TITLE: Concise mathematical definition of the fusion product on the Verlinde ring?
QUESTION [7 upvotes]: The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the loop of $G$ with the 'positive energy' condition. I'm trying to find a concise mathematical definition of the product on this ring, the so-called fusion product. In Freed-Hopkins-Teleman's Loop groups and twisted K-theory III they define an $R(G)$-module structure on this additive group using induction of representations (hence a functorial description). 
All references I'm reading just mention 'fusion rules', and cite Verlinde, whose 1988 paper is in the journal Nuclear Physics B. Surely we have a more recent discussion along the lines of the FHT construction mentioned above, and not in terms of linear combinations of coefficients of some irreps considered as generators...?

REPLY [6 votes]: For any triple $(M_0,M_1,M_\infty)$ of positive energy representations of $\widehat{LG}$ at a fixed level, restriction to small punctured discs produces a canonical (up to some scaling which doesn't matter here) action of the Lie algebra $\mathfrak{g}_X = \Gamma(X, \mathfrak{g} \otimes \mathscr{O}_X)$, where $X$ is the thrice punctured line $\mathbb{P}^1 \setminus \{0,1,\infty\}$.  The multiplicity of $M_\infty^\vee$ in $M_0 \boxtimes M_1$ (i.e., the fusion rule) is given by the dimension of the space of coinvariants.  Since you are only asking about dimension, you get the same answer from the dual space $\mathrm{Hom}_{\mathfrak{g}_X}(M_0 \otimes M_1 \otimes M_\infty, \mathbb{C})$ of conformal blocks.  Looijenga has a brief treatment of the theory.