Classification of irreducible weight modules
Let \(\mathfrak g\) be a reductive algebra over an algebraically closed field of characteristic zero. The author classifies all simple weight \(\mathfrak g\)-modules (in which each weight space is finite dimensional). By Fernando's Theorem, the problem reduces to a classification of cuspidal modules, { e.g.}, non-parabolically induced modules. For simpe Lie algebras \(\mathfrak g\), such modules exist only if \(\mathfrak g\) is of type \(A\) or \(C\). The heart of the paper is therefore an explicit classification of cuspidal modules for Lie algebras of type \(A\) and \(C\). The central technique is the study of coherent families. Write \(\mathcal A\) for the commutant of the Cartan subalgebra in the universal enveloping algebra of \(\mathfrak g\). A coherent family of degree \(d\) is a weight module \(\mathcal M\) such that each weight space, \({\mathcal M}_\lambda\), has dimension \(d\) and the function \(\lambda\rightarrow Tr(u|_{{\mathcal M}_\lambda})\) is a polynomial in \(\lambda\) for each \(u\in\mathcal A\). For any \(t\) in the dual torus, write \({\mathcal M}[t]\) for the \(\mathfrak g\)-submodule \(\bigoplus_{\lambda\in t} {\mathcal M}_\lambda\). The author shows that any cuspidal module is isomorphic to \({\mathcal M}[t]\) for a unique \(t\) in the dual torus and a unique irreducible semi-simple coherent family \(\mathcal M\). In turn, any irreducible semi-simple coherent family \(\mathcal M\) contains certain infinite dimensional simple highest weight modules that, in fact, determine \(\mathcal M\). Such highest weight modules are in fact admissible, { i.e.}, the weight multiplicities are uniformly bounded. Therefore the author first classifies all admissible highest weight modules and then uses this information to classify irreducible semi-simple coherent families. The last step is to identify all cuspidal submodules from each of these coherent families. It turns out that the complement of this set corresponds to a union of finitely many codimension one cosets of the dual torus. The author also describes concrete realizations of the coherent families and calculates the character of any simple weight module.
scientific article; zbMATH DE number 1448500
scientific article