  This is the third in a series of papers attempting to describe a uniform
geometric framework in which many integrable systems can be placed. A soliton
hierarchy can be constructed from a splitting of an infinite dimensional group
$L$ as positive and negative subgroups L_+, L_- and a commuting sequence in the
Lie algebra of L_+. Given f in L_-, there is a formal inverse scattering
solution u_f of the hierarchy. When there is a 2 co-cycle that vanishes on both
subalgebras of L_+ and L_-, Wilson constructed for each f in L_- a tau function
tau_f for the hierarchy. In this third paper, we prove the following results
for the nxn KdV hierarchy: (1) The second partials of ln(tau_f) are
differential polynomials of the formal inverse scattering solution u_f.
Moreover, u_f can be recovered from the second partials of ln(tau_f). (2) The
natural Virasoro action on ln(tau_f) constructed in the second paper is given
by partial differential operators in ln(tau_f). (3) There is a bijection
between phase spaces of the nxn KdV hierarchy and the Gelfand-Dickey (GD_n)
hierarchy on the space of order n linear differential operators on the line so
that the flows in these two hierarchies correspond under the bijection. (4) Our
Virasoro action on the nxn KdV hierarchy is constructed from a simple Virasoro
action on the negative group. We show that it corresponds to the known Virasoro
action on the GD_n hierarchy under the bijection.
