h}\otimes \mathfrak{h}{)}^W\to (\bigwedge \mathfrak{g}\otimes \mathfrak{g}{)}^{\mathfrak{g}}$ of $(\bigwedge \mathfrak{g}{)}^{\mathfrak{g}}\cong S(\mathfrak{h}{)}^W$-modules, where $\mathcal{H}$ is the space of $W$-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.

18 Pages