Quantum Hamiltonian Reduction Ii: Sheaf Level

One of the main highlights of the previous semester was an interplay between the following objects: the nilpotent cone in g, the cotangent bundle T ∗(G/B), the universal enveloping algebra U(g) (or more precisely, its central reduction Uλ(g)) and the sheaf D G/B of λ-twisted differential operators on G/B. In our present story, Symn(C) is an analog of the nilpotent cone, Hilbn(C) is an analog of T ∗(G/B). Yi has used quantum Hamiltonian reduction to quantize an affine algebraic variety Symn(C) that is obtained by classical Hamiltonian reduction. The result of quantization is a filtered algebra that should be thought as an analog of Uλ(g). What I want to do in this talk is to quantize the Hilbert scheme Hilbn(C) getting an analog of D G/B. It is obtained by a GIT Hamiltonian reduction, similarly on one hand to Hilbn(C) and on the other hand to the quantum Hamiltonian reductions constructed by Yi. The variety Hilbn(C) is not affine and so is not given by a single algebra, rather by a sheaf of algebras. A quantization is therefore should also be a sheaf. We will start by describing this new formalism. First, we will slightly generalize the quantization formalism for graded algebras, instead of Z>0-graded algebras we will consider Z-graded ones. Then we will define a quantization of an algebraic symplectic variety (although the formalism makes sense for all Poisson schemes) that will be a sheaf. Next we will see how this formalism is compatible with the usual quantization formalism for affine algebraic varieties (i.e., for algebras). To pass from a quantum algebra to a quantum sheaf we will use a version of the localization known as a microlocalization. With this in hand, we will develop the formalism of GIT quantum Hamiltonian reduction needed to quantize Hilbn(C).