Procesi bundles and Symplectic reflection algebras

TL;DR

This survey explores the relationship between Procesi bundles on symplectic resolutions and Symplectic reflection algebras, highlighting their construction, classification, and applications to representation theory and algebraic geometry.

Contribution

It constructs and classifies Procesi bundles, establishes isomorphisms between spherical Symplectic reflection algebras, and proves key positivity and localization results for related algebras.

Findings

Constructed and classified Procesi bundles.

Proved isomorphism between spherical Symplectic reflection algebras.

Established wreath Macdonald positivity and localization theorems.

Abstract

In this survey we describe an interplay between Procesi bundles on symplectic resolutions of quotient singularities and Symplectic reflection algebras. Procesi bundles were constructed by Haiman and, in a greater generality, by Bezrukavnikov and Kaledin. Symplectic reflection algebras are deformations of skew-group algebras defined in complete generality by Etingof and Ginzburg. We construct and classify Procesi bundles, prove an isomorphism between spherical Symplectic reflection algebras, give a proof of wreath Macdonald positivity and of localization theorems for cyclotomic Rational Cherednik algebras.