Mathematische Zeitschrift

, Volume 269, Issue 3, pp 809–832

Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

  • Michael Bate
  • Benjamin Martin
  • Gerhard Röhrle
  • Rudolf Tange
Article

DOI: 10.1007/s00209-010-0763-9

Cite this article as:
Bate, M., Martin, B., Röhrle, G. et al. Math. Z. (2011) 269: 809. doi:10.1007/s00209-010-0763-9

Abstract

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on Gn, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in Gn, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre’s notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.

Keywords

Conjugacy classes of n-tuples G-complete reducibility 

Mathematics Subject Classification (2000)

20G15 14L24 20E42 

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Michael Bate
    • 1
  • Benjamin Martin
    • 2
  • Gerhard Röhrle
    • 3
  • Rudolf Tange
    • 1
  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.Mathematics and Statistics DepartmentUniversity of CanterburyChristchurch 1New Zealand
  3. 3.Fakultät für MathematikRuhr-Universität BochumBochumGermany

Personalised recommendations