Semisimplification for Subgroups of Reductive Algebraic Groups

Michael Bate, Benjamin Martin, Gerhard Roehrle

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Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If 𝐺 = GL𝑛, then there is a degeneration process for obtaining from H a completely reducible subgroup 𝐻′ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup 𝐻′ of G, unique up to 𝐺(𝑘)-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for 𝐺 = GL𝑛 and with Serre’s ‘G-analogue’ of semisimplification for subgroups of 𝐺(𝑘) from [19]). We also show that under some extra hypotheses, one can pick 𝐻′ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.
Original languageEnglish
Article numbere43
Pages (from-to)1-10
Number of pages10
JournalForum of Mathematics, Sigma
Publication statusPublished - 9 Nov 2020

Bibliographical note

Acknowledgements: We are grateful to Brian Lawrence for his questions, which motivated this paper, and for his comments on an earlier draft.


  • Semisimplification
  • G-complete reducibility
  • geometric invariant theory
  • rationality
  • cocharacter-closed orbits
  • degeneration of G-orbits


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