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

The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9, 25]: given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π1(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M . In the present paper, we construct a Lagrangian submanifold of the space of representations Mg,l := HomC(πg,l, U)/U of the fundamental group πg,l of a punctured Riemann surface Σg,l into an arbitrary compact connected Lie group U . This Lagrangian submanifold 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 group-valued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.