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On a relative version of Serre's notion of G-complete reducibility

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Gerhard Röhrle
Ruhr-Universität Bochum
Thu, 22/11/2018 - 16:30 - 17:30
MPIM Lecture Hall

We first review some basic results related to Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We then discuss a relative variant of this concept where we let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^o$ of $K$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^o$ is completely reducible in the sense of Serre. This allows us to generalise a number of fundamental results from the absolute to the relative setting. This is a report on recent joint work with M. Gruchot and A. Litterick.

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