Cusps and $\D$-Modules

We study interactions between the categories of $\D$-modules on smooth and singular varieties. For a large class of singular varieties $Y$, we use an extension of the Grothendieck--Sato formula to show that $\D_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the {\em cuspidal quotients}. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\D_Y$ and $\D_X$ and a Kashiwara theorem for $\D_Y$, thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing results for complex curves and surfaces and rational Cherednik algebras). We also use this equivalence to enlarge the category of induced $\D$-modules on a smooth variety $X$ by collecting induced $\D_X$-modules on varying cuspidal quotients. The resulting {\em cusp-induced} $\D_X$-modules possess both the good properties of induced $\D$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free $\D_X$-modules.