Dierential operators and Cherednik algebras

We establish a link betweentwo geometric approaches to the representation theory of rationalCherednik algebras of type A: one based on anoncommutative Proj construction \cite{GS}; the other involving quantum hamiltonian reduction of an algebra of differential operators \cite{GG}. In this paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra.That enables us to give a direct and relatively short proof of the key result \cite[Theorem~1.4]{GS} without recourse to Haiman's deep results on the n! theorem \cite{Ha1}. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from \cite{GG}