Complete reducibility

In this note, we unify and extend various concepts in the area of \(\it G\)-complete reducibility, where \(\it G\) is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of \(\it G\)-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of \(\it G\). We show that other variations of this notion, such as relative complete reducibility and \(\sigma\)-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties