W_{1+\infty} and W(gl_N) with central charge N

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.