A complete set of intertwiners for arbitrary tensor product representations via current algebras

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\vec{V}(\vec{\lambda})$ be a tensor product of $k$ copies of finite dimensional irreducible $\mathfrak{g}$-modules. Choosing $k$ points in $\mathbb{C}$, $\vec{V}(\vec{\lambda})$ acquires a natural structure of the current algebra $\mathfrak{g}\otimes \mathbb{C}[t]$-module. Following a work of Rao [R], we produce an explicit and complete set of $\mathfrak{g}$-module intertwiners of $\vec{V}(\vec{\lambda})$ in terms of the action of the current algebra.