Affine Hecke algebra

In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.

Definition

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

• Cherednik, Ivan (2005). Double affine Hecke algebras. London Mathematical Society Lecture Note Series. 319. Cambridge University Press. ISBN 978-0-521-60918-0. 
• Nagayoshi, Iwahori; Hideya, Matsumoto (1965). "On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups". Publications Mathématiques de l'IHÉS 25: 5–48. doi:10.1007/bf02684396. http://www.numdam.org/item?id=PMIHES_1965__25__5_0. 
• Kazhdan, David; Lusztig, George (1987). "Proof of the Deligne-Langlands conjecture for Hecke algebras". Inventiones Mathematicae 87 (1): 153–21. doi:10.1007/BF01389157. Bibcode: 1987InMat..87..153K. 
• Kirillov, Alexander A. Jr (1997). "Lectures on affine Hecke algebras and Macdonald's conjectures". Bulletin of the American Mathematical Society 34 (3): 251–292. doi:10.1090/S0273-0979-97-00727-1. https://www.ams.org/bull/1997-34-03/S0273-0979-97-00727-1/home.html. 
• Lusztig, George. "Notes on affine Hecke algebras". in Cherednik, Ivan; Markov, Yavor; Howe, Roger et al.. Iwahori-Hecke Algebras and their Representation Theory: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, June 28 - July 6, 1999. Lecture Notes in Mathematics. 1804. pp. 71–103. doi:10.1007/978-3-540-36205-0_3. 
• Lusztig, George (2001). "Lectures on affine Hecke algebras with unequal parameters". arXiv:math.RT/0108172.
• Macdonald, I. G. (2003). Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics. 157. Cambridge University Press. ISBN 0-521-82472-9.