Nonsymmetric Koornwinder Polynomials and Duality

Abstract

In the fundamental work of Lusztig [L] on affine Hecke algebras, a special role is played by the root system of type Cn. The affine Hecke algebra is a deformation of the group algebra of an affine Weyl group which usually depends on as many parameters as there are distinct root lengths, i.e. one or two for an irreducible root system. However in the Cn case, the Hecke algebra H has three parameters, corresponding to the fact that there is a simple coroot which is divisible by 2. Recently, Cherednik [C1]-[C3] has introduced the notion of a double affine Hecke algebra, and has used it to prove several conjectures on Macdonald polynomials. These polynomials, and Cherednik's double affine Hecke algebra, involve two or three parameters, i.e., one more than the number of root lengths.