Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$

TL;DR

This paper classifies all simple Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$ using the $ ext{A}$-cover theory, extending previous classifications for the classical $W$-algebra.

Contribution

It provides a complete classification of simple Harish-Chandra modules over the high rank $W(2,2)$ algebra, a significant extension of prior work on the classical case.

Findings

Complete classification of simple Harish-Chandra modules over $W(2,2)$.

Extension of classical $W$-algebra module classifications.

Application of $ ext{A}$-cover theory to high rank $W$-algebras.

Abstract

In this paper, using the theory of $\A$-cover developed in \cite{B1,BF1}, we completely classify all simple Harish-Chandra modules over the high rank $W$-algebra $W(2,2)$. As a byproduct, we obtain the classification of simple Harish-Chandra modules over the classical $W$-algebra $W(2,2)$ studied in \cite{LGZ,CLW,GLZ1}.