ncts

The module theory of the Weyl algebra, known as the theory of D-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic D-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such a correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. We introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties. This is ongoing work based on joint work with Jayce Getz and Spencer Leslie.