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

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and $g=\text{Lie}\left(G\right)$. 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.