Receding polar regions of a spherical building and the center conjecture
Annales de l'Institut Fourier, Volume 63 (2013) no. 2, pp. 479-513.

We introduce the notion of a polar region of a spherical building and use some simple observations about polar regions to give elementary proofs of various fundamental properties of root groups. We combine some of these observations with results of Timmesfeld, Balser and Lytchak to give a new proof of the center conjecture for convex chamber subcomplexes of thick spherical buildings.

Nous introduisons la notion de région polaire d’un immeuble sphérique et utilisons quelques observations simples sur les régions polaires pour donner des démonstrations élémentaires de diverses propriétés fondamentales des sous-groupes radiciels. Nous combinons certaines de ces observations avec des résultats de Timmesfeld, Balser et Lytchak pour donner une nouvelle preuve de la conjecture du centre pour les sous-complexes des chambres convexes des immeubles épais sphériques.

DOI: 10.5802/aif.2767
Classification: 20E42, 20F55, 51E24
Keywords: Spherical building, root group, the center conjecture
Mot clés : immeuble sphérique, sous-groupe radiciel, la conjecture du centre
Mühlherr, Bernhard 1; Weiss, Richard M. 2

1 University of Giessen Institute for Mathematics Arndtstrasse 2 35392 Giessen (Germany)
2 Tufts University Department of Mathematics 503 Boston Avenue Medford, MA 02155 (USA)
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Mühlherr, Bernhard; Weiss, Richard M. Receding polar regions of a spherical building and the center conjecture. Annales de l'Institut Fourier, Volume 63 (2013) no. 2, pp. 479-513. doi : 10.5802/aif.2767. http://www.numdam.org/articles/10.5802/aif.2767/

[1] Aschbacher, Michael The 27-dimensional module for E 6 . IV, J. Algebra, Volume 131 (1990) no. 1, pp. 23-39 | DOI | MR | Zbl

[2] Balser, Andreas; Lytchak, Alexander Centers of convex subsets of buildings, Ann. Global Anal. Geom., Volume 28 (2005) no. 2, pp. 201-209 | DOI | MR | Zbl

[3] Bate, Michael; Martin, Benjamin; Röhrle, Gerhard On Tits’ centre conjecture for fixed point subcomplexes, C. R. Math. Acad. Sci. Paris, Volume 347 (2009) no. 7-8, pp. 353-356 | DOI | MR | Zbl

[4] Borel, A.; Tits, J. Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math., Volume 12 (1971), pp. 95-104 | DOI | MR | Zbl

[5] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[6] Cohen, Arjeh M.; Ivanyos, Gábor Root filtration spaces from Lie algebras and abstract root groups, J. Algebra, Volume 300 (2006) no. 2, pp. 433-454 | DOI | MR | Zbl

[7] Cohen, Arjeh M.; Ivanyos, Gábor Root shadow spaces, European J. Combin., Volume 28 (2007) no. 5, pp. 1419-1441 | DOI | MR | Zbl

[8] Cooperstein, Bruce N. The geometry of root subgroups in exceptional groups. I, Geom. Dedicata, Volume 8 (1979) no. 3, pp. 317-381 | DOI | MR | Zbl

[9] Cuypers, Hans The geometry of k-transvection groups, J. Algebra, Volume 300 (2006) no. 2, pp. 455-471 | DOI | MR | Zbl

[10] Dress, Andreas W. M.; Scharlau, Rudolf Gated sets in metric spaces, Aequationes Math., Volume 34 (1987) no. 1, pp. 112-120 | DOI | MR | Zbl

[11] Kempf, George R. Instability in invariant theory, Ann. of Math. (2), Volume 108 (1978) no. 2, pp. 299-316 | DOI | MR | Zbl

[12] Leeb, Bernhard; Ramos-Cuevas, Carlos The center conjecture for spherical buildings of types F 4 and E 6 , Geom. Funct. Anal., Volume 21 (2011) no. 3, pp. 525-559 | DOI | MR | Zbl

[13] Mühlherr, Bernhard; Tits, Jacques The center conjecture for non-exceptional buildings, J. Algebra, Volume 300 (2006) no. 2, pp. 687-706 | DOI | MR | Zbl

[14] Mumford, David Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin, 1965 | MR | Zbl

[15] Parker, Chris; Tent, Katrin Completely reducible subcomplexes of spherical buildings, Arch. Math. (Basel), Volume 97 (2011) no. 2, pp. 125-128 | DOI | MR | Zbl

[16] Ramos-Cuevas, C. The center conjecture for thick spherical buildings, arXiv:0909.2761

[17] Rousseau, Guy Immeubles des groupes réductifs sur les corps locaux, U.E.R. Mathématique, Université Paris XI, Orsay, 1977 (Thèse de doctorat, Publications Mathématiques d’Orsay, No. 221-77.68) | MR | Zbl

[18] Rousseau, Guy Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 5, p. A247-A250 | MR | Zbl

[19] Serre, Jean-Pierre Complète réductibilité, Astérisque (2005) no. 299, pp. Exp. No. 932, viii, 195-217 (Séminaire Bourbaki. Vol. 2003/2004) | Numdam | MR | Zbl

[20] Struyve, Koen (Non)-completeness of -buildings and fixed point theorems, Groups Geom. Dyn., Volume 5 (2011) no. 1, pp. 177-188 | DOI | MR | Zbl

[21] Timmesfeld, F. G. Subgroups of Lie type groups containing a unipotent radical, J. Algebra, Volume 323 (2010) no. 5, pp. 1408-1431 | DOI | MR | Zbl

[22] Timmesfeld, Franz Georg Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[23] Tits, J. Groupes semi-simples isotropes, Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, 1962, pp. 137-147 | MR | Zbl

[24] Tits, J. Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Invent. Math., Volume 43 (1977) no. 3, pp. 283-295 | DOI | MR | Zbl

[25] Tits, Jacques Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin, 1974 | MR | Zbl

[26] Tits, Jacques; Weiss, Richard M. Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002 | MR | Zbl

[27] Weiss, Richard M. The structure of spherical buildings, Princeton University Press, Princeton, NJ, 2003 | MR | Zbl

[28] Weiss, Richard M. The structure of affine buildings, Annals of Mathematics Studies, 168, Princeton University Press, Princeton, NJ, 2009 | MR | Zbl

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