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Complete reducibility and separability


Authors: Michael Bate, Benjamin Martin, Gerhard Röhrle and Rudolf Tange
Journal: Trans. Amer. Math. Soc. 362 (2010), 4283-4311
MSC (2000): Primary 20G15, 14L24
DOI: https://doi.org/10.1090/S0002-9947-10-04901-9
Published electronically: March 4, 2010
MathSciNet review: 2608407
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Abstract: Let $ G$ be a reductive linear algebraic group over an algebraically closed field of characteristic $ p > 0$. A subgroup of $ G$ is said to be separable in $ G$ if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of $ G$-complete reducibility for subgroups of $ G$. A separability hypothesis appears in many general theorems concerning $ G$-complete reducibility. We demonstrate that some of these results fail without this hypothesis. On the other hand, we prove that if $ G$ is a connected reductive group and $ p$ is very good for $ G$, then any subgroup of $ G$ is separable; we deduce that under these hypotheses on $ G$, a subgroup $ H$ of $ G$ is $ G$-completely reducible provided Lie $ G$ is semisimple as an $ H$-module.

Recently, Guralnick has proved that if $ H$ is a reductive subgroup of $ G$ and $ C$ is a conjugacy class of $ G$, then $ C\cap H$ is a finite union of $ H$-conjugacy classes. For generic $ p$ -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of $ n$-tuples of elements from $ H$ for $ n > 1$.


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Additional Information

Michael Bate
Affiliation: Christ Church College, Oxford University, Oxford, OX1 1DP, United Kingdom
Address at time of publication: Department of Mathematics, University of York, York, YO10 5DD, United Kingdom
Email: bate@maths.ox.ac.uk, meb505@york.ac.uk

Benjamin Martin
Affiliation: Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
Email: B.Martin@math.canterbury.ac.nz

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Email: gerhard.roehrle@rub.de

Rudolf Tange
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Address at time of publication: Department of Mathematics, University of York, York, YO10 5DD, United Kingdom
Email: rudolf.tange@rub.de, rht502@york.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-10-04901-9
Keywords: $G$-complete reducibility, separability, reductive pairs
Received by editor(s): March 24, 2008
Received by editor(s) in revised form: August 12, 2008
Published electronically: March 4, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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