Representations of the fundamental group of an l-punctured sphere generated by products of Lagrangian involutions

In this paper, we characterize unitary representations of π := π1(S \{s1, . . . , sl}) whose generators u1, . . . , ul (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions uj = σjσj+1 with σl+1 = σ1 . Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space MC := HomC(π,U(n))/U(n) . Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of MC. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of MC and give conditions on an involution defined on a quasi-Hamiltonian U -space (M,ω, μ : M → U) for it to induce an anti-symplectic involution on the reduced space M//U := μ({1})/U .