$P_{{\mathrm{af}}}^{0}=P \oplus {\mathbb{Z}}\delta$, and that $${\langle \mu,\,\alpha_{0}^{\vee} \rangle} = - {\langle \mu,\,\theta^{\vee} \rangle} \quad 
\text{for $\mu \in P_{{\mathrm{af}}}^{0}$}.$$

Let $W := \langle s_{i} \mid i \in I \rangle$ and $W_{{\mathrm{af}}} := \langle s_{i} \mid i \in I_{{\mathrm{af}}} \rangle$ be the (finite) Weyl group of ${\mathfrak{g}}$ and the (affine) Weyl group of ${\mathfrak{g}}_{{\mathrm{af}}}$, respectively, where $s_{i}$ is the simple reflection with respect to $\alpha_{i}$ for each $i \in I_{{\mathrm{af}}}$, with length function $\ell:W_{{\mathrm{af}}} \rightarrow {\mathbb{Z}}_{\ge 0}$, which gives the one on $W$ by restriction; we denote by $e \in W_{{\mathrm{af}}}$ the identity element, and by $w_{\circ} \in W$ the longest element. For each $\xi \in Q^{\vee}$, let $t_{\xi} \in W_{{\mathrm{af}}}$ denote the translation in ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ by $\xi$ (see [@Kac Sect. 6.5]); for $\xi \in Q^{\vee}$, we have $$\label{eq:wtmu}
t_{\xi} \mu = \mu - {\langle \mu,\,\xi \rangle}\delta \quad 
\text{if $\mu \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ satisfies ${\langle \mu,\,c \rangle}=0$}.$$ Then, $\bigl\{ t_{\xi} \mid \xi \in Q^{\vee} \bigr\}$ forms an abelian normal subgroup of $W_{{\mathrm{af}}}$, in which $t_{\xi} t_{\zeta} = t_{\xi + \zeta}$ holds for $\xi,\,\zeta \in Q^{\vee}$. Moreover, we know from [@Kac Proposition 6.5] that $$W_{{\mathrm{af}}} \cong 
 W \ltimes \bigl\{ t_{\xi} \mid \xi \in Q^{\vee} \bigr\} \cong W \ltimes Q^{\vee};$$ we also set $$\label{eq:Wge0}
W_{{\mathrm{af}}}^{\ge 