Annihilators of local cohomology modules and simplicity of rings of differential operators

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.
Comment: 15 pages, comments are welcome. This paper recovers and extends the results obtained by the second named author in arXiv:1501.00718 [math.AC]