The Loewy series of principal indecomposable modules in the regular nilpotent representations of 𝔰𝔩(3,k)

Let [Formula: see text] be a simple Lie algebra of type [Formula: see text] over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text] be the reduced enveloping algebra of [Formula: see text]. In this paper, when the [Formula: see text]-character [Formula: see text] is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable [Formula: see text]-modules and the dimensions for the self-extension of irreducible [Formula: see text]-modules.

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 [Formula: see text]- and [Formula: see text]-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.