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…)

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