Last updated: 26 March 2015
This is an excerpt of the paper Fusion Product of Positive Level Representations and Lie Algebra Homology by Shrawan Kumar, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA.
Let be a (finite-dimensional) complex simple Lie algebra (with the associated simply-connected complex algebraic group and let be the corresponding affine Kac-Moody Lie algebra. Fix a positive integer Let be the free Abelian group generated by the set of integrable highest weight irreducible of level (or central charge) Then there is a fusion product in making it into a commutative and associative algebra. (The associativity of this algebra follows from the so called factorization rule.) The definition of the product is in terms of the dimension of a certain space of vacua (cf. Definition 3.1). We give a new definition of a fusion product denoted in in terms of the Euler-Poincaré characteristic of certain homogeneous vector bundles on the generalised affine flag variety (cf. Definition 3.2). Our definition of the product is very simple and geometric in nature.
A comparison of the two fusion products led us to define a certain chain-complex whose terms arc finite-dimensional (cf. (2.3)). The differentials of this complex are highly non-trivial and are obtained by considering the BGG resolution for the affine Kac-Moody algebra We have made a conjecture on the homology of this complex (cf. Conjecture 2.3 and Theorem 2.4). The homology is isomorphic to the Lie algebra homology where the notation is as in §1 and Definition 2.1. Recall that if we take then is completely determined by Kostant's result" for the affine Kac-Moody algebra (proved by Garland-Lepowsky).
Validity of the above-mentioned Conjecture 2.3 will immediately imply that the two products and in are the same. In fact, a much weaker result will imply their equality (cf. Lemma 4.1). We prove this weaker result for all simple of type and (cf. Theorem 4.2). This provides an alternative (more uniform) proof of a result of Faltings (cf. Remark 4.3(b); see also Remark 4.3(c)). So far, we are able to determine the full homology only for the group (cf, Proposition 4.4).
This is only an announcement of results without proofs.