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Representations of Gan-Ginzburg algebras. (English) Zbl 1235.16015

Summary: 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 of D. Calaque, B. Enriquez and P. I. Etingof [Prog. Math. 269, 165–266 (2009; Zbl 1241.32011)] and P. Etingof, R. Freund and X. Ma [Represent. Theory 13, 33–49 (2009; Zbl 1171.20004)], respectively.

MSC:

16G20 Representations of quivers and partially ordered sets
16G10 Representations of associative Artinian rings
16S32 Rings of differential operators (associative algebraic aspects)
20C08 Hecke algebras and their representations
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References:

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