Moment map flows and the Hecke correspondence for quivers

TL;DR

This paper studies the convergence of gradient flows related to moment maps on quiver representations, providing algebraic criteria for orbit intersections and linking Morse theory to Nakajima's Hecke correspondence.

Contribution

It offers a new algebraic criterion for orbit intersection with unstable sets and interprets Nakajima's Hecke correspondence via Morse theory for quivers.

Findings

Algebraic classification of orbits intersecting unstable sets.

Morse-theoretic interpretation of Nakajima's Hecke correspondence.

Convergence properties of moment map gradient flows for quivers.

Abstract

In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical points connected by flow lines for the norm-square of a moment map. The results are valid in the general setting of finite quivers with relations.