Positive-integral-level simple affine module is integrable

In the untwisted affine realization, ${\mathrm{\Lambda }}_0(c)=1$, ${\mathrm{\Lambda }}_0(d)=0$, and ${\mathrm{\Lambda }}_0(h_i)=0$ for the finite simple coroots. Since $h_0=c-{\theta }^{\vee }$ by [F2], ${\mathrm{\Lambda }}_0(h_0)=1$. Thus $\mathrm{\ell }{\mathrm{\Lambda }}_0$ has affine simple-coroot labels $\mathrm{\ell }$ at node $0$ and $0$ at every finite node, so it is dominant integral under [F4]. Define $\widehat{P}=\{\lambda \in {\widehat{\mathfrak{h}}}^{\ast }:\lambda (h_i)\in \mathbb{Z}\text{ for every affine simple coroot }{h}_i\}$. The affine Weyl action on weights is $s_i\lambda =\lambda -\lambda (h_i){\alpha }_i$. Hence for every $\lambda \in \widehat{P}$ and every $j$, $(s_i\lambda )(h_j)=\lambda (h_j)-\lambda (h_i){\alpha }_i(h_j)\in \mathbb{Z}$, since ${\alpha }_i(h_j)$ is an entry of the affine generalized Cartan matrix. The simple reflections generate the affine Weyl group, so it preserves $\widehat{P}$.

Write $B=s\kappa$ with $s>0$, as specified by the normalized form in [F2]. For a finite root $\alpha$, let $H_{\alpha }^{\kappa }\in \mathfrak{h}$ satisfy $\kappa (H_{\alpha }^{\kappa },h)=\alpha (h)$ for all $h\in \mathfrak{h}$, and put $q_{\alpha }=\alpha (H_{\alpha }^{\kappa })>0$. By [F3], ${\alpha }^{\vee }=2H_{\alpha }^{\kappa }\mathrm{/}{q}_{\alpha }$. Invariance applied to $[e_{\alpha },f_{\alpha }]={\alpha }^{\vee }$ gives $\kappa (e_{\alpha },f_{\alpha })=2\mathrm{/}{q}_{\alpha }$. The $B$-dual representative is $H_{\alpha }^B=s^{-1}{H}_{\alpha }^{\kappa }$, so the normalized dual root form satisfies $(\alpha ,\alpha )=\alpha (H_{\alpha }^B)=q_{\alpha }\mathrm{/}s$. Consequently ${\alpha }^{\vee }=2H_{\alpha }^B\mathrm{/}(\alpha ,\alpha )$ and $B(e_{\alpha },f_{\alpha })=s\kappa (e_{\alpha },f_{\alpha })=2\mathrm{/}(\alpha ,\alpha )$. Thus the coroot and the chosen root triple are unchanged by the rescaling, while the pairing has exactly the normalization required by the affine cocycle.

Fix $\alpha \in \mathrm{\Delta }$ and set $\gamma =\delta -\alpha$. By [F3], $\gamma$ is a positive real affine root, hence $\gamma =w{\alpha }_i$ for an affine Weyl element $w$ and a simple root ${\alpha }_i$. The simple-reflection Lie automorphisms implementing a word for $w$ carry the simple-root triple $(e_i,f_i,h_i)$ to an ${\mathfrak{s}\mathfrak{l}}_2$ triple in the $\gamma ,-\gamma$ root spaces; its bracket gives the real coroot ${\gamma }^{\vee }=w{h}_i$. In the loop realization these root spaces contain $f_{\alpha }\otimes t$ and $e_{\alpha }\otimes t^{-1}$. Their affine bracket is $[f_{\alpha }\otimes t,e_{\alpha }\otimes t^{-1}]=-[e_{\alpha },f_{\alpha }]+B(f_{\alpha },e_{\alpha })c=-{\alpha }^{\vee }+\frac{2}{(\alpha ,\alpha )}c$, by [F2] and step 1.2. This bracket has $\gamma$-value $2$, so these loop root vectors also form a normalized ${\mathfrak{s}\mathfrak{l}}_2$ triple. In one-dimensional opposite root spaces, any two nonzero pairs differ by two scalars; requiring their brackets to have $\gamma$-value $2$ forces the product of those scalars to be $1$. Their brackets therefore agree, identifying the displayed bracket with ${\gamma }^{\vee }$. Since ${\alpha }^{\vee }\in \mathfrak{h}$, ${\mathrm{\Lambda }}_0({\alpha }^{\vee })=0$, and $(\mathrm{\ell }{\mathrm{\Lambda }}_0)(c)=\mathrm{\ell }$, the bracket formula gives $(\mathrm{\ell }{\mathrm{\Lambda }}_0)({\gamma }^{\vee })=2\mathrm{\ell }\mathrm{/}(\alpha ,\alpha )$. This value is an integer because the Weyl actions on the Cartan and its dual satisfy $(\mathrm{\ell }{\mathrm{\Lambda }}_0)(w{h}_i)=(w^{-1}(\mathrm{\ell }{\mathrm{\Lambda }}_0))(h_i)$ and $w^{-1}(\mathrm{\ell }{\mathrm{\Lambda }}_0)\in \widehat{P}$ by step 1.1. It is positive because the finite root form is positive definite and $\mathrm{\ell }>0$. This proves the real-coroot formula and the positive-integer assertion for every finite root.

By [F1], the module underlying $L_{\mathrm{\ell }}(\mathfrak{g})$ is the simple affine highest-weight module $L_{\widehat{A}}(\mathrm{\ell }{\mathrm{\Lambda }}_0)$. Its highest weight is dominant integral by step 1.1, and the affine GCM $\widehat{A}$ is finite, so [F5] applies and makes this module integrable in the weight-module/local-nilpotence sense of Integrable kac moody module.