Zamolodchikov Equations,” Double Affine Hecke Algebras and Bispectral Quantum Knizhnik–Zamolodchikov Equations

We use the double affine Hecke algebra of type GLN to construct an explicit consis-tent 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 of BqKZ which, upon suitable specializations of the central character, reduces to sym-metric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispec-tral problem for Ruijsenaars ’ commuting trigonometric q-difference operators. Under this correspondence, becomes a self-dual Harish-Chandra series solution + of the bispectral problem. Specializing the central character as above, we recover from + the symmetric self-dual Macdonald polynomials.