Character varieties for real forms

  • Miguel AcostaEmail author
Original Paper


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 \(\mathbb {Z}/{3}\mathbb {Z}*\mathbb {Z}/{3}\mathbb {Z}\).


Character variety Real form SL(n, C) GIT 

Mathematics Subject Classification (2000)

20C15 14L24 14D20 



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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Unité de recherche en MathématiquesUniversité du LuxembourgEsch-sur-AlzetteLuxembourg

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