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G ‐complete reducibility and semisimple modules
Author(s) -
Bate Michael,
Herpel Sebastian,
Martin Benjamin,
Röhrle Gerhard
Publication year - 2011
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdr042
Subject(s) - algebraically closed field , mathematics , algebraic group , prime (order theory) , reductive group , field (mathematics) , context (archaeology) , algebraic number , mathematical proof , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , paleontology , geometry , biology
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 > 2 dim V −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.