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 formula_1 be a Euclidean space of a finite dimension and formula_2 an affine root system on formula_1. An affine Hecke algebra is a certain associative algebra that deforms the group algebra formula_4 of the Weyl group formula_5 of formula_2 (the affine Weyl group). It is usually denoted by formula_7, where formula_8 is multiplicity function that plays the role of deformation parameter. For formula_9 the affine Hecke algebra formula_7 indeed reduces to formula_4.
Generalizations.
Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials (building on work of Eric Opdam). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ equations.