Reductive group actions

TL;DR

This paper explores the rationality properties of reductive group actions over fields of characteristic zero, unifying key theories and providing explicit compactifications for spherical varieties.

Contribution

It introduces a generalized Tits index for reductive group actions and develops a k-version of the local structure theorem, unifying and extending existing theories.

Findings

Defined a generalized Tits index for reductive actions

Constructed explicit compactifications for k-spherical varieties

Unified Luna's spherical systems with Borel-Tits' reductive group theory

Abstract

In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem.