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

TL;DR

This paper explores automorphisms of the DAHA of type heck{C_1}C_1, focusing on symmetries affecting Askey-Wilson polynomials and functions, revealing how these symmetries transform these special functions.

Contribution

It introduces and analyzes specific automorphisms of the DAHA of type heck{C_1}C_1, especially the symmetry t_4, and studies their impact on Askey-Wilson polynomials and functions.

Findings

Symmetry t_4 maps Askey-Wilson polynomials to functions.

Automorphisms induce a decomposition of the non-symmetric Askey-Wilson function.

Explicit expressions relate symmetric and anti-symmetric components.

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 $(a,b,qd^{-1},qc^{-1})$. 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.