  We formulate a general super duality conjecture on connections between
parabolic categories O of modules over Lie superalgebras and Lie algebras of
type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which
was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL)
polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be
identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish
some special cases of the BKL conjecture on the parabolic category O of
gl(m|n)-modules and additional results which support the BKL conjecture and
super duality conjecture.
