A foliation of the space of conjugacy classes of representations of a surface group. (Q5948672)

Let \(G\) be a real reductive Lie group and let \(\Pi\) be the surface group, i.e. the fundamental group of a Riemann surface \(\Sigma\) of genus \(g\). Let \(R(\Pi ,G)/G\) be the space of conjugacy classes of representations of \(\Pi\) in \(G\). Inspired by geometric quantization, the author defines a map \(\overline{\phi}: R(\Pi ,G)/G\rightarrow R(A,G)/G\), where \(A\subset \Pi\) and investigates whether the fibres of \(\overline{\phi}\) are isotropic with respect to the natural symplectic structure on \(R(\Pi ,G)/G\). Using the real polarization in the case \(g=2\) and \(G=SU(2)\), Weitsman proved that the fibres are isotropic [cf. \textit{J. Weitsman}, Commun. Math. Phys. 145, No. 3, 425--433 (1992; Zbl 0765.53021)]. The author proves that for genus one the fibres are also isotropic, but in the case of genus two of \(G=SU(3)\), the generic fibres are not isotropic.