Holonomic modules for rings of invariant differential operators

TL;DR

This paper investigates holonomic modules over rings of invariant differential operators on affine domains, establishing Bernstein inequality and analyzing filter dimension for various invariant rings under finite group actions.

Contribution

It extends Bernstein inequality and computes filter dimension for invariant differential operator rings under finite group actions, including Weyl algebra invariants.

Findings

Bernstein inequality holds for these rings

Filter dimension of all studied algebras equals 1

Results extend to quotient varieties and generalized Weyl algebras

Abstract

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.