Now choose a dominant integral weight . By yesterday, we have:
Our first aim is to prove
Proposition 1 for , the Weyl group, and the dot action. For , we have .
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 for the Kostant partition function evaluated at (inverse meaning in the group ring , where is the weight lattice of with ). Define by
I claim that
(Note that since acts on the weight lattice , it clearly acts on the group ring. Here, as usual, .)
The first claim is obvious. The second follows because the minimal expression of as a product of reflections has precisely as many terms as the number of positive roots that get sent into negative roots by , and a reflection has determinant . (more…)