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arXiv · 2407.17366

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

Tom H. Koornwinder·Marta Mazzocco
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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.

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Tom H. Koornwinder, Marta Mazzocco. 2024-07-24. Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions. https://doi.org/10.1016/j.indag.2025.05.005

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