In the representation theory of finite-dimensional simple Lie algebras $\mathfrak{g}$, two categories of modules stand out due to their contrasting nature. The first is the category of weight modules, consisting of $\mathfrak{g}$-representations where a fixed Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ acts semisimply. This category has been extensively studied over the past decades, with a classification of simple modules having finite-dimensional weight spaces obtained by O. Mathieu through the introduction of a special class of modules known as coherent families. The second category consist of modules that are freely generated by $\mathcal{U}(\mathfrak{h})$ of finte rank. Recent studies have focused on this category which includes the classification of $\mathcal{U}(\mathfrak{h})$-free modules of rank one by J. Nilsson. Interestingly, these two categories are connected through the weighting functor $\mathcal{W}$, which, as the name suggests, assigns to a $\mathfrak{h}$-free module $M$ a weight module $\mathcal{W}(M)$. This functor was a key tool in Nilsson’s classification of simple $\mathfrak{sp}(2n)$-modules that are $\mathcal{U}(\mathfrak{h})$-free of rank one.
This talk aims to explore these intriguing connections and demonstrate how Nilsson’s approach can be extended to the broader category $\mathfrak{A}$ of $\mathfrak{g}$-modules that are finitely generated by $\mathcal{U}(\mathfrak{h})$. As part of this extension, we introduce the new concept of almost-coherent families (a generalization of the standard coherent families) and the notion of almost equivalency, leading to the classification of a subclass of simple modules in $\mathfrak{A}$.