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Michael Bate, Benjamin Martin, Gerhard Röhrle

• Let \(\it G\) be a reductive algebraic group—possibly non-connected—over a field k, and let \(\it H\) be a subgroup of \(\it G\). If \(\it G\)=GL\(_n\) , then there is a degeneration process for obtaining from \(\it H\) a completely reducible subgroup \(\it H\)′ of \(\it G\); one takes a limit of \(\it H\) along a cocharacter of \(\it G\) in an appropriate sense. We generalise this idea to arbitrary reductive \(\it G\) using the notion of \(\it G\)-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a \(\it G\)-completely reducible subgroup \(\it H\)′ of \(\it G\), unique up to \(\it G\)(\(\it k\)) -conjugacy, which we call a \(\textit k-semisimplification\) of \(\it H\). This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for \(\it G\)=GLn and with Serre’s ‘\(\it G\)-analogue’ of semisimplification for subgroups of \(\it G\)(\(\it k\)) from [19]). We also show that under some extra hypotheses, one can pick \(\it H\)′ 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.