2004/09/20 by Dimitar Grantcharov, Grantcharov, Dimitar
Mathematics · #17B10 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10
paper · pdf · doi:10.48550/arxiv.math/0409371
18 pages
arxiv created 2004/09/20 · arxiv updated 2009/12/01
Let \mathfrak g be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra \bf W(n). We study weight \mathfrak g-modules using a method inspired by Mathieu's classification of the simple weight modules with finite weight multiplicities over reductive Lie algebras, \citeM. Our approach is based on the fact that every simple weight \mathfrak g-module with finite weight multiplicities is obtained via a composition of a twist and localization from a highest weight module. This allows us to transfer many results for category \cal O modules to the category of weight modules with finite weight multiplicities. As a main application of the method we reduce the problems of finding a \mathfrak g0-composition series and a character formula for all simple weight modules to the same problems for simple highest weight modules. In this way, using results of Serganova we obtain a character formula for all simple weight \bf W(n)-modules and all simple atypical nonsingular \mathfrak s\mathfrak l (m|1)-modules. Some of our results are new already in the case of a classical reductive Lie algebra \mathfrak g.