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

TL;DR

This paper demonstrates that the spherical subalgebra of the double affine Hecke algebra related to Askey-Wilson polynomials is isomorphic to Zhedanov's algebra AW(3) with a specific Casimir relation, extending previous work.

Contribution

It establishes an explicit isomorphism between the spherical subalgebra of DAHA and AW(3) with a Casimir condition, clarifying their algebraic relationship.

Findings

Spherical subalgebra of DAHA is isomorphic to AW(3) with Casimir fixed.

Antispherical subalgebra has a similar isomorphism with shifted parameters.

Derived theorems on centralizers and centers from this algebraic characterization.

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.