with ${\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2}, {\cal{D}}_{P(z)}(W))$. If $V$ is regular, i.e., any weak $V$-module is a direct sum of irreducible (ordinary) modules, then we obtain a decomposition of ${\cal{D}}_{P(z)}(W)$ into irreducible $V\otimes V$-modules with the corresponding fusion rules as the multiplicities. In the case that $W=V$, we obtain a decomposition of ${\cal{D}}_{P(z)}(V)$ of Peter-Weyl type.

This paper is intimately related to \[H3, HL0-3\] and we use certain basic principles and techniques that have been explored herein. In the tensor product theory for a vertex operator algebra \[H3, HL0-3\], for [*two*]{} given $V$-modules $W_{1}$ and $W_{2}$, a $V$-module “$W_{1}\otimes_{P(z)} W_{2}$” was constructed such that for any $V$-module $W$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ canonically gives rise to a $V$-homomorphism from “$W_{1}\otimes _{P(z)}W_{2}$” to $W'$. In the present work, by constructing a $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ for [*one*]{} given $V$-module $W$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ becomes a homomorphism in the category of (weak) [*$V\otimes V$-modules*]{} instead of the category of [*$V$-modules*]{}.

In a recent work \[FM\], Frenkel and Malikov obtained some interesting results by using Kazhdan-Lusztig tensoring and Harish-Chandra categories. The bimodule ${\cal{D}}_{P(z)}(W)$ is analogous to the Harish-Chandra bimodule associated to a module for a Lie algebra. We 