$\mathcal{U}(\mathfrak{h})$-finite modules and weight modules I: weighting functors, almost-coherent families and category $\mathfrak{A}^{\text{irr}}$
Eduardo M. Mendonça
TL;DR
The paper develops a bridge between $$-free and weight modules by introducing the weighting functor $ rak W$ and its left derived functors, proving that simple $$-finite modules of infinite dimension are $$-torsion-free and, under non-integral or singular central characters, $$-free. It then defines almost-coherent families as a natural generalization of Mathieu’s coherent families, showing that $ rak W(M)$ is almost-coherent for $$-torsion-free $M$, and establishing an almost-equivalence with irreducible coherent families. The authors classify simple objects in the category $ rk A^{ ext{irr}}$ for type $C$ entirely and obtain partial classifications for type $A$, culminating in conjectures that every simple $ rak{sl}(n+1)$-module in $ rk A^{ ext{irr}}$ is a subquotient of an exponential tensor module; they support this via parabolic-induction and translation techniques, including a surjectivity result for type $A$. The work illuminates the structure of $ rk A$-modules by leveraging parabolic Verma modules, tensor constructions, and dualities, with potential implications for full classification problems in the slope between weight and $$-free representations. In sum, the paper advances a coherent framework to classify irreducible $$-finite modules through almost-coherent weight structures and translation-equivalence machinery, especially for types $A$ and $C$.
Abstract
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.