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 $\omega_{\pm}$. 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 $\widetilde{q}$-integral identities due to Macdonald, where the deformation parameter $\widetilde{q}=\exp(-2\pi i\omega_-/\omega_+)$ is related to $q$ by modular inversion.

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Published 1 January 2008