Complete Reducibility and Conjugacy Classes of Tuples in Algebraic Groups and Lie Algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on Gn, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in Gn, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre’s notion of Gcomplete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.