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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Complete reducibility in good characteristic


Authors: Alastair J. Litterick and Adam R. Thomas
Journal: Trans. Amer. Math. Soc. 370 (2018), 5279-5340
MSC (2010): Primary 20G07, 20G41; Secondary 20G10
DOI: https://doi.org/10.1090/tran/7085
Published electronically: April 17, 2018
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Abstract: Let $ G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $ p \ge 0$. A closed subgroup $ H$ of $ G$ is called $ G$-completely reducible ($ G$-cr) if whenever $ H$ is contained in a parabolic subgroup $ P$ of $ G$, it is contained in a Levi subgroup of $ P$. In this paper we determine the $ G$-conjugacy classes of non-$ G$-cr simple connected subgroups of $ G$ when $ p$ is good for $ G$. For each such subgroup $ X$, we determine the action of $ X$ on the adjoint module $ L(G)$ and the connected centraliser of $ X$ in $ G$. As a consequence we classify all non-$ G$-cr connected reductive subgroups of $ G$, and determine their connected centralisers. We also classify the subgroups of $ G$ which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.


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Additional Information

Alastair J. Litterick
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Address at time of publication: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780 Bochum, Germany
Email: ajlitterick@gmail.com

Adam R. Thomas
Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, United Kingdom
Address at time of publication: School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK, and The Heilbronn Institute for Mathematical Research, Bristol, United Kingdom
Email: adamthomas22@gmail.com

DOI: https://doi.org/10.1090/tran/7085
Received by editor(s): September 14, 2015
Received by editor(s) in revised form: March 7, 2016, September 26, 2016, and September 30, 2016
Published electronically: April 17, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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