THE GEOMETRY OF THE HERMITIAN MATRIX MODEL AND LATTICES FOR THE NLS AND DNLS HIERARCHIES

The geometrical description of the Nonlinear Schrodinger-Toda system h ierarchy in the Sate Grassmannian with the action of the translation g roup is applied to the Hermitian one-matrix model. A family of derivat ive Nonlinear Schrodinger system hierarchies with its lattices-associa ted with the Volterra chain-which are auto-Backlund transformations, i s analyzed from a geometrical point of view. The Sate periodic flag ma nifold with the line bundles over it turns out to be the proper infini te-dimensional manifold in this case. The lattice appears as a square root of the action of the translation group; this can be understood as a reduction of the action of a translation group of a larger loop gro up. The reduction t(2n+1) = 0 Of the Hermitian one-matrix model, essen tial in the double scaling limit, is shown to be described in terms of the derivative Nonlinear Schrodinger-Volterra system hierarchy. The r ole of the heat hierarchy, self-similarity and auto-Backlund transform ations is pointed out. A characterization in Sate's Grassmannian and p eriodic flag manifold of the Hermitian one-matrix model is given. In t he latter case we are concerned with the t(2n+1) = 0 reduction.