Modules with bounded weight multiplicities for simple Lie algebras (Q1360062)

Let \({\mathfrak g}\) be a finite-dimensional simple Lie algebra over the complex numbers and \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\). Our objects of study are infinite-dimensional simple \({\mathfrak g}\)-modules \(M\) having a weight space decomposition \(M=\oplus \sum_{\mu\in{\mathfrak h}^*} M_\mu\) relative to \({\mathfrak h}\) whose weight spaces have bounded multiplicity. Thus, we assume there is some integer \(d\) such that \(\dim M_\mu\leq d\) for all \(\mu\). We argue that \({\mathfrak g}\) must be of type \(A_n\) or \(C_n\) to possess infinite-dimensional simple modules with bounded weight multiplicities, and in the course of the proof we show that the Gelfand-Kirillov dimension of such a module \(M\) must equal the rank \(n\) of the Lie algebra \({\mathfrak g}\). We then determine the \({\mathfrak g}\)-modules whose weight spaces are all one-dimensional. Results on modules for Weyl algebras enable us to explicitly realize all infinite-dimensional simple \({\mathfrak g}\)-modules with one-dimensional weight spaces in terms of multiplication and differentiation operators on ``polynomials''. In particular, all these modules arise from certain modules for Weyl algebras which we construct.