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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

12 Issues per year

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Complete reducibility and commuting subgroups

Michael Bate1 / Benjamin Martin2 / Gerhard Röhrle3

1Christ Church College, Oxford University, Oxford, OX1 1DP, UK. e-mail: bate@maths.ox.ac.uk

2Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 1, New Zealand. e-mail: B.Martin@math.canterbury.ac.nz

3Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany. e-mail: gerhard.roehrle@rub.de

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2008, Issue 621, Pages 213–235, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2008.063, July 2008

Publication History

Received:
2006-10-23
Revised:
2007-06-26
Published Online:
2008-07-01

Abstract

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≧ 0. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. Specifically, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. In our principal result we show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.

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