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

May 2012

Gold, Daniel

266a6681-ba33-4ba1-86c7-517cc3f0c6f6

Koeck, Bernhard

84d11519-7828-43a6-852b-0c1b80edeef9

(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.

## 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

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Date deposited: 21 Oct 2013 12:05

Last modified: 26 Jul 2019 00:35

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## Contributors

Author:
Daniel Gold

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