Automorphisms of DAHA type check C1 C1 map Askey-Wilson polynomials to functions and produce a symmetric plus anti-symmetric expression for the non-symmetric Askey-Wilson function.
Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
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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.
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Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions
Automorphisms of DAHA type check C1 C1 map Askey-Wilson polynomials to functions and produce a symmetric plus anti-symmetric expression for the non-symmetric Askey-Wilson function.