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In mathematics, an affine Hecke algebra is the algebra 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 refle ...
, 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 origi ...
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 origi ...
s.


Definition

Let V be a Euclidean space of a finite dimension and \Sigma an
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 f ...
on V. An affine Hecke algebra is a certain
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
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 th ...
W of \Sigma (the
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 refle ...
). 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 algebr ...
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, wh ...
(usually referred to as DAHA). Using this he was able to give a proof of Macdonald's constant term conjecture 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 origi ...
s (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 university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge became ...
, volume=157 , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, isbn=0-521-82472-9 , mr=1976581 Algebras Representation theory