Etingof conjecture for quantized quiver varieties

TL;DR

This paper calculates the number of finite-dimensional irreducible modules for algebras related to Nakajima quiver varieties, confirming Etingof's conjecture in specific cases using advanced categorical techniques.

Contribution

It provides an exact count of irreducible modules for certain quiver-related algebras, confirming Etingof's conjecture in finite and affine types.

Findings

Exact count for finite type quivers

Lower bounds for all quivers

Confirmation of Etingof's conjecture in specific cases

Abstract

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof's conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac-Moody actions and wall-crossing functors. We finish the paper outlining some future directions of research.