A review of rank one bispectral correspondence of quantum affine KZ equations and Macdonald-type eigenvalue problems

TL;DR

This paper reviews the rank one bispectral correspondence between quantum affine KZ equations and Macdonald-type eigenvalue problems, providing detailed formulas and analyzing parameter specializations that preserve this correspondence.

Contribution

It offers a detailed review of rank one bispectral correspondence and identifies the unique parameter specialization compatible with this structure.

Findings

Detailed formulas for rank one cases of bispectral correspondence

Identification of a unique parameter specialization preserving bispectrality

Clarification of the relationship between Macdonald and Askey-Wilson polynomials

Abstract

This note consists of two parts. The first part (\S 1 and \S 2) is a partial review of the works by van Meer and Stokman (2010), van Meer (2011) and Stokman (2014) which established a bispectral analogue of the Cherednik correspondence between quantum affine Knizhnik-Zamolodchikov equations and the eigenvalue problems of Macdonald type. In this review we focus on the rank one cases, i.e., on the reduced type $A_1$ and the non-reduced type $(C_1^\vee,C_1)$, to which the associated Macdonald-Koornwinder polynomials are the Rogers polynomials and the Askey-Wilson polynomials, respectively. We give detailed computations and formulas that may be difficult to find in the literature. The second part (\S 3) is a complement of the first part, and is also a continuation of our previous study (Y.-Y., 2022) on the parameter specialization of Macdonald-Koornwinder polynomials, where we found four types of specialization of the type $(C_1^\vee,C_1)$ parameters (which could be called the Askey-Wilson parameters) to recover the type $A_1$. In this note, we show that among the four specializations there is only one which is compatible with the bispectral correspondence discussed in the first part.