Reduction of Structure to Parabolic Subgroups

Abstract

Let $G$ be an affine group over a field of characteristic not two. A $G$-torsor is called isotropic if it admits reduction of structure to a proper parabolic subgroup of $G$. This definition generalizes isotropy of affine groups and involutions of central simple algebras. When does $G$ admit anisotropic torsors? Building on work of J. Tits, we answer this question for simple groups. We also give an answer for connected and semisimple $G$ under certain restrictions on its root system.