Automorphisms of the DAHA of type \checkC₁C₁ and non-symmetric Askey-Wilson functions

2024/07/24 by Tom H. Koornwinder, Koornwinder, Tom H., Marta Mazzocco +1
Computer Science · Mathematics · #16S99 #20C08 #33D45 #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2407.17366

openalex publication_date 2024/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type \checkC₁C₁ which have a relatively simple action on the generators and on the parameters, notably a symmetry t₄ 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 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.

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