Vust's theorem and higher level Schur-Weyl duality for types B, C and D

Abstract

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 in case of G = GL(V ), the algebra End Ge (V ⊗d ), where G e ⊂ G is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group S d and the linear maps {1 ⊗(i−1) ⊗ e ⊗ 1 ⊗(d−i) |i = 1, . . . , d}.In this paper, we generalize this theorem to G = O(V ) and SP(V ) for nilpotent element e with G • e being normal. As an application, we study the higher Schur-Weyl duality in the sense of [BK2] for types B, C and D, which establishes a relationship between W -algebras and degenerate affine braid algebras.