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In this paper, we construct and study some non-weight modules for the Heisenberg–Virasoro algebra and the [Formula: see text] algebra [Formula: see text]. We determine the modules, whose restriction to the universal enveloping algebra of the degree-[Formula: see text] part (modulo center) are free of rank [Formula: see text] for these two algebras. In the most interesting case, this degree-[Formula: see text] part is not the Cartan subalgebra. We also determine the simplicity of these modules, which provide new simple modules for the [Formula: see text] algebra [Formula: see text].

Abstract. The tensor product of highest weight modules with intermediate series modules over the Virasoro algebra was discussed by Zhang [Z] in 1997. Since then the irreducibility problem for the tensor products has been open. In this paper, we determine the necessary and sufficient conditions for these tensor products to be simple. From non-simple tensor products, we can get other interesting simple Virasoro modules. We also obtain that any two such tensor products are isomorphic if and only if the corresponding highest weight modules and intermediate series modules are isomorphic respectively. Our method is to develop a "shifting technique" and to widely use Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra.

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Abstract. In 1986, Shen defined a class of modules over the Witt algebra W d from irreducible modules over the general linear Lie algebra gl d , which were also given by Larsson in 1992. In 1996, Eswara Rao determined the irreducibility of these modules. In this paper, we use simpler methods to give a short and straightforward proof to the results of Eswara Rao.