Weyl’s character formula (to be proved shortly) gives an expression for the character of a finite-dimensional simple quotient of a Verma module. In here, we will express the character of the simple quotient using Verma module characters. Next time, we will calculate the coefficients involved.
Filtration on highest weight modules
Let be any highest weight module with highest weight . Then is a quotient of , so the Casimir acts on by scalar multiplication by .
Suppose we have a composition series
with successive quotients simple module . Then acts on the successive quotients by scalars that we compute in two different ways, whence by yesterday’s formula:
In fact, such a filtration exists:
Proposition 1 has a finite filtration whose quotients are isomorphic to , where (which we write as ) and satisfies the boxed formula.
In general, this follows simply because every element in has finite length, and the are the only candidates for simple modules!
Theorem 2 The category is artinian.
The only proofs I can find of this use Harish-Chandra’s theorem on characters though, so I’ll follow Sternberg in proving the proposition directly. (I hope later I’ll come back to it.)
For the proposition, induct on the sum
This sum is finite because there are only finitely many possibilities for . If it’s , then cannot have any nontrivial submodules. If it did, then any submodule is a weight module and belongs to , so has a highest weight vector , say of weight . Then, by yesterday’s remarks on Casimir elements, satisfies the condition to be in the above sum, with , contradiction. So in this case we have simplicity and the result is evident.
Now suppose this sum is . Then if , then the result is obvious, so assume we have a highest weight proper submodule , of weight satisfying the boxed condition. There is an exact sequence
and all three objects here belong to as one may easily check. Then have smaller sums as in (*), and are both highest weight modules (with weights respectively), so we can apply the inductive hypothesis.
Expression of the character of via Verma module characters
We have that
Here ranges over the finite set of integral points satisfying . Incidentally, recall that the inner product is positive-definite on .
Also since the weight space of must be one-dimensional, and for contributes nothing there.
So, we have a triangular expression for in terms of with entries one on the diagonal. Inverting:
Proposition 3 There are with
Next, we’ll actually compute these .
The value of the
This turns out to be very simple.
Proposition 4 for , the Weyl group. For , we have .
Here is the determinant of as an orthogonal transformation.
In particular, we can obtain the character of
Next time, we shall prove this proposition and associated character formulas.