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).

**Central characters **

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. (more…)