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…)