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- Their group rings have quotients called affine Hecke algebras.
- They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
- Representation theory of affine Hecke algebras was developed by Lusztig with a view towards applying it to description of representations of p-adic groups.
- It has been conjectured that all Hecke algebras that are ever needed are mild generalizations of affine Hecke algebras.
- They are known to have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
- Macdonald's conjectures were proved by using doubly affine Hecke algebras.
- Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ equations.
- He introduced double affine Hecke algebras, and used them to prove Macdonald's constant term conjecture in, He has also dealt with algebraic geometry, number theory and Soliton equations.
- Again, these were proved for general reduced root systems by, using double affine Hecke algebras, with the extension to the BC case following shortly thereafter via work of van Diejen, Noumi, and Sahi.
- Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras.
- Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra (usually referred to as DAHA).
- It turned out that the representation theory of quantum groups, modular Lie algebras and affine Hecke algebras are all tightly controlled by appropriate analogues of Kazhdan-Lusztig polynomials.
- A generalization of affine Hecke algebras, called double affine Hecke algebra, was used by Ivan Cherednik in his proof of the Macdonald conjectures.
- 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 polynomials.
- 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.
- This was conjectured by Macdonald (1982) as a generalization of the Dyson conjecture, and proved for all (reduced) root systems by Cherednik (1995) using properties of double affine Hecke algebras.
- Drinfeld also discovered a generalization of the classical Schur-Weyl duality between representations of general linear and symmetric groups that involves the Yangian of sl N and the degenerate affine Hecke algebra (graded Hecke algebra of type A, in George Lusztig's terminology).