Maximal representations of cocompact complex hyperbolic lattices, a uniform approach

Pierre-Emmanuel Chaput; Julien Maubon

doi:10.5802/jep.93

Maximal representations of cocompact complex hyperbolic lattices, a uniform approach
[Représentations maximales des réseaux hyperboliques complexes cocompacts : une approche unifiée]

Pierre-Emmanuel Chaput ¹ ; Julien Maubon ¹

¹ Institut Élie Cartan de Lorraine, Université de Lorraine Site de Nancy, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 231-281.

Résumé
Abstract

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 $Γ \subset SU (1, n)$ , avec $n > 1$ , dans un groupe hermitien $G_{ℝ}$ de type exceptionnel, alors $n = 2$ et $G_{ℝ} = E_{6 (- 14)}$ , 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 $G_{ℝ}$ . Dans le cas où $G_{ℝ}$ est de type tube, nos méthodes permettent en particulier d’établir une inégalité sur l’invariant de Toledo de la représentation $ρ : Γ \to G_{ℝ}$ qui est plus forte que l’inégalité de Milnor-Wood et qui exclut donc la possibilité d’une représentation maximale pour de tels groupes.

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 $Γ \subset SU (1, n)$ , $n > 1$ , in an exceptional Hermitian group $G_{ℝ}$ , then $n = 2$ and $G_{ℝ} = E_{6 (- 14)}$ , 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 $G_{ℝ}$ . As a by-product of our methods, when the target Hermitian group $G_{ℝ}$ has tube type, we obtain an inequality on the Toledo invariant of the representation $ρ : Γ \to G_{ℝ}$ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

Reçu le : 2017-04-27
Accepté le : 2019-04-16
Publié le : 2019-05-02

MR Zbl

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

Affiliations des auteurs :

Pierre-Emmanuel Chaput ¹ ; Julien Maubon ¹

¹ Institut Élie Cartan de Lorraine, Université de Lorraine Site de Nancy, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2019__6__231_0,
     author = {Pierre-Emmanuel Chaput and Julien Maubon},
     title = {Maximal representations of cocompact complex hyperbolic lattices, a uniform approach},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {231--281},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
     doi = {10.5802/jep.93},
     mrnumber = {3959074},
     zbl = {07057061},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.93/}
}

TY  - JOUR
AU  - Pierre-Emmanuel Chaput
AU  - Julien Maubon
TI  - Maximal representations of cocompact complex hyperbolic lattices, a uniform approach
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2019
SP  - 231
EP  - 281
VL  - 6
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.93/
DO  - 10.5802/jep.93
LA  - en
ID  - JEP_2019__6__231_0
ER  -

%0 Journal Article
%A Pierre-Emmanuel Chaput
%A Julien Maubon
%T Maximal representations of cocompact complex hyperbolic lattices, a uniform approach
%J Journal de l’École polytechnique — Mathématiques
%D 2019
%P 231-281
%V 6
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.93/
%R 10.5802/jep.93
%G en
%F JEP_2019__6__231_0

Pierre-Emmanuel Chaput; Julien Maubon. Maximal representations of cocompact complex hyperbolic lattices, a uniform approach. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 231-281. doi : 10.5802/jep.93. https://jep.centre-mersenne.org/articles/10.5802/jep.93/

Bibliographie
Cité par

[AMRT10] A. Ash, D. Mumford, M. Rapoport & Y.-S. Tai - Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010 | Zbl

[BGPG03] S. B. Bradlow, O. Garcia-Prada & P. B. Gothen - “Surface group representations and $U (p, q)$ -Higgs bundles”, J. Differential Geom. 64 (2003), p. 111-170 | DOI | MR | Zbl

[BGPG06] S. B. Bradlow, O. Garcia-Prada & P. B. Gothen - “Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces”, Geom. Dedicata 122 (2006), p. 185-213 | DOI | MR | Zbl

[BGPR17] O. Biquard, O. García-Prada & R. Rubio - “Higgs bundles, the Toledo invariant and the Cayley correspondence”, J. Topology 10 (2017) no. 3, p. 795-826 | DOI | MR | Zbl

[BH99] M. R. Bridson & A. Haefliger - Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, Berlin, 1999 | MR | Zbl

[BI07] M. Burger & A. Iozzi - “Bounded differential forms, generalized Milnor-Wood inequality and an application to deformation rigidity”, Geom. Dedicata 125 (2007), p. 1-23 | DOI | MR | Zbl

[BIW09] M. Burger, A. Iozzi & A. Wienhard - “Tight homomorphisms and Hermitian symmetric spaces”, Geom. Funct. Anal. 19 (2009) no. 3, p. 678-721 | DOI | MR | Zbl

[BIW10] M. Burger, A. Iozzi & A. Wienhard - “Surface group representations with maximal Toledo invariant”, Ann. of Math. (2) 172 (2010), p. 517-566 | DOI | MR | Zbl

[BM15] A. S. Buch & L. C. Mihalcea - “Curve neighborhoods of Schubert varieties”, J. Differential Geom. 99 (2015) no. 2, p. 255-283 | DOI | MR | Zbl

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

[Bou68] N. Bourbaki - 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, vol. 1337, Hermann, Paris, 1968 | Zbl

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

[Cor88] K. Corlette - “Flat $G$ -bundles with canonical metrics”, J. Differential Geom. 28 (1988), p. 361-382 | DOI | MR | Zbl

[Del80] P. Deligne - “La conjecture de Weil. II”, Publ. Math. Inst. Hautes Études Sci. 52 (1980), p. 137-252 | DOI | Zbl

[FH91] W. Fulton & J. Harris - Representation theory. A first course, Graduate Texts in Math., vol. 129, Springer-Verlag, New York, 1991 | Zbl

[Gro94] B. H. Gross - “A remark on tube domains”, Math. Res. Lett. 1 (1994) no. 1, p. 1-9 | DOI | MR | Zbl

[GW12] O. Guichard & A. Wienhard - “Anosov representations: domains of discontinuity and applications”, Invent. Math. 190 (2012), p. 357-438 | DOI | MR | Zbl

[Ham13] O. Hamlet - “Tight holomorphic maps, a classification”, J. Lie Theory 23 (2013) no. 3, p. 639-654 | MR | Zbl

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

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

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

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

[Hit92] N. Hitchin - “Lie groups and Teichmüller space”, Topology 31 (1992), 449–473 pages | Zbl

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

[Igu70] J.-i. Igusa - “A classification of spinors up to dimension twelve”, Amer. J. Math. 92 (1970), p. 997-1028 | DOI | MR | Zbl

[Iha67] S.-i. Ihara - “Holomorphic imbeddings of symmetric domains”, J. Math. Soc. Japan 19 (1967), p. 261-302 | DOI | MR | Zbl

[KM08] V. Koziarz & J. Maubon - “Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type”, Geom. Dedicata 137 (2008), p. 85-111 | DOI | MR | Zbl

[KM10] V. Koziarz & J. Maubon - “The Toledo invariant on smooth varieties of general type”, J. reine angew. Math. 649 (2010), p. 207-230 | MR | Zbl

[KM17] V. Koziarz & J. Maubon - “Maximal representations of uniform complex hyperbolic lattices”, Ann. of Math. (2) 185 (2017), p. 493-540 | DOI | MR | Zbl

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

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

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

[McG02] W. M. McGovern - “The adjoint representation and the adjoint action”, in Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, Encyclopaedia Math. Sci., vol. 131, Springer, Berlin, 2002, p. 159-238 | MR | Zbl

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

[Mur59] S. Murakami - “Sur certains espaces fibrés principaux différentiables et holomorphes”, Nagoya Math. J. 15 (1959), p. 171-199 | DOI | Zbl

[MX02] E. Markman & E. Z. Xia - “The moduli of flat $PU (p, p)$ -structures with large Toledo invariants”, Math. Z. 240 (2002), p. 95-109 | DOI | MR | Zbl

[NT76] H. Nakagawa & R. Takagi - “On locally symmetric Kaehler submanifolds in a complex projective space”, J. Math. Soc. Japan 28 (1976) no. 4, p. 638-667 | DOI | MR | Zbl

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

[Rat06] J. Ratcliffe - Foundations of hyperbolic manifolds, Springer, New York, 2006 | Zbl

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

[RRS92] R. Richardson, G. Röhrle & R. Steinberg - “Parabolic subgroups with abelian unipotent radical”, Invent. Math. 110 (1992) no. 3, p. 649-671 | DOI | MR | Zbl

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

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

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

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

[Sim92] C. Simpson - “Higgs bundles and local systems”, Publ. Math. Inst. Hautes Études Sci. 75 (1992), p. 5-95 | DOI | Zbl

[Siu80] Y.-T. Siu - “The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds”, Ann. of Math. (2) 112 (1980), p. 73-111 | Zbl

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

[SZ85] J. Steenbrink & S. Zucker - “Variation of mixed Hodge structure. I”, Invent. Math. 80 (1985), p. 489-542 | DOI | MR | Zbl

[SZ10] M. Sheng & K. Zuo - “Polarized variation of Hodge structures of Calabi-Yau type and characteristic subvarieties over bounded symmetric domains”, Math. Ann. 348 (2010) no. 1, p. 211-236 | DOI | MR | Zbl

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

[Xia00] E. Z. Xia - “The moduli of flat $PU (2, 1)$ structures on Riemann surfaces”, Pacific J. Math. 195 (2000), p. 231-256 | MR | Zbl

Cité par Sources :