Lecture IV: Geometric invariant theory and localization theorems

Attempts in the last decade to establish “effective” localization theorems that are broadly applicable in symplectic contexts have run up against challenging combinatorial questions regarding “singular central characters." I will explain how, in many cases of interest—especially, those related to quiver varieties—the relevant combinatorics spring from classical structures in geometric invariant theory. I will then describe how geometric invariant theory provides the organizing principles to tame the challenges, yielding precise algorithms characterizing the regular and singular values of central character-type parameters.