Finite-dimensional representations of W-algebras

Abstract

$W$-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this article, we prove a conjecture of Premet that gives an almost complete classification of finite-dimensional irreducible modules for $W$-algebras. A key ingredient in our proof is a relationship between Harish-Chandra bimodules and bimodules over $W$-algebras that is also of independent interest.