Now choose a dominant integral weight . By yesterday, we have:

Our first aim is to prove

Proposition 1for , 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 .

Also, if , we have a “geometric series” result

Here the second term in parentheses must be considered in the same way that a character of an infinite-dimensional Verma module is considered: as a function vanishing outside a finite union of sets of the form .

The reason this is important is that

as is easily checked. In particular, it follows that So, if is the Verma module,

If we apply this to the inverted formula, we find

**Computation of **

*Suppose now that is dominant integral.*

First, I claim that is a dominant integral weight. We need to check that for all . There is a lemma:

Lemma 2Suppose is such that for all ; then is in the weight lattice, i.e. for all .

This relies on some general facts about root systems, and I’m not particularly inclined to prove it here. I should probably have made this clearer earlier. It follows that if we choose **fundamental weights** with where ranges over , then the weight lattice is spanned by the .

Anyway, we do know that for any ,

So . By the lemma, it follows that is dominant integral.

However, we actually don’t even need the lemma. All we need to know for the sequel is that for all , which is now clear because for all .

Now, in the formula, , we find that is invariant under any (this is true for any finite-dimensional representation—the weights are invariant under the Weyl group, which may be seen using the representation theory of ). Also, applying to changes the sign by . As a result, we find that

It follows that if , then . This, incidentally, is the same as saying that in view of the definition of the dot action.

I claim now that this accounts for all the nonzero . If not, since the set of weights is invariant under , we can find some with and in the first Weyl chamber, that satisfying if .

Now by the computation for the Casimir element, we know that:

We can write this as

since is a sum of positive roots ( being the highest weight) and is dominant.

Then since brackets nonnegatively with all positive roots and brackets positively with all positive roots, we have that and , with equality holding in the latter iff . But this means equality must hold, and .

In particular, we have proved the formula

**Weyl character formula **

We can rewrite the boxed formula as

whence taking (since is independent of !) and using (this is the trivial representation, everything acting by zero), we find:

and thus:

Theorem 3 (Weyl)For dominant integral,

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