Stability and disconnected groups

We study the notion of semistability for principal bundles over curves with possibly disconnected reductive structure group. We establish a new characterization of the behavior of semistability under change of group, novel even in the connected case, and prove that all existing notions of semistability are equivalent, thus settling a question by Biswas-Gomez. The key ingredients for our results include a study of cocharacters and characters of disconnected linear algebraic groups, and an extension of the recursive description of Kirwan stratifications in Geometric Invariant Theory to the case of disconnected groups.