Automorphisms of the DAHA of type Cˇ₁C₁ and non-symmetric Askey–Wilson functions

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type Cˇ₁C ₁ which have a relatively simple action on the generators and on the parameters, notably a symmetry t 4 which sends the Askey–Wilson (AW) 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 AW polynomials and functions. Interestingly t₄ 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.