Inventiones mathematicae

, Volume 161, Issue 1, pp 177–218

A geometric approach to complete reducibility


DOI: 10.1007/s00222-004-0425-9

Cite this article as:
Bate, M., Martin, B. & Röhrle, G. Invent. math. (2005) 161: 177. doi:10.1007/s00222-004-0425-9


Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G.

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.Mathematics and Statistics DepartmentUniversity of CanterburyChristchurch 1New Zealand

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