The University of Southampton
University of Southampton Institutional Repository

Aspects of G-Complete reducibility

Aspects of G-Complete reducibility
Aspects of G-Complete reducibility
Let G be a connected reductive algebraic group, and ? a Frobenius morphism of G. Corresponding to the notion of G complete reducibility, due to J.-P. Serre, we introduce a new notion of (G; ?)-complete reducibility. We show that a ?-stable subgroup of G is (G; ?)-completely reducible if and only if it is G-completely reducible. We also strengthen this result in one direction to show that if H is a ?-stable non G-completely reducible subgroup of G, then it is contained in a proper ?-stable parabolic subgroup P of G, and in no Levi subgroup of P. We go on to introduce another new notion, that of G?- complete reducibility for subgroups of G?. We show that a subgroup of G? is G? completely reducible if and only if it is (G; ?)-completely reducible. Finally, we introduce the notion of strong ?-reductivity in G for ?-stable subgroups of G, and show that this is an analogue to the notion of strong reductivity in G in the setting of ?-stability. We discuss a notion of G-complete reducibility for Lie subalgebras of Lie(G), which was introduced by McNinch. We show that if H is a subgroup of G that is contained in C (S), where S is a maximal torus of CG(Lie(H)), then H is G-completely reducible if and only if Lie(H) is G-completely reducible. We give criteria for a Lie subalgebra of Lie(G) to be G-completely reducible. For example, an ideal in Lie(G) is G-completely reducible if it isinvariant under the adjoint action of G.
Gold, Daniel
266a6681-ba33-4ba1-86c7-517cc3f0c6f6
Gold, Daniel
266a6681-ba33-4ba1-86c7-517cc3f0c6f6
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9

Gold, Daniel (2012) Aspects of G-Complete reducibility. University of Southampton, Mathematics, Doctoral Thesis, 181pp.

Record type: Thesis (Doctoral)

Abstract

Let G be a connected reductive algebraic group, and ? a Frobenius morphism of G. Corresponding to the notion of G complete reducibility, due to J.-P. Serre, we introduce a new notion of (G; ?)-complete reducibility. We show that a ?-stable subgroup of G is (G; ?)-completely reducible if and only if it is G-completely reducible. We also strengthen this result in one direction to show that if H is a ?-stable non G-completely reducible subgroup of G, then it is contained in a proper ?-stable parabolic subgroup P of G, and in no Levi subgroup of P. We go on to introduce another new notion, that of G?- complete reducibility for subgroups of G?. We show that a subgroup of G? is G? completely reducible if and only if it is (G; ?)-completely reducible. Finally, we introduce the notion of strong ?-reductivity in G for ?-stable subgroups of G, and show that this is an analogue to the notion of strong reductivity in G in the setting of ?-stability. We discuss a notion of G-complete reducibility for Lie subalgebras of Lie(G), which was introduced by McNinch. We show that if H is a subgroup of G that is contained in C (S), where S is a maximal torus of CG(Lie(H)), then H is G-completely reducible if and only if Lie(H) is G-completely reducible. We give criteria for a Lie subalgebra of Lie(G) to be G-completely reducible. For example, an ideal in Lie(G) is G-completely reducible if it isinvariant under the adjoint action of G.

Text
Thesis.pdf - Other
Download (925kB)

More information

Published date: May 2012
Organisations: University of Southampton, Pure Mathematics

Identifiers

Local EPrints ID: 354409
URI: http://eprints.soton.ac.uk/id/eprint/354409
PURE UUID: cbe14101-c8e9-4613-bf3f-2d1ceccbaa23
ORCID for Bernhard Koeck: ORCID iD orcid.org/0000-0001-6943-7874

Catalogue record

Date deposited: 21 Oct 2013 12:05
Last modified: 15 Mar 2024 03:10

Export record

Contributors

Author: Daniel Gold
Thesis advisor: Bernhard Koeck ORCID iD

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×