Quantized multiplicative quiver varieties

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Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra Dq=Dq(Matd(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Matd(Q). The algebra Dq 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 Aˡambdad(Q) of Dq with moment parameter λ. We show that Aλd(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 Aλd(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 Aₙ₋₁, and its generalization constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the category of equivariant Dq-modules by a Serre sub-category of aspherical modules.