  We study the representation theory of finite W-algebras. After introducing
parabolic subalgebras to describe the structure of W-algebras, we define the
Verma modules and give a conjecture for the Kac determinant. This allows us to
find the completely degenerate representations of the finite W-algebras. To
extract the irreducible representations we analyse the structure of singular
and subsingular vectors, and find that for W-algebras, in general the maximal
submodule of a Verma module is not generated by singular vectors only.
Surprisingly, the role of the (sub)singular vectors can be encapsulated in
terms of a `dual' analogue of the Kazhdan-Lusztig theorem for simple Lie
algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support
our conjectures with some examples, and briefly discuss applications and the
generalisation to infinite W-algebras.
