A note on the Loewy lengths of baby Verma modules for modular Lie algebras

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Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and g=Lie(G). We study the representations of the reductive Lie algebra g with p-character χ of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. [10, II.Lem.7.20]). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.