I’m now aiming to get to the major character formulas for the (finite-dimensional) simple quotients for dominant integral. They will follow from formal manipulations with character symbols and a bit of reasoning with the Weyl group. First, however, it is necessary to express as a sum of characters of Verma modules. We will do this by considering any highest weight module of weight for integral but not necessarily dominant, and considering a filtration on it whose quotients are simple modules where there are only finitely many possibilities for . Applying this to the Verma module, we will then get an expression for in terms of , which we can then invert.
First, it is necessary to study the action of the Casimir element (w.r.t. the Killing form). Recall that this is defined as follows: consider a basis for the semisimple Lie algebra and its dual basis under the Killing form isomorphism . Then the Casimir element is
for dual to . As we saw, this is a central element. I claim now that the Casimir acts by a constant factor on any highest weight module, and that the constant factor is determined by the weight in such a sense as to give information about the preceding filtration (this will become clear shortly).
Let and let be the Verma module. Then is also a vector with weight , so it is a constant multiple of . Since generates and is central, it follows that acts on by a scalar . Then becomes a character , i.e. an algebra-homomorphism. There is in fact a theorem of Harish-Chandra that states that determines the weight up to “linkage” (i.e. up to orbits of the dot action of the Weyl group: ), though I shall not prove this here.
The goal is to compute for the Casimir element as above. We shall first find a general expression.
By the PBW basis theorem, if we choose a basis of the Cartan subalgebra and a -type basis of (with being paired with and spanning a subalgebra isomorphic to with , etc.), then we can find a basis for via
There is a vector subspace , and any weight extends to a character , still denoted . With respect to the above basis, there is a projection .
This computes the character .
Write as a sum of terms as in (*). Those where all the are zero contribute precisely ; it must be seen that the others contribute nothing. Any term with some must annihilate the vector because these correspond to positive weights. I claim now that no term of the form
can even appear in the expansion for in this basis. Then bracketing with any is an operator on diagonalizable with respect to this basis. Moreover:
if ranges over the weights corresponding to . If we choose so that the thing in brackets is nonzero, we find that this is impossible, because anything in the kernel of bracketing with (e.g. in the center of ) cannot have any term in this basis with nonzero eigenvalue.
This proves the proposition.
Action of the Casimir
It is now necessary to compute for the Casimir.
First, choose a basis for and let be the dual basis. Next, choose for each root some and a dual ; then , of course. We can arrange things such that .
by the way we arranged things. However, the term is not in the right form for the basis. So we write this as
Now, in the projection , the middle term disappears, so we find that
where is the dual to under the identification . (This is a general fact about brackets between ; I proved it here.)
Consider the bilinear form on obtained by duality, which makes the dual bases of in biorthogonal; then
because this is checked for those dual bases in . Moreover, by assumption , so in total we find:
for one-half the sum of the positive roots.
Next: to show that the character of simple quotients is a sum of Verma module characters, for integral weights anyway.