Quantized multiplicative quiver varieties

TL;DR

This paper constructs a quantized algebra related to quiver varieties, showing it as a flat deformation of classical structures, and applies it to describe representations of certain Hecke algebras.

Contribution

It introduces a new algebra D_q as a flat q-deformation of differential operators on quiver varieties and relates its Hamiltonian reduction to multiplicative quiver varieties, providing a novel quantization framework.

Findings

A flat q-deformation algebra D_q constructed from quiver data.

The Hamiltonian reduction yields a Fedosov quantization of multiplicative quiver varieties.

Representation categories of DAHA are described via equivariant D_q-modules.

Abstract

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.