Coherent sheaves on varieties arising in Springer theory, and category 0

Abstract

In this thesis, we will study three topics related to Springer theory (specifically, the geometry of the exotic nilpotent cone, and two-block Springer fibers), and stability conditions for category 0. In the first chapter, we will be studying the geometry of the exotic nilpotent cone (which is a variant of the nilpotent cone in type C introduced by Kato). Bezrukavnikov has established a bijection between A+, the dominant weights for an arbitrary simple algebraic group H, and 0, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit (as originally conjectured by Lusztig and Vogan). Here we prove an analogous statement for the exotic nilpotent cone. In the second chapter (which is based on joint work with Rina Anno), we study the exotic t-structure for a two-block Springer fibre (i.e. for a nilpotent matrix of type (m + n, n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using techniques developed by Cautis and Kamnitzer, we show that the irreducible objects in the heart of the exotic t-structure are indexed by crossingless (m, m + 2n) matchings. We also show that the resulting Ext algebras resemble Khovanov's arc algebras (but placed on an annulus). In the third chapter, we study stability conditions on certain sub-quotients of category 0. Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a "real variation of stability conditions" (which are related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example, by studying certain sub-quotients of category 0 with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category 0, and certain leading coefficient polynomials coming from translation functors.