Schur-Weyl Duality For Type B On V^（?）3

Let G be a complex linear algebraic group,g= Lie(G)its Lie algebra and e ∈g a nilpotent element.Vust’s theorem says that EndGe(V(?)d)=(?)d[e]in the case of G = GL(V),where Ge(?)C G is the stabilizer of e under the adjoint action and(?)d[e]is generated by the image of the natural actions of d-th symmetric group(?)d and the linear maps {1(?)(i-1)(?)e(?)1(?)(d-i)|i= 1＝2,...,d}.Under certain conditions Li Luo and Husileng Xiao[1]generalized this theorem to G = O(V)or SP(V)and showed the Vust’s theorem and double centralizer property for its Lie algebra.This paper aims at simplifying the double centralizer property in[1].Yu Zhang,a student of Li Luo,considered the case that d = 2 and e ∈ g is a regular nilpotent element in her master’s paper.In this paper,we provide a simpler approach for her case first and the double centralizer property in[1]in the case that d = 3 and e ∈ g is regular.