Topics in Representation Theory: The Weyl Integral and Character Formulas

We have seen that irreducible representations of a compact Lie group G can be constructed starting from a highest weight space and applying negative roots to a highest weight vector. One crucial thing that this construction does not easily tell us is what the character of this irreducible representation will be. The character would tell us not just which weights occur in the representation, but with what multiplicities they occur (this multiplicity is one for the highest weight, but in general can be larger). The importance of the knowing the characters of the irreducibles is that, given an arbitrary representation, we can then compute its decomposition into irreducibles. As a vector space, the character ring R(G) has a distinguished basis given by the characters χi(g) of the irreducible representations. Recall that the orthogonality relations for characters are essentially the same as in the finite group case, with the sum over group elements replace by an integral ∫ χi(g)χj(g)dg = δ