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 EndGe(V⊗d), where Ge⊂G is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group Sd and the linear maps {1⊗(i−1)⊗e⊗1⊗(d−i)|i=1,…,d}. In this paper, we give an analogue of Vust's Theorem for G=O(V) and SP(V) when the nilpotent elements e satisfy that G⋅e‾ is normal. As an application, we study the higher Schur–Weyl duality in the sense of [4] for types B, C and D, which establishes a relationship between W-algebras and degenerate affine braid algebras.