Hamiltonian reduction and nearby cycles for mirabolic DD-modules

We study holonomic DD-modules on SLn(C)×CnSLn(C)×Cn, 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 Verdier's specialization functor for representations of Cherednik algebras which agrees, on category OO, with the restriction functor of Bezrukavnikov and Etingof. In type A  , we also consider a Verdier specialization functor on mirabolic DD-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic DD-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.