# Affine Hecke algebra

﻿
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 $mathbb\left\{C\right\} \left[W\right]$ of the Weyl group $W$ of $Sigma$ (the affine Weyl group). It is usually denoted by $H\left(Sigma,q\right)$, where $q:Sigma ightarrow mathbb\left\{C\right\}$ is multiplicity function that plays the role of deformation parameter. For $qequiv 1$ the affine Hecke algebra $H\left(Sigma,q\right)$ indeed reduces to $mathbb\left\{C\right\} \left[W\right]$.

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Double affine Hecke algebra — In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik,… …   Wikipedia

• Hecke algebra — is the common name of several related types of associative rings in algebra and representation theory. The most familiar of these is the Hecke algebra of a Coxeter group , also known as Iwahori Hecke algebra, which is a one parameter deformation… …   Wikipedia

• Macdonald polynomial — In mathematics, Macdonald polynomials P λ are a two parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987). They generalize several other families of orthogonal… …   Wikipedia

• Yangian — is an important structure in modern representation theory, a type of a quantum group with origins in physics. Yangians first appeared in the work of Ludvig Faddeev and his school concerning the quantum inverse scattering method in the late 1970s… …   Wikipedia

• Kazhdan–Lusztig polynomial — In representation theory, a Kazhdan–Lusztig polynomial P y,w ( q ) is a member of a family of integral polynomials introduced in work of David Kazhdan and George Lusztig Harv|Kazhdan|Lusztig|1979. They are indexed by pairs of elements y , w of a… …   Wikipedia

• List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

• Building (mathematics) — In mathematics, a building (also Tits building, Bruhat–Tits building) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.… …   Wikipedia

• N!-conjecture — In mathematics, the n! conjecture is the conjecture that the dimension of a certain bi graded module of diagonal harmonics is n!. It was made by A. M. Garsia and proved by M. Haiman. It implies Macdonald s positivity conjecture about his… …   Wikipedia

• Macdonald polynomials — In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal polynomials in several variables, introduced by Macdonald (1987). Macdonald originally associated his polynomials with weights λ of finite root systems and used just …   Wikipedia

• Ian Macdonald — Ian Grant Macdonald (* 11. Oktober 1928 in London) ist ein englischer Mathematiker. Ian Macdonald in Oberwolfach 1977 Inhaltsverzeichnis 1 Leben …   Deutsch Wikipedia