Semisimplification for Subgroups of Reductive Algebraic Groups

Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If 𝐺 = GL𝑛, then there is a degeneration process for obtaining from H a completely reducible subgroup 𝐻′ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup 𝐻′ of G, unique up to 𝐺(𝑘)-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for 𝐺 = GL𝑛 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.