Symplectic reflection algebras and affine Lie algebras

TL;DR

This paper explores the potential categorification of affine Lie algebra structures through the representation theory of symplectic reflection algebras, proposing conjectures based on geometric and algebraic insights.

Contribution

It introduces conjectures linking symplectic reflection algebra representations to affine Lie algebra decompositions, inspired by geometric representation theory insights.

Findings

Formulation of conjectures relating symplectic reflection algebras to affine Lie algebra structures

Insights into the connection between quantum connections and symplectic reflection algebras

Proposals for categorification in representation theory

Abstract

The goal of this paper is to present some results and (more importantly) state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory of affine Lie algebras (namely, decompositions of the restriction of the basic representation to finite dimensional and affine subalgebras). These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections for Hilbert schemes of resolutions of Kleinian singularities and representations of symplectic reflection algebras.