Tau Functions and Virasoro Symmetries for Drinfeld-Sokolov Hierarchies

For each Drinfeld-Sokolov integrable hierarchy associated to affine Kac-Moody algebra, we obtain a uniform construction of tau function by using tau-symmetric Hamiltonian densities, moreover, we represent its Virasoro symmetries as linear/nonlinear actions on the tau function. The relations between the tau function constructed in this paper and those defined for particular cases of Drinfeld-Sokolov hierarchies in the literature are clarified. We also show that, whenever the affine Kac-Moody algebra is simply-laced or twisted, the tau functions of the Drinfeld-Sokolov hierarchy coincide with the solutions of the corresponding Kac-Wakimoto hierarchy from the principal vertex operator realization of the affine algebra.