DOUBLE COSET CONSTRUCTION OF MODULI SPACE OF HOLOMORPHIC BUNDLES AND HITCHIN SYSTEMS

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