the space of intertwining operators and the space of $P(z)$-intertwining maps of the same type we obtain a canonical linear isomorphism from the space ${\cal{V}}^{W'}_{W_{1}W_{2}}$ of intertwining operators of the indicated type to $\Hom _{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In the case that $W=V$, we obtain a decomposition of Peter-Weyl type for ${\cal{D}}_{P(z)}(V)$, which are what we call the regular representations of $V$.

Let us start with the classical Peter-Weyl theory. Let $G$ be a group $G$ and $F(G,{\C})$ be the space of complex-valued functions on $G$. Then $F(G,{\C})$ is a $G\times G$-module with $$\begin{aligned}
\label{egbim}
((g_{1},g_{2})f)(g)=f(g_{1}^{-1}gg_{2})\end{aligned}$$ for $g_{1}, g_{2}, g\in G, f\in F(G,{\C})$. Furthermore, if $G$ is a topological (Lie, algebraic) group, all continuous (${\C}^{\infty}$, algebraic) functions form a $G\times G$-submodule. Now let $G$ be a compact Lie group and $C^{0}(G)$ be the space of continuous functions on $G$. A representative function on $G$ is a continuous function which generates a finite-dimensional $1\otimes G$-submodule of $C^{0}(G)$, and the regular representation of $G$ is the $G\times G$-module ${\cal{J}}(G,{\C})$ of representative functions on $G$ (cf. \[BD\]). Let $U$ be an irreducible $G$-module. For $u\in U,\; u^{*}\in U^{*}$, we can view $u^{*}\otimes u$ as a function on $G$ by $$(u^{*}\otimes u)(g)=\<u^{*},gu\>\;\;\;\mbox{ for }g\in G.$$ 