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 𝔤 = Lie(G). In this paper, we study representations of the reductive Lie algebra 𝔤 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 Uχ(𝔤)-module is projective if and only if it is a strong "tilting" module, i.e., admitting both ${\mathcal Z}_Q$- and ${\mathcal Z}^{w^I}_Q$-filtrations. Then by an analogy of the arguments in [2] for G1T-modules, we construct so-called Andersen–Kaneda filtrations associated with each projective 𝔤-module of p-character χ, and finally obtain sum formulas from those filtrations.