Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations

We use 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 Φ 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 Ruijsenaars’ commuting trigonometric q-difference operators. Under this correspondence, Φ becomes a self-dual Harish-Chandra series solution Φ+of the bispectral problem. Specializing the central character as above, we recover from Φ+the symmetric self-dual Macdonald polynomials.