thank Mirko Primc for his help on the topic and valuable suggestions.

Affine Lie algebras and basic modules {#algebra}
=====================================

For $\ell \geq 4$ let ${\mathfrak{g}}$ be a simple Lie algebra of type $D_{\ell}$, ${\mathfrak{h}}$ a Cartan subalgebra of ${\mathfrak{g}}$ and $R$ the corresponding root system. Fix a basis $\Pi = \{\alpha_1, \dots, \alpha_l \}$ of $R$ where $\alpha_1 = \epsilon_1 - \epsilon_2, \dots , \alpha_{\ell-1} = \epsilon_{\ell-1} - \epsilon_{\ell}$, $\alpha_{\ell} = \epsilon_{\ell-1} + \epsilon_{\ell}$ and $\epsilon_i, \ i=1, \dots, \ell$ are the usual orthonormal unit vectors. Then we have the corresponding triangular decomposition, ${\mathfrak{g}}= {\mathfrak{n}}_{-} \oplus {\mathfrak{h}}\oplus {\mathfrak{n}}_+$.

We denote by $\langle \cdot,\cdot\rangle$ the Killing form on ${\mathfrak{g}}$ which enables us to identify ${\mathfrak{h}}\cong {\mathfrak{h}}^*$. Let $\theta$ be the maximal root, i.e. $\theta = \epsilon_1 + \epsilon_2$. Then the Killing form can be normalized so that $\langle \theta,\theta\rangle = 2$.

For each root $\alpha$ we fix a root vector $x_{\alpha}$. Let $Q=Q(R)$ and $P=P(R)$ be the root and weight lattice of ${\mathfrak{g}}$ respectively.

Denote by $\omega_1, \dots, \omega_{\ell}$ the fundamental weights and define, for convenience, $\omega_0 = 0$. In this paper we fix $\omega = \omega_1=\epsilon_1$. Notice that $\omega$ is a minuscule weight, i.e. $$\{\langle \omega, \alpha \rangle \ 