Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions

Abstract

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}{C}_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,q{d}^{-1},q{c}^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. We take the rank one case of Stokman's Cherednik kernel for $B{C}_n$ as the definition of the non-symmetric Askey--Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term.

v5: 39 pages; errors on p.7 (line 6 and first diagram) and in formulas (4.9), (6.2), (6.3), (7.24), (7.25), still present in published version in Indag. Math., have neen corrected here

References in corpus (4)

• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• The Askey-Wilson algebra and its avatars