Nilpotent Orbits in the Dual of Classical Lie Algebras in Characteristic 2 and the Springer Correspondence

Throughout this paper, let k be a field of characteristic 2. Let G be a classical group over k and g be its Lie algebra. Let g be the dual vector space of g. Fix a Borel subgroup B and a maximal torus T ⊂ B of G. Let B = TU be the Levi decomposition of B. Let b, t and n be the Lie algebra of B, T and U respectively. We have a natural action of G on g, g.ξ(x) = ξ(Ad(g)x) for g ∈ G, ξ ∈ g and x ∈ g. Let n = {ξ ∈ g∗|ξ(b) = 0}. An element ξ in g is called nilpotent if G.ξ∩n′ 6= ∅([4]). We classify the nilpotent orbits in g under the action of G in the cases when k is algebraically closed and when k is a finite field Fq. In particular, we obtain the number of nilpotent orbits over Fq and the structure of groups of components of the centralizers. Let Gad be an adjoint algebraic group of type B,C or D over k (assume k algebraically closed) and gad be the Lie algebra of Gad. In [7], we have constructed a Springer correspondence for gad(see the references therein). In Section 5, we use the Deligne-Fourier transform to construct a Springer correspondence for gad. Let Aad be the set of all pairs (c ′,F ) where c is a nilpotent Gad-orbit in gad and F ′ is an irreducible Gad-equivariant local system on c ′ (up to isomorphism). We construct a bijective map from the set of isomorphism classes of irreducible representations of the Weyl group of Gad to the set A ′ ad.