$\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$

This paper builds upon J. Nilsson's classification of rank one $\mathcal{U}(\mathfrak{h})$-free modules by extending the analysis to modules without rank restrictions, focusing on the category $\mathfrak{A}$ of $\mathcal{U}(\mathfrak{h})$-finite $\mathfrak{g}$-modules. A deeper investigation of the weighting functor $\mathcal{W}$ and its left derived functors, $\mathcal{W}_*$, led to the proof that simple $\mathcal{U}(\mathfrak{h})$-finite modules of infinite dimension are $\mathcal{U}(\mathfrak{h})$-torsion free. Furthermore, it is shown that these modules are $\mathcal{U}(\mathfrak{h})$-free if they possess non-integral or singular central characters. It is concluded that the existence of $\mathcal{U}(\mathfrak{h})$-torsion-free $\mathfrak{g}$-modules is restricted to Lie algebras of types A and C. The concept of an almost-coherent family, which generalizes O. Mathieu's definition of coherent families, is introduced. It is proved that $\mathcal{W}(M)$, for a $\mathcal{U}(\mathfrak{h})$-torsion-free module $M$, falls within this class of weight modules. Furthermore, a notion of almost-equivalence is defined to establish a connection between irreducible semi-simple almost-coherent families and O. Mathieu's original classification. Progress is also made in classifying simple modules within the category $\mathfrak{A}^{\text{irr}}$, which consists of $\mathcal{U}(\mathfrak{h})$-finite modules $M$ with the property that $\mathcal{W}(M)$ is an irreducible almost-coherent family. A complete classification is achieved for type C, with partial classification for type A. Finally, a conjecture is presented asserting that all simple $\mathfrak{sl}(n+1)$-modules in $\mathfrak{A}^{\text{irr}}$ are isomorphic to simple subquotients of exponential tensor modules, and supporting results are proved.