  Finite W-algebras are certain associative algebras arising in Lie theory.
Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our
base field is algebraically closed and of characteristic 0) and its nilpotent
element e. In this paper we classify finite dimensional irreducible modules
with integral central character over W-algebras. In more detail, in a previous
paper the first author proved that the component group A(e) of the centralizer
of the nilpotent element under consideration acts on the set of finite
dimensional irreducible modules over the W-algebra and the quotient set is
naturally identified with the set of primitive ideals in U(g) whose associated
variety is the closure of the adjoint orbit of e. In this paper for a given
primitive ideal with integral central character we compute the corresponding
A(e)-orbit. The answer is that the stabilizer of that orbit is basically a
subgroup of A(e) introduced by G. Lusztig. In the proof we use a variety of
different ingredients: the structure theory of primitive ideals and
Harish-Chandra bimodules of semisimple Lie algebras, the representation theory
of W-algebras, the structure theory of cells and Springer representations, and
multi-fusion monoidal categories.
