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We study certain representations of quantum toroidal $\mathfrak{gl}_1$algebra for $q=t$. We construct explicit bosonization of the Fock modules$\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, itis naturally identified with the basic level 1 representation of affine$\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$acting on these Fock modules. As an application, we prove the relation on$q$-deformed conformal blocks which was conjectured in the study of$q$-deformation of isomonodromy/CFT correspondence.

Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks