In this paper, we characterize unitary representations of $\pi:=\piS$ whose generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions $u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. 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 $\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.

We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time $\pi$ on the boundary. Furthermore, it is shown that on $\Real^n$ the asymptotics of certain short-range perturbations of $\Delta^k$ can be recovered from the scattering matrix at a finite number of energies.