A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Abstract

Let Π be the fundamental group of a closed orientable surface of genus g ≥ 1, and let R(Π, G)/G be the space of conjugacy classes of representations of Π into a connected real reductive Lie group G. Motivated by the theory of geometric quantization, we define a map φ¯ on R(Π, G)/G and investigate whether the fibres of φ¯ are isotropic with respect to the natural symplectic structure on R(Π, G)/G. If g = 2 and G = SU(2), then the foliation given by the fibres of φ¯ is equivalent to a real polarization defined by Weitsman, and we reprove his result that the fibres are isotropic in this case. If g = 1 then the fibres of φ¯ are also isotropic, but we give an example to show that in general they are not.