Classification of finite dimensional irreducible modules over W-algebras

TL;DR

This paper classifies finite dimensional irreducible modules over W-algebras associated with semisimple Lie algebras and nilpotent elements, linking module classification to primitive ideals and group actions.

Contribution

It provides a detailed classification of modules over W-algebras, computing stabilizers of A(e)-orbits for primitive ideals with integral central character.

Findings

Computed A(e)-orbit stabilizers for primitive ideals

Linked module classification to primitive ideal structure

Used diverse algebraic and geometric tools

Abstract

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.