Journal of Lie Theory 34 (2024), No. 2, 453--468
Copyright Heldermann Verlag 2024
Strong Integrality of Inversion Subgroups of Kac-Moody 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
[Abstract-pdf]
\def\Q{{\mathbb Q}} \def\Z{{\mathbb Z}} \DeclareMathOperator{\End}{End} Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody 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 Kac-Moody 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 Chevalley-Serre 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: Kac-Moody groups, Chevalley groups, integrality.
MSC: 20G44, 81R10; 22F50, 17B67.
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