 Order characteristic variety of a Weyl algebra module

The algebraic set defined by the annihilator of a module of the associated graded module for a good filtration for the order filtration is the order characteristic variety. It lies in affine space with coordinates $x_1,\dots ,x_n,{\xi }_1,\dots ,{\xi }_n$ and is invariant under scaling the ${\xi }_i$. Its dimension is the dimension of a Weyl algebra module for the order filtration. The order filtration of a Weyl algebra assigns degrees zero to the $x_i$ and one to the ${\xi }_i$; a characteristic variety formed with the Bernstein filtration need not be the same algebraic set.