Finite dimensional representations of W-algebras

TL;DR

This paper classifies finite-dimensional irreducible modules of finite type W-algebras, establishing a key conjecture and exploring their relation to Harish-Chandra bimodules, advancing understanding of their representation theory.

Contribution

It proves Premet's conjecture on the classification of finite-dimensional irreducible modules for W-algebras and investigates their connection to Harish-Chandra bimodules.

Findings

Complete classification of finite-dimensional irreducible modules for W-algebras.

Established a relation between Harish-Chandra bimodules and W-algebra bimodules.

Confirmed a key conjecture of Premet.

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 paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish-Chandra bimodules and bimodules over $W$-algebras.