On The Representations Of Modular Lie Algebras SL And Lie Superalgebras OSP Type

In Part 1 in this paper, we classify all irreducible representations of sl3 forp= 3, which completes the classification of the irreducible representations of sl3 over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants. Hence we give a counterexample to show the failure of Kac-Weisfeiler conjecture. Also inspired of the sl3 case, we get the character formulae for all simple modules and the Cartan invariants for sln in general for regular nilpotent case.In Part 2, we proved a conjecture doesn’t hold in general such that we fail to translate the problem of solving the character formulae for simple modules in BGG category for osp into the combinatorial problems via the L-KL theory in [3] like the gl case. Since from the quantum aspect to compute the bar involution on Fock space is extreme difficult, we used a more directly way to compute the specific character formulae and translation functors actions for simple modules in finite dimensional category for osp of low ranks.