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.

 

To see this, note that {M} is finitely generated over {U \mathfrak{g}} by some vectors {v_1, \dots, v_n}, which wlog we assume to be weight vectors with weights {\lambda_1, \dots, \lambda_n}. Let {\lambda_1}, say, be the highest one (according to bracketing with some {\gamma \in E} used to define the splitting {\Phi = \Phi^+ \cup \Phi^-}); then {v_1} is a highest weight vector, and the submodule {M'} generated by {v} is a highest weight module. By induction on the dimension of {\mathfrak{n}^+ v_1 + \dots + \mathfrak{n}^+ v_n}, we can assume that there are appropriate filtrations on {M'} and {M/M'}; thus we get one on {M}. (The case {n=1} is immediate.)

By the definition of {\mathrm{ch}(M)}, we also have:

Proposition 2 If {0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0} is an exact sequence in {\mathcal{O}}, then {\mathrm{ch}(M) = \mathrm{ch}(M') + \mathrm{ch}(M'')}.

 

As a result, it follows that the character of any object in category {\mathcal{O}} is supported on a finite union of sets of the form

\displaystyle \lambda - L

where {L} is the {\mathbb{Z}_{\geq 0}}-lattice generated by the positive roots.

Tensor products

Let {V = \bigoplus_\beta V_{\beta}, W = \bigoplus_{\gamma} W_{\gamma}} be {\mathfrak{g}}-modules with decomposition into their weight spaces. Then

\displaystyle V \otimes W = \bigoplus_{\beta, \gamma} V_{\beta} \otimes W_{\gamma}

where {\mathfrak{h}} acts on {V_{\beta} \otimes W_{\gamma} } with weight {\beta+\gamma}. In particular,

\displaystyle (V \otimes W)_{\alpha} = \bigoplus_{\beta + \gamma = \alpha} V_{\beta} W_{\gamma}

and we have

\displaystyle \mathrm{ch}(V \otimes W) = \mathrm{ch}(V) \ast \mathrm{ch}(W)

where {\mathrm{ch}(V) \ast \mathrm{ch}(W)} is the convolution of the two characters, defined by

\displaystyle (\mathrm{ch}(V) \ast \mathrm{ch}(W)) (\alpha) = \sum_{\beta} \mathrm{ch}(V)(\beta) \mathrm{ch}(W)(\alpha - \beta),

where we consider the characters as functions. For this to make sense, we may assume that {V \in \mathcal{O}} and {W} is finite -dimensional, say. If we let {e(\lambda): \mathfrak{h}^{\vee} \rightarrow \mathbb{C}} be the characteristic function of {\{\lambda\}}, then {e(\lambda) \ast e(\mu) = e(\lambda+\mu)}, which explains our rules. Often we will drop the {\ast} sign for convolution though.

In particular, on finite-dimensional representations, we have a ring-homomorphism from the Grothendieck ring {K(Rep(\mathfrak{g}))} to the space of finitely supported functions on {\mathfrak{h}^{\vee}} with convolution.

The character of the Verma module

The character of the Verma module is easily computed. Let {\alpha_i, 1 \leq i \leq r \in \Phi^+} be the positive roots, and let {Y_i \in \mathfrak{g}_{-\alpha_i}} be nonzero. Recall that {V(\lambda)} has a basis consisting of

\displaystyle Y_1^{j_1} \dots Y_r^{j_r} v

where {v} is the initial highest weight vector, and that this displayed quantity has weight {\lambda - \sum j_i \alpha_i}. It follows that {\dim(V(\lambda)_{\beta})} is the number of ways one can express {\lambda - \beta} as a sum {\sum j_i \alpha_i} for each {j_i \in \mathbb{Z}_{\geq 0}}.

So, define the Kostant partition function

\displaystyle p(v) := \mathrm{number} \ \mathrm{of} \ \mathrm{expressions} \ \ v = \sum j_i \alpha_i.

It now follows that

\displaystyle \mathrm{ch}(V(\beta)) = \sum_{\mu} p(\lambda - \mu) e(\mu) = p \ast e(\lambda).

Here {p} corresponds to the sum {\sum p(\mu) e(\mu)}.

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