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

SIGMA 4 (2008), 052, 17 pages      arXiv:0711.2320
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra

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

Received November 15, 2007, in final form June 03, 2008; Published online June 10, 2008

This paper builds on the previous paper by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra.

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

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  1. Berest Y., Etingof P., Ginzburg V., Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279-337, math.QA/0111005.
  2. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. 1992 (1992), no. 9, 171-180.
  3. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, math.AG/0011114.
  4. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Cambridge University Press, 2004.
  5. Granovski Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  6. 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,
  7. Koornwinder T.H., The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063, 15 pages, math.QA/0612730.
  8. Macdonald I.G., Affine Hecke algebra and orthogonal polynomials, Cambridge University Press, 2003.
  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. Oblomkov A., Double affine Hecke algebras of rank 1 and affine cubic surfaces, Int. Math. Res. Not. 2004 (2004), 877-912, math.RT/0306393.
  11. Opdam E.M., Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1-18.
  12. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150 (1999), 267-282, q-alg/9710032.
  13. Sahi S., Some properties of Koornwinder polynomials, in q-Series from a Contemporary Perspective, Contemp. Math. 254 (2000), 395-411.
  14. Sahi S., Raising and lowering operators for Askey-Wilson polynomials, SIGMA 3 (2007), 002, 11 pages, math.QA/0701134.
  15. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.

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