Macdonald polynomials are associated with an irreducible affine root system, and are of two types: symmetric and nonsymmetric. The former are orthogonal polynomials with rational function coefficients in q,t, which are invariant under the corresponding finite Weyl group; upon setting q=t=0, they specialize to the irreducible characters of semisimple Lie algebras, in particular to Schur polynomials in type A. Macdonald polynomials have deep connections with: double affine Hecke algebras (DAHA), p-adic groups, integrable systems, conformal field theory, statistical mechanics, Hilbert schemes etc. This series of lectures will explore two closely related sides of the story of Macdonald polynomials: their central role in the representation theory of affine Lie algebras, and combinatorial constructions.

I will start with the definition of Macdonald polynomials, and their construction in terms of the DAHA. I will continue with two classes of combinatorial formulas for Macdonald polynomials and the connection between them: the type-independent Ram-Yip formula, based on the so-called alcove model, and tableau formulas in classical types. Then I will discuss the way in which various specializations of Macdonald polynomials occur in representation theory, particularly as graded characters of certain modules for affine Lie algebras (Demazure modules, Kirillov-Reshetikhin modules, and several variations of them). The mentioned alcove model leads to a combinatorial model for the corresponding Kashiwara crystals; these are colored directed graphs encoding representations of quantum algebras in the limit of the quantum parameter going to 0. I will conclude with several recent developments in the area.

The lectures contain joint work with my collaborators: Satoshi Naito, Daisuke Sagaki, Anne Schilling, Travis Scrimshaw, and Mark Shimozono, as well as my students Arthur Lubovsky and Adam Schultze. They will be largely self-contained, and only basic knowledge of the representation theory of Lie algebras is assumed.