Abstract:
This is joint work with Paolo Papi. Let W be an irreducible finite reflection group, h its (complexified) reflection module. H=C[h]/I, where I is the ideal generated by polynomial invariants of positive degree. A=(Λ(h)⊗H)W is an exterior algebra and we completely determine the A-module structure of N:=homW(h,Λ(h)⊗H).
When h is the Cartan subalgebra of a simple Lie algebra g, it is well known and easy that A is canonically isomorphic to (Λ(g))g and we verify that N=homg(g,Λ(g) as an A-module.
Finally if V is an irreducible g-module whose zero weight space we denote by V0, we construct a degree preserving map homg(V,Λ(g))→homW(V0,Λ(h)⊗H) which we conjecture to be injective. This conjecture implies a well known conjecture by Reeder.