Quasifinite Representations of Classical Lie Subalgebras of [formula omitted]1+∞

Abstract

We show that there are precisely two, up to conjugation, anti-involutionsσ±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±of the Lie subalgebra of this algebra fixed by −σ±, 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]/(um+1) and its classical Lie subalgebras ofB,CandDtypes. Character formulas forpositive primitiverepresentations of D±(including all the unitary ones) are obtained. We also realize a class of primitive representations of D±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 moduleVcof D±carries a vertex algebra structure and establish a relationship betweenVcforc∈12Z and W-algebras.