In this chapter, we study representations of complex semisimple Lie algebras. Recall that by results of Section 6.3, every finite-dimensional representation is completely reducible and thus can be written in the form V = ⊕niVi, where Vi are irreducible representations and ni ∈ ℤ+ are the multiplicities. Thus, the study of representations reduces to classification of irreducible representations and finding a way to determine, for a given representation V, the multiplicities ni. Both of these questions have a complete answer, which will be given below.

Throughout this chapter, g is a complex finite-dimensional semisimple Lie algebra. We fix a choice of a Cartan subalgebra and thus the root decomposition g = h ⊕⊕Rgα (see Section 6.6). We will freely use notation from Chapter 7; in particular, we denote by αi, i = 1 … r, simple roots, and by si ∈ W corresponding simple reflections. We will also choose a non-degenerate invariant symmetric bilinear form (,) on g.

All representations considered in this chapter are complex and unless specified otherwise, finite-dimensional.

Weight decomposition and characters

As in the study of representations of sl(2, ℂ) (see Section 4.8), the key to the study of representations of g is decomposing the representation into the eigenspaces for the Cartan subalgebra.

Definition 8.1. Let V be a representation of g. A vector u ∈ V is called a vector of weight λ ∈ h* if, for any h ∈ h, one has hu = 〈λ, h〉u.