Symplectic double for moduli spaces of G-local systems on surfaces

A decorated surface  SS is a topological oriented surface with punctures and holes, equipped with a finite set of special points on the boundaries of holes, considered modulo isotopy. Each hole boundary has at least one special point.Let G be a split semi-simple algebraic group over QQ. We introduce a moduli space DG,SDG,S, and define a collection of special rational coordinate systems on it.The moduli space DG,SDG,S is the symplectic double of the Poisson moduli space XG,SXG,S of framed G-local systems on S  . Its dimension is dimDG,S=2dimXG,SdimDG,S=2dimXG,S. Its symplectic form is upgraded to a K2K2-symplectic structure for which the special coordinates are K2K2-Darboux coordinates.