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 1If , then there is a finite filtration on whose quotients are highest weight modules. (more…)