Double Coset Construction of Moduli Space of Holomorphic Bundles and Hitchin Systems

We present a description of the moduli space of holomorphic vector bundles over Riemann curves as a double coset space which is differ from the standard loop group construction. Our approach is based on equivalent definitions of holomorphic bundles, based on the transition maps or on the first order differential operators. Using this approach we present two independent derivations of the Hitchin integrable systems. We define a 'superfree" upstairs system from which Hitchin systems are obtained by three step hamiltonian reductions. Special attention is given to the Schottky parameterization of curves.