Quantum Affine Knizhnik–Zamolodchikov Equations and Quantum Spherical Functions, I

Cherednik's quantum affine Knizhnik–Zamolodchikov (KZ) equations associated to an affine Hecke algebra module M form a holonomic system of q-difference equations acting on M-valued functions on a complex torus T. In this paper the quantum affine Knizhnik–Zamolodchikov equations are related to the Cherednik–Macdonald theory when M is induced from a character of a standard parabolic subalgebra of the affine Hecke algebra. We set up correspondences between solutions of the quantum affine KZ equations and, on the one hand, solutions to the spectral problem of the Cherednik–Dunkl q-difference reflection operators (generalizing work of Kasatani and Takeyama) and, on the other hand, solutions to the spectral problem of the Cherednik–Macdonald q-difference operators (generalizing work of Cherednik). The correspondences are applicable to all relevant spaces of functions on T and for all parameter values, including the cases that q and/or the Hecke algebra parameters are roots of unity.