  We give a dimension bound on the irreducible components of the characteristic
variety of a system of linear partial differential equations defined from a
suitable filtration of the Weyl algebra $A_{n}(k)$. This generalizes an
important consequence of the fact that a characteristic variety defined from
the order filtration is involutive.
  More explicitly, we consider a filtration of $A_{n}(k)$ induced by any vector
$(u,v) \in {\mathbb Z}^{n}\times {\mathbb Z}^{n}$ such that the associated
graded algebra is the commutative polynomial ring in $2n$ indeterminates. The
order filtration is the special case $(u,v) = (0,1)$. Any finitely generated
left $A_{n}(k)$-module $M$ has a good filtration with respect to $(u,v)$ and
this gives rise to a characteristic variety $\Ch_{(u,v)}(M)$ which depends only
on $(u,v)$ and $M$. When $(u,v) = (0,1)$, the characteristic variety is
involutive and this implies that its irreducible components have dimension at
least $n$. In general, the characteristic variety may fail to be involutive,
but we are still able to prove that each irreducible component of
$\Ch_{(u,v)}(M)$ has dimension at least $n$.
