Normality Of Classical Nilpotent Orbit Closures In Positive Characteristic An Higher Level Schur-Weyl Duality Of Type BCD

The present thesis consist of two parts.In the first chapter we investigate the nor-mality of closures of Orthogonal and Symplectic nilpotent orbits in positive characteristic.We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure,and conversely if the closure is normal,then any type e minimal irreducible degeneration does not occur in it.Here,the minimal irreducible degenerations of a nilpotent orbit are introduced by W.Hesselink in[Hes](or see[KP2]from which we take Table 1.1 for the complete list of all minimal irreducible degenerations).Our result is a weak version in positive characteristic of[KP2,Theorem 16.2(ii)],one of the main results of[KP2]over complex numbers.The results of this part was contained in[XS].In the second chapter we will give type BCD Vust theorem and it’s application to higher Schur-Weyl duality.Let G be a complex linear algebraic group,g = Lie(G)be its Lie algebra and e ∈ g a nilpotent element.Vust theorem says that in the 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(?)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.This part is based on the work[LX].