RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

Forum of Mathematics Sigma, vol. 8

Abstract

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

Authors 3

1. Maike Gruchot Aachen Lehrstuhl für Algebra und Zahlentheorie

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, D-52062Aachen, Germany;

   Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, D-52062Aachen, Germany

2. Alastair Litterick

   University of Essex

   Affiliation as printed

   Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, EssexCO4 3SQ, UK;

   Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, EssexCO4 3SQ, UK

3. Gerhard Röhrle

   Ruhr University Bochum

   Affiliation as printed

   Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780Bochum, Germany;

   Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780Bochum, Germany