Quasifinite representations of classical Lie subalgebras of W1+∞

We show that there are precisely two, up to conjugation, anti-involutions sigma(+/-) of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension (D) over cap(+/-) of the Lie subalgebra of this algebra fixed by -sigma(+/-), and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]/(u(m+1)) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of (D) over cap(+/-) (including all the unitary ones) are obtained. We also realize a class of primitive representations of (D) over cap(+/-) in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V-c of (D) over cap(+) carries a vertex algebra structure and establish a relationship between V-c for c is an element of 1/2Z and W-algebras. (C) 1998 Academic Press