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Quantization of the Marsden-Weinstein reduction for extended Dynkin quivers. (English) Zbl 1036.16024
Summary: Let \(\Gamma\) be a finite subgroup of \(\text{SL}_2(k)\), for \(k\) an algebraically closed field of characteristic zero. W. Crawley-Boevey and the author [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)] have introduced some noncommutative quantizations \({\mathcal O}^\lambda\) of the coordinate ring of the associated Kleinian singularity \(k^2/\Gamma\) indexed by those \(\lambda\) in \(Z(k\Gamma)\) which have trace one on the regular representation. Let \(Q\) be the quiver obtained by orienting the extended Dynkin graph associated to \(\Gamma\) by the McKay correspondence. Let \(\text{Rep}(Q,\delta)\) denote the space of representations of \(Q\) with dimension vector equal to the minimal imaginary root \(\delta\) of the corresponding affine root system. The group \(\text{GL}(\delta)=\prod_i\text{GL}(\delta_i)\) acts naturally on \(\text{Rep}(Q,\delta)\). It is shown that each \({\mathcal O}^\lambda\) may be realised as a certain quotient of the algebra of \(\text{GL}(\delta)\)-invariant differential operators on \(\text{Rep}(Q,\delta)\).

MSC:
16S32 Rings of differential operators (associative algebraic aspects)
16S30 Universal enveloping algebras of Lie algebras
16G20 Representations of quivers and partially ordered sets
16S80 Deformations of associative rings
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