Microlocalization of rational Cherednik algebras

TL;DR

This paper constructs a microlocalization of rational Cherednik algebras of type S_n via quantization of the Hilbert scheme, establishing an equivalence of module categories under specific parameters.

Contribution

It introduces a microlocalization framework for rational Cherednik algebras using Hilbert scheme quantization, linking algebraic and geometric perspectives.

Findings

Established a microlocalization of rational Cherednik algebras.

Proved category equivalence between H-modules and microlocalized modules.

Connected algebraic structures with geometric quantization methods.

Abstract

We construct a microlocalization of the rational Cherednik algebras $H$ of type $S_n$. This is achieved by a quantization of the Hilbert scheme $\Hilb^n\C^2$ of $n$ points in $\C^2$. We then prove the equivalence of the category of $H$-modules and the one of modules over its microlocalization under certain conditions on the parameter.