Character varieties for real forms

Abstract

Let \(\varGamma \) be a finitely generated group and G a real form of \(\mathrm {SL}_n(\mathbb {C})\). We propose a definition for the G-character variety of \(\varGamma \) as a subset of the \(\mathrm {SL}_n(\mathbb {C})\)-character variety of \(\varGamma \). We consider two anti-holomorphic involutions of the \(\mathrm {SL}_n(\mathbb {C})\) character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of \(\mathrm {SL}_n(\mathbb {C})\). We study in detail an example: the \(\mathrm {SL}_n(\mathbb {C})\), \(\mathrm {SU}(2,1)\) and \(\mathrm {SU}(3)\) character varieties of the free product \(\mathbb {Z}/{3}\mathbb {Z}*\mathbb {Z}/{3}\mathbb {Z}\).

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Notes

  1. 1.

    Florentino and Lawton give a first proof of this fact in the appendix of [10]; the proof is corrected in Remark 4.7 of [11].

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Acknowledgements

The author would like to thank his advisors Antonin Guilloux and Martin Deraux, as well as Pierre Will, Elisha Falbel and Julien Marché for many discussions about this article. He would also like to thank Maxime Wolff, Joan Porti, Michael Heusener, Cyril Demarche and the PhD students of IMJ-PRG for helping him to clarify many points of the paper, as well as the anonymous referee for many improvements of the article.

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Correspondence to Miguel Acosta.

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This work was carried out while the author was at IMJ-PRG (UPMC, Paris, France) and IECL (Université de Lorraine, Nancy, France).

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Acosta, M. Character varieties for real forms. Geom Dedicata 203, 257–277 (2019). https://doi.org/10.1007/s10711-019-00435-3

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Keywords

  • Character variety
  • Real form
  • SL(n, C)
  • GIT

Mathematics Subject Classification (2000)

  • 20C15
  • 14L24
  • 14D20