Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations

TL;DR

This paper constructs and analyzes the bispectral quantum KZ equations using double affine Hecke algebras, providing explicit solutions that connect to Macdonald polynomials and bispectral problems.

Contribution

It introduces the bispectral quantum KZ equations within the double affine Hecke algebra framework and constructs explicit self-dual solutions linking to Macdonald polynomials.

Findings

Constructed explicit q-difference equations called BqKZ.

Established a correspondence between BqKZ solutions and bispectral problems.

Recovered Macdonald polynomials as special solutions.

Abstract

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik's quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution \Phi of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars' commuting trigonometric q-difference operators. Under this correspondence \Phi becomes a self-dual Harish-Chandra series solution \Phi^+ of the bispectral problem. Specializing the central character as above, we recover from \Phi^+ the symmetric self-dual Macdonald polynomials.