no. 6
Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity
Marquis, Timothée1
1FAU Erlangen-Nuernberg Department Mathematik Cauerstrasse 11 91058 Erlangen (Germany)
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2519-2576

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3301
Classification: 20G44, 20E42, 20E18
Keywords: Kac–Moody groups, Lie correspondence, Gabber–Kac simplicity, Linearity problem, Isomorphism problem
Mots-clés : Groupes de Kac–Moody, Correspondance de Lie, Simplicité au sens de Gabber–Kac, Problème de linéarité, Problème d’isomorphisme

Marquis, Timothée  1

1 FAU Erlangen-Nuernberg Department Mathematik Cauerstrasse 11 91058 Erlangen (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Marquis, Timothée. Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2519-2576. doi: 10.5802/aif.3301
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