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

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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 G n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G n, 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.

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Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, UK

    Michael Bate & Rudolf Tange

  2. Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 1, New Zealand

    Benjamin Martin

  3. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    Gerhard Röhrle

Authors
  1. Michael Bate
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  2. Benjamin Martin
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  3. Gerhard Röhrle
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  4. Rudolf Tange
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Correspondence to Gerhard Röhrle.

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Bate, M., Martin, B., Röhrle, G. et al. Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras. Math. Z. 269, 809–832 (2011). https://doi.org/10.1007/s00209-010-0763-9

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  • DOI: https://doi.org/10.1007/s00209-010-0763-9

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