- Affine Hecke algebra
**Definition**Let $V$ be a Euclidean space of a finite dimension and $Sigma$ an

affine root system on $V$. An**affine Hecke algebra**is a certainassociative algebra that deforms thegroup algebra $mathbb\{C\}\; [W]$ of theWeyl group $W$ of $Sigma$ (theaffine Weyl group ). It is usually denoted by $H(Sigma,q)$, where $q:Sigma\; ightarrow\; mathbb\{C\}$ ismultiplicity function that plays the role of deformation parameter. For $qequiv\; 1$ the affine Hecke algebra $H(Sigma,q)$ indeed reduces to $mathbb\{C\}\; [W]$.**Generalizations**Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-calleddouble affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a prove of Macdonald's constant term conjecture forMacdonald polynomial s (building on work ofEric Opdam ). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ-equations.**References***Iwahori, Nagayoshi; Matsumoto, Hideya [

*http://www.numdam.org/item?id=PMIHES_1965__25__5_0 "On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups."*] Publications Mathématiques de l'IHÉS, 25 (1965), p. 5-48

*Kazhdan, David, Lusztig, George "Proof of the Deligne-Langlands conjecture for Hecke algebras." Invent. Math. 87 (1987), no. 1, 153--215. MathSciNet|id=88d:11121

*A. A. Kirillov [*http://www.ams.org/bull/1997-34-03/S0273-0979-97-00727-1/home.html Lectures on affine Hecke algebras and Macdonald's conjectures*] Bull. Amer. Math. Soc. 34 (1997), 251-292.

*Lusztig, George "Notes on affine Hecke algebras." Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999), 71--103, Lecture Notes in Math., 1804, Springer, Berlin, 2002. MathSciNet|id=2004d:20006

*G. Lusztig [*http://arXiv.org/math.RT/0108172 "Lectures on affine Hecke algebras with unequal parameters"*]

*Macdonald, I. G. "Affine Hecke algebras and orthogonal polynomials." Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, 2003. x+175 pp. ISBN 0-521-82472-9 DOI|10.2277/0521824729 MathSciNet|id=2005b:33021

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