Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras

TL;DR

This paper establishes a geometric connection between rational Cherednik algebra representations, sheaves on Nakajima quiver varieties, and Hecke algebra parameters, providing new insights into their interrelations.

Contribution

It introduces a Z-algebra construction linking Cherednik algebra representations to sheaves on Nakajima quiver varieties and interprets category O ordering geometrically.

Findings

Relates Cherednik algebra representations to sheaves on quiver varieties

Provides a geometric interpretation of category O ordering

Connects the geometry to the a-function for Hecke algebras

Abstract

We relate the representations of the rational Cherednik algebras associated with the complex reflection group G(m,1,n) to sheaves on Nakajima quiver varieties associated with extended Dynkin gaphs via a Z-algebra construction. As the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. We interpret the ordering on category O geometrically using this relationship; we also relate the geometry to the a-function for Hecke algebras with unequal parameters.