Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity
(2019) Institut Fourier. Annales — Vol. 69, n° 6, p. 2519-2576 (2019)
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- 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 functorial 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 ${\mathfrak G}_A^{pma}(k)$. In addition, we explain how the problematic of providing a unified theory of complete Kac--Moody groups is related to the problem of Gabber--Kac simplicity of ${\mathfrak G}_A^{pma}(k)$, asking whether every normal subgroup of ${\mathfrak G}_A^{pma}(k)$ that is contained in ${\mathfrak U}_A^{ma+}(k)$ must be trivial. We contribute to this problem by giving the first counter-examples to Gabber--Kac simplicity. We further present several motivations for the study of this problem, 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.
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Marquis, T. (2019). Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity. Institut Fourier. Annales, 69(6), 2519-2576. https://doi.org/10.5802/aif.3301 (Original work published 2019)