Ben Wilson (University of Sydney)

Friday 10th March, 12.05-12.55pm, Carslaw 159

Verma modules for affine sl(2)

We review the notion of Verma modules for Kac-Moody Lie algebras, and proceed to survey the state-of-the-art of the theory of Verma modules for the most basic non-finite type case, affine sl(2). We shall see that all the modules of the theory are well understood, except for one -- the imaginary Verma module of level zero -- about which almost nothing is known.

Recent results in the study of this module shall then be presented. In particular, we will present a classification of the irreducible subquotients and a beautiful family of singular vectors. The classification follows from a realization in terms of the symmetric Laurent polynomials.

If time permits, we shall discuss a family of finite rank approximations of our module (rank in terms of another polynomial realization). These approximations were first constructed to facilitate the study of the original module using a computational algebra package, such as Magma, but have turned out to be of theoretical importance. The approximations yield a simple reproof of a difficult result of Futorny and may be used to construct finite dimensional modules for Kac-Moody Lie algebras.