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

Marquis, Timothée

doi:10.5802/aif.3301

Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity
[Autour de la correspondance de Lie pour les groupes de Kac–Moody maximaux et de la simplicité au sens de Gabber–Kac]

Marquis, Timothée ¹

¹ FAU Erlangen-Nuernberg Department Mathematik Cauerstrasse 11 91058 Erlangen (Germany)

Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2519-2576.

Résumé
Abstract

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.

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.

Reçu le : 2017-03-30
Révisé le : 2018-04-26
Accepté le : 2019-01-17
Publié le : 2019-10-29

DOI : 10.5802/aif.3301

Classification : 20G44, 20E42, 20E18
Keywords: Kac–Moody groups, Lie correspondence, Gabber–Kac simplicity, Linearity problem, Isomorphism problem
Mot clés : Groupes de Kac–Moody, Correspondance de Lie, Simplicité au sens de Gabber–Kac, Problème de linéarité, Problème d’isomorphisme

Affiliations des auteurs :

Marquis, Timothée ¹

¹ FAU Erlangen-Nuernberg Department Mathematik Cauerstrasse 11 91058 Erlangen (Germany)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Around the {Lie} correspondence for complete {Kac{\textendash}Moody} groups and {Gabber{\textendash}Kac} simplicity},
     journal = {Annales de l'Institut Fourier},
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Marquis, Timothée. Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2519-2576. doi : 10.5802/aif.3301. https://aif.centre-mersenne.org/articles/10.5802/aif.3301/

Bibliographie
Cité par

[1] Abramenko, Peter; Brown, Kenneth S. Buildings. Theory and applications, Graduate Texts in Mathematics, 248, Springer, 2008, xxii+747 pages | Zbl

[2] Capdeboscq, Inna; Rémy, Bertrand On some pro- $p$ groups from infinite-dimensional Lie theory, Math. Z., Volume 278 (2014) no. 1-2, pp. 39-54 | DOI | MR | Zbl

[3] Caprace, Pierre-Emmanuel “Abstract” homomorphisms of split Kac–Moody groups, Mem. Am. Math. Soc., Volume 198 (2009) no. 924, xvi+84 pages | MR | Zbl

[4] Caprace, Pierre-Emmanuel; Reid, Colin D.; Willis, George A. Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups, Forum Math. Sigma, Volume 5 (2017), e12, 89 pages | DOI | MR | Zbl

[5] Caprace, Pierre-Emmanuel; Rémy, Bertrand Simplicity and superrigidity of twin building lattices, Invent. Math., Volume 176 (2009) no. 1, pp. 169-221 | DOI | MR | Zbl

[6] Caprace, Pierre-Emmanuel; Stulemeijer, Thierry Totally disconnected locally compact groups with a linear open subgroup, Int. Math. Res. Not. (2015) no. 24, pp. 13800-13829 | DOI | MR | Zbl

[7] Carbone, Lisa; Chung, Sjuvon; Cobbs, Leigh; McRae, Robert; Nandi, Debajyoti; Naqvi, Yusra; Penta, Diego Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, J. Phys. A, Math. Gen., Volume 43 (2010) no. 15, 155209, 30 pages | DOI | MR | Zbl

[8] Carbone, Lisa; Garland, Howard Existence of lattices in Kac–Moody groups over finite fields, Commun. Contemp. Math., Volume 5 (2003) no. 5, pp. 813-867 | DOI | MR | Zbl

[9] Dixon, John D.; du Sautoy, Marcus P. F.; Mann, Avinoam; Segal, Dan Analytic pro- $p$ groups, Cambridge Studies in Advanced Mathematics, 61, Cambridge University Press, 1999, xviii+368 pages | DOI | MR | Zbl

[10] Hainke, Guntram; Köhl, Ralf; Levy, Paul Generalized spin representations, Münster J. Math., Volume 8 (2015) no. 1, pp. 181-210 (With an appendix by Max Horn and Ralf Köhl) | MR | Zbl

[11] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, 1990, xxii+400 pages | Zbl

[12] Kang, Seok-Jin; Melville, Duncan J. Rank $2$ symmetric hyperbolic Kac–Moody algebras, Nagoya Math. J., Volume 140 (1995), pp. 41-75 | DOI | MR | Zbl

[13] Kumar, Shrawan Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser, 2002, xvi+606 pages | MR | Zbl

[14] Lyndon, Roger C.; Schupp, Paul E. Combinatorial group theory, Classics in Mathematics, Springer, 2001, xiv+339 pages (Reprint of the 1977 edition) | Zbl

[15] Marquis, Timothée Topological Kac–Moody groups and their subgroups, Université Catholique de Louvain (Belgium) (2013) (Ph. D. Thesis) | Zbl

[16] Marquis, Timothée Abstract simplicity of locally compact Kac-Moody groups, Compos. Math., Volume 150 (2014) no. 4, pp. 713-728 | DOI | MR | Zbl

[17] Marquis, Timothée An introduction to Kac–Moody groups over fields, EMS Textbooks in Mathematics, European Mathematical Society, 2018, 343 pages | DOI | MR | Zbl

[18] Mathieu, Olivier Construction du groupe de Kac–Moody et applications, C. R. Math. Acad. Sci. Paris, Volume 306 (1988) no. 5, pp. 227-230 | MR | Zbl

[19] Moody, Robert A simplicity theorem for Chevalley groups defined by generalized Cartan matrices (preprint)

[20] Morita, Jun Root strings with three or four real roots in Kac-Moody root systems, Tôhoku Math. J., Volume 40 (1988) no. 4, pp. 645-650 | DOI | MR | Zbl

[21] Mühlherr, Bernhard Locally split and locally finite twin buildings of $2$ -spherical type, J. Reine Angew. Math., Volume 511 (1999), pp. 119-143 | DOI | MR | Zbl

[22] Mühlherr, Bernhard Twin buildings, Tits buildings and the model theory of groups (Würzburg, 2000) (London Mathematical Society Lecture Note Series), Volume 291, Cambridge University Press, 2002, pp. 103-117 | DOI | MR | Zbl

[23] Nikolov, Nikolay; Segal, Dan On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. Math., Volume 165 (2007) no. 1, pp. 171-238 | DOI | MR | Zbl

[24] Rémy, Bertrand Groupes de Kac–Moody déployés et presque déployés, Astérisque, 277, Société Mathématique de France, 2002, viii+348 pages | Zbl

[25] Rémy, Bertrand Topological simplicity, commensurator super-rigidity and non-linearities of Kac–Moody groups, Geom. Funct. Anal., Volume 14 (2004) no. 4, pp. 810-852 (With an appendix by P. Bonvin) | DOI | MR | Zbl

[26] Rémy, Bertrand; Ronan, Mark Topological groups of Kac–Moody type, right-angled twinnings and their lattices, Comment. Math. Helv., Volume 81 (2006) no. 1, pp. 191-219 | DOI | MR | Zbl

[27] Riehm, Carl The congruence subgroup problem over local fields, Am. J. Math., Volume 92 (1970), pp. 771-778 | DOI | MR | Zbl

[28] Rousseau, Guy Groupes de Kac–Moody déployés sur un corps local, II Masures ordonnées, Bull. Soc. Math. Fr., Volume 144 (2016) no. 4, pp. 613-692 | DOI | MR | Zbl

[29] Tits, Jacques Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra, Volume 105 (1987) no. 2, pp. 542-573 | DOI | MR | Zbl

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