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

Abstract

This paper builds on the previous paper arXiv:math/0612730 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.

This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/ In v2 remarks about duality anti-algebra isomorphism and about shift operators added

References in corpus (3)

• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Double affine Hecke algebras of rank 1 and affine cubic surfaces
• Raising and Lowering Operators for Askey-Wilson Polynomials

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• The Universal Askey-Wilson Algebra
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• Totally bipartite tridiagonal pairs
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• The universal DAHA of type $(C_1^{\vee },C_1)$ and Leonard pairs of $q$-Racah type
• On the shape of a tridiagonal pair
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions
• An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform