
Journal of Lie Theory 34 (2024), No. 2, 453468 Copyright Heldermann Verlag 2024 Strong Integrality of Inversion Subgroups of KacMoody Groups Abid Ali Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A. abid.ali@rutgers.edu Lisa Carbone Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A. carbonel@math.rutgers.edu Dongwen Liu School of Mathematical Sciences, Zhejiang University, Hangzhou, P. R. China maliu@zju.edu.cn Scott H. Murray Department of Mathematics, Rutgers University, Piscataway, New Jersey, U.S.A. scotthmurray@gmail.com [Abstractpdf] \def\Q{{\mathbb Q}} \def\Z{{\mathbb Z}} \DeclareMathOperator{\End}{End} Let $A$ be a symmetrizable generalized Cartan matrix with corresponding KacMoody algebra $\frak{g}$ over $\Q$. Let $V=V^{\lambda}$ be an integrable highest weight $\frak{g}$module with dominant regular integral weight $\lambda$ and representation $\rho: \frak{g}\to \End(V)$, and let $V_\Z=V^{\lambda}_\Z$ be a $\Z$form of $V$. Let $G_V(\Q)$ be the associated minimal KacMoody group generated by the automorphisms $\exp(t\rho(e_{i}))$ and $\exp(t\rho(f_{i}))$ of $V$, where $e_i$ and $f_i$ are the ChevalleySerre generators and $t\in\Q$. Let $G(\Z)$ be the group generated by $\exp(t\rho(e_{i}))$ and $\exp(t\rho(f_{i}))$ for $t\in\Z$. Let $\Gamma(\Z)$ be the Chevalley subgroup of $G_V(\Q)$, that is, the subgroup that stabilizes the lattice $V_{\Z}$ in $V$. For a subgroup $M$ of $G_V(\Q)$, we say that $M$ is integral if $M\cap G(\Z) = M\cap \Gamma(\Z)$ and that $M$ is strongly integral if there exists $v\in V_\Z$ such that $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G({\mathbb{Z}})$ for all $g\in M$. We prove strong integrality of inversion subgroups $U_{(w)}$ of $G_V(\Q)$ for $w$ in the Weyl group, where $U_{(w)}$ is the group generated by positive real root groups that are flipped to negative root groups by $w^{1}$. We use this to prove strong integrality of subgroups of the unipotent subgroup $U$ of $G_V(\Q)$ that are 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. Keywords: KacMoody groups, Chevalley groups, integrality. MSC: 20G44, 81R10; 22F50, 17B67. [ Fulltextpdf (168 KB)] for subscribers only. 