$\mathcal{U}(\mathfrak{h})$-free modules and weight representations

The classification of simple modules for a simple Lie algebra $\mathfrak{g}$ seems beyond reach: it is complete only for $\mathfrak{sl}(2)$. However, some classes of simple $\mathfrak{g}$-modules are well understood, such the category of weight modules with finite dimensional weight spaces. Irreducible weight representations were classified due to the effort of S. Fernando and O. Mathieu. Recently non-weight modules have been studied and the category of $\mathfrak{h}$-free modules (i.e., $\mathfrak{g}$-modules on which the Cartan subalgebra $\mathfrak{h}$ acts freely) has draw attention in the community. Still, the complete classification of $\mathfrak{h}$-free modules of finite-rank seems to be a hard project and the only known case is when the rank equals one, due to J. Nilsson). Surprisingly, in spite of being of a completely opposite nature, weight modules and $\mathfrak{h}$-free modules carry interesting connections. These connections are mostly obtained due to the weighting functor $\mathcal{W}$ that, as the name suggests, assigns to a $\mathfrak{h}$-free module $M$ a weight module $\mathcal{W}(M)$.  Such functor was the main tool to the Nilsson's classification of simple $\mathfrak{sp}(2n)$-module $\mathfrak{h}$-free of rank one.
This talk intends to illustrate the connection between both categories by introducing Nillson's approach. Furthermore, we intend to show how we can use the weighting functor (and its left derived functors) to study the larger category of $\mathfrak{g}$-modules that are $\mathcal{U}(\mathfrak{h})$-finitely generated.