Poincaré series and an application to Weyl algebras

Let An be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over An. By further exploring the relationship between the Poincaré series and the dimension and the multiplicity of M, we are able to prove that the tensor product of two finitely generated modules over An has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A.