ABSTRACT
Let g be a complex simple Lie algebra and let g˜ be the corresponding affine Kac-Moody Lie algebra. Fix a positive integer ℓ. Let R1(G) be the free Abelian group generated by the set of integrable highest weight irreducible g˜-modules of level ℓ. Then there is a fusion product in R1(G) making it into a commutative and associative algebra. The definition of the product is in terms of the dimension of a certain space of vacua. This chapter gives a new definition of a fusion product in terms of the Euler-Poincare characteristic of certain homogeneous vector bundles on the generalised affine flag variety X. A comparison of the two fusion products led us to define a certain chain-complex F^ whose terms are finite-dimensional G-modules. The differentials of this complex are highly non-trivial and are obtained by considering the BGG resolution for the affine Kac-Moody algebra g˜.
