Some special functions associated to infinite root systems (Third Talk in the Series)
In the first part, we will consider representations of classical finite dimensional simple Lie algebras over the complex field. Dimensions of weight spaces of representations are key numbers of interest. They are determined by two pieces of data : a special function (the Kostant partition function) on the root lattice, and the action of a group (the Weyl group) on this lattice. We describe how to construct some ``polynomial analogues'' of weight multiplicities, and relate them to some invariant functions called Hall-Littlewood polynomials. These provide a continuous deformation between two other interesting bases of invariant functions - the Weyl characters and Weyl orbit sums.
In the second part, we define affine Kac-Moody Lie algebras. These are infinite dimensional Lie algebras can be obtained from the so called ``loop algebra'' over a finite dimensional Lie algebra. We describe their structure and representations. Finally, we will define the affine versions of Hall-Littlewood polynomials.
In the last part, we will study the behaviour of polynomial analogues of weight multiplicities along strings in irreducible representations of affine Lie algebras. We will obtain interesting generating functions for these in special cases. Finally, we relate these to the so called ``constant term identities'' that appear in Macdonald's theory of orthogonal polynomials associated to root systems.