2021/02/28 by Yiyang Li, Li, Yi-Yang, Bin Shu +3
Mathematics · #17B10 #17B20 #17B35 #17B50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2103.00431
openalex publication_date 2021/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 χ of standard Levi form associated with a given subset I of the simple root system Π of \frak g. Let Uχ(\frak g) be the reduced enveloping algebra of \frak g. A notion "quasi-simple module" (denoted by \mathcal Lχ(λ)) is introduced. The properties of such a module turn out to be better than those of the corresponding simple module \widehat Lχ(λ). It enables us to investigate the Uχ(\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 Lχ(λ) is zero, and the projective dimension of \mathcal Lχ(λ) is finite when λ 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 Uχ(\frak g)-modules) by replacing \widehat Lχ(λ) by \mathcal Lχ(λ). Second, with the aid of quasi-simple modules, we get a formula on the Loewy lengths of standard modules and proper standard modules over Uχ(\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.