Let be a representation of a semisimple Lie algebra , a Cartan subalgebra , and some choice of splitting on the roots.
Recall from the representation theory of finite groups that to each representation of a finite group one can associate a character function , and that the ring generated by the characters is the Grothendieck ring of the semisimple tensor category . There is something similar to be said for semisimple Lie algebras. So, assume acts semisimply on and that the weight spaces are finite-dimensional, and set formally
In other words, we define the character so as to include all the information on the size of the weight spaces at once.
It is necessary, however, to define what for . Basically, it is just a formal symbol; can more rigorously be thought of as a function . Nevertheless, we want to think of as a formal exponential in a sense; we want to have . The reason is that we can tensor two representations, and we want to talk about multiplying to characters.
I now claim that the above condition on makes sense for , the BGG category. This will follow because it is true for highest weight modules and we have:
Proposition 1 If , then there is a finite filtration on whose quotients are highest weight modules.
To see this, note that is finitely generated over by some vectors , which wlog we assume to be weight vectors with weights . Let , say, be the highest one (according to bracketing with some used to define the splitting ); then is a highest weight vector, and the submodule generated by is a highest weight module. By induction on the dimension of , we can assume that there are appropriate filtrations on and ; thus we get one on . (The case is immediate.)
By the definition of , we also have:
Proposition 2 If is an exact sequence in , then .
As a result, it follows that the character of any object in category is supported on a finite union of sets of the form
where is the -lattice generated by the positive roots.
Let be -modules with decomposition into their weight spaces. Then
where acts on with weight . In particular,
and we have
where is the convolution of the two characters, defined by
where we consider the characters as functions. For this to make sense, we may assume that and is finite -dimensional, say. If we let be the characteristic function of , then , which explains our rules. Often we will drop the sign for convolution though.
In particular, on finite-dimensional representations, we have a ring-homomorphism from the Grothendieck ring to the space of finitely supported functions on with convolution.
The character of the Verma module
The character of the Verma module is easily computed. Let be the positive roots, and let be nonzero. Recall that has a basis consisting of
where is the initial highest weight vector, and that this displayed quantity has weight . It follows that is the number of ways one can express as a sum for each .
So, define the Kostant partition function
It now follows that
Here corresponds to the sum .