mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an affine Hecke algebra is the

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

associated to an

affine Weyl group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflec ...

, and can be used to prove

Macdonald's constant term conjecture In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald orig ...

for

Macdonald polynomial In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald orig ...

Definition

Let

V

be a Euclidean space of a finite dimension and

\Sigma

affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple ''p''-adic algebraic groups, and correspond to fa ...

V

. An affine Hecke algebra is a certain

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

that deforms the group algebra

\mathbb /math> of the

Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...

W

\Sigma

(the

). It is usually denoted by

H(\Sigma,q)

, where

q:\Sigma\rightarrow \mathbb

is multiplicity function that plays the role of deformation parameter. For

q\equiv 1

the affine Hecke algebra

H(\Sigma,q)

indeed reduces to

\mathbb /math>.

Generalizations

Ivan Cherednik Ivan Cherednik (Иван Владимирович Чередник) is a Russian-American mathematician. He introduced double affine Hecke algebras, and used them to prove Macdonald's constant term conjecture in . He has also dealt with algebrai ...

introduced generalizations of affine Hecke algebras, the so-called

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 ...

(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 Eric Marcus Opdam (born 1960) is a Dutch mathematician, specializing in algebra and harmonic analysis. He is one of the two namesakes of Heckman–Opdam polynomials. Opdam received his PhD from Leiden University in 1988 under the supervision of G ...

). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the

quantum KZ equations In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a co ...

References

* * * * * * *{{cite book , last1=Macdonald , first1=I. G. , year=2003 , title=Affine Hecke Algebras and Orthogonal Polynomials , series=

Cambridge Tracts in Mathematics Cambridge ( ) is a city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 United Kingdom census, the population of t ...

, volume=157 , publisher=

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, isbn=0-521-82472-9 , mr=1976581 Algebras Representation theory