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Previously we showed that the tensor product of a weight module over ageneralized Weyl algebra (GWA) with a weight module over another GWA is aweight module over a third GWA. In this paper we compute tensor products ofsimple and indecomposable weight modules over generalized Weyl algebrassupported on a finite orbit. This allows us to give a complete presentation bygenerators and relations of the Grothendieck ring of the categories of weightmodules over a tower of generalized Weyl algebras in this setting. We alsoobtain partial results about the split Grothendieck ring. We described the caseof infinite orbits in previous work.

Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case