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A geometric approach to complete reducibility

A geometric approach to complete reducibility
A geometric approach to complete reducibility
Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G.
Linear algebraic groups, G-completely reducible subgroups, strongly reductive groups, spherical buildings
0020-9910
177-218
Bate, Michael
26927955-b184-4b80-90c9-f3dde9cdba18
Martin, Benjamin
67bee23e-9ae4-47c3-b9f1-b6f08e6eb5b3
Roehrle, Gerhard
85f9d4eb-d522-4a95-bde9-300e8e3e7886
Bate, Michael
26927955-b184-4b80-90c9-f3dde9cdba18
Martin, Benjamin
67bee23e-9ae4-47c3-b9f1-b6f08e6eb5b3
Roehrle, Gerhard
85f9d4eb-d522-4a95-bde9-300e8e3e7886

Bate, Michael, Martin, Benjamin and Roehrle, Gerhard (2005) A geometric approach to complete reducibility. Inventiones Mathematicae, 161 (1), 177-218. (doi:10.1007/s00222-004-0425-9).

Record type: Article

Abstract

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G.

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More information

Published date: July 2005
Keywords: Linear algebraic groups, G-completely reducible subgroups, strongly reductive groups, spherical buildings

Identifiers

Local EPrints ID: 38372
URI: http://eprints.soton.ac.uk/id/eprint/38372
ISSN: 0020-9910
PURE UUID: cbc55f5f-48e7-4eca-ac80-7897769bcd0b

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Date deposited: 07 Jun 2006
Last modified: 15 Jul 2019 19:03

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Contributors

Author: Michael Bate
Author: Benjamin Martin
Author: Gerhard Roehrle

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