# Semisimplification for Subgroups of Reductive Algebraic Groups

Michael Bate, Benjamin Martin, Gerhard Roehrle

Research output: Contribution to journalArticle

## Abstract

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 H0 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 H0 of G, unique up to G(k)-conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction which 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 [21]). We also show that under some extra hypotheses, one can pick H0 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.
Original language English Forum of Mathematics, Sigma Accepted/In press - 21 Jul 2020

## Keywords

• Semisimplification
• G-complete reducibility
• geometric invariant theory
• rationality
• cocharacter-closed orbits
• degeneration of G-orbits