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G-complete reducibility and semisimple modules

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JournalBulletin of the london mathematical society
DateE-pub ahead of print - 25 Jul 2011
DatePublished (current) - Dec 2011
Issue number6
Volume43
Number of pages10
Pages (from-to)1069-1078
Early online date25/07/11
Original languageEnglish

Abstract

Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic p > 0. Our first aim in this note is to give concise and uniform proofs for two fundamental and deep results in the context of Serre’s notion of G-complete reducibility, at the cost of less favourable bounds. Here are some special cases of these results: Suppose that the index (H : H¿) is prime to p and that p > 2dimV - 2 for some faithful G-module V . Then the following hold: (i) V is a semisimple H-module if and only if H is G-completely reducible;
(ii) H¿ is reductive if and only if H is G-completely reducible. We also discuss two new related results. (i) If p dim V for some G-module V and H is a G-completely reducible subgroup of G, then V is a semisimple H-module; this generalizes Jantzen’s semisimplicity theorem (which is the case H = G). (ii) If H acts semisimply on V ¿ V* for some faithful G-module V , then H
is G-completely reducible.

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