Gregory G. Smith - Initial ideals in the Weyl algebra

Let A_n(k) be the Weyl algebra over a field of characteristic zero and let M be a finitely generated left A_n(k)-module. If A_n(k) is equipped with a filtration such that the associated graded algebra is the commutative polynomial ring in 2nindeterminates, we prove that each irreducible component of the characteristic variety of M has dimension at least n. In particular, this generalizes an important consequence of the fact that characteristic variety is involutive when A_n(k) has the order filtration. We also establish, for certain skew polynomial rings including the Weyl algebra and universal enveloping algebras for finite dimensional Lie algebras, an equidimensionality theorem for the characteristic variety, extending known results to non-Zariskian filtrations.