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

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.