On a relative version of Serre's notion of G-complete reducibility

We first review some basic results related to Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We then discuss a relative variant of this concept where we let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^o$ of $K$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^o$ is completely reducible in the sense of Serre. This allows us to generalise a number of fundamental results from the absolute to the relative setting. This is a report on recent joint work with M. Gruchot and A. Litterick.