Decompositions in categories of highest weight modules (Q1080935)

Let \({\mathfrak g}\) be a complex semisimple Lie algebra, \({\mathfrak p}={\mathfrak m}\oplus {\mathfrak u}\) its parabolic subalgebra with reductive component \({\mathfrak m}\) and nil radical \({\mathfrak u}\). The authors study the decomposition into indecomposable modules and the classification of indecomposable modules in the category of \({\mathfrak p}\)-locally finite, \({\mathfrak m}\)-completely reducible, \({\mathfrak g}\)-modules with a nondegenerate, \({\mathfrak g}\)-invariant bilinear form. Then they prove that a nondegenerate \({\mathfrak g}\)-invariant bilinear form on this type of module is defined, essentially, by its ''signature''.