I am trying to figure out representations of loop groups in order to understand conformal blocks. I am currently trying to figure out are the weight spaces of a representation with a given lowest weight and get my hands dirty with one. To build the representation I use the following construction. Let $\hat{\mathfrak{g}}=\mathbb{C}\langle d\rangle \oplus \mathbb{C} \langle k\rangle \oplus \mathfrak{g}\[ z, z^{-1} \]$ be the affine Lie algebra associated to the extended Affine group $\mathbb{T}\ltimes \tilde{L}G$. It has Cartan sub-algebra $\hat{\mathfrak{h}}=\mathbb{C}\langle d\rangle \oplus \mathbb{C} \langle k\rangle \oplus \mathfrak{h}$. Let $\delta$ and $\Lambda$ be dual to $d$ and $k$ respectively.
In the notes I am using it builds a representation of weight $\lambda+l\Lambda$ in the following way. Let $V_0$ be an irreducible representation of $S^1 \times G$ of weight $\lambda+l\Lambda$. Then consider: $$ \mathfrak{U}(\hat{\mathfrak{g}})\otimes_{\mathfrak{U}(\hat{\mathfrak{b}}_- + \mathfrak{g})\ }V_0$$ It then goes on to say that this has a maximal proper sub-module and that the quotient is the correct representation. I've been staring at this for a while and I cannot figure out what it the sub-module is. Can anyone give me a set of generators or point me to somewhere I may find them. All the literature I have relates to this is on normal Verma modules and google has't turned up anything.
