Classification of finite-dimensional irreducible modules over W-algebras

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(9) 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 for semisimple Lie algebras, the representation theory of W-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.