Filtrations in Modular Representations of Reductive Lie Algebras

Let G be a connected reductive group G over an algebraically closed field k of prime characteristic p, and g = Lie(G). In this paper, we study modular 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 support variety theory we prove a theorem that a Uχ(g)-module is projective if and only if it is a strong “tilting” module, i.e. admitting both ZQand Z w Q -filtrations (to see Theorem 4.1). Then by 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.