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
     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.

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

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

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

[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, Tome 5 (2017), e12, 89 pages | Article | Zbl 1401.22003

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

[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 | Article

[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., Tome 43 (2010) no. 15, 155209, 30 pages | Article

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

[9] Dixon, John D.; Du Sautoy, Marcus P. F.; Mann, Avinoam; Segal, Dan Analytic pro-p groups, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 61 (1999), xviii+368 pages | Article | Zbl 0934.20001

[10] Hainke, Guntram; Köhl, Ralf; Levy, Paul Generalized spin representations, Münster J. Math., Tome 8 (2015) no. 1, pp. 181-210 (With an appendix by Max Horn and Ralf Köhl) | Zbl 1402.17032

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

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

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

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

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

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

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

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

[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., Tome 40 (1988) no. 4, pp. 645-650 | Article

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

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

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

[24] Rémy, Bertrand Groupes de Kac–Moody déployés et presque déployés, Société Mathématique de France, Astérisque, Tome 277 (2002), viii+348 pages | Zbl 1001.22018

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

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

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

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

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