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

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.