Let {X} be a representation of a semisimple Lie algebra {\mathfrak{g}}, a Cartan subalgebra {\mathfrak{h}}, and some choice of splitting {\Phi = \Phi^+ \cup \Phi^-} on the roots.

Recall from the representation theory of finite groups that to each representation of a finite group {G} one can associate a character function {\chi}, and that the ring generated by the characters is the Grothendieck ring of the semisimple tensor category {Rep(G)}. There is something similar to be said for semisimple Lie algebras. So, assume {\mathfrak{h}} acts semisimply on {X} and that the weight spaces are finite-dimensional, and set formally

\displaystyle \mathrm{ch}(X) := \sum_{\lambda} \dim X_{\lambda} e(\lambda).

In other words, we define the character so as to include all the information on the size of the weight spaces at once.

It is necessary, however, to define what {e(\lambda)} for {\lambda \in \mathfrak{h}^{\vee}}. Basically, it is just a formal symbol; {\mathrm{ch}(X)} can more rigorously be thought of as a function {\mathfrak{h}^{\vee} \rightarrow \mathbb{Z}_{\geq 0}}. Nevertheless, we want to think of {e(\lambda)} as a formal exponential in a sense; we want to have {e(\lambda) e(\lambda') = e(\lambda + \lambda')}. The reason is that we can tensor two representations, and we want to talk about multiplying to characters.

I now claim that the above condition on {X} makes sense for {X \in \mathcal{O}}, the BGG category. This will follow because it is true for highest weight modules and we have:

Proposition 1 If {M \in \mathcal{O}}, then there is a finite filtration on {M} whose quotients are highest weight modules. (more…)