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

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