Affine Hecke algebra
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In mathematics, an affine Hecke algebra is the Hecke algebra of an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.
[edit] Definition
Let V be a Euclidean space of a finite dimension and Σ an affine root system on V. An affine Hecke algebra is a certain associative algebra that deforms the group algebra ![\mathbb{C}[W]](http://upload.wikimedia.org/math/f/3/2/f3211f3b114d6fc547da3a238d069742.png)
of the Weyl group W of Σ (the affine Weyl group). It is usually denoted by H(Σ,q), where 
is multiplicity function that plays the role of deformation parameter. For 
the affine Hecke algebra H(Σ,q) indeed reduces to .
[edit] 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.
[edit] References
- Cherednik, Ivan (2005), Double affine Hecke algebras, London Mathematical Society Lecture Note Series, 319, Cambridge University Press, MR2133033, ISBN 978-0-521-60918-0
- Iwahori, Nagayoshi; Matsumoto, Hideya 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. MR88d:11121
- A. A. Kirillov 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. MR2004d:20006
- G. Lusztig 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 MR2005b:33021
