Decomposable Representations and Lagrangian Subspaces of Moduli Spaces Associated to Surface Groups

In this paper, we construct a lagrangian subspace of the space of representations Mg,l := HomC(πg,l, U)/U of the fundamental group πg,l of a punctured Riemann surface into an arbitrary compact connected Lie group U . This lagrangian subspace is obtained as the fixed-point set of an anti-symplectic involution β̂ defined on Mg,l. We show that the involution β̂ is induced by a form-reversing involution β defined on the quasi-hamiltonian space (U × U) × C1 × · · · × Cl. The fact that β̂ has a non-empty fixed-point set is a consequence of the real convexity theorem for groupvalued momentum maps announced in [27]. The notion of decomposable representation provides a geometric interpretation of the lagrangian subspace thus obtained.