Stability and disconnected groups

We study the notion of semistability for principal bundles over curves with possibly disconnected structure group. We establish a new characterization of semistability under change of group which is novel even in the connected case. A key ingredient is our identification of the rational characters of any linear algebraic group with the Weyl-invariant rational characters of a maximal torus. In the reductive case, we prove an analogous statement for integral cocharacters. As an application, we extend the recursive description of Kirwan stratifications in Geometric Invariant Theory to disconnected groups, and use it in our study semistability for principal bundles.