Study On Representations Of Reductive Algebraic Groups,of Their Lie Algebras And Of Some Generalized Structure

This article consists of two parts.Part 1:Let K be an algebraically closed field of prime characteristic p.If p divides n,the information of irreducible representations over sl_n（K）are still not completely clear.In this article,we consider this case,and investigate the categories of regular and subregular nilpotent representations,obtain fundamental results.All these results are new.Part 2:The second part of this article is a sequel to Shu-Xue-Yao’s paper[53]where the author studied the so-called enhanced groups and related dualities for degenerate double Hecke algebras over C.In addition to proving a conjecture in[53],we introduce an enhanced Brauer algebra,and investigate the enhanced dualities for classical groups of type B,C,and D.Moreover,we give the parabolic Schur-Weyl duality for general linear supergroups.The precise description of our results are as follows:（1）For K is an algebraically closed field of prime characteristic p,I will precisely describe simple modules of sl_nwith p|n for regular and subregular nilpotent representa-tions.（2）I will prove a conjecture in[53],which is related to the invariants of a maximal parabolic subgroup of GL_n+1over C.Consequently,the natural tensor invariants of this typical maximal parabolic subgroup of GL_n+1are determined when n≥r.（3）I will present the restricted and Levi dualities for orthogonal and symplectic groups over C,hence the parabolic duality will naturally be described.Moreover,a special case will be listed in§6.5 by some explicit calculations.（4）By the dualities for Schur superalgebras given by Brundan-Kujawa（see[9,Re-mark 5.8（iii）]）,I will give the parabolic Schur-Weyl duality for general linear supergroups（when d≤m+n）.