Poincaré series and an application to Weyl algebras

Let A_n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A_n. 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 A_n 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.