On relative complete reducibility

Let K be a reductive subgroup of a reductive group G over an algebraically closed field k. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed K-orbits in Gn, where K acts by simultaneous conjugation on n-tuples of elements from G. This extends work of Richardson and is also a natural generalization of Serre's notion of G-complete reducibility. In this paper we revisit this idea, giving a characterization of relative G-complete reducibility which directly generalizes equivalent formulations of G-complete reducibility. If the ambient group G is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange