Abstract
We construct explicitly a large family of new simple modules for an arbitrary finite W-algebra of type A. A basis of these modules is given by the Gelfand-Tsetlin tableaux whose entries satisfy certain sets of relations. Characterization and an effective method of constructing such admissible relations are given. In particular we describe a family of simple infinite dimensional highest weight relation modules. We also prove a sufficient condition for the simplicity of tensor product of two highest weight relation modules and establish the simplicity of the tensor product any number of relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest weight modules.
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