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 correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and 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.
In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.