Generalized Cherednik-Macdonald identities

We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual Multiplicity function, symmetrically on two additional parameters omega +/-. They are natural analogues of the Cherednik-Macdonald constant term q-identities in which the deformation parameter q = exp(2 pi i omega+/omega-) is allowed to have modulus one. They unite the Cherednik-Macdonald constant term q-identities with closely related Jackson (q) over tilde -integral identities due to Macdonald, where the deformation parameter (q) over tilde = exp (-2 pi i omega-/omega+) is related to q by modular inversion.