Higher Level Schur-Weyl Duaity For Types B And C On V^（?）2

Let G be a complex linear algebraic group,g=Lie（G）its Lie algebra and e∈g is a nilpotent element.Let G=GL（V）and G_e?G is the stabilizer of e under the adjoint action.Vust’s theorem says that here is a double centralizer property between G_eand S_d[e]=σ（S_d）∪{1^{（?）（i-1）}（?）e（?）1^{（?）（d-i）}:i=1,···,d}on V^（?）d.Let G=O（V）or SP（V）,g=so（V）or sp（V）,nilpotent element e∈g with （?） being normal,Luo and Xiao[15]showed that a double centralizer property between G_eand B_d[e].Particularly,let g=so_2n+1or sp_2nis types B or C,e∈g is a regular nilpotent element and d takes2,I make a simplification of the Vust’s theorem in[15]and show a double centralizer property between g_eand B_d[e]on V^（?）2in this paper.As an application,I study the higher Schur-Weyl duality in the sense of[3]and show the higher level Schur-Weyl duaity for types B and C on V^（?）2,find a relationship between W-algebras and degenerate affine braid algebras.