Differential operators and Cherednik algebras

TL;DR

This paper connects two geometric methods for studying rational Cherednik algebras of type A, showing their equivalence and simplifying proofs of key results without relying on complex prior theorems.

Contribution

It unifies noncommutative Proj and quantum Hamiltonian reduction approaches, demonstrating their compatibility and confirming a conjecture about characteristic cycles.

Findings

Hamiltonian reduction intertwines geometric and algebraic functors.

Characteristic cycles from two approaches are proven to be equal.

Provides a shorter proof of a key theorem without deep prior results.

Abstract

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present 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, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].