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.

We denote ${R}_{\ell}\left(G\right)$ equipped with the fusion product ${\otimes}^{\u2022}$ (resp. ${\otimes}^{F}\text{)}$ by $({R}_{\ell}\left(G\right),{\otimes}^{\u2022})$ (resp. $({R}_{\ell}\left(G\right),{\otimes}^{F})\text{).}$ Recall that the associativity of $({R}_{\ell}\left(G\right),{\otimes}^{\u2022})$ follows from the factorization rule (cf. [TUY1989]) for ${\mathbb{P}}^{1}$ with punctures.

Set (for $\lambda ,\mu ,\nu \in {P}_{\ell}^{+}\text{)}$ $${\stackrel{\u203e}{\chi}}_{\lambda}(\nu ,\mu )\sum _{i\ge 1}{{\left(-1\right)}^{i}}\text{dim}\left({\text{Hom}}_{\U0001d524}(V\left(\lambda \right),{H}_{i}({\stackrel{\sim}{\U0001d532}}^{-},L(V\left(\nu \right),\ell )\otimes V(\mu ;1)))\right)\text{.}$$

For any $1\le i\le \text{rank}$ $\U0001d524,$ let ${\omega}_{i}\in {P}^{+}$ be the $i\text{-th}$ fundamental weight corresponding to $\U0001d524\text{.}$ We have the following lemma.

The products ${\otimes}^{\u2022}$ and ${\otimes}^{F}$ in ${R}_{\ell}\left(G\right)$ coincide if and only if for all $\lambda ,\mu ,\nu \in {P}_{\ell}^{+},$ ${\stackrel{\u203e}{\chi}}_{\lambda}(\nu ,\mu )=0\text{.}$ In fact, the products ${\otimes}^{\u2022}$ and ${\otimes}^{F}$ in ${R}_{\ell}\left(G\right)$ coincide if and only if for all $\lambda ,{\omega}_{i},\nu \in {P}_{\ell}^{+},$ ${\stackrel{\u203e}{\chi}}_{\lambda}(\nu ,{\omega}_{i})=0\text{.}$

As a consequence of the above lemma, together with some results of [Agr1995, Corollary 4.3], [BMi1995] and some partial determination of ${H}_{i}({\stackrel{\sim}{u}}^{-},L(V\left(\nu \right),\ell )\otimes V({\omega}_{i};1))$ for those ${\omega}_{i}$ such that $\u27e8{\omega}_{i},{\theta}^{\vee})\le 2,$ we obtain the following result.

For any simple (simply-connected) group $G$ of type ${A}_{n},$ ${B}_{n},$ ${C}_{n},$ ${D}_{n}$ or ${G}_{2},$ the products ${\otimes}^{\u2022}$ and ${\otimes}^{F}$ in ${R}_{\ell}\left(G\right)$ coincide. In particular, for these groups, the $\mathbb{Z}\text{-linear}$ map $\beta :R\left(G\right)\to {R}_{\ell}\left(G\right)$ (cf. Definition 3.2) is an algebra homomorphism with respect to the product ${\otimes}^{\u2022}$ in ${R}_{\ell}\left(G\right)\text{.}$

(a) Of course ${\stackrel{\u203e}{\chi}}_{\lambda}(\nu ,\mu )=0,$ for all $\lambda ,\mu ,\nu \in {P}_{\ell}^{+},$ if Conjecture 2.3 is true. In particular, the validity of Conjecture 2.3 will imply that ${\otimes}^{\u2022}$ and ${\otimes}^{F}$ coincide in ${R}_{\ell}\left(G\right)\text{.}$ (b) The "in particular" statement of the above theorem is due to Faltings [Fal1994, Appendix], proved via case by case computation. In fact, this result of Faltings was the main motivation behind our work. (c) It is likely that we can prove Theorem 4.2 for all $G$ and all positive integer $\ell $ by combining our proof of the above Theorem and some results of Finkelberg [Fin1993]. (Some details in [Fin1993] are not clear to me as yet.)

Finally, we can prove the validity of Conjecture (2.3) for $G=SL\left(2\right)\text{.}$

Conjecture 2.3 is true for the group $G=SL\left(2\right)\text{.}$ In particular. Theorem 2.4 is true in this case. In fact, in this case, one has a rather precise description of ${H}_{0}({\stackrel{\sim}{u}}^{-},L(V\left(\nu \right),\ell )\otimes V(\mu ;1))$ and hence of ${H}_{i}({\stackrel{\sim}{u}}^{-},L(V\left(\nu \right),\ell )\otimes V(\mu ;1))$ for all $i\ge 0\text{.}$