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

Abstract

In this paper, using the theory of $A$-cover developed in [2], [3], we completely classify all simple Harish-Chandra modules over the higher rank W-algebra $W(2,2)$. According to this result, we give a classification of all simple Harish-Chandra modules over some related higher rank Lie (super)algebras, such as higher rank deformed ${\mathrm{bms}}_3$ algebras, the subalgebra of higher rank super-Galilean conformal algebras and so on. Moreover, we obtain the classification of simple Harish-Chandra modules over the classical W-algebra $W(2,2)$ studied in [8], [14], [19] as a byproduct.