${\mathfrak{g}}$, and set $$\label{eq:P-fin}
P:=\bigoplus_{i \in I} {\mathbb{Z}}{\varpi}_{i}, \qquad 
P^{+} := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}.$$

Let ${\mathfrak{g}}_{{\mathrm{af}}} = \bigl({\mathfrak{g}}\otimes {\mathbb{C}}[z,z^{-1}]\bigr) \oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$ be the untwisted affine Lie algebra over ${\mathbb{C}}$ associated to ${\mathfrak{g}}$, where $c$ is the canonical central element, and $d$ is the scaling element (or the degree operator), with Cartan subalgebra ${\mathfrak{h}}_{{\mathrm{af}}} = {\mathfrak{h}}\oplus {\mathbb{C}}c \oplus {\mathbb{C}}d$. We regard an element $\mu \in {\mathfrak{h}}^{\ast}:={\mathop{\rm Hom}\nolimits}_{{\mathbb{C}}}({\mathfrak{h}},\,{\mathbb{C}})$ as an element of ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ by setting ${\langle \mu,\,c \rangle}={\langle \mu,\,d \rangle}:=0$, where ${\langle \cdot\,,\,\cdot \rangle}:{\mathfrak{h}}_{{\mathrm{af}}}^{\ast} \times {\mathfrak{h}}_{{\mathrm{af}}} \rightarrow {\mathbb{C}}$ is the canonical pairing of ${\mathfrak{h}}_{{\mathrm{af}}}^{\ast}:={\mathop{\rm Hom}\nolimits}_{{\mathbb{C}}}({\mathfrak{h}}_{{\mathrm{af}}},\,{\mathbb{C}})$ and ${\mathfrak{h}}_{{\mathrm{af}}}$. Let $\{ \alpha_{i}^{\vee} \}_{i \in I_{{\mathrm{af}}}} \subset {\mathfrak{h}}_{{\mathrm{af}}}$ and $\{ \alpha_{i} \}_{i \in I_{{\mathrm{af}}}} \subset {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ be the set of simple coroots and simple roots of ${\mathfrak{g}}_{{\mathrm{af}}}$, respectively, where $I_{{\mathrm{af}}}:=I \sqcup \{0\}$; note that ${\langle \alpha_{i},\,c \rangle}=0$ and ${\langle \alpha_{i},\,d \rangle}=\delta_{i0}$ for $i \in I_{{\mathrm{af}}}$. Denote by $\delta \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$ the null root of ${\mathfrak{g}}_{{\mathrm{af}}}$; recall that $\alpha_{0}=\delta-\theta$. Also, let $\Lambda_{i} \in {\mathfrak{h}}_{{\mathrm{af}}}^{\ast}$, $i \in I_{{\mathrm{af}}}$, denote the fundamental weights for ${\mathfrak{g}}_{{\mathrm{af}}}$ such that ${\langle \Lambda_{i},\,d \rangle}=0$, and set $$\label{eq:P}
P_{{\mathrm{af}}} := 
  \left(\bigoplus_{i \in I_{{\mathrm{af}}}} {\mathbb{Z}}\Lambda_{i}\right) \oplus 
   {\mathbb{Z}}\delta \subset {\mathfrak{h}}^{\ast}, \qquad 
P_{{\mathrm{af}}}^{0}:=\bigl\{\mu \in P_{{\mathrm{af}}} \mid {\langle \mu,\,c \rangle}=0\bigr\};$$ notice that 