Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 063, 15 pages      math.QA/0612730      http://dx.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.

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