weights $\Lambda_i \subset ({\mathfrak{h}}^e)^*$ by $\langle \Lambda_i, \alpha_j \rangle = \delta_{ij}$ and $\Lambda_i (d)=0$ where $\langle \cdot, \cdot \rangle$ is the standard extension of Killing form onto ${\mathfrak{h}}^e$.

Let $L(\Lambda)$ be a standard ${\tilde{\mathfrak{g}}}$-module, i.e. an irreducible highest weight module with dominant integral highest weight

$$\begin{aligned}
\label{tezina}
\Lambda = k_0 \Lambda_0 + k_1 \Lambda_1 + \dots + k_l \Lambda_{\ell}, \quad k_i \in {\mathbb{Z}}_{+}, \ i=1, \dots, \ell.\end{aligned}$$

The central element $c$ acts on $L(\Lambda)$ as a scalar $k$ which is called the level of module $L(\Lambda)$. Modules $L(\Lambda_0), L(\Lambda_1), L(\Lambda_{\ell-1})$ and $L(\Lambda_{\ell})$ are of level $1$, whilst $L(\Lambda_i), i=2, \dots, \ell -2$ are level $2$ modules. In general, module $L(\Lambda)$ with $\Lambda$ as in is of the level $$k=k_0 + k_1 + 2k_2 + \dots + 2k_{\ell-2} + k_{\ell-1} + k_{\ell}.$$

Feigin-Stoyanovsky’s type subspaces $W(\Lambda)$ {#feigin}
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Let $L(\Lambda)$ be a standard ${\tilde{\mathfrak{g}}}$-module with a dominant integral highest weight $$\Lambda = k_0 \Lambda_0 + k_1 \Lambda_1 + \dots + k_{\ell} \Lambda_{\ell}.$$ Then $L(\Lambda)$ is of level $k=k_0 + k_1 + 2 k_2 + \dots +2 k_{\ell-2} + k_{\ell-1} + k_{\ell}$. Fix a highest weight vector $v_{\Lambda}$.

Recall the induced ${\mathbb{Z}}$-gradation we have on ${\tilde{\mathfrak{g}}}$ with respect to fundamental weight $\omega=\omega_1$: 