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Given two linearly reductive algebraic groups, is their semidirect product reductive again? By linearly reductive, I mean that any rational representation of the group is completely reducible. In particular, do you know any references?

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Suppose it is not reductive. Look at the unipotent radical under the projection maps. I think that's enough to get a contradiction. – M Turgeon Aug 20 '12 at 13:53
Aw, I'm sorry, I should have said "linearly reductive". I'll edit the question. – Jesko Hüttenhain Aug 21 '12 at 7:26

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