Strong integrality of inversion subgroups of Kac-Moody groups

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac–Moody algebra $\mathfrak{g}$ over $\mathbb{Q}$. Let $V=V^{\lambda }$ be an integrable highest weight $\mathfrak{g}$-module and let $V_{\mathbb{Z}}=V_{\mathbb{Z}}^{\lambda }$ be a $\mathbb{Z}Z$-form of $V$. Let $G$ be an associated minimal representation-theoretic Kac–Moody group and let $G(\mathbb{Z})$ be its integral subgroup. Let $\mathrm{\Gamma }(\mathbb{Z})$ be the Chevalley subgroup of $G$, that is, the subgroup that stabilizes the lattice $V_{\mathbb{Z}}$ in $V$. For a subgroup $M$ of $G$, we say that $M$ is integral if $M\cap G(\mathbb{Z})=M\cap \mathrm{\Gamma }(\mathbb{Z})$ and that $M$ is strongly integral if there exists $v\in V_{\mathbb{Z}}^{\lambda }$ such that, for all $g\in M$, $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G(\mathbb{Z})$. We prove strong integrality of inversion subgroups $U_{(w)}$ of $G$ where, for $w\in W$, $U_{(w)}$ is the the group generated by positive real root groups that are flipped to negative roots by $w^{-1}$. We use this to prove strong integrality of subgroups of the unipotent subgroup $U$ of $G$ generated by commuting real root groups. When $A$ has rank 2, this gives strong integrality of subgroups $U_1$ and $U_2$ where $U=U_1\ast U_2$ and each $U_i$ is generated by `half’ the positive real roots.