Geometric category O and symplectic duality

The purpose of this proposal is to study algebraic symplectic varieties, which arise naturally in algebraic geometry (Hilbert schemes), representation theory (quiver varieties, Springer theory), combinatorics and polyhedral geometry (hypertoric varieties), and string theory (moduli spaces of gauge theories and of Higgs bundles). Our primary interest will be a certain category of sheaves on these varieties, which we introduce in Section 1.3. The center of this category will be isomorphic to the cohomology ring of the variety, and its Grothendieck group will be isomorphic to a deformation of this cohomology ring. When our variety is a quiver variety, these cohomology groups carry geometrically-defined actions of Kac-Moody algebras, and these actions lift to actions on our category by endo-functors. Thus this proposal touches on both “geometric representation theory” (realizing representations as cohomology groups of varieties) and “higher representation theory” (actions of algebras on categories). In Sections 1 and 2 we discuss geometric category O and symplectic duality, all of which is joint work with Tom Braden, Anthony Licata, and Ben Webster. Section 3 is devoted to four other, smaller projects, each with a different set of collaborators.