Quantum Groups and Symplectic Reductions

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra ${\mathfrak{d}}_{\mathfrak{g},V}$ and the associated affine Kac-Moody algebra. In this paper, we show that this Lie algebra can be made into a sheaf of Lie algebras over $T^{\ast }[V\mathrm{/}G]=[{\mu }^{-1}(0)\mathrm{/}G]$, where $\mu :T^{\ast }V\to {\mathfrak{g}}^{\ast }$ is the moment map. We identify this sheaf of Lie algebras with the tangent Lie algebra of the stack $T^{\ast }[V\mathrm{/}G]$. Moreover, we show that there is an equivalence of braided tensor categories between the bounded derived category of graded modules of ${\mathfrak{d}}_{\mathfrak{g},V}$ and graded perfect complexes of $[{\mu }^{-1}(0)\mathrm{/}G]$.