Bispectral quantum Knizhnik-Zamolodchikov equations for arbitrary root systems

TL;DR

This paper generalizes the bispectral quantum Knizhnik-Zamolodchikov equations to arbitrary root systems, providing explicit solutions and linking them to Macdonald's $q$-difference operators, broadening their applicability.

Contribution

It extends the BqKZ equations from type A to arbitrary root systems and constructs explicit solutions, establishing a bispectral correspondence with Macdonald operators.

Findings

Extended BqKZ to arbitrary root systems.

Constructed explicit solutions of BqKZ.

Linked BqKZ solutions to Macdonald's $q$-difference operators.

Abstract

The bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra $H$ of type $A_{N-1}$ is a consistent system of $q$-difference equations which in some sense contains two families of Cherednik's quantum affine Knizhnik-Zamolodchikov equations for meromorphic functions with values in principal series representations of $H$. In this paper we extend this construction of BqKZ to the case where $H$ is the affine Hecke algebra associated to an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald's $q$-difference operators.