Strong integrality of inversion subgroups of Kac-Moody groups

TL;DR

This paper proves the strong integrality of certain subgroups within Kac-Moody groups, specifically inversion subgroups and parts of the unipotent subgroup, enhancing understanding of their algebraic structure.

Contribution

It establishes the strong integrality property for inversion subgroups and certain unipotent subgroups of Kac-Moody groups, a novel result in the theory of these infinite-dimensional groups.

Findings

Proves strong integrality of inversion subgroups $U_{(w)}$.

Establishes strong integrality of subgroups generated by commuting real root groups.

Demonstrates strong integrality for specific subgroups in rank 2 cases.

Abstract

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac--Moody algebra $\frak{g}$ over ${\mathbb Q}$. Let $V=V^{\lambda}$ be an integrable highest weight $\frak{g}$-module and let $V_{\mathbb Z}=V^{\lambda}_{\mathbb Z}$ 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 $\Gamma({\mathbb Z})$ 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 $M\cap G({\mathbb Z})=M\cap \Gamma({\mathbb Z})$ and that $M$ is strongly integral if there exists $v\in V^{\lambda}_{\mathbb Z}$ 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 $U=U_{1}{\Large{*}}\ U_{2}$ and each $U_{i}$ is generated by `half' the positive real roots.