Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity

TL;DR

This paper investigates the structure and functoriality of complete Kac-Moody groups, explores their relation to the Gabber-Kac simplicity conjecture, and provides examples of non-dense minimal Kac-Moody groups.

Contribution

It introduces a functoriality dependence of these groups on their Lie algebra and relates the theory to the Gabber-Kac simplicity conjecture, with applications to non-linearity and isomorphism problems.

Findings

Examples of non-dense minimal Kac-Moody groups are constructed.

Functoriality dependence of groups on their Lie algebra is established.

Connections to the Gabber-Kac simplicity conjecture are clarified.

Abstract

Let $k$ be a field and $A$ be a generalised Cartan matrix, and let ${\mathfrak G}_A(k)$ be the corresponding minimal Kac-Moody group of simply connected type over $k$. Consider the completion ${\mathfrak G}_A^{pma}(k)$ of ${\mathfrak G}_A(k)$ introduced by O. Mathieu and G. Rousseau, and let ${\mathfrak U}_A^{ma+}(k)$ denote the unipotent radical of the positive Borel subgroup of ${\mathfrak G}_A^{pma}(k)$. In this paper, we exhibit some functoriality dependence of the groups ${\mathfrak U}_A^{ma+}(k)$ and ${\mathfrak G}_A^{pma}(k)$ on their Lie algebra. We also produce a large class of examples of minimal Kac-Moody groups ${\mathfrak G}_A(k)$ that are not dense in their Mathieu-Rousseau completion ${\mathfrak G}_A^{pma}(k)$. Finally, we explain how the problematic of providing a unified theory of complete Kac-Moody groups is related to the conjecture of Gabber-Kac simplicity of ${\mathfrak G}_A^{pma}(k)$, stating that every normal subgroup of ${\mathfrak G}_A^{pma}(k)$ that is contained in ${\mathfrak U}_A^{ma+}(k)$ must be trivial. We present several motivations for the study of this conjecture, as well as several applications of our functoriality theorem, with contributions to the question of (non-)linearity of ${\mathfrak U}_A^{ma+}(k)$, and to the isomorphism problem for complete Kac-Moody groups over finite fields. For $k$ finite, we also make some observations on the structure of ${\mathfrak U}_A^{ma+}(k)$ in the light of some important concepts from the theory of pro-$p$ groups.