Semistability of G-torsors and parabolic subgroups in positive characteristic

Let k be a field and X be a k-curve. Let also G be a reductive group scheme over X. Semistability for G-torsors can be defined by several ways that depend on assumptions on k and G. These approaches are both well defined and equivalent when k is of characteristic zero. In this talk I will explain in which generalities it is possible to extend some of these approaches to the positive characteristic framework and compare them. This requires to investigate whether some well known results in representation theory in characteristic zero still hold true in characteristic p > 0. More specifically, an analogous statement of a theorem of Morozov (which classifies, in characteristic 0, parabolic subalgebras of the Lie algebra of a reductive group by means of their nilradical) is a cornerstone of all this unification attempt.

In the first part of the talk, I will provide an overview of the geometric content and emphasize
the role of parabolic subgroups in all this theory of semistability. The second part of the talk
will be dedicated to the extension of Morozov’s theorem to positive characteristics, and the way it allows one to get a more uniform vision of the different historical definitions of semistability of G-torsors.