D-SETS AND BG-FUNCTORS IN KAZHDAN-LUSZTIG THEORY

By using Deodhar's combinatorial setting and Bernstein Gelfand project ive functors, this paper provides some necessary and sufficient condit ions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Ka zhdan-Lusztig polynomials.