Filtrations in modular representations of reductive lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and = Lie(G). In this paper, we study representations of the reductive Lie algebra with p-character χ of standard Levi form associated with an index subset I of simple roots. With aid of the support variety theory, we prove that a U_χ(g)-module is projective if and only if it is a strong "tilting" module, i.e., admitting both Z_Q- and Z^wI_Q- filtrations. Then by an analogy of the arguments in [2] for G ₁T-modules, we construct so-called AndersenKaneda filtrations associated with each projective -module of p-character χ, and finally obtain sum formulas from those filtrations.