Reduction of Structure to Parabolic Subgroups

TL;DR

This paper investigates conditions under which affine groups over fields admit anisotropic torsors, extending the concept of isotropy and building on Tits' work for simple groups and certain semisimple groups.

Contribution

It provides new criteria for the existence of anisotropic torsors in affine groups, generalizing previous notions of isotropy for affine groups and involutions.

Findings

Characterization of when simple affine groups admit anisotropic torsors.

Extension of results to connected semisimple groups with specific root systems.

Generalization of isotropy concepts to broader classes of algebraic groups.

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.