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Reductive group actions

Friedrich Knop, Bernhard Krötz

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.

Reductive group actions

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.

Paper Structure

This paper contains 20 sections, 109 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Invariants of quasi-elementary groups
  4. The local structure theorem
  5. Invariant valuations
  6. Valuations and geometry
  7. Toroidal embeddings
  8. Horospherical varieties
  9. The Weyl group
  10. The root system
  11. Wonderful varieties
  12. Boundary degenerations
  13. The minimal boundary orbits
  14. Boundary degenerations in direction of a one-parameter subgroup
  15. Boundary degenerations in direction of the valuation cone
  16. ...and 5 more sections

Key Result

Proposition 1

For a connected $k$-group $H$, the following are equivalent:

Theorems & Definitions (225)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Definition 4
  • Theorem 3.1: Kempf*Cor. 4.3
  • Corollary 1: Kempf*Remark after Cor. 4.4
  • proof
  • Theorem 3.2
  • ...and 215 more