Cusps and $\mathcal{D}$-modules

We study interactions between the categories of $\mathcal{D}$-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 $\mathcal{D}_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the cuspidal quotients. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\mathcal{D}_Y$ and $\mathcal{D}_X$and a Kashiwara theorem for $\mathcal{D}_Y$, 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 $\mathcal{D}$-modules on a smooth variety $X$ by collecting induced $\mathcal{D}_X$-modules on varying cuspidal quotients. The resulting cusp-induced $\mathcal{D}_X$-modules possess both the good properties of induced $\mathcal{D}$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free $\mathcal{D}_X$-modules.