Lie Algebra Modules with Finite Dimensional Weight Spaces, I

Let g denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let I) denote a Cartan subalgebra of g. In this paper we study finitely generated g-modules that decompose into direct sums of finite dimensional l)-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 is a simple Lie algebra that admits a torsion free module, then g is of type A or C.