Manchester Geometry Seminar 2000/2001

In 1988 I.G.Macdonald introduced a remarkable family of symmetric polynomials which attracted a huge attention over past decade. This family depends on two parameters $q$ and $t$ and specialises in various cases to Schur functions, Hall-Littlewood functions and Jack polynomials. This explains the resonance they had in representation theory, combinatorics and mathematical physics. It is quite natural to consider more general polynomials related to an arbitrary root system (or simple Lie algebra). Macdonald proposed a number of conjectures about these polynomials, some of them are open yet. A considerable progress in this theory came in 1994-95 with the works by I.Cherednik, who proved some of the Macdonald's conjectures using the so-called double affine Hecke algebras. In my talk I will give an introduction to Macdonald's theory and will present a new approach which helps to recover Cherednik's results on a more elementary level and leads to some new results in this area.

1. Macdonald's lecture at ICM98 in Berlin (available at http://www.math.uiuc.edu/documenta)
2. Nicely written Kirillov's survey of Cherednik's theory (Bull AMS, 1997, available at http://front.math.ucdavis.edu/math.QA/9501219)
3. Cherednik's papers in Invent.Math. (1994) and Ann.Math. (1995) (available at http://front.math.ucdavis.edu/q-alg/9503006)