Small representations in the exterior algebra of a simple Lie algebra: problems

In the talk we shall review various problems and recall few results regarding the graded dimension of the space ${\rm Hom}_{\mathfrak g}(V,\bigwedge{\mathfrak g})$, ${\mathfrak g}$ a simple Lie algebra over the complex numbers, $V$ a small irreducible ${\mathfrak g}$ module, and its structure as a module over the invariants $\bigwedge{\mathfrak g}^{\mathfrak g}$.