Lie algebra modules with finite-dimensional weight spaces. I

Abstract

Let g \mathfrak {g} denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let h \mathfrak {h} denote a Cartan subalgebra of g \mathfrak {g} . In this paper we study finitely generated g \mathfrak {g} -modules that decompose into direct sums of finite dimensional h \mathfrak {h} -weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if g \mathfrak {g} is a simple Lie algebra that admits a torsion free module, then g \mathfrak {g} is of type A A or C C .