Representations of Gan-Ginzburg algebras

TL;DR

This paper constructs a functor linking D-modules on quiver representation spaces to modules over Gan-Ginzburg algebras, providing explicit representations for affine Dynkin and star-shaped quivers, generalizing previous functors.

Contribution

It introduces a new functor connecting D-modules and Gan-Ginzburg algebras, extending prior type A and BC functors to broader quiver classes.

Findings

Explicit construction of representations for affine Dynkin quivers.

Lie theoretic construction of rational generalized double affine Hecke algebra representations.

Generalization of existing functors to more complex quiver types.

Abstract

Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of rank n. When the quiver is affine Dynkin we obtain an explicit construction of representations of the corresponding wreath-product symplectic reflection algebra of rank n. When the quiver is star-shaped, but not finite Dynkin, we use this functor to obtain a Lie theoretic construction of representations of a "spherical" subalgebra of the Gan-Ginzburg algebra isomorphic to a rational generalized double affine Hecke algebra of rank n. Our functors are a generalization of the type A and type BC functors from arXiv:math/0702670 and arXiv:0801.1530 respectively.