A combinatorial approach to relative complete reducibility

For a reductive subgroup K of a reductive group G, the notion of relative complete reducibility gives an algebraic description of the closed K-orbits in Gn, where K acts by simultaneous conjugation. In this thesis we show that questions about reductive groups acting on arbitrary affine varieties can be translated to the setting of relative GL(V)-complete reducibility. Furthermore, we present characterizations of relative GL(V )-complete reducibility in terms of certain subsets of flags of V . These characterizations lead to combinatorial descriptions of closed orbits, which may assist in proving Tits’ Centre Conjecture for convex subsets of spherical buildings.