Maximal representations of cocompact complex hyperbolic lattices, a uniform approach
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 231-281.

We complete the classification of maximal representations of cocompact complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups E 6(-14) and E 7(-25) . We prove that if ρ is a maximal representation of a cocompact complex hyperbolic lattice ΓSU(1,n), n>1, in an exceptional Hermitian group G , then n=2 and G =E 6(-14) , and we describe completely the representation ρ. The case of classical Hermitian target groups was treated by Vincent Koziarz and the second named author [KM17]. However we do not focus immediately on the exceptional cases and instead we provide a more unified perspective, as independent as possible of the classification of the simple Hermitian Lie groups. This relies on the study of the cominuscule representation of the complexification G of the target group G . As a by-product of our methods, when the target Hermitian group G has tube type, we obtain an inequality on the Toledo invariant of the representation ρ:ΓG which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

Nous complétons la classification des représentations maximales des réseaux hyperboliques complexes dans les groupes de Lie hermitiens en traitant le cas des groupes exceptionnels E 6(-14) et E 7(-25) . Nous montrons que si ρ est une représentation maximale d’un réseau hyperbolique complexe cocompact ΓSU(1,n), avec n>1, dans un groupe hermitien G de type exceptionnel, alors n=2 et G =E 6(-14) , et nous décrivons complètement la représentation ρ. Le cas des groupes hermitiens classiques avait été traité par Vincent Koziarz et le deuxième auteur cité [KM17]. Cependant, nous ne nous restreignons pas immédiatement aux groupes exceptionnels : nous proposons au contraire une approche unifiée, aussi indépendante que possible de la classification des groupes de Lie hermitiens simples. Cette approche repose sur une étude de la représentation cominuscule de la complexification du groupe d’arrivée G . Dans le cas où G est de type tube, nos méthodes permettent en particulier d’établir une inégalité sur l’invariant de Toledo de la représentation ρ:ΓG qui est plus forte que l’inégalité de Milnor-Wood et qui exclut donc la possibilité d’une représentation maximale pour de tels groupes.

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Accepted:
Published online:
DOI: 10.5802/jep.93
Classification: 53C35, 22E40, 32L05, 32Q15, 17B10, 20G05
Keywords: Complex hyperbolic lattices, Milnor-Wood inequality, maximal representations, cominuscule representations, exceptional Lie groups, harmonic Higgs bundles, holomorphic foliations
Mot clés : Réseaux hyperboliques complexes, inégalité de Milnor-Wood, représentations maximales, représentations cominuscules, groupes de Lie exceptionnels, fibrés de Higgs harmoniques, feuilletages holomorphes
Chaput, Pierre-Emmanuel 1; Maubon, Julien 1

1 Institut Élie Cartan de Lorraine, Université de Lorraine Site de Nancy, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex
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Chaput, Pierre-Emmanuel; Maubon, Julien. Maximal representations of cocompact complex hyperbolic lattices, a uniform approach. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 231-281. doi : 10.5802/jep.93. http://www.numdam.org/articles/10.5802/jep.93/

[AMRT10] Ash, Avner; Mumford, David; Rapoport, Michael; Tai, Yung-Sheng Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010 | Zbl

[BGPG03] Bradlow, Steven B.; Garcia-Prada, Oscar; Gothen, Peter B. Surface group representations and U(p,q)-Higgs bundles, J. Differential Geom., Volume 64 (2003), pp. 111-170 | DOI | MR | Zbl

[BGPG06] Bradlow, Steven B.; Garcia-Prada, Oscar; Gothen, Peter B. Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata, Volume 122 (2006), pp. 185-213 | DOI | MR | Zbl

[BGPR17] Biquard, Olivier; García-Prada, Oscar; Rubio, Roberto Higgs bundles, the Toledo invariant and the Cayley correspondence, J. Topology, Volume 10 (2017) no. 3, pp. 795-826 | DOI | MR | Zbl

[BH99] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren Math. Wiss., 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[BI07] Burger, Marc; Iozzi, Alessandra Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity, Geom. Dedicata, Volume 125 (2007), pp. 1-23 | DOI | MR | Zbl

[BIW09] Burger, Marc; Iozzi, Alessandra; Wienhard, Anna Tight homomorphisms and Hermitian symmetric spaces, Geom. Funct. Anal., Volume 19 (2009) no. 3, pp. 678-721 | DOI | MR | Zbl

[BIW10] Burger, Marc; Iozzi, Alessandra; Wienhard, Anna Surface group representations with maximal Toledo invariant, Ann. of Math. (2), Volume 172 (2010), pp. 517-566 | DOI | MR | Zbl

[BM15] Buch, Anders S.; Mihalcea, Leonardo C. Curve neighborhoods of Schubert varieties, J. Differential Geom., Volume 99 (2015) no. 2, pp. 255-283 | DOI | MR | Zbl

[Bor69] Borel, Armand Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969

[Bou68] Bourbaki, Nicolas Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, 1337, Hermann, Paris, 1968 | Zbl

[Che97] Chevalley, Claude The algebraic theory of spinors and Clifford algebras, Collected works, 2, Springer-Verlag, Berlin, 1997 | MR | Zbl

[Cor88] Corlette, Kevin Flat G-bundles with canonical metrics, J. Differential Geom., Volume 28 (1988), pp. 361-382 | DOI | MR | Zbl

[Del80] Deligne, Pierre La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci., Volume 52 (1980), pp. 137-252 | DOI | Zbl

[FH91] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Math., 129, Springer-Verlag, New York, 1991 | Zbl

[Gro94] Gross, Benedict H. A remark on tube domains, Math. Res. Lett., Volume 1 (1994) no. 1, pp. 1-9 | DOI | MR | Zbl

[GW12] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012), pp. 357-438 | DOI | MR | Zbl

[Ham13] Hamlet, Oskar Tight holomorphic maps, a classification, J. Lie Theory, Volume 23 (2013) no. 3, pp. 639-654 | MR | Zbl

[HC56] Harish-Chandra Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math., Volume 78 (1956), pp. 564-628 | DOI | MR | Zbl

[Hel01] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Math., 34, American Mathematical Society, Providence, RI, 2001 (Corrected reprint of the 1978 original) | MR | Zbl

[Her91] Hernández, Luis Maximal representations of surface groups in bounded symmetric domains, Trans. Amer. Math. Soc., Volume 324 (1991), pp. 405-420 | DOI | MR | Zbl

[Hit87] Hitchin, Nigel The self-duality equations on a Riemann surface, Proc. London Math. Soc., Volume 55 (1987), p. 59–126 | MR | Zbl

[Hit92] Hitchin, Nigel Lie groups and Teichmüller space, Topology, Volume 31 (1992), p. 449–473 | Zbl

[Hum75] Humphreys, James E. Linear algebraic groups, Graduate Texts in Math., 21, Springer, New York, 1975 | MR | Zbl

[Igu70] Igusa, Jun-ichi A classification of spinors up to dimension twelve, Amer. J. Math., Volume 92 (1970), pp. 997-1028 | DOI | MR | Zbl

[Iha67] Ihara, Shin-ichiro Holomorphic imbeddings of symmetric domains, J. Math. Soc. Japan, Volume 19 (1967), pp. 261-302 | DOI | MR | Zbl

[KM08] Koziarz, Vincent; Maubon, Julien Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type, Geom. Dedicata, Volume 137 (2008), pp. 85-111 | DOI | MR | Zbl

[KM10] Koziarz, Vincent; Maubon, Julien The Toledo invariant on smooth varieties of general type, J. reine angew. Math., Volume 649 (2010), pp. 207-230 | MR | Zbl

[KM17] Koziarz, Vincent; Maubon, Julien Maximal representations of uniform complex hyperbolic lattices, Ann. of Math. (2), Volume 185 (2017), pp. 493-540 | DOI | MR | Zbl

[Kna02] Knapp, Anthony W. Lie groups beyond an introduction, Progress in Math., 140, Birkhäuser Boston, Inc., Boston, MA, 2002 | MR | Zbl

[Kos12] Kostant, Bertram The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group, Moscow Math. J., Volume 12 (2012) no. 3, pp. 605-620 | DOI | MR | Zbl

[Man06] Manivel, Laurent Configurations of lines and models of Lie algebras, J. Algebra, Volume 304 (2006) no. 1, pp. 457-486 | DOI | MR | Zbl

[McG02] McGovern, William M. The adjoint representation and the adjoint action, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action (Encyclopaedia Math. Sci.), Volume 131, Springer, Berlin, 2002, pp. 159-238 | MR | Zbl

[Mok89] Mok, Ngaiming Metric rigidity theorems on Hermitian locally symmetric manifolds, Series in Pure Mathematics, 6, World Scientific, Teaneck, NJ, 1989 | MR | Zbl

[Mur59] Murakami, Shingo Sur certains espaces fibrés principaux différentiables et holomorphes, Nagoya Math. J., Volume 15 (1959), pp. 171-199 | DOI

[MX02] Markman, Eyal; Xia, Eugene Z. The moduli of flat PU(p,p)-structures with large Toledo invariants, Math. Z., Volume 240 (2002), pp. 95-109 | DOI | MR | Zbl

[NT76] Nakagawa, Hisao; Takagi, Ryoichi On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Japan, Volume 28 (1976) no. 4, pp. 638-667 | DOI | MR

[PS69] Piatetski-Shapiro, Ilya Automorphic functions and the geometry of classical domains, Mathematics and its applications, 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969 | MR | Zbl

[Rat06] Ratcliffe, John Foundations of hyperbolic manifolds, Springer, New York, 2006 | Zbl

[Roy80] Royden, Halsey L. The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv., Volume 55 (1980) no. 4, pp. 547-558 | DOI | MR | Zbl

[RRS92] Richardson, Roger; Röhrle, Gerhard; Steinberg, Robert Parabolic subgroups with abelian unipotent radical, Invent. Math., Volume 110 (1992) no. 3, pp. 649-671 | DOI | MR | Zbl

[Sam78] Sampson, Joseph H. Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4), Volume 11 (1978), pp. 211-228 | DOI | MR | Zbl

[Sat80] Satake, Ichiro Algebraic structures of symmetric domains, Kanô Memorial Lectures, 4, Princeton University Press, Princeton, NJ, 1980 | MR | Zbl

[Sel60] Selberg, Atle On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp. 147-164 | MR | Zbl

[Sim88] Simpson, Carlos Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988), pp. 867-918 | DOI | MR | Zbl

[Sim92] Simpson, Carlos Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci., Volume 75 (1992), pp. 5-95 | DOI | Zbl

[Siu80] Siu, Yum-Tong The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2), Volume 112 (1980), pp. 73-111 | Zbl

[SV00] Springer, Tonny A.; Veldkamp, Ferdinand D. Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000 | Zbl

[SZ85] Steenbrink, Joseph; Zucker, Steven Variation of mixed Hodge structure. I, Invent. Math., Volume 80 (1985), pp. 489-542 | DOI | Zbl

[SZ10] Sheng, Mao; Zuo, Kang Polarized variation of Hodge structures of Calabi-Yau type and characteristic subvarieties over bounded symmetric domains, Math. Ann., Volume 348 (2010) no. 1, pp. 211-236 | DOI | MR | Zbl

[Wol72] Wolf, Joseph A. Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970) (Pure and App. Math.), Volume 8, Dekker, New York, 1972, pp. 271-357 | MR | Zbl

[Xia00] Xia, Eugene Z. The moduli of flat PU(2,1) structures on Riemann surfaces, Pacific J. Math., Volume 195 (2000), pp. 231-256 | MR | Zbl

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