Semisimplification for subgroups of reductive algebraic groups

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.