Automorphisms of the DAHA of type $\check{C_1}C_1$ and their action on Askey-Wilson polynomials and functions. I. The flip $(a,b,c,d)\mapsto(a,b,qd^{-1},qc^{-1})$

Tom H. Koornwinder, Marta Mazzocco

Published 2024-07-24Version 1

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 $(a,b,qd^{-1},qc^{-1})$. 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. In the second part of the paper we look for a version of non-symmetric AW functions that behave well under these symmetries.