  We study holonomic D-modules on SL_n(C)xC^n, called mirabolic modules,
analogous to Lusztig's character sheaves. We describe the supports of simple
mirabolic modules. We show that a mirabolic module is killed by the functor of
Hamiltonian reduction from the category of mirabolic modules to the category of
representations of the trigonometric Cherednik algebra if and only if the
characteristic variety of the module is contained in the unstable locus.
  We introduce an analogue of the Verdier specialization functor for
representations of Cherednik algebras which agrees, on category O, with the
restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a
Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian
reduction intertwines specialization functors on mirabolic D-modules with the
corresponding functors on representations of the Cherednik algebra. This allows
us to apply known purity results for nearby cycles in the setting considered by
Bezrukavnikov and Etingof.
