  Beginning with the data of a quiver Q, and its dimension vector d, we
construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the
algebra of differential operators on the affine space Mat_d(Q). The algebra D_q
is equivariant for an action by a product of quantum general linear groups,
acting by conjugation at each vertex. We construct a quantum moment map for
this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of
D_q with moment parameter \lambda. We show that A^\lambda_d(Q) is a flat formal
deformation of Lusztig's quiver varieties, and their multiplicative
counterparts, for all dimension vectors satisfying a flatness condition of
Crawley-Boevey: indeed the product on A^\lambda_d(Q) yields a Fedosov
quantization the of symplectic structure on multiplicative quiver varieties. As
an application, we give a description of the category of representations of the
spherical double affine Hecke algebra of type A_{n-1}, and its generalization
constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the
category of equivariant D_q-modules by a Serre sub-category of aspherical
modules.
