Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B ⊂ G be a Borel subgroup. Then B acts with finitely many orbits on the variety N 2 ⊂ g of the nilpotent elements whose height is at most 2. We provide a parametrization of the B-orbits in N 2 in terms of subsets of pairwise orthogonal roots, and we provide a complete description of the inclusion order among the B-orbit closures in terms of the Bruhat order on certain involutions in the affine Weyl group of g.
Introduction.Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let N ⊂ g be the nilpotent cone. It is well known that G acts with finitely many orbits on N : for instance, when G is a classical group, the nilpotent G-orbits are parametrized in terms of partitions, and the partial order defined by the inclusions of their closures is nicely expressed in terms of the dominance order of partitions.Given e ∈ N a natural index of nilpotency is the height, defined as ht(e) = max{n ∈ N | ad(e) n = 0}.