\vert \ \alpha \in R \} = \{-1, 0 ,1 \}.$$ This gives us the induced ${\mathbb{Z}}$-gradation of ${\mathfrak{g}}$: $$\begin{aligned}
\label{grad}
{\mathfrak{g}}= {\mathfrak{g}}_{-1} + {\mathfrak{g}}_0 + {\mathfrak{g}}_1.\end{aligned}$$ Set $\Gamma = \{ \alpha \in R \ \vert \ \langle \omega, \alpha  \rangle = 1\}.$ Then $$\Gamma = \{ \gamma_{2}, \dots, \gamma_{\ell}, \gamma_{\underline{\ell}}, \dots, \gamma_{\underline{2}} \}$$ where $\gamma_i = \epsilon_1 + \epsilon_i$ and $\gamma_{\underline{i}} = \epsilon_1 - \epsilon_i$, $i=1, \dots, \ell$. Notice that $\gamma_2$ is the maximal root, $\gamma_2 = \theta$.

Denote by ${\tilde{\mathfrak{g}}}$ the associated affine Lie algebra (see, for example, [@K]), $${\tilde{\mathfrak{g}}}= {\mathfrak{g}}\otimes {\mathbb{C}}[t, t^{-1}] \oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$$ where $c$ is the canonical central element and $d$ the degree operator. Set $x(j)=x \otimes t^{j}$ for $x \in {\mathfrak{g}}$, $j \in {\mathbb{Z}}$ and let $x(z)=\sum_{n \in {\mathbb{Z}}} x(n)z^ {-n-1}$ be the formal Laurent series in formal variable $z$. Set ${\mathfrak{h}}^e = {\mathfrak{h}}\oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$, ${\tilde{\mathfrak{n}}}_{\pm} = {\mathfrak{g}}\otimes t^{\pm 1} {\mathbb{C}}[t^{\pm 1}] \oplus {\mathfrak{n}}_{\pm}$. We have a triangular decomposition: ${\tilde{\mathfrak{g}}}= {\tilde{\mathfrak{n}}}_{-} \oplus {\mathfrak{h}}^e \oplus {\tilde{\mathfrak{n}}}_{+}$ and the induced ${\mathbb{Z}}$-gradation with respect to $\omega$:

$${\tilde{\mathfrak{g}}}={\tilde{\mathfrak{g}}}_{-1} + {\tilde{\mathfrak{g}}}_0 + {\tilde{\mathfrak{g}}}_1.$$

Let $\widetilde{\Pi} = \{\alpha_0, \alpha_1, \cdots, \alpha_{\ell} \} \subset ({\mathfrak{h}}^e)^*$ be the set of simple roots of ${\tilde{\mathfrak{g}}}$. We define the fundamental 