Quantum D-modules, Elliptic Braid Groups and Double Affine Hecke Algebras

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U . The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid [Ly], [LyMa], and also certain geometric constructions of Calaque, Enriquez, and Etingof [CEE] concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U = Ut(slN ), as t → 1. In the latter case, we produce representations of the double affine Hecke algebra of type An−1, for each n.