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There is a general method for constructing a soliton hierarchy from a splitting  of a loop group as positive and negative sub-groups together with a commuting linearly independent sequence in the positive Lie algebra . Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each  in the negative subgroup  a solution  of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function  for each element . In this paper, we give integral formulas for variations of  and second partials of , discuss whether we can recover solutions  from , and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the -hierarchy.

Tau Functions and Virasoro Actions for soliton Hierarchies