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…)