Year: 2020
Type: article
Abstract: LetGbe a reductive algebraic group—possibly non-connected—over a fieldk, and letHbe a subgroup ofG. If $G= {GL }_n$ , then there is a degeneration process for obtaining fromHa completely reducible subgroup $H'$ ofG; one takes a limit ofHalong a cocharacter ofGin an appropriate sense. We generalise this idea to arbitrary reductiveGusing the notion ofG-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces aG-completely reducible subgroup $H'$ ofG, unique up to $G(k)$ -conjugacy, which we call ak-semisimplificationofH. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.
Source: Forum of Mathematics Sigma
DOI: 10.1017/fms.2020.30
Authors: Michael Bate, Benjamin Martin, Gerhard Röhrle
Institutions: University of York, University of Aberdeen, Ruhr University Bochum
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,078
OpenAlex ID: W3016327315