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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

Abstract: 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.

Keywords: Zhedanov's algebra $AW(3)$; double affine Hecke algebra in rank one; Askey–Wilson polynomials; spherical subalgebra.

MSC: 33D80

Received: November 15, 2007; in final form June 3, 2008; Published online June 10, 2008

Language: English

DOI: 10.3842/SIGMA.2008.052

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ArXiv: 0711.2320