Stability conditions for Gelfand-Kirillov subquotients of category O

TL;DR

This paper constructs a new example of stability conditions in derived categories of certain sub-quotients of category O, revealing a geometric structure related to the dual Cartan and braid group actions.

Contribution

It introduces a novel construction of stability conditions for Gelfand-Kirillov sub-quotients of category O using braid group actions and translation functors, expanding the understanding of stability manifolds.

Findings

Explicit description of a sub-manifold in the space of Bridgeland stability conditions.

Identification of this sub-manifold as a covering space of a hyperplane complement.

Connection between stability conditions and Gelfand-Kirillov dimension in category O.

Abstract

Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a "real variation of stability conditions" (which is related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example of representation theoretic significance, by studying certain sub-quotients of category O with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category O, and certain leading coefficient polynomials coming from translation functors. Consequently, we use this to explicitly describe a sub-manifold in the space of Bridgeland stability conditions on these sub-quotient categories, which is a covering space of a hyperplane complement in the dual Cartan.