Cusps and -Modules

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We study interactions between the categories of -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 -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 and ₓ and a Kashiwara theorem for, 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 -modules on a smooth variety X by collecting induced ₓ-modules on varying cuspidal quotients. The resulting cusp-induced ₓ-modules possess both the good properties of induced -modules (in particular, a Riemann-Hilbert description) and, when X is a curve, a simple characterization as the generically torsion-free ₓ-modules.