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We investigate the free fields realization of the twisted Heisenberg-Virasoroalgebra $\mathcal{H}$ at level zero. We completely describe the structure ofthe associated Fock representations. Using vertex-algebraic methods andscreening operators we construct singular vectors in certain Verma modules asSchur polynomials. We completely solve the irreducibility problem for tensorproduct of irreducible highest weight modules with intermediate series. We alsodetermine the fusion rules for an interesting subcategory of$\mathcal{H}$-modules. Finally, as an application we present a free fieldrealization of the $W(2,2)$-algebra and interpret the $W(2,2)$-singular vectorsas $\mathcal{H}$-singular vectors in Verma modules.

Free field realization of the twisted Heisenberg-Virasoro algebra at level zero and its applications