SYMPLECTIC STRUCTURE OF THE MODULI SPACE OF FLAT CONNECTION ON A RIEMANN SURFACE

We consider the canonical symplectic structure on the moduli space of flat g-connections on a Riemann surface of genus g with a marked points. For a being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum of a copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie group G* and g copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group G (the pair (G,G*) corresponds to the Lie algebra g).