p̲-reduced Multicomponent KP Hierarchy and Classical W-algebras W(glN,p̲)

For each partition p ̲ of an integer N ≥ 2 , consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p ̲ -reduced r -component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W -algebra W ( gl N , p ̲ ) , and write down explicit formulas for evolution of these generators along the commuting Hamiltonian flows.