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

Marquis, Timothée

Annales de l'Institut Fourier, to appear, 58 p.

Abstract
Résumé

Let $k$ be a field and $A$ be a generalised Cartan matrix, and let $𝔊_{A} (k)$ be the corresponding minimal Kac–Moody group of simply connected type over $k$ . Consider the completion $𝔊_{A}^{p m a} (k)$ of $𝔊_{A} (k)$ introduced by O. Mathieu and G. Rousseau, and let $𝔘_{A}^{m a +} (k)$ denote the unipotent radical of the Borel subgroup of $𝔊_{A}^{p m a} (k)$ . In this paper, we exhibit a functorial dependence of the groups $𝔘_{A}^{m a +} (k)$ and $𝔊_{A}^{p m a} (k)$ on their Lie algebra. We also provide several contributions to fundamental questions in the general theory of maximal Kac–Moody groups: (non-)Gabber–Kac simplicity over certain finite fields, (non-)density of a minimal Kac–Moody group in its Mathieu–Rousseau completion, (non-)linearity of maximal pro- $p$ subgroups, and the isomorphism problem.

Soit $𝔊_{A} (k)$ le groupe de Kac–Moody minimal simplement connexe associé à un corps $k$ et à une matrice de Cartan généralisée $A$ . On note $𝔊_{A}^{p m a} (k)$ la complétion de $𝔊_{A} (k)$ introduite par O. Mathieu et G. Rousseau, et $𝔘_{A}^{m a +} (k)$ le radical unipotent du sous-groupe de Borel de $𝔊_{A}^{p m a} (k)$ . Dans cet article, nous mettons en évidence une dépendance fonctorielle des groupes $𝔘_{A}^{m a +} (k)$ et $𝔊_{A}^{p m a} (k)$ en leur algèbre de Lie. Nous apportons en outre plusieurs contributions à certaines questions fondamentales de la théorie générale des groupes de Kac–Moody maximaux : (non-)densité du groupe de Kac–Moody minimal dans sa complétion de Mathieu–Rousseau, (non-)Gabber–Kac simplicité sur certains corps finis, (non-)linéarité des sous-groupes pro- $p$ maximaux, et problème d’isomorphisme.

Received : 2017-03-30
Revised : 2018-04-26
Accepted : 2019-01-17
Classification: 20G44, 20E42, 20E18
Keywords: Kac–Moody groups, Lie correspondence, Gabber–Kac simplicity, Linearity problem, Isomorphism problem

@unpublished{AIF_0__0_0_A5_0,
     author = {Marquis, Timoth\'ee},
     title = {Around the Lie correspondence for complete Kac--Moody groups and Gabber--Kac simplicity},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity. Annales de l'Institut Fourier, to appear, 58 p.

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