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In this paper we discuss the structure of the tensor productV'_{\alpha,\beta}\otimes L(c,h) of irreducible module from intermediate seriesand irreducible highest weight module over the Virasoro algebra. We generalizeZhang's irreducibility criterion, and show that irreducibility depends on theexistence of integral roots of a certain polynomial, induced by a singularvector in the Verma module V(c,h). A new type of irreducible Vir-module withinfinite-dimensional weight subspaces is found. We show how the existence ofintertwining operator for modules over vertex operator algebra yieldsreducibility of V'_{\alpha ,\beta}\otimes L(c,h) which is a completely newpoint of view to this problem. As an example, the complete structure of thetensor product with minimal models c=-22/5 and c=1/2 is presented.

Application of vertex algebras to the structure theory of certain representations over the Virasoro algebra