W1+∞ and W(glN) with central charge N

We study representations of the central extension of the Lie algebra of differential operators on the circle, the W1+∞ 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(glN) with the same central charge and that all irreducible representations of W(glN) with central charge N arise in this way. We also establish a duality between ''integral'' modules of W(glN) and finite-dimensional irreducible modules of glN, and conjecture their fusion rules. (orig.)