Aspects of G-Complete reducibility

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.

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ORCID for Bernhard Koeck: orcid.org/0000-0001-6943-7874

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