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It is shown how Darboux coordinates on a reduced symplectic vector space maybe used to parametrize the phase space on which the finite gap solutions ofmatrix nonlinear Schr\"odinger equations are realized as isospectralHamiltonian flows. The parametrization follows from a moment map embedding ofthe symplectic vector space, reduced by suitable group actions, into the dual$\tilde\grg^{+*}$ of the algebra $\tilde\grg^+$ of positive frequency loops ina Lie algebra $\grg$. The resulting phase space is identified with a Poissonsubspace of $\tilde\grg^{+*}$ consisting of elements that are rational in theloop parameter. Reduced coordinates associated to the various Hermitiansymmetric Lie algebras $(\grg,\grk)$ corresponding to the classical Liealgebras are obtained.Comment: 17 pgs, plus 4 Tables Preprint CRM-183

Matrix nonlinear Schrödinger equations and moment maps into loop algebras