Affine Hecke algebra


In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.

Definition

Let be a Euclidean space of a finite dimension and an affine root system on. An affine Hecke algebra is a certain associative algebra that deforms the group algebra of the Weyl group of . It is usually denoted by, where is multiplicity function that plays the role of deformation parameter. For the affine Hecke algebra indeed reduces to.

Generalizations

Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra. Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials. Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ equations.