Maximal representations of uniform complex hyperbolic lattices

Abstract

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset\mathrm{SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $G$. We prove that necessarily $G=\mathrm{SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $\rho$-equivariant map from the complex hyperbolic $n$-space to the symmetric space associated to $\mathrm{SU}(p,q)$. This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of $\mathrm{SU}(p,q)$, the representation $\rho$ extends to a representation of $\mathrm{SU}(n,1)$ in $\mathrm{SU}(p,q)$.

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  • [You] Go to document D. C. Youla, "A normal form for a matrix under the unitary congruence group," Canad. J. Math., vol. 13, pp. 694-704, 1961.
    @ARTICLE{You, mrkey = {0132754},
      issn = {0008-414X},
      author = {Youla, D. C.},
      mrclass = {15.30},
      doi = {10.4153/CJM-1961-059-8},
      journal = {Canad. J. Math.},
      zblnumber = {0103.25201},
      volume = {13},
      mrnumber = {0132754},
      fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      mrreviewer = {O. Taussky-Todd},
      title = {A normal form for a matrix under the unitary congruence group},
      year = {1961},
      pages = {694--704},
      }

Authors

Vincent Koziarz

Univ. Bordeaux, IMB, CNRS, UMR 5251, F-33400 Talence, France

Julien Maubon

Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France