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 certain associative algebra that deforms the group algebra $mathbb\{C\}\; [W]$ of the Weyl group $W$ of $Sigma$ (the affine Weyl group). It is usually denoted by $H(Sigma,q)$, where $q:Sigma\; ightarrow\; mathbb\{C\}$ is multiplicity 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-called double affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a prove 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

*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