With T. Nevins. **``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.