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. We take the rank one case of Stokman's Cherednik kernel for $BC_n$ 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.
Categories: math.CA math.RT