The Hyperbolic Moduli Space of Flat Connections and the Isomorphism of Symplectic Multiplicity Spaces

Let G be a simple complex Lie group, g be its Lie algebra, K be a maximal compact form of G and k be a Lie algebra of K. We denote by X → X the anti-involution of g which singles out the compact form k. Consider the space of flat g-valued connections on a Riemann sphere with three holes which satisfy the additional condition A(z) = −A(z). We call the quotient of this space over the action of the gauge group g(z) = g(z) a hyperbolic moduli space of flat connections. We prove that the following three symplectic spaces are isomorphic: 1. The hyperbolic moduli space of flat connections. 2. The symplectic multiplicity space obtained as symplectic quotient of the triple product of co-adjoint orbits of K. 3. The Poisson-Lie multiplicity space equal to the Poisson quotient of the triple product of dressing orbits of K.