SPEAKER: Nicholas Proudfoot

TITLE: Quantizations of conical symplectic resolutions

ABSTRACT: The most studied example of a conical symplectic resolution is the cotangent bundle M of the flag manifold G/B, which resolves the nilpotent cone in Lie(G). Much of what goes under the name "geometric representation theory" is the study of this resolution, called the Springer resolution. Here are two cool features of this subject: - If you construct a deformation quantization of M and take global sections, you get the ring of global (twisted) differential operators on the flag variety, which is isomorphic to a central quotient of the universal enveloping algebra of Lie(G). This allows you to study representations of Lie(G) in terms of sheaves on M. - There is a natural action of "convolution operators" on the cohomology of M which provides a geometric construction of the regular representation of the Weyl group of G. This action can be promoted to a braid group action on a category by replacing cohomology classes with sheaves. I will make the case that these two phenomena fit neatly into a theory that applies to arbitrary conical symplectic resolutions, including (for example) quiver varieties, hypertoric varieties, and Hilbert schemes of points on ALE spaces. This is joint work with Braden, Licata, and Webster