 Bernstein growth dimension of a Weyl algebra module

For a nonzero finitely generated Weyl algebra module, choose a finite-dimensional generating subspace $M_0$. Under the Bernstein filtration, $\dim (F_j{A}_n{M}_0)$ is eventually polynomial because the graded module is finite over a polynomial ring. Its degree is independent of the generating subspace, since two generating filtrations bound each other after fixed shifts. It equals the Gelfand–Kirillov dimension of a module.