Fusion Product of Positive Level Representations and Lie Algebra Homology

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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.

Introduction

Let $𝔤$ be a (finite-dimensional) complex simple Lie algebra (with the associated simply-connected complex algebraic group $G\text{)}$ and let $\stackrel{\sim }{𝔤}$ 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 $\stackrel{\sim }{𝔤}\text{-modules}$ of level (or central charge) $\ell \text{.}$ Then there is a fusion product ${\otimes }^{•}$ 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 ${\otimes }^{•}$ 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 $\hat{F}$ 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 $\stackrel{\sim }{𝔤}\text{.}$ We have made a conjecture on the homology $H_{•}\left(\hat{F}\right)$ of this complex (cf. Conjecture 2.3 and Theorem 2.4). The homology $H_{•}\left(\hat{F}\right)$ is isomorphic to the Lie algebra homology $H_{•}({\stackrel{\sim }{u}}^{-}L(V\left(\nu \right),\ell )\otimes V(\mu ;1)),$ where the notation is as in §1 and Definition 2.1. Recall that if we take $\mu =0,$ then $H_{•}({\stackrel{\sim }{u}}^{-}L(V\left(\nu \right),\ell )\otimes V(\mu ;1))$ is completely determined by Kostant's $\text{"}n\text{-homology}$ result" for the affine Kac-Moody algebra $\stackrel{\sim }{𝔤}$ (proved by Garland-Lepowsky).

Validity of the above-mentioned Conjecture 2.3 will immediately imply that the two products ${\otimes }^{•}$ 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 $𝔤$ 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 $H_{•}\left(\hat{F}\right)$ only for the group $G=SL\left(2\right)$ (cf, Proposition 4.4).

This is only an announcement of results without proofs.

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