SIGMA 3 (2007), 063, 15 pages      math.QA/0612730      https://doi.org/10.3842/SIGMA.2007.063
Contribution to the Vadim Kuznetsov Memorial Issue

The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case

Tom H. Koornwinder
Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Received December 22, 2006, in final form April 23, 2007; Published online April 27, 2007
A slight error in formula (2.8) for Q₀ is corrected November 07, 2007

Abstract
Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faithful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to symmetric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.

Key words: Zhedanov's algebra AW(3); double affine Hecke algebra in rank one; Askey-Wilson polynomials; non-symmetric Askey-Wilson polynomials.

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References

1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. (1985), no. 319.
2. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. (1992), no. 9, 171-180.
3. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Cambridge University Press, 2004.
4. Grünbaum F.A., Haine L., On a q-analogue of the string equation and a generalization of the classical orthogonal polynomials, in Algebraic Methods and q-Special Functions, Editors J.F. van Diejen and L. Vinet, CRM Proc. Lecture Notes, Vol. 22, Amer. Math. Soc., 1999, 171-181.
5. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1998, http://aw.twi.tudelft.nl/~koekoek/askey/.
6. Koornwinder T.H., The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math. (2007), article in press, doi:10.1016/j.cam.2006.10.015, math.CA/0601303.
7. Macdonald I.G., Affine Hecke algebra and orthogonal polynomials, Cambridge University Press, 2003.
8. NCAlgebra: a "Non Commutative Algebra" package running under Mathematica, http://www.math.ucsd.edu/~ncalg/.
9. Noumi M., Stokman J.V., Askey-Wilson polynomials: an affine Hecke algebraic approach, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Nova Sci. Publ., Hauppauge, NY, 2004, 111-144, math.QA/0001033.
10. Rains E.M., A difference integral representation of Koornwinder polynomials, in Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math. 417 (2006), 319-333, math.CA/0409437.
11. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267-282, q-alg/9710032.
12. Sahi S., Some properties of Koornwinder polynomials, in q-Series from a Contemporary Perspective, Contemp. Math. 254 (2000), 395-411.
13. Sahi S., Raising and lowering operators for Askey-Wilson polynomials, SIGMA 3 (2007), 002, 11 pages, math.QA/0701134.
14. Stokman J.V., Koornwinder polynomials and affine Hecke algebras, Int. Math. Res. Not. (2000), no. 19, 1005-1042, math.QA/0002090.
15. Terwilliger P., Vidunas R., Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), 411-426, math.QA/0305356.
16. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.