[@LNSSS2 Introduction]).

In order to explain our results, theories, and ideas more precisely, we need some notation. Let ${\mathfrak{g}}$ denote the Lie algebra of $G$, and let ${\mathfrak{g}}_{\mathrm{af}}$ denote the associated untwisted affine Lie algebra; we fix a Borel subgroup $B$ of $G$ and a maximal torus $H \subset B$, and set $N := [B,B]$. Let $W = N_{G}(H)/H$ be the Weyl group, which is generated by the simple reflections $s_{i}$, $i \in I$; $W$ can be thought of as acting on the dual space ${\mathfrak{h}}^{\ast}$ of the Cartan subalgebra ${\mathfrak{h}}:= {\mathop{\rm Lie}\nolimits}(H)$. We set $W_{{\mathrm{af}}} := W \ltimes Q^{\vee}$, with $Q^{\vee, +} := \sum_{i \in I} {\mathbb{Z}}_{\geq 0} \alpha_{i}^{\vee} \subset 
Q^{\vee} := \bigoplus_{i \in I} {\mathbb{Z}}\alpha_{i}^{\vee}$ (the coroot lattice). Let $P = \bigoplus_{i \in I} {\mathbb{Z}}{\varpi}_{i} \subset {\mathfrak{h}}^{\ast}$ be the weight lattice generated by the fundamental weights ${\varpi}_{i}$, $i \in I$, and set $P^+ := \sum_{i \in I} {\mathbb{Z}}_{\ge 0} {\varpi}_{i}$.

For an algebraic group $E$ over ${\mathbb{C}}$, we denote by $E{(\hspace{-1pt}(z)\hspace{-1pt})}$ and $E{[\hspace{-1pt}[z]\hspace{-1pt}]}$ the space of ${\mathbb{C}}{(\hspace{-1pt}(z)\hspace{-1pt})}$-valued points and the space of ${\mathbb{C}}{[\hspace{-1pt}[z]\hspace{-1pt}]}$-valued points of $E$, respectively, viewed as an (ind-)scheme over ${\mathbb{C}}$. Let ${\mathop{\tt ev}\nolimits}_{0} : G{[\hspace{-1pt}[z]\hspace{-1pt}]} \rightarrow G$ be the evaluation map at $z = 0$, 