Monogamous subvarieties of the nilpotent cone

Let G be a reductive algebraic group over an algebraically closed field k of prime characteristic not $2$, whose Lie algebra is denoted g. We call a subvariety X of the nilpotent cone N ⊂ g monogamous if for every e∈ X, the sl₂-triples $(e,h,f)$ with f∈ X are conjugate under the centraliser CG(e). Building on work by the first two authors, we show there is a unique maximal closed G-stable monogamous subvariety V ⊂ N and that it is an orbit closure, hence irreducible. We show that V can also be characterised in terms of Serre's G-complete reducibility.