Characters of representations (of semisimple Lie algebras)

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 ${j_i \in \mathbb{Z}_{\geq 0}}$ .