Quasi-simple modules and Loewy lengths in modular representations of reductive Lie algebras

TL;DR

This paper introduces quasi-simple modules in the modular representation theory of reductive Lie algebras, revealing their advantageous properties and deriving new formulas for Loewy lengths and implications for Lusztig's conjecture.

Contribution

It defines quasi-simple modules with better properties than simple modules, enabling new insights into module structures and formulas for Loewy lengths in modular Lie algebra representations.

Findings

Quasi-simple modules have zero first self-extension.

Finite projective dimension for p-regular weights.

New formulas for Loewy lengths of modules.

Abstract

Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic $p>0$. In this paper, we study the representations of $\frak g$ with a $p$-character $\chi$ of standard Levi form associated with a given subset $I$ of the simple root system $\Pi$ of $\frak g$. Let $U_\chi({\frak g})$ be the reduced enveloping algebra of $\frak g$. A notion "quasi-simple module" (denoted by $\mathcal L_\chi(\lambda)$) is introduced. The properties of such a module turn out to be better than those of the corresponding simple module $\widehat L_\chi(\lambda)$. It enables us to investigate the $U_\chi({\frak g})$-modules from a new point of view, and correspondingly gives rise new consequences. First, we show that the first self extension of $\mathcal L_\chi(\lambda)$ is zero, and the projective dimension of $\mathcal L_\chi(\lambda)$ is finite when $\lambda$ is $p$-regular. These properties make it significant to rewrite the formula of Lusztig's Hope (Lusztig's conjecture on the irreducible characters in the category of $U_\chi({\frak g})$-modules) by replacing $\widehat L_\chi(\lambda)$ by $\mathcal L_\chi(\lambda)$. Second, with the aid of quasi-simple modules, we get a formula on the Loewy lengths of standard modules and proper standard modules over $U_\chi({\frak g})$. And by studying some examples, we formulate some conjectures on the Loewy lengths of indecomposable projective $\frak g$-modules, standard modules and proper standard modules.