Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence

Abstract

Abstract. Let G be a simply connected algebraic group of type B, C or D over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of G. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.

Affine braid group actions on derived categories of Springer resolutions

“…Observe, however, that it is satisfied if g is a simple G-module (because in this case N ′ identifies with the usual nilpotent cone in g), for example if G is of type G 2 and p = 3 (see [Hu,§0.13]). This property is also proved if G is of type B, C or D and p = 2 in [Xu,§5.6].…”

“…[J3]). However, these facts have to be checked directly (without using g) if we want to relax the assumptions on p. Some of these facts are proved in [Xu,§5], using the same arguments as those of [J3] and some results of [KW].…”

Nilpotent pieces in the dual of odd orthogonal lie algebras

“…Let {u i , i ∈ [1, m − 1]} be the unique set of vectors (see [8,Lemma 3.6]) such that [8,Lemma 3.8]) and β| W is nondegenerate. Define…”

“…Then for any x ∈ W and any v ∈ V, β ξ (x, v) = β(T ξ x, v). Moreover T ξ ∈ o(W) is nilpotent (see [8,Lemma 3.11]). Let π W : V → W denote the natural projection.…”

“…be such that β ξ 2 (e i , e j ) = 0 except β ξ 2 (e −8 , e 3 ) = β ξ 2 (e −7 , e 2 ) = β ξ 2 (e −6 , e 0 ) = β ξ 2 (e −5 , e 1 ) = β ξ 2 (e −4 , e −1 ) = β ξ 2 (e −3 , e −2 ) = 1. We have m = 1, v 1 = e 0 , v 0 = e 6 , (can choose) u 0 = e −6 , W = span{e ±i , i ∈ [1,8]…”