A review of rank one bispectral correspondence of quantum affine KZ equations and Macdonald-type eigenvalue problems (Recent developments in Combinatorial Representation Theory)

Abstract

This note consists of two parts. The first part (§1 and §2) is a partial revie 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 𝘈1 and the non-reduced type (𝘊₁ᵛ, 𝘊₁), to which the associated MacdonaldKoornwinder 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 (§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 (𝘊₁ᵛ, 𝘊₁) parameters (which could be called the Askey-Wilson parameters) to recover the type 𝘈1. 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.