Irreducible components of characteristic varieties

Gregory G. Smith

Published 1999-12-08Version 1

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 $n$.