It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic, $$ \chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G $$
Is there a generalization of this for chiral algebras? If we let $\chi_\lambda(z,\tau)$ be the characters at modulus $\tau$ and fugacities $z\in \mathfrak g$ (i.e., the torus trace of a primary $\lambda$ in a WZW model), is there an expression of the form $$ \chi_\lambda(z,\tau)\chi_\mu(z,\tau)\overset?=\chi_{\mu\times\lambda}(z,\tau) $$ where $\mu\times\lambda$ denotes fusion? (I tried to check this equality in a simple example and it seems to fail; is there a version of this equality that is true? After all, the characters obviously "know" the fusion rules via the Verlinde formula. On the other hand, it is not clear to me how this equality could possibly hold if $\lambda,\mu$ are not mutually local...)
