Quantization of the Marsden-Weinstein reduction for extended dynkin quivers

Abstract

Let Γ be a finite subgroup of SL 2 ( k), for k an algebraically closed field of characteristic zero. W. Crawley-Boevey and the author [8] have introduced some noncommutative quantizations O λ of the coordinate ring of the associated Kleinian singularity k 2/ Γ indexed by those λ in Z( kΓ) which have trace one on the regular representation. Let Q be the quiver obtained by orienting the extended Dynkin graph associated to Γ by the McKay correspondence. Let Rep( Q, δ) denote the space of representations of Q with dimension vector equal to the minimal imaginary root δ of the corresponding affine root system. The group GL( δ) = Π i GL( δ i ) acts naturally on Rep( Q, δ). It is shown that each O λ may be realised as a certain quotient of the algebra of GL(δ)-invariant differential operators on Rep( Q, δ).