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Double Hecke algebras or double affine Hecke algebras (DAHA) or __Cherednik algebra__s are a particular class of 2-parametric families of associative algebras. These families are flat deformations of certain crossed product algebras involving Coxeter groups. Ivan Cherednik introduced DAHA in proving Macdonald conjectures about orthogonal polynomials attached to reduced root systems.
I should add the following remark to our discussion in the other thread: while there is no standard mathematical difference in saying double Hecke algebra versus double affine Hecke algebra there is of course a difference between saying Hecke algebra and affine Hecke algebra. The latter maybe deserves a new entry.
Notice that rational Cherednik algebras defined by Etingof and Ginzburg are not literally DAHAs, but certain degenerations of those and can be considered a special case of symplectic reflection algebras.
Edit: for this reason I have moved some of the references into the new entry rational Cherednik algebra.
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