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We build representations of the elliptic braid group from the data of aquantum D-module M over a ribbon Hopf algebra U. The construction is modelledon, and generalizes, similar constructions by Lyubashenko and Majid, and alsocertain geometric constructions of Calaque, Enriquez, and Etingof concerningtrigonometric Cherednik algebras. In this context, the former construction isthe special case where M is the basic representation, while the latterconstruction can be recovered as a quasi-classical limit of U=U_t(sl_N), as tlimits 1. In the latter case, we produce representations of the double affineHecke algebra of type A_{n-1}, for each n.

Quantum D-modules, elliptic braid groups, and double affine Hecke algebras