Representations of finite W-algebras

by Simon Goodwin, University of Birmingham.

To each nilpotent element e in a complex semisimple Lie algebra \g, one can associate a finite W-algebra denoted by U(\g,e). This algebra can be viewed as the enveloping of the Slodowy slice through the adjoint orbit of e, and has many connections to other areas of Lie theory. After presenting some history and motivation we will present an approach, due to Brundan, Kleshchev and the author, to highest weight representation theory of finite W-algebras. There is not a natural comultiplication on finite W-algebras; however, it is possible to give the tensor product of a U(\g,e)-module with a finite dimensional U(\g)-module the structure of a U(\g,e)-module. We will discuss properties of these tensor products, which are expected to be of importance in understanding the representation theory of U(\g,e).