  We study representations of the central extension of the Lie algebra of
differential operators on the circle, the W-infinity algebra. We obtain
complete and specialized character formulas for a large class of
representations, which we call primitive; these include all quasi-finite
irreducible unitary representations. We show that any primitive representation
with central charge N has a canonical structure of an irreducible
representation of the W-algebra W(gl_N) with the same central charge and that
all irreducible representations of W(gl_N) with central charge N arise in this
way. We also establish a duality between "integral" modules of W(gl_N) and
finite-dimensional irreducible modules of gl_N, and conjecture their fusion
rules.
