and set ${\mathbf{I}}:= {\mathop{\tt ev}\nolimits}_{0}^{-1}(B)$, an Iwahori subgroup of $G{[\hspace{-1pt}[z]\hspace{-1pt}]}$; we also set $\tilde{{\mathbf{I}}} := {\mathbf{I}}\rtimes {\mathbb{C}}^{\ast}$, the semi-direct product group, where the group ${\mathbb{C}}^{\ast}$ (of loop rotations) acts on ${\mathbf{I}}$ as the dilation on $z$. Now, we define ${\mathbf{Q}_{G}^{\mathrm{rat}}}:= G{(\hspace{-1pt}(z)\hspace{-1pt})} / H N{(\hspace{-1pt}(z)\hspace{-1pt})}$, which is a pure ind-scheme of infinite type. Then the set of ${\mathbf{I}}$-orbits is in natural bijection with $W_{{\mathrm{af}}}$; let ${\mathbf{Q}_{G}}( x )$ denote the ${\mathbf{I}}$-orbit closure corresponding to $x \in W_{{\mathrm{af}}}$. We define ${\mathbf{Q}_{G}}:= {\mathbf{Q}_{G}}( e )$.

Our first main result is the following:

\[fnormal\] For each $x \in W_{\mathrm{af}}$, the scheme ${\mathbf{Q}_{G}}( x )$ is normal. In addition, there is an explicit $P^{+}$-graded algebra $R_G$ such that ${\mathbf{Q}_{G}}= {\mathop{\rm Proj}\nolimits}R_G$; here our ${\mathop{\rm Proj}\nolimits}$ is the $P^+$-graded one.

We remark that Theorem \[fnormal\] affirmatively answers [@BF14c Conjecture 2.1] and relevant speculations therein. Also, as we see below, the scheme ${\mathbf{Q}_{G}}$ is far from being “compact” (cf. [@Kat17 Theorem A] and [@FGT (7.1)]). In order to prove Theorem \[fnormal\] naturally, we introduce a “semi-infinite" Bott-Samelson-Demazure-Hansen tower that yields a normal ring $R_G$. From the construction, $R_G$ contains the projective coordinate ring of ${\mathbf{Q}_{G}}$. Moreover, on the basis of the fact that $R_G$ is generated by 