Now choose a dominant integral weight ${\lambda}$. By yesterday, we have:

$\displaystyle \mathrm{ch} L(\lambda) = \sum_{\mu < \lambda} b(\lambda, \mu) \mathrm{ch} V(\mu).$

Our first aim is to prove

Proposition 1 ${b(\lambda, w \cdot \lambda) = (-1)^w}$ for ${w \in W}$, the Weyl group, and ${\cdot}$ the dot action. For ${\mu \notin W\lambda}$, we have ${b(\lambda, \mu)=0}$.

After this, it will be relatively easy to obtain WCF using a few formal manipulations. To prove it, though, we use a few such formal manipulations already.

Manipulations in the group ring

I will now define something that is close to an “inverse” of the Verma module character ${p \ast e(\lambda)}$ for ${p(\lambda)}$ the Kostant partition function evaluated at ${-\lambda}$ (inverse meaning in the group ring ${\mathbb{Z}[L]}$, where ${L}$ is the weight lattice of ${\beta}$ with ${<\beta, \delta> \in \mathbb{Z} \ \forall \delta \in \Delta}$). Define ${q}$ by

$\displaystyle q = \prod_{\alpha \in \Phi^+} \left( e(\alpha/2) - e(-\alpha/2) \right).$

I claim that

$\displaystyle q = e(\rho) \prod_{\alpha \in \Phi^+} (1 - e(-\alpha)), \ \ wp = (-1)^w p, \quad \forall w \in W.$

(Note that since ${w}$ acts on the weight lattice ${L}$, it clearly acts on the group ring. Here, as usual, ${\rho = \frac{1}{2} \sum_{\gamma \in \Phi^+} \gamma}$.)

The first claim is obvious. The second follows because the minimal expression of ${w}$ as a product of reflections has precisely as many terms as the number of positive roots that get sent into negative roots by ${w}$, and a reflection has determinant ${-1}$. (more…)

Today’s is going to be a long post, but an important one.  It tells us precisely what weights are allowed to occur as highest weights in finite-dimensional representations of a semisimple Lie algebra.

Dominant integral weights

Let ${V}$ be a finite-dimensional simple representation of a semisimple Lie algebra ${\mathfrak{g}}$, with Cartan subalgebra ${\mathfrak{h}}$, and root space decomposition ${\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha}}$. Suppose given a base ${\Delta}$ and a corresponding division ${\Phi = \Phi^+ \cup \Phi^-}$.

For each ${\alpha \in \Phi^+}$, choose ${X_{\alpha} \in \mathfrak{g}_{\alpha}, Y_{\alpha} \in \mathfrak{g}_{-\alpha}}$ such that ${[X_{\alpha},Y_{\alpha}] = H_{\alpha}}$ and ${X_{\alpha}, Y_{\alpha}, H_{\alpha}}$ generate a subalgebra ${\mathfrak{s}_{\alpha}}$ isomorphic to ${\mathfrak{sl}_2}$.

Consider the weight space decomposition

$\displaystyle V = \bigoplus_{\beta \in \Pi} V_{\beta}$

where ${\Pi}$ denotes the set of weights of ${V}$. Then if ${\beta \in \Pi}$ is the weight associated to a highest weight vector, ${\beta(H_{\alpha})}$ is necessarily a nonnegative integer by the representation theory of ${\mathfrak{sl}_2}$. In other words,

$\displaystyle <\beta, \alpha> := 2 \frac{ (\beta, \alpha)}{(\alpha, \alpha)} \in \mathbb{Z}_{\geq 0}.$

Any weight ${\beta}$ satisfying that identity for all ${\alpha \in \Phi^+}$ is called dominant integral. We have shown that the highest weight of a finite-dimensional simple ${\mathfrak{g}}$-representation is necessarily dominant integral. In fact, given a dominant integral weight, we can actually construct such a finite-dimensional simple module.

The set of merely integral weights—those ${\beta}$ with ${<\beta, \alpha> \in \mathbb{Z}}$ for ${\alpha \in \Phi}$—form a lattice, spanned by vectors ${\lambda_i}$ such that ${<\lambda_i, \delta_j> = \delta_{ij}}$, where the last ${\delta_{ij}}$ is the Kronecker delta.

Theorem 1 The unique simple quotient of the Verma module ${V(\beta)}$ is finite-dimensional if and only if ${\beta}$ is dominant integral. (more…)