Generalized finite and affine $W$-algebras in type $A$

We construct a new family of affine $W$-algebras $W^k(\lambda,\mu)$ parameterized by partitions $\lambda$ and $\mu$ associated with the centralizers of nilpotent elements in $\mathfrak{gl}_N$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $\lambda$ we recover the affine $W$-algebras $W^k(\mathfrak{gl}_N,f)$ of Kac, Roan and Wakimoto, associated with nilpotent elements $f\in\mathfrak{gl}_N$ of type $\mu$. Our construction is based on a version of the BRST complex of the quantum Drinfeld-Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras $W^k(\lambda,\mu)$ yields a family of generalized finite $W$-algebras $U(\lambda,\mu)$ which we also describe independently as associative algebras.