ar X iv : h ep - t h / 94 05 12 1 v 2 4 O ct 1 99 4 W 1 + ∞ and W ( gl N ) with central charge N

The Lie algebra D̂, which is the unique non-trivial central extension of the Lie algebra D of differential operators on the circle [KP1], has appeared recently in various models of two-dimensional quantum field theory and integrable systems, cf., e.g., [BK, FKN, PRS, IKS, CTZ, ASvM]. A systematic study of representation theory of the Lie algebra D̂, which is often referred to as W1+∞ algebra, was initiated in [KR]. In that paper irreducible quasi-finite highest weight representations of D̂ were classified and it was shown that they can be realized in terms of irreducible highest weight representations of the Lie algebra of infinite matrices. In the first part of the present paper we recall some of the results of [KR] and, as an immediate corollary, obtain complete and specialized character formulas for an arbitrary primitive representation of D̂. (A primitive representation of D̂ is an “analytic continuation” of a quasi-finite irreducible unitary representation of D̂.) The results of [KR] were used previously in [Mat,AFOQ] to derive character formulas of primitive representations of central charge c = 1. In the second part of the paper we exhibit a connection between D̂ and the W– algebra W(glN) at the central charge N . Our main result is that any primitive Supported by a Junior Fellowship from Harvard Society of Fellows and in part by NSF grant DMS-9205303. Supported in part by NSF grant DMS-9103792.