Maximal representations of cocompact complex hyperbolic lattices, a uniform approach

Chaput, Pierre-Emmanuel; Maubon, Julien

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

Nous complétons la classification des représentations maximales des réseaux hyperboliques complexes dans les groupes de Lie hermitiens en traitant le cas des groupes exceptionnels ${\mathrm{E}}_{6(-14)}$ et ${\mathrm{E}}_{7(-25)}$. Nous montrons que si $\rho$ est une représentation maximale d’un réseau hyperbolique complexe cocompact $\Gamma \subset \mathrm{SU}(1,n)$, avec $n>1$, dans un groupe hermitien $G_{ℝ}$ de type exceptionnel, alors $n=2$ et $G_{ℝ}={\mathrm{E}}_{6(-14)}$, et nous décrivons complètement la représentation $\rho$. 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 $\rho :\Gamma \to G_{ℝ}$ 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 ${\mathrm{E}}_{6(-14)}$ and ${\mathrm{E}}_{7(-25)}$. We prove that if $\rho$ is a maximal representation of a cocompact complex hyperbolic lattice $\Gamma \subset \mathrm{SU}(1,n)$, $n>1$, in an exceptional Hermitian group $G_{ℝ}$, then $n=2$ and $G_{ℝ}={\mathrm{E}}_{6(-14)}$, and we describe completely the representation $\rho$. 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 $\rho :\Gamma \to G_{ℝ}$ which is stronger than the Milnor-Wood inequality (thereby excluding maximal representations in such groups).

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

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