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 $\U0001d524$ be a (finite-dimensional) complex simple Lie algebra (with the associated simply-connected complex algebraic group $G\text{)}$ and let $\stackrel{\sim}{\U0001d524}$ be the corresponding affine Kac-Moody Lie algebra. Fix a positive integer $\ell \text{.}$ Let ${R}_{\ell}\left(G\right)$ be the free Abelian group generated by the set of integrable highest weight irreducible $\stackrel{\sim}{\U0001d524}\text{-modules}$ of level (or central charge) $\ell \text{.}$ Then there is a fusion product ${\otimes}^{\u2022}$ in ${R}_{\ell}\left(G\right)$ 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 ${\otimes}^{\u2022}$ 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 ${\otimes}^{F}$ in ${R}_{\ell}\left(G\right)$ in terms of the Euler-Poincaré characteristic of certain homogeneous vector bundles on the generalised affine flag variety $X$ (cf. Definition 3.2). Our definition of the product ${\otimes}^{F}$ is very simple and geometric in nature.

A comparison of the two fusion products led us to define a certain chain-complex $\stackrel{\u02c6}{F}$ whose terms arc finite-dimensional $G\text{-modules}$ (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 $\stackrel{\sim}{\U0001d524}\text{.}$ We have made a conjecture on the homology ${H}_{\u2022}\left(\stackrel{\u02c6}{F}\right)$ of this complex (cf. Conjecture 2.3 and Theorem 2.4). The homology ${H}_{\u2022}\left(\stackrel{\u02c6}{F}\right)$ is isomorphic to the Lie algebra homology ${H}_{\u2022}({\stackrel{\sim}{u}}^{-}L(V\left(\nu \right),\ell )\otimes V(\mu ;1)),$ where the notation is as in §1 and Definition 2.1. Recall that if we take $\mu =0,$ then ${H}_{\u2022}({\stackrel{\sim}{u}}^{-}L(V\left(\nu \right),\ell )\otimes V(\mu ;1))$ is completely determined by Kostant's $\text{"}n\text{-homology}$ result" for the affine Kac-Moody algebra $\stackrel{\sim}{\U0001d524}$ (proved by Garland-Lepowsky).

Validity of the above-mentioned Conjecture 2.3 will immediately imply that the two products ${\otimes}^{\u2022}$ and ${\otimes}^{F}$ in ${R}_{\ell}\left(G\right)$ 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 $\U0001d524$ of type ${A}_{n},$ ${B}_{n},$ ${C}_{n},$ ${D}_{n}$ and ${G}_{2}$ (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 ${H}_{\u2022}\left(\stackrel{\u02c6}{F}\right)$ only for the group $G=SL\left(2\right)$ (cf, Proposition 4.4).

This is only an announcement of results without proofs.