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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 polynomial]]s.

==Definition==
Let <math>V</math> be a Euclidean space of a finite dimension and <math>\Sigma</math> an [[affine root system]] on <math>V</math>. An '''affine Hecke algebra''' is a certain [[associative algebra]] that deforms the [[group algebra]] <math>\mathbb{C}[W]</math> of the [[Weyl group]] <math> W</math> of <math>\Sigma</math> (the [[affine Weyl group]]). It is usually denoted by <math> H(\Sigma,q)</math>, where <math>q:\Sigma\rightarrow \mathbb{C}</math> is [[multiplicity function]] that plays the role of deformation parameter. For <math>q\equiv 1</math> the affine Hecke algebra <math> H(\Sigma,q)</math> indeed reduces to <math>\mathbb{C}[W]</math>.

==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 polynomial]]s (building on work of [[Eric Opdam]]). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the [[quantum KZ equations]].

==References==
*{{Cite journal| last1=Cherednik | first1=Ivan | year=2005 | title=Double affine Hecke algebras | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | volume=319 | isbn=978-0-521-60918-0 | mr=2133033}}
*{{cite journal |last1=Nagayoshi |first1=Iwahori |last2=Hideya |first2=Matsumoto |year=1965 |title=On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups |url=http://www.numdam.org/item?id=PMIHES_1965__25__5_0 |journal=[[Publications Mathématiques de l'IHÉS]] |volume=25 |pages=5–48 |mr=185016 |zbl=0228.20015 }}
*{{cite journal |last1=Kazhdan |first1=David |last2=Lusztig |first2=George |year=1987 |title=Proof of the Deligne-Langlands conjecture for Hecke algebras |journal=[[Inventiones Mathematicae]] |volume=87 |issue=1 |pages=153–21 |mr=862716 |doi=10.1007/BF01389157 }}
*{{cite journal |last1=Kirillov |first1=Alexander A., Jr |year=1997 |url=http://www.ams.org/bull/1997-34-03/S0273-0979-97-00727-1/home.html |title=Lectures on affine Hecke algebras and Macdonald's conjectures |journal=[[Bulletin of the American Mathematical Society]] |volume=34 |pages=251–292 |doi=10.1090/S0273-0979-97-00727-1 |mr=1441642 |issue=3}}
*{{cite journal |last1=Lusztig |first1=George |title=Notes on affine Hecke algebras |journal=[[Lecture Notes in Mathematics]] |volume=1804 |pages=71–103 |doi=10.1007/978-3-540-36205-0_3 |mr=1979925}}
*{{cite arxiv |last1=Lusztig |first1=George |year=2001 |title=Lectures on affine Hecke algebras with unequal parameters |class=math.RT |eprint=math.RT/0108172}}
*{{cite book |last1=Macdonald |first1=I. G. |year=2003 |title=Affine Hecke Algebras and Orthogonal Polynomials |series=[[Cambridge Tracts in Mathematics]] |volume=157 |publisher=[[Cambridge University Press]] |doi=10.2277/0521824729 |isbn=0-521-82472-9 |mr=1976581}}

[[Category:Algebras]]
[[Category:Representation theory]]

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en>Cydebot: Robot - Speedily moving category Air dispersion modeling to :Category:Atmospheric dispersion modeling per CFDS.