Classification of finite-dimensional irreducible modules over \(W\)-algebras (Q2877512)

In this paper the authors classify finite dimensional irreducible modules with integral central character over finite \(W\)-algebras associated to the choice of a nilpotent element \(e\) in a semi-simple finite dimensional Lie algebra \(\mathfrak{g}\) over an algebraically closed field of characteristic zero. From a previous work of the first author it was known that the component group \(A(e)\) acts on the set of isomorphism classes of modules in question and the corresponding orbit set can be naturally identified with the set of primitive ideals in the universal enveloping algebra of \(\mathfrak{g}\) whose associated variety is the closure of the adjoint orbit of \(e\). The present paper computes the \(A(e)\)-orbit associated to a fixed primitive ideal with integral central character. It turns out that the stabilizer associated to this orbit is basically a subgroup of \(A(e)\) introduced by Lusztig. The arguments in the paper use various techniques ranging from Harish-Chandra bimodules, representations of finite \(W\)-algebras to Springer representation and multi-fusion monoidal categories.NEWLINENEWLINEAs a bonus, the authors obtain an alternative proof for the description of the multi-fusion monoidal category associated to a special orbit as was conjectured by Lusztig. The results of the paper together with the used techniques also allow to compute dimensions of simple finite dimensional modules as specified above and relate them to the Goldie rank of the corresponding primitive ideals.