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Title: Aspects of G-Complete reducibility
Author: Gold, Daniel
Awarding Body: University of Southampton
Current Institution: University of Southampton
Date of Award: 2012
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Let G be a connected reductive algebraic group, and σ 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; σ)-complete reducibility. We show that a σ-stable subgroup of G is (G; σ)-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 σ-stable non G-completely reducible subgroup of G, then it is contained in a proper σ-stable parabolic subgroup P of G, and in no Levi subgroup of P. We go on to introduce another new notion, that of Gσ- complete reducibility for subgroups of Gσ. We show that a subgroup of Gσ is Gσ completely reducible if and only if it is (G; σ)-completely reducible. Finally, we introduce the notion of strong σ-reductivity in G for σ-stable subgroups of G, and show that this is an analogue to the notion of strong reductivity in G in the setting of σ-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.
Supervisor: Koeck, Bernhard Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics