  We study the representations of the W-algebra W(g) associated to an arbitrary
finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov
reductions. The characters of irreducible representations with non-degenerate
highest weights are expressed by Kazhdan-Lusztig polynomials. The irreduciblity
conjecture of Frenkel, Kac and Wakimoto is proved completely for the "-"
reduction and partially for the "+" reduction. In particular, the existence of
the minimal series representations (= the modular invariant representations) of
W(g) is proved.
