Representation Theoretical Meaning of the Initial Value Problem for the Toda Lattice Hierarchy II

Abstract

The Toda lattice hierarchy is shown to have the virtual Bruhat decomposition of the $A_{\infty }$ group as its parameter space, instead of the Grassmann manifold for the KP hierarchy. The small cells of the Bruhat decomposition are examined. The parametrization of the KP hierarchy is utilized for this purpose with the help of the Miura transformation.