By A. Campillo

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**Sample text**

8), but the particular scalar by which it acts is somewhat more difficult to determine by that lemma alone. 5 Kac Character Formula In this section, we finally prove the Kac Character formula, which is the generalization of the Weyl character formula for a Kac-Moody algebra g. 8), deduce the Jacobi Triple Product identity (which appears everywhere from physics to number theory) and its generalizations the MacDonald Identities. Of course, the calculation of the formal character of an irreducible module L(Λ) from the category O is an interesting representation-theoretic result in its own right.

2) α∈Φ+ i=1 for nonnegative integers kα,i and an appropriate choice of ordering on the e−α,i . 3) α=µ i=1 i=1 40 α(x) v. 3) holds. We may rewrite the formal character of U (n− ) as follows ch U (n− ) = dim(U (n− )−α )e−α = α∈Φ+ 1 + e−α + (e−α )2 + . . 3) holds. Because U (n− )−µ is in bijection with M (Λ)Λ−µ , these weight spaces have the same dimension and we may relate the characters by ch U (n− ) = eΛ ch M (Λ). 4) that 1 + e−α + (e−α )2 + . . ch M (Λ) = eΛ mult α . α∈Φ+ But the element 1+e−α +(e−α )2 +.

3]. So as a formal series, we write ch M (Λ) = eΛ . −α )mult α α∈Φ+ (1 − e This formula is a precursor to the Kac character formula, which gives an explicit method of computing the character of the irreducible module L(Λ) in the case that Λ is what is called a dominant integral weight. In the finite dimensional case, the analogue of the Kac character formula, called the Weyl character formula, is proven using multiplicities [V : L(Λ)] for V ∈ O. These multiplicities are usually found using composition series of modules V ∈ O: [V : L(Λ)] should be the number of composition factors isomorphic to L(Λ), which would be independent of the choice of series by the Jordan¨ Holder Theorem.