  We show that there are precisely two, up to conjugation, anti-involutions
sigma_{\pm} 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 \hat{D}^{\pm} of the Lie subalgebra of
this algebra fixed by - sigma_{\pm}, 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 truncated polynomial algebra C[u] / (u^{m+1}) and its
classical Lie subalgebras of B, C and D types. Character formulas for positive
primitive representations of \hat{D}^{\pm} (including all the unitary ones) are
obtained. We also realize a class of primitive representations of \hat{D}^{\pm}
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 \hat{D}^+ carries a vertex algebra structure and establish a
relationship between V_c for half-integral central charge c and W-algebras.
