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

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 pma (k) of 𝔊 A (k) introduced by O. Mathieu and G. Rousseau, and let 𝔘 A ma+ (k) denote the unipotent radical of the Borel subgroup of 𝔊 A pma (k). In this paper, we exhibit a functorial dependence of the groups 𝔘 A ma+ (k) and 𝔊 A pma (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 pma (k) la complétion de 𝔊 A (k) introduite par O. Mathieu et G. Rousseau, et 𝔘 A ma+ (k) le radical unipotent du sous-groupe de Borel de 𝔊 A pma (k). Dans cet article, nous mettons en évidence une dépendance fonctorielle des groupes 𝔘 A ma+ (k) et 𝔊 A pma (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
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     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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