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  (the affine Weyl group). 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 (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. References

Algebras
Representation theory