Moment map flows and the Hecke correspondence for quivers

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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.
Original languageEnglish
Pages (from-to)730-794
Number of pages65
JournalAdvances in Mathematics
Early online date15 Sept 2017
Publication statusPublished - 7 Nov 2017

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