Grant: Discovery Grants Program - Individual

As indicated  in the section on contributions, the history of the classification of simple infinite dimensional modules with a finite dimensional weight space is long and has several contributors inculding marjor contributions by Suran Fernando and Olivier Mathieu. We continue to write a monograph to put the solution in one source.   We hope to bring to a definite conclusion the work that we began with our graduate student on a basis-operator presentation of simple torsion free modules of  finite degree when the lie algebra is of type A.  We, with Lemire and Molev,  have begun to study A. Molev's construction of  Gel'fand-Tseitlin type basis  for simple finite dimensional modules for algebras of type C with the aim of modifing this construction to produce a basis for torsion freee modules for these algebras. We seek to show that a Chari-Pressley type construction yields all torsion free modules of finite degree for affine Lie algebras. Although we believe that we have a reasonable approach, via Roa's, construction,  to solving this there are technical difficulties to over come. Our work on Generalized Verma Modules with S. Futorny should lend itself to answering the question of when is a Generalized Verma Modules as described in Contributisions contained in another and other such lattice questions whose answer is know in the Classical Verma case. Moreover, it might lend itself to obtaining multiplicity formulas for simple infinite dimenional modules with a finite dimension weight space. There is only a jerm of an idea here but it might bear fruit.