The purpose of this grant is to study the structure of certain categories of sheaves on symplectic algebraic varieties. When the variety in question is the cotangent bundle to a flag variety, then the category that one obtains is equivalent to a block of the "Category O" of representations of a Lie algebra, originally introduced by Bernstein, Gelfand, and Gelfand. For this reason, this category is referred to as "Geometric Category O". This category is conjectured to have many beautiful properties, which have been verified in certain special cases that arise naturally from representation theory and polyhedral geometry. Chief among these conjectures is that the categories are Koszul, and that each symplectic variety has a "symplectic dual" whose associated category is Koszul dual to that of the original variety. This phenomenon is expected to form a bridge between various approaches to categorification of Lie algebra representations and link invariants that were previously thought to be unrelated.
Symplectic algebraic varieties arise naturally from many different areas of mathematics. Geometric and topological properties of hypertoric varieties have shed new light on the topology of hyperplane arrangements and the combinatorics of matroids. Quiver varieties provide geometric realizations of actions of infinite dimensional Lie algebras, leading to canonical bases and to actions on categories of sheaves. This grant plays a key role in this picture, providing new geometric insight to such phenomena as Gale duality in combinatorics and level-rank duality in representation theory. This project will contribute to each of these endeavors independently, and will also advance a common treatment that unifies our understanding of the various individual phenomena. In addition to the research component, the grant includes an annual workshop for graduate students and postdocs that bridges these various fields.