2022/11/24 by Kohei Yamaguchi, Yamaguchi, Kohei, Shintarou Yanagida +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Quantum Mechanics and Non-Hermitian Physics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2211.13671
openalex publication_date 2022/11/24 · openalex created_date 2022/11/30 · openalex updated_date 2026/07/28
This note consists of two parts. The first part (§1 and §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 A1 and the non-reduced type (C1^\vee,C1), 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 (§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 (C1^\vee,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.