Nilpotent orbits of height 2 and involutions in the affine Weyl group

Abstract

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}.

Parametrization, structure and Bruhat order of certain spherical quotients

“…The problem of describing the inclusion relations between the orbit closures has been addressed in [1,5] in certain cases. Recently, [12] associate to any nilpotent element of height 2 an involution in the affine Weyl group, and show that the orbit closures are described restricting the Bruhat order on the affine Weyl group. To the best of our knowledge, there is no general approach to these problems.…”