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MORSE THEORY OF THE MOMENT MAP FOR REPRESENTATIONS OFQUIVERS

MEGUMI HARADA AND GRAEME WILKIN

ABSTRACT. The main results of this manuscript concern the Morse theory associated tothe norm-square of a Kahler moment map f = ‖Φ − α‖2 on the space of representationsRep(Q,v) of a quiver; these are the first steps in a larger research program concerningthe hyperkahler analogue of the well-known Kirwan surjectivity theorem in symplecticgeometry. The first main result is that, although ‖Φ − α‖2 is not necessarily proper, thenegative gradient flow with respect to f converges to a critical point of f . Hence we ob-tain a Morse stratification of Rep(Q,v). We also give explicit descriptions of the criticalsets of f in terms of subrepresentations. The second main result concerns the relation-ship between the analysis and the algebraic geometry: the Morse stratification is equiv-alent to the algebro-geometric Harder-Narasimhan stratification on Rep(Q,v), and thelimit of the negative gradient flow is isomorphic to the associated graded object of theHarder-Narasimhan-Jordan-Holder filtration of the initial condition. Finally, we explicitlyconstruct local coordinates around the Morse strata, an important tool for future Morse-theoretic applications. The last section contains some immediate applications of these re-sults. First, in the hyperkahler setting of Nakajima quiver varieties, we give a linearizeddescription of the negative normal bundle to the critical sets of f when restricted to thezero set of the complex moment map (under a technical hypothesis on the stability param-eter). Second, we prove Kirwan surjectivity theorems in rational cohomology and integralK-theory for moduli spaces of representations of quivers. Finally, we observe that theMorse theory developed in this manuscript immediately generalizes to certain equivariantcontexts.

CONTENTS

1. Introduction 22. Preliminaries 62.1. Lie group actions, moment maps, and Kahler quotients 62.2. Moduli spaces of representations of quivers and quiver varieties 83. Morse theory with ‖Φ− α‖2 113.1. Convergence of the gradient flow 123.2. Critical points of ‖Φ− α‖2 153.3. Distance decreasing formula along the flow 184. The Harder-Narasimhan stratification 244.1. Slope and stability for representations of quivers 244.2. Comparison of stratifications 295. An algebraic description of the limit of the flow 44

Date: August 5, 2008.2000 Mathematics Subject Classification. Primary: 53D20; Secondary: 53C26 .Key words and phrases. quiver, representations of quivers, hyperkahler quotient, quiver variety, equivari-

ant cohomology, Morse theory.1

2 MEGUMI HARADA AND GRAEME WILKIN

6. Fibre bundle structure of strata 487. Applications 537.1. Hyperkahler quotients 537.2. Kirwan surjectivity for representation varieties in cohomology and K-theory 547.3. Equivariant Morse theory and equivariant Kirwan surjectivity 56References 57

1. INTRODUCTION

The main purpose of this manuscript is to lay the groundwork for an equivariantMorse-theoretic approach to the study of the topology of hyperkahler quotients in thespecific case of Nakajima quiver varieties. More specifically, the motivation for our workis the following long-standing question. Suppose a Lie group G acts hyperhamiltonianly1

on a hyperkahler manifold (M,ωI , ωJ , ωK) with moment maps ΦI ,ΦJ ,ΦK from M to g∗

respectively, and consider the G-equivariant inclusion

(1.1) (ΦI ,ΦJ ,ΦK)−1(α) ↪→M

where α is a central value in (g∗)3. The long-standing question is: When is the inducedring homomorphism from the G-equivariant cohomology2 H∗

G(M) of the original hyper-hamiltonian G-space M to the ordinary cohomology H∗(M////G) of the hyperkahler quo-tient M////G := (ΦI ,ΦJ ,ΦK)−1(α)/G (induced by (1.1)) a surjection of rings?

In the symplectic category, the analogous result is the well-known Kirwan surjectivitytheorem [14]. This powerful theorem was the impetus for much subsequent work in equi-variant symplectic geometry which computes the topology of symplectic quotients, sinceit reduces the problem of computing H∗(M//G) to that of computing an equivariant co-homology ring H∗

G(M) and the kernel of κ (see e.g. [11, 20, 26, 7]). Thus it is quite naturalto ask whether a similar theory can be developed in the case of hyperkahler quotients.

In the symplectic setting, the proof of the Kirwan surjectivity theorem [14] uses theMorse theory of the norm-square ‖Φ‖2 of the moment map. Although ‖Φ‖2 is not Morse inthe classical sense, Kirwan showed that it has sufficiently good properties (in modern ter-minology, ‖Φ‖2 is a“Morse-Kirwan” function) to produce aG-equivariantly perfect strati-fication of the original HamiltonianG-space for which the minimal stratum is the level setΦ−1(0). From this one can then deduce that the induced ring map H∗

G(M) → H∗G(Φ−1(0))

is surjective. This Morse-theoretic point of view has yielded many other results in addi-tion to this fundamental surjectivity theorem: e.g. there are the computations in equivari-ant cohomology via localization by ‖Φ‖2 as done by Paradan [23] (see also Woodward[28]), as well as explicit computations of the kernel of the Kirwan map κ using Morse-Kirwan theory of ‖Φ‖2 as carried out by Tolman and Weitsman [26] (see also Goldin [7]).

One of the implicit goals in the present manuscript is to develop some tools towardthe development of hyperkahler analogues of this “Morse theory of the moment map” insymplectic geometry; in this sense, this paper ought to be viewed as a few initial stepsin a larger research program. At present, we restrict our attention to the case of moduli

1A Lie group G acts hyperhamiltonianly on a hyperkahler manifold (M,ωI , ωJ , ωK) if it is hamiltonianwith respect to each of the symplectic structures ωI , ωJ , ωK .

2We assume throughout the paper that the coefficient ring is Q.

MORSE THEORY OF THE MOMENT MAP FOR REPRESENTATIONS OF QUIVERS 3

spaces of representations of quivers and Nakajima quiver varieties for two simple rea-sons. Firstly, we have found that the theory for this case has much in common with thatof the moduli spaces of Higgs bundles and the moduli spaces of vector bundles whicharise in gauge theory. This includes, for instance, the presence of algebraic-geometrictools such as a Harder-Narasimhan stratification on the space of representations of thequiver which is compatible with the Morse theory of the moment map. Secondly, thestudy of quiver varieties is already a vast and active subject with many connections toother areas of mathematics, including geometric representation theory, gauge theory, andmirror symmetry (see e.g. [3, 9] and references therein).

We emphasize that although our results are motivated by, and analogues of, similar re-sults from gauge theory, it is not at all always straightforward to translate to the setting ofquivers these gauge-theoretic tools – most particularly because the linear algebra associ-ated to a quiver representation can quickly get quite complicated. Hence we develop newideas and constructions in order to carry through the analogous program. The strength ofour work lies in the fact that we are able to deal with the general setting of a moment mapassociated to any representation space of any quiver, making no restrictive assumptionsabout the underlying graph or dimension vector.

We now turn to a more in-depth outline of the contents of our paper. Let Q = (I, E)be a quiver, i.e. an oriented graph with vertices I and edges E. We always assume Q isfinite. For an edge (also called an “arrow”) a ∈ E, let t(a), h(a) ∈ I denote the head andtail of the arrow respectively. Suppose given a dimension vector v = (v`)`∈I ∈ ZI≥0. Fromthis data we may build the affine space of representations of the quiver,

Rep(Q,v) =(⊕a∈E Hom(Vt(a), Vh(a))

)⊕ (⊕`∈I Hom(V`,W`))

where each V`,W` for ` ∈ I is a hermitian inner product space, with dimension dim(V`) =v`.Then Rep(Q,v) is naturally equipped with the componentwise conjugation action ofthe compact Lie group G :=

∏`∈I U(V`), where U(V`) is the unitary group associated to

V` for each vertex ` ∈ I, and the moduli space of representations of the quiver Q (at acentral parameter α) is the Kahler (or GIT) quotient Rep(Q,v)//αG. Similarly, the cotan-gent bundle T ∗ Rep(Q,v) is naturally a quaternionic affine space and is hyperhamiltonianwith respect to the induced action of G. The hyperkahler quotient T ∗ Rep(Q,v)////(α,0)G ofT ∗ Rep(Q,v) byG (at (α, 0) ∈ g∗⊕g∗

Ca central value) is then called a Nakajima quiver va-

riety. It is important for our purposes to note that since the cotangent bundle T ∗ Rep(Q,v)can be also viewed as a representation space of a quiver Q obtained from Q by “doublingarrows”, i.e. Q := (I, E ∪ E) where for each oriented edge a ∈ E in the original quiver,we also include an edge a ∈ E between the same vertices but in the opposite direction.Thus our results concerning spaces of representations of quivers also may be applied totheir cotangent bundles, and hence are applicable also to the study of Nakajima quivervarieties.

In the gauge-theoretic study of the moduli of holomorphic bundles, there are notionsof stability and semi-stability coming from geometric invariant theory, from which one ob-tains the Harder-Narasimhan stratification of the space of holomorphic structures on a vec-tor bundle. In the setting of quivers, Reineke [24] showed that analogous notions alsohold for the space Rep(Q,v) of representations of quivers, and in particular there existsa Harder-Narasimhan stratification of T ∗ Rep(Q,v) (associated to a choice of stability pa-rameter α). On the other hand, just as in Kirwan’s original work, we may also consider


the Morse-type stratification obatined from considering the flow along the negative gra-dient vector field of the norm-square of the Kahler moment map ‖Φ − α‖2 on Rep(Q,v).That such a stratification even makes sense is non-trivial, since Rep(Q,v) is non-compact,so it is not immediate that the gradient flow even converges. Our first main theorem (The-orem 3.1 in Section 3) is that the flow does converge, and hence the Morse stratificationexists; in fact, our result is valid for any linear action on a hermitian vector space, not justthis specific case of quivers.

Theorem 1.1. Let V be a hermitian vector space and suppose that a compact connected Lie groupG acts linearly on V via an injective homomorphism G → U(V ). Let Φ : V → g∗ ∼= g denote amoment map for this action. For α ∈ g, define f := ‖Φ− α‖2 : V → R and denote by γ(x, t) thenegative gradient flow on V with respect to f . Then for any initial condition x0 ∈ V, the gradientflow γ(x0, t) : R→ V exists for all time t ∈ R and converges to a critical point x∞ of f .

From our proof of Theorem 1.1, it is also straightforward to conclude that the negativegradient flow preserves closed GC-invariant subsets of V , which in particular impliesthat the Morse strata obtained via Theorem 1.1 restricts to the level set of a holomorphicmoment map ΦC

−1(0) in the case of the hyperkahler construction of Nakajima quivervarieties (see Corollary 3.8). Moreover, an analysis of the properties of the representationswhich are critical with respect to the norm-square ‖Φ− α‖2 yields explicit descriptions ofthe connected components of the critical sets in terms of smaller-rank quiver varieties(see Section 3.2), thus providing us with an avenue for inductive computations of thecohomology rings or Poincare polynomials (e.g. Proposition 3.14).

Our next main result (Theorem 4.15) is that the Morse stratification on Rep(Q,v) ob-tained by Theorem 1.1 and the Harder-Narasimhan stratification mentioned above are infact equivalent (when specified with respect to the same parameter α). Thus there is atight relationship between the algebraic geometry and the Morse theory; in particular, inany given situation, we may use whichever viewpoint is more convenient. For details,see Section 4.

Theorem 1.2. Let Q = (I, E) be a quiver and v ∈ ZI≥0 a choice of dimension vector. LetRep(Q,v) be the associated representation space and Φ : Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I u(V`) be

the moment map for the standard Hamiltonian action of G =∏

`∈I U(V`) on Rep(Q,v). Then thealgebraic stratification of Rep(Q,v) by Harder-Narasimhan type with respect to α coincides withthe analytic stratification of Rep(Q,v) by the negative gradient flow of f = ‖Φ− α‖2.

Our third main theorem (Theorem 5.4) is an algebraic description of the limit of thenegative gradient flow with respect to ‖Φ − α‖2. There is a refinement of the Harder-Narasimhan filtration of a representation of a quiver which uses stable subrepresentationsinstead of semistable subrepresentations; this is called the Harder-Narasimhan-Jordan-Holder (H-N-J-H) filtration (in fact, it is a double filtration). We show that the limit ofthe flow is isomorphic to the associated graded object of this H-N-J-H filtration of theinitial condition. This is a quiver analogue of results for holomorphic bundles (see [5])and Higgs bundles (see [27]).

Theorem 1.3. Let Q = (I, E),v ∈ ZI≥0, and Rep(Q,v) be as in Theorem 4.15. Let A0 ∈Rep(Q,v), and let A∞ be the limit of the negative gradient flow of ‖Φ−α‖2. Then A∞ is isomor-phic to the graded object of the H-N-J-H filtration of A0.


Our last main result is an explicit construction of local coordinates (built in terms ofthe complex group action on the Harder-Narasimhan strata) near any representation ofa given Harder-Narasimhan type. This provides a useful tool for local computations inneighborhoods of strata, for example in standard Morse-theoretic arguments which builda manifold by inductively gluing Morse strata. For the following, ρCA denotes the in-finitesmal complex group action on Rep(Q,v,w), and Rep(Q,v)∗,µ denotes the subset ofa Harder-Narasimhan stratum Rep(Q,v)µ which preserves a fixed Harder-Narasimhanfiltration.

Proposition 1.4. Let Q = (I, E),v ∈ ZI≥0, and Rep(Q,v) be as in Theorem 4.15. Let µ bea non-minimal Harder-Narasimhan type of Rep(Q,v). Suppose A ∈ Rep(Q,v)∗,µ and let A∞denote the limit point of the negative gradient flow with respect to ‖Φ−α‖2 with initial conditionA. Then there exists ε > 0 and g ∈ GC such that for any B ∈ Rep(Q,v)µ with ‖B − A‖ < ε,there exist unique elements u ∈ (ker ρCA∞)⊥ and δa ∈ ker(ρCA∞)∗ ∩ Rep(Q,v)∗ such that B =g−1 exp(u) · (A∞ + δa).

Using these local coordinates, we conclude in Proposition 6.9 that the Harder-Narasimhanstrata Rep(Q,v) have well-behaved tubular neighborhoods which may also be identifiedwith the disk bundle of their normal bundle in Rep(Q,v); it is also straightforward tocompute their codimensions (Proposition 6.5).

Finally, in the last section, we present some immediate applications of the above results.More specifically, we first prove in Section 7.1 that although the level set ΦC

−1(0) may besingular, at least near the critical sets of ‖ΦR−α‖2, the level set can be described locally interms of the linearized data ker dΦC (Theorem 7.1). Secondly, In Theorems 7.2 and 7.3 weprove Kirwan surjectivity results in both rational cohomology and integral topologicalK-theory for moduli spaces of representations of quivers. Thirdly, we observe that theMorse theory developed in this manuscript immediately generalizes to certain equivari-ant settings in Theorem 7.5, which yields as immediate corollaries equivariant Kirwansurjectivity theorems in both rational cohomology and integral topological K-theory formoduli spaces of representations of quivers; here the equivariance is with respect to anyclosed subgroup of U(Rep(Q,v)) which commutes with the group G =

∏` U(V`).

As mentioned earlier, this manuscript contains the initial steps in a larger research pro-gram. The first and most obvious next step is to use the Morse-theoretic tools developedin this manuscript to derive surjectivity arguments in rational cohomology or K-theoryfor some class of Nakajima quiver varieties. This will be the subject of a sequel to this pa-per. Secondly, for any such cases, it would be of interest to compute explicitly the kernelker(κ) of the Kirwan map κ to obtain new and concrete descriptions of the cohomology orK-theory rings of Nakajima quiver varieties. As in the case of many Kahler quotients, aswell as for hypertoric varieties and hyperpolygon spaces [16, 17], it would be of interestto find geometric or representation-theoretic interpretations of the generating classes, as,say, characteristic classes of certain vector bundles, as well as to expore the relationshipsbetween the combinatorial data specifying the quiver variety and its cohomology or K-theory rings. Thirdly, each of the above questions may be repeated in the equivariant case,in which we wish to consider a residual group action (obtained from some subgroup ofU(Rep(Q,v))) on the quotient Nakajima quiver variety. Finally, it is of course still com-pletely open whether there exists a hyperkahler Kirwan surjectivity theorem in a generalsetting, not restricted to either Nakajima quiver varieties or even to (say) hyperkahler


quotients of affine quaternionic space. We hope to address these and related questions infuture work.

Acknowledgements. We thank Georgios Daskalopoulos, Lisa Jeffrey, Jonathan Weits-man, and Richard Wentworth for their constant support and encouragement, and toTamas Hausel and Hiroshi Konno for their interest in our work. We are grateful to ReyerSjamaar for pointing out to us his work in [25]. Both authors thank the hospitality of theBanff Research Centre, which hosted a Research-in-Teams workshop on our behalf, aswell as the American Institute of Mathematics, where part of this work was conducted.The first author was supported in part by an NSERC Discovery Grant.

2. PRELIMINARIES

In this section, we set up the notation and sign conventions to be used throughout.We have chosen our conventions so that our moment map formulae agree with those ofNakajima in [21].

2.1. Lie group actions, moment maps, and Kahler quotients. Let G be a compact con-nected Lie group. Let g denote its Lie algebra and g∗ its dual, and let 〈·, ·〉 denote a fixedG-invariant inner product on g. We will always identify g∗ with g using this inner prod-uct; by abuse of notation, we also denote by 〈·, ·〉 the natural pairing between g and itsdual.

Suppose that G acts on the left on a manifold M . Then we define the infinitesmal actionρ : M × g → TM by

(2.1) ρ : (x, u) ∈M × g 7→ d

dt

∣∣∣∣t=0

(exp tu) · x ∈ TpM,

where {exp tu} denotes the 1-parameter subgroup of G corresponding to u ∈ g and g · xthe group action. We also denote by ρ(u) the vector field on M generated by u, specifiedby ρ(u)(x) = ρ(x, u) as above. Similarly, for a fixed x ∈ M, we denote by ρx : g → TxMthe restriction of the infinitesmal action to the point x, i.e. ρx(u) := ρ(x, u) for u ∈ g. If Mis a Riemannian manifold, we use ρ∗ to denote the adjoint ρ∗x : TxM → g with respect tothe Riemannian metric on M and the fixed inner product on g.

In the special case in which M = V is a vector space, the tangent bundle TM may benaturally identified with V × V. In this situation, we denote by δρ(u) the restriction toTxV ∼= V of the derivative of π2 ◦ ρ(u) : V → V where ρ(u) : V → TV is as above andπ2 : TV ∼= V × V → V is the projection to the second (fiber) factor; the basepoint x ∈ Vof δρ(u) : TxV ∼= V → V is understood by context. Similarly let δρ(X) : g → V denote thelinear map defined by δρ(X)(v) := δρ(v)(X).

Now let (M,ω) be a symplectic manifold and suppose G acts preserving ω. Then Φ :M → g∗ is a moment map for thisG-action if Φ isG-equivariant with respect to the givenG-action onM and the coadjoint action on g∗, and in addition, for all x ∈M,X ∈ TxM,u ∈ g,

(2.2) 〈dΦx(X), u〉 = −ωx(ρ(x, u), X),

where we have identified the tangent space at Φ(x) ∈ g∗ with g∗. By the identification g∗ ∼=g using the G-invariant inner product, we may also consider Φ to be a g-valued map; byabuse of notation we also denote by Φ the G-equivariant composition Φ : M → g∗

∼=→ g. In


particular, by differentiating the condition that Φ is G-equivariant we obtain the relation

d

dt

∣∣∣∣t=0

etuΦ(x)e−tu =d

dt

∣∣∣∣t=0

Φ(etu · x

)⇔ [u,Φ(x)] = dΦx (ρ(x, u)) ,

for all u ∈ g, x ∈M. Hence we may conclude that

(2.3) 〈[u,Φ(x)], v〉g = −ωx(ρ(x, v), ρ(x, u)),

for all u, v ∈ g, x ∈M.IfM is additionally Kahler, the relationship between the complex structure I , the metric

g, and the symplectic form ω on M is that

(2.4) ωx(X, Y ) = gx(IX, Y ) = −gx(X, IY )

for all x ∈ M,X, Y ∈ TxM. The above equations imply that the metric gM is I-invariant.We say aG-action on a Kahler manifold is Hamiltonian if it preserves the Kahler structureand is Hamiltonian with respect to ω.

In the setting of Hamiltonian actions on Kahler manifolds, we additionally have the fol-lowing. We continue to treat Φ as a map M → g. The first observation links the derivativeof the moment map with the adjoint of the infinitesmal action.

Lemma 2.1. Let (M,ω, I, g) be a Kahler manifold equipped with a Hamiltonian G-action and amoment map Φ : M → g. Then

dΦ(x) = ρ∗xI

for all x ∈M.

Proof. For any X ∈ TxM,u ∈ g,

〈dΦx(X), u〉 = −ωx(ρ(x, u), X) by definition of Φ

= ωx(X, ρ(x, u)) by skew-symmetry of ω= gx(IX, ρ(x, u)) by (2.4)= gx(IX, ρx(u)) by definition of ρx

= 〈ρ∗xIX, u〉g by definition of ρ∗x.

�

The second links the derivative of the G-equivariance condition to the adjoint action.

Lemma 2.2. Let (M,ω, I, g) be a Kahler manifold equipped with a Hamiltonian G-action and amoment map Φ : M → g. Then

(2.5) ρ∗xIρx(u) = [u,Φ(x)]

for all x ∈M,u ∈ g.


Proof. We have

〈[u,Φ(x)], v〉 = −ωx(ρ(x, v), ρ(x, u)) by (2.3)= −gx(Iρ(x, v), ρ(x, u)) by (2.4)= gx(ρ(x, v), Iρ(x, u)) by I-invariance of g= gx(ρx(v), Iρx(u)) by definition of ρx

= 〈v, ρ∗xIρx(u)〉 by definition of ρ∗xfor all u, v ∈ g, x ∈M. The result follows. �

Now assume that M = V is a hermitian vector space.

Lemma 2.3. Let (V, ω, I, g) be a hermitian vector space (with the standard Kahler structure)equipped with a Hamiltonian G-action and a moment map Φ : V → g. Then

(2.6) ρ∗xIδρ(u)(X) = [u, ρ∗xIX]− (δρ(X))∗(Iρx(u)),

for all x ∈ V, u ∈ g, X ∈ TxV.

Proof. From Lemma 2.2, we know that for all u, v ∈ g, we have

〈ρ∗xIρx(u), v〉 = 〈[u,Φ(x)], v〉.From the definition of ρ∗x we obtain

gx(Iρx(u), ρx(v)) = 〈[u,Φ(x)], v〉.Taking a derivative with respect to the variable x in V , evaluating in the direction X ∈TxV, and recalling that for M = V the metric gx is constant in x, we obtain

gx(Iδρ(u)(X), ρx(v)) + gx(Iρx(u), δρ(v)(X)) = 〈[u, dΦx(X)], v〉= 〈[u, ρ∗xIX], v〉.

Again the definition of ρ∗x gives us

〈ρ∗xIδρ(u)(X), v〉+ 〈(δρ(X))∗Iρx(u), v〉 = 〈[u, ρ∗xIX], v〉

for all u, v ∈ g, X ∈ TxV. The result follows. �

2.2. Moduli spaces of representations of quivers and quiver varieties. In this section,we quickly recall the construction of the spaces which are the main objects of study inthis manuscript, namely, the moduli spaces of representations of quivers. They are builtfrom the combinatorial data of a finite oriented graph Q (a quiver) and a dimension vec-tor v (which specifies the underlying vector space of the representation) and are KahlerHamiltonian quotients of an affine space of representations Rep(Q,v). We also recall theconstruction of Nakajima quiver varieties, which are hyperkahler quotients of the holo-morphic cotangent bundles T ∗ Rep(Q,v) of Rep(Q,v). Although the main results of thispaper are stated in terms of Rep(Q,v), our results have applications also to Nakajimaquiver varieties; we mention these as appropriate throughout the paper.

We refer the reader to [21] for details on what follows. Let Q = (I, E) be a finiteoriented graph with vertices I and oriented edges a ∈ E, where we denote by out(a) ∈ Ithe outgoing vertex of the arrow a, and by in(a) the incoming vertex. Suppose given a


finite-dimensional hermitian vector space V` for each vertex ` ∈ I, with dimC(V`) = v`.Assembling this data gives us the dimension vector v = (v`)`∈I ∈ (Z≥0)

I . The space ofrepresentations of the quiver Q with dimension vector v is defined as

Rep(Q,v) :=⊕a∈E

Hom(Vout(a), Vin(a)),

Here Hom(−,−) denotes the hermitian vector space of C-linear homomorphisms. We alsodenote by

(2.7) Vect(Q,v) :=⊕`∈I

V`

the underlying vector space of the representation, and let rank(Q,v) := dim(Vect(Q,v))denote its dimension.

The notion of a subrepresentation is straightforward. Suppose given representationsA ∈ Rep(Q,v), A′ ∈ Rep(Q,v′) with corresponding vector spaces {V`}`∈I , {V ′

` }`∈I respec-tively. We say A′ is a subrepresentation of A, denoted A′ ⊆ A, if V ′

` ⊆ V` for all ` ∈ I, the V ′`

are invariant under A, i.e. Aa(V′out(a)) ⊆ V ′

in(a) for all a ∈ E, and A′ is the restriction of A,i.e. A′a = Aa|V ′

out(a)for all a ∈ E.

For each ` ∈ I, let U(V`) denote the unitary group associated to V`. The group G =∏`∈I U(V`) acts on Rep(Q,v) by conjugation, i.e.

(g`)`∈I · (Aa)a∈E =(gin(a)Aag

∗out(a)

)a∈E

.

Hence the infinitesmal action of an element u = (u`)`∈I ∈∏

`∈I u(V`) is given by

ρ((A,B,C,D), u) =(uin(a)Aa − Aauout(a)

)a∈E

.

Moreover, the Kahler form ω on Rep(Q,v) is given as follows: for δA1 = ((δA1)a)a∈E , δA2 =((δA2)a)a∈E , two tangent vectors at a point in Rep(Q,v),

(2.8) ω(δA1, δA2) =∑a∈E

Im(tr(δA1)∗a(δA2)a),

where Im denotes the imaginary part of an element in C.We now explicitly compute the moment map Φ for the G-action on Rep(Q,v). Denote

by Φ` the `-th component, and identify u(V`) ∼= u(V`)∗ using the invariant pairing

〈u, v〉 := tr(u∗v).

With these conventions, the natural action of U(V ) on Hom(V ′, V ) and Hom(V, V ′) givenby, for A ∈ Hom(V ′, V ), B ∈ Hom(V, V ′),

g · A = gA, g ·B = Bg∗

(where the right hand side of the equations is ordinary matrix multiplication), has mo-ment maps

Φ(A) =i

2AA∗ ∈ u(V ), Φ(B) = − i

2B∗B ∈ u(V )

respectively. From here we conclude that for A = (Aa)a∈E ∈ Rep(Q,v),

Φ`(A) =i

2

∑a:in(a)=`

AaA∗a +

∑a′:out(a′)=`

−A∗a′Aa′

.


Hence we have

(2.9) Φ(A) =∑a∈E

[Aa, A∗a]

where AaA∗a is understood to be valued in u(Vin(a)) and A∗aAa in u(Vout(a)). Henceforth we

often simplify the notation further and write (2.9) as

(2.10) Φ(A) = [A,A∗],

where the summation over the arrows a ∈ E is understood.

Remark 2.4. There is a central S1 ⊆ G =∏

`∈I U(V`), given by the diagonal embeddinginto the scalar matrices in each U(V`), which acts trivially on Rep(Q,v). For the purposesof taking quotients, it is sometimes more convenient to consider the action of the quotientgroup PG := G/S1, which then acts effectively on Rep(Q,v). From this point of view,the moment map naturally takes values in the subspace (pg)∗ ⊆ g∗, where the inclusion isinduced by the quotient g → pg := g/Lie(S1), so we may consider (ΦR,ΦC) as a functionfrom M to (pg)∗ ⊕ (pgC)

∗. On the other hand, the moment map (being a commutator)takes values in the traceless matrices in g ∈ u(Vect(Q,v)), the orthogonal complement ofLie(S1), which may be identified with pg∗. In this sense there is no substantial distinctionbetween G and PG for the purpose of taking quotients. However, the difference doesbecome relevant when computing equivariant cohomology.

We now construct the relevant moduli spaces. Let α = (α`)`∈I ∈ (iR)I . Then thisuniquely specifies a central element in g by (α` idV`

)`∈I , which by abuse of notation wealso denote by α. We always assume that tr(α) = 0, i.e. α ∈ pg∗ (see Remark 2.4), and thatG acts freely on the level set Φ−1(α). Then the Hamiltonian quotient

(2.11) Xα(Q,v) := Φ−1(α)/G

is often called in the literature the moduli space of α-stable representations of the quiver Q withdimension vector v. We also call it a representation variety of the quiver Q.

Remark 2.5. There is a generalization of the construction just outlined which also appearsin the literature and which merits mention. Although we do not explicitly address thiscase in the exposition here and to follow, in fact our results also apply to this more gen-eral construction, as we now explain. Suppose given two dimension vectors v,w ∈ ZI≥0

specifying hermitian vector spaces V`,W` with dimC(V`) = v`, dimC(W`) = w` for all ` ∈ I.We may define

Rep(Q,v,w) :=

(⊕a∈E

Hom(Vout(a), Vin(a))

)⊕

(⊕`∈I

Hom(V`,W`)

)and the associated Kahler quotient Xα(Q,v,w) := Φ−1(α)/G for the analogous momentmap Φ on Rep(Q,v,w). Hence Rep(Q,v) and Xα(Q,v) is the special case when w = 0.In the literature, the case w = 0 is often called the “unframed” case, while w 6= 0 is the“framed” case. As was pointed out by Crawley-Boevey [2, p. 261], however, a framedrepresentation variety Xα(Q,v,w) can also be realized as an unframed representationvariety for a different quiver. The details of this “Crawley-Boevey trick” do not concernus here; suffice it to say that our results may also be extended to the framed case with nofurther work.


We now introduce a variant on the construction given above which yields the Naka-jima quiver varieties mentioned in the introduction. The first step is to consider theholomorphic cotangent bundle T ∗ Rep(Q,v) of Rep(Q,v), with the identification

(2.12) T ∗ Rep(Q,v) ∼= ⊕a∈E

(Hom(Vout(a), Vin(a))⊕ Hom(Vin(a), Vout(a))

).

Here, for any two complex vector spaces V and V ′, we consider Hom(V ′, V ) to be thecomplex dual of Hom(V, V ′) by the C-linear pairing

(A,B) ∈ Hom(V, V ′)× Hom(V ′, V ) 7→ tr(BA) ∈ C.

Remark 2.6. The identification (2.12) makes it evident that T ∗ Rep(Q,v) is identified withRep(Q,v) where Q is quiver obtained from Q by setting I = I and, for the edges E,adding for each a ∈ E in Q an extra arrow a with the reverse orientiation.

We equip T ∗ Rep(Q,v) with a Kahler structure ωR given by the standard structure oneach Hom(−,−) in the summand as for Rep(Q,v). This has a corresponding real momentmap ΦR derived as before. However, being a holomorphic cotangent bundle, T ∗ Rep(Q,v)in addition has a canonical holomorphic symplectic structure ωC given by, for (δAi, δBi) =((δAi)a, (δBi)a)a∈E for i = 1, 2,

(2.13) ωC ((δA1, δB1), (δA2, δB2)) =∑a∈E

tr((δA1)a(δB2)a − (δA2)a(δB1)a) ∈ C.

The complex group GC =∏

`∈I GL(V`) also acts naturally by conjugation on T ∗ Rep(Q,v).Using the complex bilinear pairing

〈u, v〉 = − tr(uv)

for u, v ∈ gl(V`), we may identify gl(V`) with its complex dual gl(V`)∗. It is then straight-

forward to compute the holomorphic moment map ΦC : T ∗ Rep(Q,v) → g∗C∼= gC for the

GC-action with respect to ωC. We have for (A,B) ∈ T ∗ Rep(Q,v) that

ΦC(A,B)` =∑

a:in(a)=`

AaBa −∑

a′:out(a′)=`

Ba′Aa′ ,

which again we may write using the same simplified notation as in (2.10) as

(2.14) ΦC(A,B) = [A,B],

where the summation over arrows a ∈ E is understood.Suppose given a parameter α = (α`)`∈I and corresponding central element α = (α` idV`

)`∈I ∈Z(g) as above, and assume that G acts freely on the intersection Φ−1

R(α)∩Φ−1

C(0). Then we

define the Nakajima quiver variety associated to (Q,v, α) as

(2.15) Mα(Q,v) := Φ−1R

(α) ∩ Φ−1C

(0)/G.

This is a special case of a hyperkahler quotient.

3. MORSE THEORY WITH ‖Φ− α‖2

Let V be a hermitian vector space and suppose a compact connected Lie group G actslinearly on V via an injective homomorphism G→ U(V ). Let Φ denote the correspondingmoment map. For α ∈ g, let f := ‖Φ − α‖2. Denote by γ(x, t) the negative gradient flowon V with respect to grad(f), i.e. γ(x, t) satisfies

(3.1) γ(x, 0) = x, γ′(x, t) = − grad(f)γ(x,t).


In this section we prove that the gradient flow of f exists for all time t, and moreover,converges to a critical point of f . This general situation of a linear action on a vector spacecontains the main case of interest in this paper, namely, when V is the space Rep(Q,v).

Theorem 3.1. Let V be a hermitian vector space and suppose that a compact connected Lie groupG acts linearly on V via an injective homomorphism G → U(V ). Let Φ : V → g∗ ∼= g denote amoment map for this action. For α ∈ g, define f := ‖Φ− α‖2 : V → R and denote by γ(x, t) thenegative gradient flow on V with respect to f . Then for any initial condition x0 ∈ V, the gradientflow γ(x0, t) : R→ V exists for all time t ∈ R and converges to a critical point x∞ of f .

Remark 3.2. Although in the case Rep(Q,v) we always restrict to central parameters α ∈Z(g), this hypothesis is unecessary for Theorem 3.1.

In the special case when V = Rep(Q,v), we also explicitly describe in Proposition 3.11the critical sets of f in terms of lower-rank representation spaces. We also define a Morse-theoretic stratification of Rep(Q,v) by decomposing into strata Sν parametrized by theα-types ν ∈ Rrank(Q,v) (defined in Definition 3.12) of the critical sets of f . As discussedin detail in Sections 4 and 5 below, both the notion of α-type as well as the Morse strat-ification by the Sν turn out to be intimately related to the algebraically defined Harder-Narasimhan and Harder-Narasimhan-Jordan-Holder filtrations of a representation of aquiver.

Throughout the section, we also mention the applications of our Morse theory to thespecial case of the construction of Nakajima quiver varieties; in particular, as we recordin Corollary 3.8 and Definition 3.15, the Morse constructions of this section restrict in theappropriate sense to the level set of the holomorphic moment map (2.14), which is theintermediate space used to define Mα(Q,v) in (2.15).

3.1. Convergence of the gradient flow. We first recall that the gradient of the norm-square of any moment map Φ with respect to a Kahler symplectic structure ω, compatiblemetric g and complex structure I is given as follows. For any x ∈ V,X ∈ TxV,

dfx(X) = 2〈(dΦ)x(X),Φ(x)− α〉= −2(ω)x(ρ(x,Φ(x)− α), X)

= −2gx(Iρ(x,Φ(x)− α), X)

= gx(−2Iρ(x,Φ(x)− α), X),

so the negative gradient vector field for f = ‖Φ− α‖2 is given by

(3.2) − grad(f)(x) = 2Iρ(x, (Φ(x)− α)).

We now proceed to the proof of Theorem 3.1, the argument for which is contained in Lem-mas 3.3 to 3.6 below. We begin by showing that the negative gradient flow is defined forall time t ∈ R. Note that since G ⊆ U(V ) acts complex-linearly on V , its complexificationGC ⊆ GL(V ) also acts naturally on V .

Lemma 3.3. Let (V,G,Φ, α) be as in the statement of Theorem 3.1, and let γ(x, t) denote thenegative gradient flow (3.1) of f = ‖Φ − α‖2. Then for any initial condition x0 ∈ V, γ(x0, t) isdefined for all t ∈ R.


Proof. Local existence for ODEs shows that for any x0 ∈ V the gradient flow γ(x0, t) existson (−εx0 , εx0) for some εx0 > 0 depending continuously on x0 (see for example Lemmas1.6.1 and 1.6.2 in [12]).

By construction, along the negative gradient flow x = γ(x0, t) of f(x) = ‖Φ(x) − α‖2,the function f is decreasing. Therefore ‖Φ(x) − α‖ is bounded along the flow, and thereexists a constant C such that ‖Φ(γ(x0, t))‖ ≤ C.

Lemma 4.10 of [25] shows that the set of points

KC := {x ∈ GC · x0 ⊂ V : ‖Φ(x)‖ ≤ C}is a compact subset of V . Equation (3.2) shows that the finite-time gradient flow is con-tained in a GC-orbit (see for example Section 4 of [14]) so we conclude that γ(x0, t) ∈ KC

for any value of t for which γ(x0, t) is defined. Since KC is compact and εx > 0 is a con-tinuous function on KC , there exists ε > 0 such that εx ≥ ε > 0 on KC . Therefore we caniteratively extend the flow so that it exists for all time (see for example the proof of [12,Theorem 1.6.2]). �

Next we show that the flow converges on a subsequence to a critical point of f .

Lemma 3.4. Let (V,G,Φ, α) be as in the statement of Theorem 3.1, and let γ(x, t) : V × R → R

denote the negative gradient flow of f = ‖Φ − α‖2. Then for any initial condition x0 ∈ V, thereexists a sequence {tn}∞n=0 with limn→∞ tn = ∞ and a critical point x∞ ∈ V of f such that

limn→∞

γ(x0, tn) = x∞.

Proof. The proof of Lemma 3.3 shows that the negative gradient flow {γ(x0, t)}t≥0 is con-tained in a compact setKC . Hence there exists a sequence {tn}∞n=0 such that limn→∞ γ(x0, tn) =x∞ for some x∞ ∈ KC . To see that x∞ is a critical point of f , firstly note that sincef(γ(x0, tn)) is bounded below and nonincreasing as a function of n,

(3.3) limn→∞

df

dt(γ(x0, tn)) = 0.

Moreover, equation (3.2) shows that the gradient vector field grad f is continuous on V ,so

(3.4) ‖ grad f(x∞)‖2 = limn→∞

‖ grad f(γ(x0, tn))‖2

= limn→∞

df(grad f(γ(x0, t))) = − limn→∞

df

dt(γ(x0, tn)) = 0.

Therefore x∞ is a critical point of f . �

Even if the negative gradient flow γ(x0, t) converges along a subsequence {tn}∞n=0, it isstill possible that the flow {γ(x0, t)}t≥0 itself does not converge; for instance, γ(x0, t) may“spiral” around a critical point x∞ (cf. [18, Example 3.1]). The next lemma shows that thiskind of behavior does not happen, i.e. the flow γ(x0, t) does indeed converge.

Lemma 3.5. Let (V,G,Φ, α) be as in the statement of Theorem 3.1, and let γ(x, t) : V × R → R

denote the negative gradient flow of f = ‖Φ − α‖2. Then for any initial condition x0 ∈ V, thereexists a critical point x∞ ∈ V of f such that

limt→∞

γ(x0, t) = x∞.


The main step in the proof of Lemma 3.5 is the Lojasiewicz gradient inequality of [19](see also [18] for an exposition), which in the notation of this paper can be stated in thefollowing form.

Lemma 3.6. Let (V,G,Φ, α) be as in the statement of Theorem 3.1. Then for every critical pointx∞ of f there exists δ > 0, C > 0, and 0 < γ < 1 such that

(3.5) ‖ grad f(x)‖ ≥ C |f(x)− f(x∞)|γ

for any x ∈ V with ‖x− x∞‖ < δ.

The rest of the proof of Lemma 3.5 follows exactly that in [18], so we omit the details.The main idea is to show that once the gradient flow gets close to a critical point thenit either converges to a nearby critical point or it flows down to a lower critical point.The proof given in [18] assumes that ‖Φ − α‖2 is proper; however, we do not need thiscondition, since we have the result of Lemma 3.4 which allows us to reduce to the casestudied in [18].

Proof. (Proof of Theorem 3.1) Lemma 3.3 shows that the negative gradient flow γ(x0, t)exists for all time t ∈ R and for any initial condition x0 ∈ V. Lemma 3.5 shows that{γ(x0, t)} converges to a limit point x∞ ∈ V . Of necessity, this limit point x∞ agrees withthe limit point of the subsequence of Lemma 3.4, which shows in addition that x∞ is acritical point of f . The theorem follows. �

We close with some straightforward applications of Theorem 3.1 and its proof, in par-ticular to the case of the cotangent bundle T ∗ Rep(Q,v). We begin with the followingobservation.

Corollary 3.7. Let (V,G,Φ, α) be as in the statement of Theorem 3.1, and let γ(x, t) : V ×R→ R

denote the negative gradient flow of f = ‖Φ − α‖2. Let Y ⊆ V be any closed GC-invariantsubset of V . Then the negative gradient flow γ(x, t) preserves Y , i.e. for any initial conditiony0 ∈ Y, γ(y0, t) ∈ Y for all t ∈ R, and the limit point y∞ := limt→∞ γ(y0, t) is also contained inY .

Proof. As we already saw in the proof of Lemma 3.3, equation (3.1) implies that the neg-ative gradient vector field is always contained in the image of the infinitesmal GC actionon V . Therefore, the finite-time gradient flow is contained in a GC-orbit, and hence in Y ,since Y is GC-invariant. Moreover, since Y is closed, the limit y∞ = limt→∞ γ(y0, t) is alsocontained in Y , as desired. �

Now we return to the case of quivers and their representations. Recall from Remark 2.6that we may also view the cotangent bundle T ∗ Rep(Q,v) as a representation space Rep(Q,v),where Q is a quiver with the same vertices as Q but with edges “doubled”. The real mo-ment map ΦR on T ∗ Rep(Q,v) is the usual Kahler moment map for the linear G-actionon Rep(Q,v), so the results in this section apply. On the other hand, the definition ofMα(Q,v) makes clear that we need to analyze the negative gradient flow of ‖ΦR−α‖2 noton all of T ∗ Rep(Q,v) but on ΦC

−1(0). We have the following.

Corollary 3.8. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let ΦR :T ∗ Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I u(V`) and ΦC : T ∗ Rep(Q,v) → g∗

C∼= gC ∼=

∏`∈I gl(V`)

be the real and holomorphic moment maps, respectively, for the action of G =∏

`∈I U(V`) onT ∗ Rep(Q,v). Let γ(x, t) : T ∗ Rep(Q,v)× R→ T ∗ Rep(Q,v) denote the negative gradient flow


on T ∗ Rep(Q,v) with respect to ‖ΦR−α‖2 for a choice of parameter α ∈ Z(g). Then the negativegradient flow γ(x, t) preserves ΦC

−1(0).

Proof. The result follows immediately from Corollary 3.7 upon observing that ΦC−1(0), by

the definition of ΦC in (2.14), is a GC-invariant subset of T ∗ Rep(Q,v), and also that ΦC iscontinuous, so ΦC

−1(0) is closed. �

3.2. Critical points of ‖Φ− α‖2. In this section we analyze properties of the componentsof the critical set Crit(‖Φ − α‖2) in the quiver case. Let Q = (I, E) be a quiver as inSection 2.2, and Rep(Q,v) the space of representations of Q for a choice of dimensionvector v. For notation, let β := Φ(A)− α for a central parameter α.

We begin with some observations. First, in this special case, a representation A =(Aa)a∈E ∈ Rep(Q,v) is a critical point of f = ‖Φ − α‖2, i.e. the negative gradient (3.2)vanishes at A, if and only if

(3.6) i(βin(a)Aa − Aa(βout(a))

)= 0

for all a ∈ E.Secondly, since iβ ∈

∏`∈I iu(V`) is Hermitian, it can be diagonalized, and the eigenval-

ues are all purely real. In particular, since the action of g on V` is by left multiplication, wehave an eigenvalue decomposition

V` =⊕

λ

V`,λ

where the sum is over distinct eigenvalues of iβ. Let Vλ denote the λ-eigenspace of iβ inVect(Q,v) = ⊕`∈IV`, and let vλ = (dimC(V`,λ))`∈I ∈ ZI≥0 denote the associated dimensionvector.

Thirdly, (3.6) implies that Aa preserves Vλ for all eigenvalues λ and all edges a ∈ E.In particular, the restrictions Aλ := A|Vλ

are subrepresentations of A. Hence we get adecomposition of the representation

(3.7) A =⊕

λ

Aλ,

where again the sum is over distinct eigenvalues of iβ.It turns out that there is a convenient way to describe these eigenvalues λ of iβ in terms

of the rank and the α-slope of the subrepresentations Aλ. A reader familiar with the lan-guage of slope stability in the case of holomorphic bundles will immediately recognizethe definitions below as the natural analogues in this context. We will explore these con-nections further in Section 4. For the moment, we have the following.

Definition 3.9. LetQ = (I, E) be a quiver, with associated hermitian vector spaces {V`}`∈Iand dimension vector v = (v`)`∈I ∈ ZI≥0. Let α = (α`)`∈I ∈ (iR)I . We define the α-degreeof (Q,v) by

(3.8) degα(Q,v) :=∑`∈I

iα`v`.

We also define the rank of (Q,v) as

(3.9) rank(Q,v) :=∑`∈I

v` ∈ Z = dimC(Vect(Q,v)).


Finally, we define the α-slope µα(Q,v) of (Q,v) by

(3.10) µα(Q,v) :=degα(Q,v)

rank(Q,v).

At a critical point A of f = ‖Φ− α‖2, the α-slope of the subrepresentation Aλ turns outto exactly correspond to the eigenvalue of iβ on Vλ.

Lemma 3.10. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0, Rep(Q,v) itsassociated representation space, and Φ : Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I U(V`) a moment map for

the standard Hamiltonian action of G =∏

`∈I U(V`) on Rep(Q,v). Suppose A is a critical pointof f = ‖Φ − α‖2, and further suppose λ is an eigenvalue of iβ = i(Φ(A) − α), with associatedsubrepresentation Aλ. Then

λ = −µα(Aλ).

Proof. Equation (3.6) implies thatA preserves the eigenspace Vλ for each λ. Hence for eachedge a ∈ E, Aa decomposes as a sum Aa = ⊕λAa,λ according to (3.7), and in turn iΦ(A)may also be written as a sum

iΦ(A) =⊕

λ

iΦλ(Aλ) =⊕

λ

∑a∈E

−1

2[Aa,λ, A

∗a,λ] ∈ u(Vλ),

where each summand in last expression is considered as an element in the appropriateu(V`,λ) and Φλ := Φ|Rep(Q,vλ). It is evident that trace(Φλ(Aλ)) = 0 for each λ since it is asum of commutators. On the other hand, by definition, Vλ is the λ-eigenspace of iβ =i(Φ(A)− α), so we have

i(Φ(A)− α)|Vλ= λ idVλ

.

Taking the trace of the above equation, we obtain

−i trace(α|Vλ) = λ rank(Vλ).

By definition, the α-degree of the representation Aλ is degα(Q,vλ) = i trace(α|Vλ), and so

λ = −degα(Q,vλ)

rank(Vλ)= −µα(Q,vλ),

as desired. �

In fact, the converse also holds.

Proposition 3.11. Let Q = (I, E), v ∈ ZI≥0, G =∏

`inI U(V`), and Φ : Rep(Q,v) → g∗ ∼= g beas in Lemma 3.10. Then A ∈ Rep(Q,v) is a critical point of f = ‖Φ− α‖2 if and only if A splitsinto orthogonal subrepresentations A = ⊕λAλ as in (3.7), where each Aλ satisfies

i (Φλ(Aλ − α|Vλ)) = −µα(Q,vλ) · idVλ

.

Proof. The proof of Lemma 3.10 shows that if A is a critical point of f = ‖Φ − α‖2 withassociated splitting (3.7), then for each eigenvalue λ we have Φλ(Aλ) = α|Vλ

− iλ · idVλ. To

show the other direction, we see that if iβ|Vλ= iΦλ(Aλ) − iα|Vλ

is a scalar multiple of theidentity on each Vλ, then (3.6) holds for all a ∈ E since (Q,A) splits as in (3.7). Thereforethe negative gradient vector field vanishes and A is a critical point of ‖Φ− α‖2. �

The above proposition indicates clearly that the α-slopes of the subrepresentations inthe splitting A = ⊕λAλ of a critical representation encodes crucial information about thecritical point. This leads us to the following definition.


Definition 3.12. Let A ∈ Rep(Q,v) be a representation. Let

(3.11) A =m⊕

s=1

As, V` =m⊕

s=1

V`,s

be a splitting of A into subrepresentations. For each s, 1 ≤ s ≤ m, let vs be the associateddimension vector

(3.12) vs = (dimC(V`,s)`∈I) ∈ ZI≥0

of Vs. Then the α-type associated to the splitting (3.11) is the vector ν ∈ Rrank(Q,v) given by

ν = (µα(Q,v1), . . . , µα(Q,v1), µα(Q,v2), . . . , µα(Q,v2), . . . , µα(Q,vm), . . . , µα(Q,vm)) ,

where there are rank(Q,vs) terms equal to µα(Q,vs) for each 1 ≤ s ≤ m. We alwaysassume that the subrepresentations are ordered so that

µα(Q,vi) ≥ µα(Q,vj)

for all i < j, i.e. the slopes are non-increasing.

In the special case when A ∈ Rep(Q,v) is a critical point of f = ‖Φ − α‖2, Proposi-tion 3.11 shows that there exists a canonical splitting of A, given by the eigenspace de-composition associated to iβ = i(Φ(A)− α). In this case, we say that the α-type of A is theα-type of this canonical splitting.

Let T be the set of all possible α-types for elements of Rep(Q,v). The description of thecritical points of ‖Φ−α‖2 by their α-types, along with the convergence of the gradient flowproven in Theorem 3.1, leads to the following Morse stratification of the space Rep(Q,v).

Definition 3.13. Let γ(x, t) denote the negative gradient flow of f = ‖Φ−α‖2 on Rep(Q,v).Let Cν denote the set of critical points of f of α-type ν. We define the analytic (or Morse-theoretic) stratum of α-type ν to be

(3.13) Sν :={A ∈ Rep(Q,v) : lim

t→∞γ(A, t) ∈ Cν

},

the set of points in Rep(Q,v) that “flow down” to Cν .

By Theorem 3.1, every point limits to some critical point, so⋃ν∈T

Sν = Rep(Q,v). We call

this the analytic (or Morse-theoretic) stratification of the space Rep(Q,v). In Sections 4and 6 we show that this analytic stratification has good local properties, in the sense of [1,Proposition 1.19].

The negative gradient flow also allows us to compute the topology of the analytic strataof Rep(Q,v) in terms of that of the critical sets.

Proposition 3.14. Let Q = (I, E),v ∈ ZI≥0, G =∏

`∈I U(V`), and Φ : Rep(Q,v) → g∗ ∼=g ∼=

∏`∈I u(V`) be as in Lemma 3.10. Let γ(x, t) denote the negative gradient flow with respect to

f = ‖Φ−α‖2, and Cν and Sν the critical set and analytic stratum of a fixed α-type ν, respectively.Then the flow γ(x, t) restricted to Sν defines a G-equivariant deformation retract of Sν onto Cν .In particular,

H∗G(Sν) ∼= H∗

G(Cν).


Proof. Theorem 3.1 in this section shows that the convergence of γ(x, t) also holds in ourcase, where the Morse function is the norm-square of a moment map associated to a linearG-action on a vector space V (even if Φ is not proper). From here, the same argument asin [18] allows us to conclude that taking the limit of the negative gradient flow defines adeformation retract Sν → Cν . Since the gradient flow equations (3.2) are G-invariant, theflow is G-equivariant, as is the retract Sν → Cν . The theorem follows. �

We close this section with an observation about the case of hyperkahler quotients andNakajima quiver varieties. As we have seen in Corollary 3.7, the negative gradient flow off = ‖Φ−α‖2 restricts to any closed GC-invariant subset of Rep(Q,v), and in Corollary 3.8we dealt with a special case of interest, namely, the level set ΦC

−1(0) ⊆ T ∗ Rep(Q,v). Sinceγ(x, t) restricts to ΦC

−1(0), the strata Sν also restrict to ΦC−1(0), and we have the following.

Definition 3.15. Let Q = (I, E), v ∈ ZI≥0, G, ΦR, and ΦC be as in Corollary 3.8. Then theanalytic (or Morse-theoretic) stratum of ΦC

−1(0) of α-type ν is defined as

Zν :={y ∈ ΦC

−1(0) | limt→∞

γ(y, t) ∈ Cν

}and is equal to Sν ∩ ΦC

−1(0). We call the decomposition

ΦC−1(0) =

⋃ν

Zν

the analytic (or Morse-theoretic) stratification of ΦC−1(0).

3.3. Distance decreasing formula along the flow. In this section we prove a distancedecreasing formula for the distance between the G-orbits of two solutions A1(t), A2(t) tothe gradient flow equations (3.1), whose initial conditions A1(0), A2(0) are related by anelement of GC (see Remark 3.24). This result is crucial for the proof of Proposition 4.39,which is in turn a key step toward the proof of one of our main theorems, Theorem 4.15,in Section 4. It should also be viewed as a quiver analogue of the distance decreasingformula in [6], for Hermitian metrics on a holomorphic vector bundle that vary accordingto the Yang-Mills flow, except that here we do our analysis directly on the groupGC ratherthan on the space of metrics as in [6].

Throughout, let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. GivenA(0) ∈ Rep(Q,v), let g = g(t) be a curve in GC that satisfies the following flow equationwith initial condition:

∂g

∂tg−1 = i (Φ(g(t) · A(0))− α) ,

g(0) = id .(3.14)

Equation (3.14) is an ODE, so solutions are unique and exist locally. We will show laterthat they exist for all time.

Remark 3.16. By definition, ∂g∂tg−1 is self-adjoint, so (g∗)−1 ∂g∗

∂t= ∂g

∂tg−1 wherever defined.

This will be used in the sequel.

Lemma 3.17. Let A(0) ∈ Rep(Q,v) as above and g(t) a curve satisfying (3.14). Let A(t) :=g(t) · A(0). Then the curve A(t) in Rep(Q,v) satisfies the negative gradient flow equation (3.2),i.e.

(3.15)∂A

∂t= IρA (Φ(A(t))− α) .


Proof. Differentiating each component of g(t) · A(0) gives us

(3.16)∂Aa

∂t=

∂

∂t

(g(t)Aa(0)g(t)−1

)=∂g

∂tAa(0)g(t)

−1 − g(t)Aa(0)g(t)−1∂g

∂tg(t)−1.

This can be rewritten as∂Aa

∂t=

[∂g

∂tg(t)−1, Aa(t)

]= [i (Φ(A)− α) , Aa(t)]

= IρA (Φ(A)− α) .

�

We now define a function which measures, in a sense, the distance between a positiveself-adjoint matrix and the identity matrix. This will be the key ingredient allowing us toanalyze the distance between G-orbits of different negative gradient flows in Section 4.For a positive self-adjoint h ∈ GC, let

(3.17) σ(h) := trh+ trh−1 − 2 rank(Q,v).

Note that σ(h) ≥ 0 for all h, with equality if and only if h = id.It is convenient to also define a shifted version of the moment map Φ. This is analogous

to Donaldson’s change of metric in [6].

Definition 3.18. Let h ∈ GC be self-adjoint. Then we define for A ∈ Rep(Q,v)

(3.18) Φh(A) := i∑a∈E

[Aa, hA

∗ah

−1].

In the special case when h = g−1(g∗)−1 for some g ∈ GC, a computation shows that

(3.19) gΦh(A)g−1 = Φ(g · A).

Now letA1(0) andA2(0) be two initial conditions related byGC, soA2(0) = g0·A1(0). Letg1(t) and g2(t) be the corresponding solutions to equation (3.14). Then the correspondingsolutions to the gradient flow equations areA1(t) = g1(t) ·A1(0) andA2(t) = g2(t) ·A2(0) =g2(t) · g0 · A1(0). Let g(t) = g2(t) · g0 · g1(t)

−1 be the element of GC that connects the twoflows, i.e. g(t) · A1(t) = A2(t), and let h(t) := g(t)−1 (g(t)∗)−1.

The following is the crucial result of this section; it states that the discrepancy betweenh(t) and the identity matrix, measured by the function σ in (3.17), is non-increasing alongthe flow.

Theorem 3.19. Let h(t) be as above. Then ∂∂tσ(h(t)) ≤ 0.

Proof. Differentiating g = g2g0g−11 and h = g−1(g∗)−1 yields

(3.20)∂g

∂tg−1 =

∂g2

∂tg−12 − g

(∂g1

∂tg−11

)g−1

and∂h

∂t= −g−1

(∂g2

∂tg−12

)(g∗)−1 +

(∂g1

∂tg−11

)g−1(g∗)−1

− g−1

((g∗2)

−1∂g∗2

∂t

)(g∗)−1 + g−1(g∗)−1

((g∗1)

−1∂g∗1

∂t

).

(3.21)


The observation in Remark 3.16, (3.14), and (3.21) in turn yields

∂h

∂t= −2g−1i (Φ(A2(t))− α) gh(t) + i (Φ(A1(t))− α)h(t) + h(t)i (Φ(A1(t))− α)

= −2i(Φh(t)(A1(t))− α

)h(t) + i (Φ(A1(t))− α)h(t) + h(t)i (Φ(A1(t))− α) .

(3.22)

Furthermore, recalling the following formula in gC

(3.23) [u, vw] = v[u,w] + [u, v]w

which in particular implies [u, v−1] = −v−1[u, v]v−1 and [u, v] = −v[u, v−1]v for v ∈ gCinvertible, we have

iΦh(A)− iΦ(A) = −∑a∈E

([Aa, hA

∗ah

−1]− [Aa, A∗a])

= −∑a∈E

([Aa, hA

∗ah

−1]− [Aa, hh−1A∗a]

)= −

∑a∈E

[Aa, h[A

∗a, h

−1]]

= −∑a∈E

(h[Aa, [A

∗a, h

−1]]+ [Aa, h][A

∗a, h

−1])

= −h∑a∈E

([Aa, [A

∗a, h

−1]]− [Aa, h

−1]h[A∗a, h−1]).

(3.24)

A similar computation yields

(3.25) i (Φh(A)− Φ(A)) =∑a∈E

([Aa, [A

∗a, h]]− [A∗a, h]h

−1[Aa, h])h−1.

Differentiating σ(h(t)) gives us

(3.26)∂

∂tσ(h(t)) = tr

∂h

∂t− tr

(h−1∂h

∂th−1

).

Equation (3.22) then shows that

(3.27) tr∂h

∂t= −2i tr

((Φh(t)(A1(t))− Φ(A1(t))

)h(t)

)and

(3.28) − tr

(h−1∂h

∂th−1

)= 2i tr

(h(t)−1

(Φh(t)(A1(t))− Φ(A1(t))

)).


Combined with equation (3.25) we see that

tr∂h

∂t= −2

∑a∈E

tr([Aa, [A

∗a, h]]− [A∗a, h]h

−1[Aa, h])

= 2∑a∈E

tr ([A∗a, h]g∗g[Aa, h])

= −2∑a∈E

tr ((g[Aa, h])∗g[Aa, h])

≤ 0

Similarly, equation (3.24) shows that

− trh−1∂h

∂th−1 = −2

∑a∈E

tr([Aa, [A

∗a, h

−1]]− [Aa, h

−1]h[A∗a, h−1])

= 2∑a∈E

tr([Aa, h

−1]g−1(g∗)−1[A∗a, h−1])

= −2∑a∈E

tr((

(g∗)−1[A∗a, h−1])∗

(g∗)−1[A∗a, Bh−1])

≤ 0.

The result follows.�

We next show that a bound on the value of σ yields a bound on the distance to theidentity matrix.

Proposition 3.20. Let g ∈ GC and h = g−1(g∗)−1. For each ε > 0 there exists δ such that ifσ(h) < δ then

(3.29) ‖h− id ‖+ ‖h−1 − id ‖ < ε.

Proof. Since h = g−1(g∗)−1, h is positive, and unitarily diagonalizable. Let hd := u∗hu bethe diagonalisation of h by a unitary matrix u (with eigenvalues nonincreasing). Sinceboth σ and the inner product are invariant under unitary conjugation, then σ(h) = σ(hd),‖hd− id ‖ = ‖h− id ‖ and ‖h−1

d − id ‖ = ‖h−1− id ‖. Hence the problem reduces to studyingdiagonal, positive, self-adjoint matrices.

Let λ1, . . . , λn be the eigenvalues of h. Then

‖hd − id ‖ =

(n∑

j=1

(λj − 1)2

) 12

≤n∑

j=1

|λj − 1| ,

and similarly

(3.30) ‖h−1d − id ‖ ≤

n∑j=1

∣∣λ−1j − 1

∣∣ .


Let δ > 0 such that if λj + λ−1j − 2 < δ then |λj − 1|+ |λ−1

j − 1| < 1nε. If σ(h) < δ, from the

definition of σ it follows that λj + λ−1j − 2 < δ for each j and we obtain

(3.31) ‖hd − id ‖+ ‖h−1d − id ‖ ≤

n∑j=1

|λj − 1|+∣∣λ−1

j − 1∣∣ < ε.

The result follows. �

For any g ∈ GC, the next result shows that if σ(g−1(g∗)−1) is small, then there existssome g in the G-orbit of g that is close to the identity. Let h = g−1(g∗)−1. Then as above,h is positive and unitarily diagonalizable; let hd = u∗hu for u unitary and hd diagonal.Since the eigenvalues are positive and real, there exists a well-defined positive squareroot, denoted

√hd. Let g := u(

√hd)

−1u∗. We have the following.

Lemma 3.21. Let g ∈ GC and g as above. Then g ∈ G · g. Moreover, for any ε > 0 there exists aδ > 0 such that if σ(g−1(g∗)−1) < δ then ‖g − id ‖ < ε.

Proof. A calculation shows that g−1(g∗)−1 = h. Since g−1(g∗)−1 = h = g−1(g∗)−1, then wehave

gg−1(g∗)−1g∗ = id → (gg−1)−1 = (gg−1)∗,

and so gu := gg−1 is unitary and g ∈ G·g. Therefore, it only remains to show that ‖g−id ‖ <ε if σ(h) is small enough.

Observe g is self-adjoint and positive, with eigenvalues λ1, . . . , λn equal to those of(√hd)

−1. Therefore

(3.32) ‖g − id ‖ = ‖u−1gu− id ‖ = ‖√hd

−1− id ‖ =

(n∑

j=1

(λj − 1)2

) 12

≤n∑

j=1

∣∣∣λj − 1∣∣∣ .

Since each λj is positive, then∣∣∣λj − 1

∣∣∣ ≤ ∣∣∣λ2j − 1

∣∣∣ and we have

(3.33) ‖g − id ‖ ≤n∑

j=1

∣∣∣λ2j − 1

∣∣∣where λ2

j is the jth eigenvalue of h−1. Now the same argument as in Proposition 3.20shows that for any given ε > 0 there exists a δ > 0 such that if σ(h−1) = σ(h) < δ then∑n

j=1

∣∣∣λ2j − 1

∣∣∣ < ε. The result follows. �

We may now show that the solutions to (3.14) exist for all time.

Corollary 3.22. For any initial condition A(0) ∈ Rep(Q,v), the solution to equation (3.14)exists for all t.

Proof. Let A(0) ∈ Rep(Q,v), and let g1(t) denote the associated solution to (3.14). Localexistence for ODEs shows that g1(t) exists for t ∈ [0, T ) for some T > 0, so it remains toshow that the solution can be extended past t = T .

For ε > 0, let δ > 0 be as in Lemma 3.21. Since the solution g1(t) is continuous, there ex-ists t0 ∈ [0, T ) with σ (g1(t0)

∗g1(t0)) < δ. Let g2(t) be the solution to (3.14) with initial con-dition g1(t0) ·A0. This is just a translation of the first solution, so g2(t) = g1(t+ t0)g1(t0)

−1,and the problem reduces to extending the solution for g2(t) past T − t0.


Since g1(0), g2(0) are related by an element of GC, then as in the above analysis we maydefine g(t) so that g(t) = g2(t)g1(t0)g1(t)

−1, and by Theorem 3.19, σ(g(t)−1(g(t)∗)−1) ≤σ(g1(t0)

−1(g1(t0)∗)−1) = σ(g1(t0)

∗g1(t0)) < δ for t ∈ [0, T − t0). Here we use that σ(h) =σ(h−1) by definition of σ for any positive-definite h. Then Lemma 3.21 shows that for anyt ∈ [0, T − t0), the G-orbit of g(t) under left multiplication intersects a compact set in GC.Since G itself is compact, we may conclude g(t) remains in a compact set. Therefore wecan extend the solution for g(t) and hence g2(t) past T − t0, as desired.

�

Finally, we show that estimates on σ(h) = σ(g−1(g∗)−1) yields distance estimates be-tween moment map values of g · A and g−1

u · A. Such an estimate is crucial in the nextsection.

Proposition 3.23. Let g ∈ GC, h = g−1(g∗)−1, and gu as above. Suppose A ∈ Rep(Q,v). Thenfor every ε > 0 there exists δ > 0 such that if σ(h) < δ then

(3.34) ‖Φ(g · A)− Φ(g−1u · A)‖ < ε.

Proof. Since Φ is G-equivariant and g = g−1u g,

Φ(g · A) = g−1u Φ(g · A)gu and Φ(gu · A) = g−1

u Φ(A)gu.

Moreover, since the inner product is invariant under the conjugate action of G, it is suffi-cient to find a bound on ‖Φ(g ·A)−Φ(A)‖, or equivalently ‖gΦh(A)g−1−Φ(A)‖, by (3.19).First, using Proposition 3.20 and Lemma 3.21, let δ > 0 be such that if σ(h) < δ then g− idand h are both bounded in norm by some constant. We have then

‖gΦh(A)g−1 − Φh(A)‖ = ‖g[Φh(A), g−1

]‖

= ‖g[Φh(A), g−1 − id

]‖

≤ 2 (‖g − id ‖+ ‖ id ‖) ‖Φh(A)‖‖g−1 − id ‖≤ C1‖g−1 − id ‖

for some constant C1 > 0. A similar computation using (3.24) yields

‖Φh(A)− Φ(A)‖ ≤ C2‖h−1 − id ‖for a constant C2 > 0. Hence the triangle inequality yields

‖Φ(g · A)− Φ(A)‖ ≤ C1‖g−1 − id ‖+ C2‖h−1 − id ‖.Again using Proposition 3.20 and 3.21 and that σ(h) = σ(h−1) for any positive-definite h,by possibly shrinking δ we conclude that if σ(h) < δ then

(3.35) C1‖g−1 − id ‖+ C2‖h−1 − id ‖ < ε,

as desired. �

Remark 3.24. In the notation of Theorem 3.19, given initial conditionsA1(0),A2(0) = g0·A1,solutions A1(t), A2(t) to the gradient flow with initial conditions A1(0), A2(0) respectively,and g(t) ∈ GC the group action that connects the two flows, let gu(t) ∈ G be the unitaryelement associated to g(t) as above. Then Theorem 3.19 and Proposition 3.23 togetherimply that for any ε > 0 there exists δ > 0 such that if σ(g∗0g0) < δ, then ‖Φ(A2(t)) −Φ(gu(t) · A1(t))‖ < ε for all t. In other words, the G-orbits of the two solutions remainclose. This fact is crucial for the results of the next section.


4. THE HARDER-NARASIMHAN STRATIFICATION

In this section, we relate the Morse-theoretic stratification of Rep(Q,v) obtained in Sec-tion 3 to the Harder-Narasimhan stratification (recalled below) of Rep(Q,v) with respectto the same parameter α. This latter stratification is defined in terms of slope stabilityconditions similar to the case of holomorphic bundles; in this sense, the content of thissection is to exhibit a tight relationship between the algebraic-geometric description ofthis stratification (via stability) and the analytic description (via gradient flow). We re-strict the discussion to the unframed case Rep(Q,v), which (see Remarks 2.5 and 2.6)covers all other cases of interest.

To set our notation, we first briefly recall in Section 4.1 some preliminary definitions.

4.1. Slope and stability for representations of quivers. Let Q = (I, E) be a quiver withspecified dimension vector v ∈ ZI≥0 and α ∈ (iR)I a choice of parameter. In order todefine the slope stability condition with respect to the parameter α, it will be necessary tocompare the α-slope (as in Definition 3.9) of a representation A ∈ Rep(Q,v) with that ofits invariant subspaces. We make this notion more precise below.

Using the notion of α-degree and α-slope in Definition 3.9, we may define the followingslope stability condition for quivers with respect to α (see [24, Definition 2.1]).

Definition 4.1. A representationA ∈ Rep(Q,v) is called α-stable (respectively α-semistable)if for every proper subrepresentation A′ ∈ Rep(Q,v′) of A we have

(4.1) µα(Q,v′) < µα(Q,v) ( respectively µα(Q,v′) ≤ µα(Q,v)) .

We denote by Rep(Q,v)α−st (respectively Rep(Q,v)α−ss) the subset in Rep(Q,v) of α-stable (respectively α-semistable) representations.

We now briefly recall the notion of stability arising from geometric invariant theory.Assume now that the parameter α = (α`)`∈I associated to the quiver Q is integral, i.e.satisfies iα` ∈ Z for all ` ∈ I. Let χα : GC =

∏`∈I GL(V`) → C be the character of GC given

byχα(g) :=

∏`∈I

det(g`)iα` .

Using χα, we define a lift of the action of GC from Rep(Q,v) to the trivial complex linebundle L := Rep(Q,v)× C by

(4.2) g ·((Aa)a∈E , ξ

):=((gin(a)Aag

−1out(a)

)a∈E

, χα(g)ξ).

Definition 4.2 ([13]). Let A = (Aa)a∈E ∈ Rep(Q,v). Then we say A is χα-semistable if forany ξ ∈ C \ {0}, the GC-orbit closure GC · (x, ξ) in L is disjoint from the zero section of L.We say A is χα-stable if the orbit GC · (x, ξ) is closed.

The main result of [13] is that when α is integral, the χα-(semi)stability condition aboveis equivalent to the α-(semi)stability conditions of Definition 4.1 when α is integral. Inthis manuscript, however, we wish to also analyze Hamiltonian quotients of Rep(Q,v),so we now recall the relationship between the above α-stability conditions and momentmap level sets. The following lemma appears in [22], but we provide here a differentproof, which comes from our results in Section 3 on the gradient flow of ‖ΦR−α‖2 on thespace Rep(Q,v,w).


Lemma 4.3. ([22, Corollary 3.22]) Let Q = (I, E) be a quiver with specified dimension vectorv ∈ ZI≥0. Let α be an integral central parameter. If a representation A ∈ Rep(Q,v) is α-stablethen there exists g ∈ GC such that Φ(g ·A) = α. If a representationA ∈ Rep(Q,v) is α-semistablethen the orbit closure GC · A has non-trivial intersection with Φ−1(α).

Proof. The results of the previous section show that for any initial conditionA ∈ Rep(Q,v),the negative gradient flow γ(A, t) of ‖Φ−α‖2 is contained in a compact set, and generatedby the action of a path g(t) ∈ GC. In this situation, if A is χα-semistable then for any ξ 6= 0,ξ(t) = χα(g(t))ξ is bounded away from zero (note that this is only true if the gradient flowis contained in a compact set). For (A, ξ) ∈ L, define

F (A, ξ) =1

2‖A‖2 +

1

2log ‖ξ‖2.

(This F is the analogue of the Donaldson functional in this situation.) A calculation showsthat ∂F

∂t= −‖Φ− α‖2, so the lemma follows from the fact that if A is semistable, then ξ(t)

is bounded away from zero along the gradient flow, and hence F is bounded below. �

Having established the connection with GIT, we now turn our attention to the definitionof the Harder-Narasimhan stratification of Rep(Q,v) with respect to α. The constructionuses the definition of slope stability in Definition 4.1. We first need some preliminaries.

Definition 4.4. Let Q = (I, E) be a quiver. Suppose A ∈ Rep(Q,v) is a representationof Q with associated hermitian vector spaces {V`}`∈I , and similarly A′ ∈ Rep(Q,v′) with{V ′

` }`∈I . We say a collection of linear homomorphisms ψ` : V` → V ′` is a homomorphism

of representations of quivers if ψ` intertwines the actions of A and A′, i.e. for all a ∈ E,ψin(a)Aa = A′aψout(a).

We may also define a quotient representation in the standard manner.

Definition 4.5. Let A ∈ Rep(Q,v) be a representation of a quiver Q = (I, E) with associ-ated hermitian vector spaces {V`}`∈I , and let A′ ∈ Rep(Q,v′) be a subrepresentation of Awith {V ′

` }`∈I . Then the quotient representation A = A/A′ ∈ Rep(Q,v − v′) is defined tobe the collection of linear maps on the quotient vector spaces {V`/V

′` }`∈I induced by the

Aa, i.e.Aa : Vout(a)/V

′out(a) → Vin(a)/V

′in(a).

This is well-defined since A preserves the V ′` .

Using these definitions we can make sense of an exact sequence of representations ofquivers. We will use the usual notation

0 // A // B // C // 0

for A ∈ Rep(Q,v′), B ∈ Rep(Q,v), C ∈ Rep(Q,v′′) where A → B is an inclusion withimage a subrepresentation of B and C ∼= B/A. The following lemma appears in [24], andgives a quiver analogue of well-known results for the case of holomorphic bundles.

Lemma 4.6. ([24, Lemma 2.2]) Let Q = (I, E) be a quiver. Suppose that

0 // A // B // C // 0

is a short exact sequence of representations of the quiverQ, withA ∈ Rep(Q,v′), B ∈ Rep(Q,v), C ∈Rep(Q,v′′). Then

µα(Q,v′) ≤ µα(Q,v) ⇔ µα(Q,v) ≤ µα(Q,v′′)


andµα(Q,v′) ≥ µα(Q,v) ⇔ µα(Q,v) ≥ µα(Q,v′′),

withµα(Q,v′) = µα(Q,v) ⇔ µα(Q,v) = µα(Q,v′′).

Similarly, the following Proposition is contained in [24, Proposition 2.5]. (See also [15,V.1.13, V.7.17] for a discussion in the case of holomorphic bundles.)

Proposition 4.7. ([24, Proposition 2.5]) Let Q = (I, E) be a quiver and A ∈ Rep(Q,v) for achoice of dimension vector v ∈ ZI≥0. Let α = (α`)`∈I be a central parameter. Then there exists aunique sub-representation A′ ∈ Rep(Q,v′) such that

(1) µα(Q, v) ≤ µα(Q,v′) for all proper sub-representations A ∈ Rep(Q, v) of A, and(2) if A is a proper subrepresentation of A with µα(Q, v) = µα(Q,v), then rank(Q, v) <

rank(Q,v′).Such an A′ ∈ Rep(Q,v′) is called the maximal α-semistable subrepresentation of A ∈Rep(Q,v).

With Proposition 4.7 in hand, the proof of the following is also standard (see also [15,V.1.13, V.7.15] for the case of holomorphic bundles).

Theorem 4.8. (The Harder-Narasimhan filtration, cf. [24, Prop 2.5]) Let Q = (I, E) be a quiverand A ∈ Rep(Q,v) for a choice of dimension vector v ∈ ZI≥0. Let α = (α`)`∈I be a centralparameter. Then there exists a canonical filtration of A by subrepresentations

(4.3) 0 = A0 ( A1 ( A2 . . . ( AL = A,

where Aj ∈ Rep(Q,vj), and Aj/Aj−1 is the maximal α-semistable subrepresentation of A/Aj−1

for all 1 ≤ j ≤ L.

We will call the length L of this sequence (4.3) the Harder-Narasimhan α-length of A.In particular, an α-semistable representation has Harder-Narasimhan α-length 1. We willoften abbreviate “Harder-Narasimhan” as “H-N”.

From the H-N filtration we may read off the following parameters.

Definition 4.9. The Harder-Narasimhan (H-N) type of the filtration (4.3) is the orderedrank(Q,v)-tuple(4.4)(µα(Q,v1), . . . , µα(Q,v1), µα(Q,v2), . . . , µα(Q,v2), . . . , µα(Q,vL), . . . , µα(Q,vL)) ∈ Rrank(Q,v),

where there are rank(Q,vj) terms equal to µα(Q,vj) for all 1 ≤ j ≤ L. The Harder-Narasimhan (H-N) type of a representation of a quiver with respect to a parameter αis the H-N type of its Harder-Narasimhan filtration.

Definition 4.10. Given a fixed parameter α, let Rep(Q,v)ν ⊆ Rep(Q,v) denote the subsetof representations of H-N type ν ∈ Rrank(Q,v) with respect to α. We call this the Harder-Narasimhan (H-N) stratum of Rep(Q,v) of H-N type ν. The Harder-Narasimhan strat-ification is the decomposition of Rep(Q,v) indexed by the set of all Harder-Narasimhantypes:

(4.5) Rep(Q,v) =⋃ν

Rep(Q,v)ν .


In [24, Definition 3.6], Reineke describes a partial ordering on the set of H-N typesanalogous to the H-N partial ordering in [1], and [24, Proposition 3.7] shows that theclosures of the subsets behave well with respect to this partial order, i.e.

(4.6) Rep(Q,v)ν ⊂⋃

ν′≥ν

Rep(Q,v)ν′ .

Moreover, this decomposition isGC-invariant under the natural action ofGC on Rep(Q,v).

Lemma 4.11. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let α =(α`)`∈I be a stability parameter. Let Rep(Q,v)ν be the H-N stratum of Rep(Q,v) of H-N typeν ∈ Rrank(Q,v). Then Rep(Q,v)ν is invariant under the natural GC =

∏`∈I GL(V`)-action on

Rep(Q,v).

Proof. Let g ∈ GC, A ∈ Rep(Q,v), and 0 = A0 ( A1 ( · · · ( AL = A the canonicalHarder-Narasimhan filtration of A. To prove the claim, it would suffice to show that

(4.7) 0 = g · A0 ( g · A1 ( · · · ( g · AL = g · Ais the canonical Harder-Narasimhan filtration of g · A. Since g acts by conjugation, g · Aj

is also a subrepresentation of g · A, and since conjugation preserves dimension vectors,the α-slope of the subrepresentation g ·A1 is equal to that of A1. Furthermore, any propersubrepresentation B ( g · A corresponds to a proper subrepresentation of g−1B ( A,where again both the α-slopes and the ranks ofB and g−1·B are equal. From the definitionof maximal α-semistability, it follows that g ·A1 ( g ·A is the unique maximal α-semistablesubrepresentation of g · A. Proceeding similarly, we conclude that (4.7) is the canonicalHarder-Narasimhan filtration of g · A, as desired. �

We close this section with two observations. The first concerns an induced stratificationassociated to the level set ΦC

−1(0) ⊆ T ∗ Rep(Q,v) appearing in the hyperkahler quotientconstruction of Nakajima quiver varieties. The second concerns a convenient choice ofstability parameter, with respect to which computations with Harder-Narasimhan stratacan be simplified; we discuss further some consequences of such a convenient choice inSection 7.1.

First, recall that when constructing Nakajima quiver varieties, we start with T ∗ Rep(Q,v)for a quiver Q and dimension vector v, and take a level set Φ−1(α) ∩ Φ−1

C(0). Viewing

T ∗ Rep(Q,v) as Rep(Q,v) as in Remark 2.6, we may also stratify T ∗ Rep(Q,v) into H-Nstrata with respect to α. Let Φ−1

C(0)ν := Φ−1

C(0) ∩ T ∗ Rep(Q,v)ν be the intersection of

ΦC−1(0) with the H-N stratum of H-N type ν. Then we call the decomposition

(4.8) ΦC−1(0) =

⋃ν

ΦC−1(0)ν

the H-N stratification of ΦC−1(0), and since ΦC

−1(0) is closed in T ∗ Rep(Q,v), the closureproperty (4.6) with respect to the partial ordering is still satisfied, i.e.

(4.9) ΦC−1(αC)ν ⊆

⋃ν′≥ν

ΦC−1(αC)ν′ .

The point here is that the H-N stratification of T ∗ Rep(Q,v) induces one on ΦC−1(0).

Secondly, as mentioned in the introduction, one of the main motivations for the Morse-theoretic results of the present manuscript is the study of the (ordinary or equivariant)


cohomology rings of the Kahler and hyperkahler quotients of Rep(Q,v) and T ∗ Rep(Q,v),respectively. It turns out that for the cohomology computations, it is often useful to choosea stability parameter α with respect to which the H-N α-length of the H-N strata arealways less than or equal to 2. This greatly simplifies, for example, explicit computationsof Morse indices at the critical sets. Moreover, we may additionally choose α such thaton Rep(Q,v), α-semistability is equivalent to α-stability; this is also convenient since itimplies that the resulting GIT quotient is smooth. We have the following.

Definition 4.12. A triple (Q,v, α), consisting of a quiver Q, associated dimension vectorv, and a choice of stability parameter α ∈ Z(g), is called 2-filtered if for any representationA ∈ Rep(Q,v) its associated canonical H-N filtration has H-N α-length less than or equalto 2.

The following proposition shows that for a large class of quivers (Q,v), there exists achoice of stability parameter α such that (Q,v, α) is 2-filtered.

Proposition 4.13. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Supposethat there exists ` ∈ I with v` = dim(V`) = 1. Then there exists a choice of stability parameterα ∈ Z(g) such that

• (Q,v, α) is 2-filtered, and• Rep(Q,v)α−ss = Rep(Q,v)α−st.

Remark 4.14. As already noted in Section 2.2, we have restricted attention in our expositionto the unframed quotients Rep(Q,v)//G because any framed quotient Rep(Q,v,w)//G forQ = (I, E) can be realized, by the construction by Crawley-Boevey in [2, p. 261] (seealso Remark 2.5), as an unframed quotient associated to a different quiver Q′ = (I ′, E ′).We recall (see [2] for details) that the construction involves an addition of a “vertex atinfinity”, i.e. I ′ = I ∪ {∞}, with v∞ = dimC(V∞) = 1. Hence the hypothesis of Propo-sition 4.13 is always satisfied for an unframed quiver representation space associated viathis Crawley-Boevey trick to a framed one.

Proof of Proposition 4.13. Let I = {1, 2, . . . , n,∞} denote the vertices of the quiver, where∞ ∈ I denotes a vertex with v∞ = dimC(V∞) = 1.

The proof is by explicit construction. Let α ∈ iR be a pure imaginary parameter withiα < 0. Define α by

(4.10) α` =

{α, for 1 ≤ ` ≤ n,−α (

∑n`=1 v`) , for ` = ∞.

Let A ∈ Rep(Q,v). It is straightforward from the definition of α-slope that the maximalα-semistable subrepresentation A′ ∈ Rep(Q,v′) with associated hermitian vector spaces{V ′

` ⊆ V`}`∈I is the maximal subrepresentation of A with V ′∞ = V∞. Now consider the

quotient representation A/A′, with associated dimension vector v′′. As just seen, v′′∞ = 0.Since the α-parameter is constant on all other vertices ` 6= ∞, the α-slope of any subrep-resentation of A/A′ is equal to the α-slope µα(Q,v′′) of A/A′. This implies A/A′ is alreadyα-semistable, and we conclude thatA has Harder-Narasimhan α-length less than or equalto 2, as desired.

To prove the last claim, we must show that if A ∈ Rep(Q,v) is α-semistable, then thereare no proper subrepresentations A′ ∈ Rep(Q,v′) of A with µα(Q,v′) = µα(Q,v). We firstobserve from the definition of α that µα(Q,v) = 0. Now we take cases: if dimC(V

′∞) =


v′∞ = 0, then µα(Q,v′) = iα < 0, hence µα(Q,v′) < µα(Q,v). On the other hand, ifdimC(V

′∞) = v′∞ = 1, and if A′ is a proper subrepresentation, then there exists ` ∈ I

with v′` < v`, so in particular∑n

`=1 v′` <

∑n`=1 v`. From this it follows that µα(Q,v′) <

µα(Q,v). Hence α-semistability implies α-stability. The reverse implication follows fromthe definitions, so we conclude Rep(Q,v)α−ss = Rep(Q,v)α−st, as desired. �

4.2. Comparison of stratifications. The main result of this section is that the analyticMorse stratification of Rep(Q,v) obtained in Section 3 and the H-N stratification obtainedin Section 4.1 are equivalent. In this sense, there is a tight relationship between the analy-sis and the algebraic geometry; in particular, we may use either perspective for any givenproblem, as is convenient.

Theorem 4.15. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0, Rep(Q,v)its associated representation space, and Φ : Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I U(V`) a moment

map for the standard Hamiltonian action of G =∏

`∈I U(V`) on Rep(Q,v). Then the algebraicstratification of Rep(Q,v) by Harder-Narasimhan type (as in Definition 4.10) coincides with theanalytic Morse stratification of Rep(Q,v) by the negative gradient flow of f = ‖Φ − α‖2 (as inDefinition 3.13).

The first step in the proof of Theorem 4.15 is to prove some background results aboutthe relationship between convex invariant functions on Lie algebras and the moment mapΦ : Rep(Q,v) → g∗ (see also [1, Section 12]). These will allow us to to relate the α-type ofthe canonical splitting at a critical pointA of f = ‖Φ−α‖2 to the Harder-Narasimhan typeof A, thus extending to the case of quivers the results of Atiyah and Bott in [1, Section8] for the Yang-Mills functional. For notation, in the following let H = U(Vect(Q,v))denote the full unitary group of Vect(Q,v), h = u(Vect(Q,v)) its Lie algebra, and hC =gl(Vect(Q,v),C) = h⊗R C its complexification.

Definition 4.16. A convex invariant function is a smooth function p : h → R satisfying thefollowing conditions:

(1) p is invariant under the adjoint action of H , and(2) for any straight line β : (−ε, ε) → h, we have d2

dt2p(β(t)) ≥ 0 for all t ∈ (−ε, ε).

Remark 4.17. (1) The prime example of a convex invariant function on h is the norm-square function ‖ · ‖2 : h → R with respect to an Ad-invariant metric. However,the norm-square function alone does not yield fine enough information for ouranalysis of the H-N stratification; this is why we must introduce general convexinvariant functions (see e.g. Lemma 4.24).

(2) Unless stated otherwise, we always normalize convex invariant functions so thatp(0) = 0.

(3) Though we state the definition of convex invariant functions in terms of the Liealgebra h = u(Vect(Q,v)), it is evident that the same definition can be given forany Lie algebra of a compact Lie group. Moreover, if p : h → R is convex invariantandG ⊆ H is a Lie subgroup, then the restriction of p to g ⊆ h is naturally a convexG-invariant function on g. In our specific setting, we may hence obtain convexinvariant functions associated to the subgroup G =

∏`∈I U(V`) ⊆ U(Vect(Q,v)).

We will use this many times.


For technical reasons to be explained below, we also wish to study convex functionson the complexification hC that are invariant under the adjoint action of H (extended C-linearly to hC).

Definition 4.18. Let p : h → R be a convex invariant function. Then the complexification ofp, denoted pC : hC → R, is defined by pC(u+ iv) := p(u) + p(v) for all u, v ∈ h.

The defining properties of a convex invariant function hold also for its complexification.

Lemma 4.19. Let pC : hC → R be the complexification of a convex invariant function p : h → R.Then

(1) pC(Adg(u) + iAdg(v)) = pC(u+ iv) for all g ∈ H and u, v ∈ h.(2) For any straight line βC : (−ε, ε) → hC, we have d2

dt2pC(β(t)) ≥ 0.

Proof. The first assertion follows from the computation

pC(Adg(u) + iAdg(v)) = p(Adg(u)) + p(Adg(v)) = p(u) + p(v) = pC(u+ iv).

For the second, let βC(t) = β1(t) + iβ2(t). Since both β1, β2 are straight lines in h, wehave d2

dt2pC(βC(t)) = d2

dt2p(β1(t)) + d2

dt2p(β2(t)) ≥ 0, since both terms are indiviually non-

negative. �

We mainly use these convex invariant functions by composing them with the momentmap. Indeed, given a convex invariant function p : h → R, we may define an associatedfunction P : Rep(Q,v) → R by

(4.11) P (A) := p ◦ ι ◦ (Φ(A)− α),

i.e. precomposing with the shifted moment map Φ−α : Rep(Q,v) → g ⊆ h and the linearinclusion ι : g =

∏`∈I u(V`) ↪→ h. Hence the norm-square of the moment map f = ‖Φ−α‖2

is a special case, and will play a distinguished role throughout. On occasion, for technicalreasons it is necessary to consider also the complexification pC of p and think of P as thecomposition of Φ− α with the restriction to h ⊆ hC = h⊕ ih of pC : hC → R, but wheneverpossible we will suppress explicit use of pC.

Since P is a composition involving p, the local data of P near a point A ∈ Rep(Q,v) canbe related to that of p. We begin with a computation of gradients. LetA ∈ Rep(Q,v), recallthat we use δA to denote a tangent vector in TA Rep(Q,v) ∼= Rep(Q,v), and let πg : h → gdenote the orthogonal projection to g with respect to the standard invariant inner producton h = u(Vect(Q,v)). We have

gRep(Q,v) (gradP (A), δA) = dPA(δA) by definition of gradient= (dp)Φ(A)−α ◦ dιΦ(A)−α ◦ dΦA(δA) by definition of P

=⟨grad p(Φ(A)− α), dιΦ(A)−α ◦ dΦA(δA)

⟩h

by definition of gradient

= 〈πg grad p(Φ(A)− α), dΦA(δA)〉g since ι is a linear inclusion

= 〈πg grad p(Φ(A)− α), ρ∗AI(δA)〉g by Lemma 2.1

= gRep(Q,v) (−IρA(πg grad p(Φ(A− α))), δA)

by definition of ρ∗A, I-invariance of 〈, 〉

(4.12)


for any δA ∈ TA Rep(Q,v), so we conclude

(4.13) gradP (A) = −IρA(πg grad p(Φ(A)− α)).

In the special case f(A) = ‖Φ(A)− α‖2, the equation (4.13) becomes equation (3.2).Since Rep(Q,v) is an affine space, its tangent bundle is trivial, and we may view the gra-

dient vector field of (4.13) as a function Rep(Q,v) → Rep(Q,v) (indeed, the formula (4.13)already makes this identification implicitly). Taking yet another derivative and againidentifying tangent spaces with Rep(Q,v), we obtain at each A ∈ Rep(Q,v) the “secondderivative of P” at A, denoted HP (A) : Rep(Q,v) → Rep(Q,v). We use the same nota-tion for the entirely analogous second derivative Hp(Φ(A)−α) : h → h of p at the pointΦ(A)− α ∈ g.

A computation yields

HP (A)(δA) = −IδρA(πg grad p(Φ(A)− α))(δA)− IρA

(πgHp(Φ(A)−α)(dΦA(δA))

)= −IδρA(πg grad p(Φ(A)− α))(δA)− IρA

(πgHp(Φ(A)−α)(ρ

∗AIδA)

).

(4.14)

The linear map HP (A) : Rep(Q,v) → Rep(Q,v) is the derivative of the gradient vectorfield, and so at a point A ∈ Rep(Q,v) where gradP (A) = 0, HP (A) is exactly the Hessianof P at A. Condition (2) in Lemma 4.19 guarantees that for a convex invariant function,HpC(u) is non-negative definite. A computation shows that the complexification pC onhC = h⊕ ih satisfies

(4.15) grad pC(u+ iv) = grad p(u) + i grad p(v),

and

(4.16) HpC(u+iv)(w1 + iw2) = Hp(u)(w1) + iHp(v)(w2),

for all u, v, w1, w2 ∈ h. In particular, HpC(u+iv) is also non-negative-definite.As mentioned above, we introduce general convex invariant functions in order to bet-

ter analyze the relationship between the Morse strata of f = ‖Φ − α‖2 and the Harder-Narasimhan strata with respect to α. The first step is Lemma 4.22, which compares criticalpoints of f to that of a general P . We first need some preliminary computations.

Lemma 4.20. Let p : h → R be a convex invariant function. Then(1) grad p(Adg(u)) = Adg(grad p(u)), for all g ∈ H, u ∈ h,(2) Hp(u)([u, v]) = [grad p(u), v] , for all u, v ∈ h,

(3)[Hp(u)(v), w

]= Hp(u)([v, w]) + d

dt

∣∣t=0

Hp(u+tv)([u,w]), for all u, v, w ∈ h.

Proof. (1) Since p is H-invariant under the adjoint action, p(Adg(u)) = p(u) for all g ∈H, u ∈ h. Differentiating this identity with respect to the variable u, we have for anyg ∈ H and any u and v in h,

dpAdg(u)(Adg(v)) = dpu(v)

where we use that Adg : h → h is linear, and identify Tuh ∼= TAdg(u)h ∼= h. By definition ofgradients, we conclude

〈grad p(Adg(u)),Adg(v)〉 = 〈grad p(u), v〉

for all g ∈ H, u, v ∈ h. Replacing v with Adg−1(v) and noting that the inner product isAd-invariant, we obtain the desired result.


(2) Differentiate the previous formula with respect to the group variable g ∈ H. Letv ∈ h and {exp(−tv)} a 1-parameter subgroup of H . Then

d

dt

∣∣∣∣t=0

grad p(Adexp(−tv)(u)) =d

dt

∣∣∣∣t=0

Adexp(−tv)(grad p(u))

⇒ Hp(u)([u, v]) = [grad p(u), v]

for all u, v ∈ h, as desired.(3) Returning again to (1), we now differentiate with respect to both the variables u ∈ h

and g ∈ H. Let v, w ∈ h. For s, t ∈ R real parameters, the formula (1) gives us

grad p(Adexp(−sw)(u+ tv)) = Adexp(−sw)(grad p(u+ tv)).

We compute the derivative with respect to s and t of the left hand side first. We have

d2

dtds

∣∣∣∣s,t=0

grad p(Adexp(−sw)(u+ tv) =d

dt

∣∣∣∣t=0

Hp(u+tv)([−w, u+ tv])

= Hp(u)([v, w]) +d

dt

∣∣∣∣t=0

Hp(u+tv)([u,w]).

On the other hand, the right hand side is

d2

dsdt

∣∣∣∣s,t=0

Adexp(−sw)(grad p(u+ tv)) =d

ds

∣∣∣∣s=0

Adexp(−sw)(Hp(u)(v))

= [−w,Hp(u)(v)]

= [Hp(u)(v), w],

giving the desired result. �

From formulas (4.15) and (4.16) we also derive the following.

Lemma 4.21. Let pC be the complexification of a convex invariant function p : h → R. Then(1) HpC(u+iu)([u,w1 + iw2]) = [grad p(u), w1 + iw2] for all u,w1, w2 ∈ h, and(2) [Hp(u)(v), w1 + iw2] = HpC(u+iu)([v, w1 + iw2]) + d

dt

∣∣t=0

HpC(u+iu+t(v+iv))([u,w1 + iw2])for all u, v, w1, w2 ∈ h.

Proof. (1) is immediate from (4.16) and the C-linearity of the commutator. To see (2), wecompute

[Hp(u)(v), w1 + iw2] = [Hp(u)(v), w1] + i[Hp(u)(v), w2] by C-linearity of [ , ]

= Hp(u)([v, w1]) +d

dt

∣∣∣∣t=0

Hp(u+tv)([u,w1])

+ iHp(u)([v, w2]) + id

dt

∣∣∣∣t=0

Hp(u+tv)([u,w2]), by Lemma 4.20 (3)

= HpC(u+iu)([v, w1 + iw2]) +d

dt

∣∣∣∣t=0

HpC(u+iu+t(v+iv))([u,w1 + iw2]) by (4.16),

(4.17)

thus obtaining the desired result.�


Now we observe that critical points of ‖Φ− α‖2 are also critical points of P = pC ◦ (Φ−α) for any convex invariant function p : h → R. This is a quiver analogue of a similarstatement for holomorphic bundles in [1].

Lemma 4.22. Let p : h → R be a convex invariant function. Then a critical point of f = ‖Φ−α‖2

is also a critical point of P = p◦(Φ−α).Moreover, the converse holds if the Hessian of p is strictlypositive definite.

Proof. Equation (3.2) shows that for A ∈ Rep(Q,v),

(4.18) grad f(A) = −IρA(Φ(A)− α) = −I⊕a∈E

[Φ(A)− α,Aa],

so A is a critical point of f if and only if ρA(Φ(A)− α) = 0.On the other hand, at A = (Aa)a∈E ∈ Rep(Q,v) we have

gradP (A) = −IρA (πg grad p(Φ(A)− α)) by (4.13)

= −I⊕a∈E

[πg grad p(Φ(A)− α), Aa].(4.19)

Hence we wish to show that for all a ∈ E, [πg grad p(Φ(A) − α), Aa] = 0. Note that sinceAa ∈ Hom(Vt(a), Vh(a)), the component of [grad p(Φ(A)− α), Aa] (computed as a commuta-tor in gl(Vect(Q,v))) in Hom(Vt(a), Vh(a)) is equal to [πg grad p(Φ(A)− α), Aa]. In particular,it suffices to show that [grad p(Φ(A) − α), Aa] = 0 for all a ∈ E, where the commutator isinterpreted in hC = gl(Vect(Q,v)). By Lemma 4.21 (1), we have

[grad p(Φ(A)− α), Aa] = HpC(Φ(A)−α+i(Φ(A)−α))([Φ(A)− α,Aa]),

but by assumption A ∈ Crit(f) so [Φ(A) − α,Aa] = 0 for all a ∈ E. Hence the right handside is equal to 0, as desired.

Now suppose that the Hessian of p is strictly positive definite, and that A is a criticalpoint of P . Then gradP (A) = 0, so each summand [πg grad p(Φ(A)−α), Aa] in (4.19) is indi-vidually 0. As just observed, [πg grad p(Φ(A)−α), Aa] is the Hom(Vt(a), Vh(a))-component of[grad p(Φ(A)−α), Aa]. In particular, since all other components Hom(V`, Vk) in gl(Vect(Q,v))for ` 6= t(a) or k 6= h(a) are orthogonal in the standard metric to Hom(Vt(a), Vh(a)), and be-cause [Φ(A)− α,Aa] ∈ Hom(Vt(a), Vh(a)), we conclude

ggl(Vect(Q,v))([πg grad p(Φ(A)− α), [Φ(A)− α,Aa]) =

ggl(Vect(Q,v))([grad p(Φ(A)− α), Aa], [Φ(A)− α,Aa]) =

ggl(Vect(Q,v))(HpC(Φ(A)−α+i(Φ(A)−α))([Φ(A)− α,Aa]), [Φ(A)− α,Aa]).

(4.20)

Since [πg grad p(Φ(A)− α), Aa] = 0 for each a, this means that the last expression in (4.20)is equal to 0. On the other hand, HpC(Φ(A)−α+i(Φ(A)−α)) is strictly positive-definite. Hencewe must have [Φ(A)− α,Aa] = 0 for all a ∈ E, i.e. ρA(Φ(A)− α) = 0, and A is critical forf . �

A calculation similar to that in the previous proof leads to the following lemma. Heref : Rep(Q,v) → R is the norm-square f = ‖Φ− α‖2.

Lemma 4.23. Let p : h → R be a convex invariant function, and P = p◦ι◦(Φ−α) : Rep(Q,v) →R. Then

(4.21) gRep(Q,v) (gradP (A), grad f(A)) ≥ 0


Proof. From previous computations (4.18) and (4.19), to show (4.21) it suffices to check

gRep(Q,v)

(−I⊕a∈E

[πg grad p(Φ(A)− α), Aa],−I⊕a∈E

[Φ(A)− α,Aa]

)

= gRep(Q,v)

(⊕a∈E

[πg grad p(Φ(A)− α), Aa],⊕a∈E

[Φ(A)− α,Aa]

)by I-invariance

=∑a∈E

ga([πg grad p(Φ(A)− α), Aa], [Φ(A)− α,Aa]) ≥ 0,

(4.22)

where ga denotes the standard metric on Hom(Vt(a), Vh(a)). Fix a ∈ E. By the same argu-ment as in the proof of Lemma 4.22, we conclude that

ga([πg grad p(Φ(A)−α), Aa], [Φ(A)−α,Aa]) = ggl(Vect(Q,v))([grad p(Φ(A)−α), Aa], [Φ(A)−α,Aa]).

Lemma 4.21 (1) then saysggl(Vect(Q,v))([grad p(Φ(A)− α), Aa], [Φ(A)− α,Aa]) =

ggl(Vect(Q,v))(HpC(Φ(A)−α+i(Φ(A)−α))([Φ(A)− α,Aa]), [Φ(A)− α,Aa]),(4.23)

and the right hand side is ≥ 0 since HpC(u+iu) is non-negative-definite for any u ∈ h. Thisis true for all a ∈ E, so the result follows.

�

Having established some basic relationships between convex invariants and the norm-square function f = ‖Φ − α‖2, we now make the connection between convex invariantfunctions and the Harder-Narasimhan strata of Rep(Q,v) by first defining convex invari-ant functions on rank(Q,v)-tuples of real numbers (and hence in particular on Harder-Narasimhan types) as follows (cf. [1, p.572]). Given µ ∈ Rrank(Q,v), let Λµ denote thediagonal rank(Q,v) × rank(Q,v) matrix with entries µ = (µ1, µ2, . . . , µrank(Q,v)) alongthe diagonal. Then iΛµ is evidently an element of u(Vect(Q,v)), and given any convexU(Vect(Q,v))-invariant function p on u(Vect(Q,v)), we may define

p(µ) := p(iΛµ).

Since any element in u(Vect(Q,v)) is diagonalizable, and since p isU(Vect(Q,v))-invariant,the function p is determined by its values on µ ∈ Rrank(Q,v) for µ = (µ1 ≥ µ2 ≥ · · · ≥µrank(Q,v)). Conversely, an element µ = (µ1 ≥ µ2 ≥ · · · ≥ µrank(Q,v)) is entirely determinedby the values of the convex U(Vect(Q,v))-invariant functions on µ; this is essential towhat follows.

Lemma 4.24. Suppose µ, ν ∈ Rrank(Q,v) with µ = (µ1 ≥ µ2 ≥ · · · ≥ µrank(Q,v)), ν = (ν1 ≥ν2 ≥ · · · ≥ νrank(Q,v)). If p(µ) = p(ν) for all convex U(Vect(Q,v))-invariant functions p :u(Vect(Q,v)) → R, then µ = ν.

Proof. Immediate from [1, (8.21)]. �

We next show that the value of p at ν is a lower bound on the pullback P restricted tothe Harder-Narasimhan stratum of type ν.

Proposition 4.25. Let A ∈ Rep(Q,v)ν . Then for any convex invariant function p on h we haveP (A) ≥ p(ν).


Proof. First consider the case when the Harder-Narasimhan filtration of A has two steps,i.e. the representation A is an extension of the form

(4.24) 0 → A1 → A→ A2 → 0

where A1 ∈ Rep(Q,v1), A2 ∈ Rep(Q,v2), v1 + v2 = v. Thus the Harder-Narasimhantype is of the form ν = (ν1, . . . , νrank(Q,v) with ν1 = ν2 = · · · = νrank(Q,v1) > νrank(Q,v1)+1 =· · · = νrank(Q,v), and νk = µα1(Q,v1) for all 1 ≤ k ≤ rank(Q,v1), and νk = µα2(Q,v2) forrank(Q,v1) + 1 ≤ k ≤ rank(Q,v). Here α1 = α|Vect(Q,v1) and α2 = α|Vect(Q,v2). For eachedge a ∈ E we can write

(4.25) Aa =

(A1

a ηa

0 A2a

)with respect to the H-N filtration; here we think of eachAa as taking values in gl(Vect(Q,v))although for a fixed edge a the homomorphism is non-zero only in the component Hom(Vt(a), Vh(a)).Then

(4.26) [Aa, A∗a] =

([A1

a, (A1a)∗] + ηaη

∗a ηa(A

2a)∗ − (A1

a)∗ηa

A2aη

∗a − η∗aA

1a [A2

a, (A2a)∗]− η∗aηa

),

so Φ(A)− α has the form

(4.27) Φ(A)− α =

(i∑

a∈E ([A1a, (A

1a)∗] + ηaη

∗a)− α1 i

∑a∈E (ηa(A

2a)∗ − (A1

a)∗ηa)

i∑

a∈E (A2aη

∗a − η∗aA

1a) i

∑a∈E ([A2

a, (A2a)∗]− η∗aηa)− α2

).

Let β1 = i∑

a∈E ([A1a, (A

1a)∗] + ηaη

∗a) − α1 and β2 = i

∑a∈E ([A2

a, (A2a)∗]− η∗aηa) − α2, and

note that

−i tr β1 =∑a∈E

tr ηaη∗a + i trα1 =

∑a∈E

tr ηaη∗a + degα(Q,v1) ≥ degα(Q,v1),

with equality if and only if ηa = 0 for all a ∈ E (i.e. the extension (4.24) splits). A similarcalculation shows that −i tr β2 ≤ degα(Q,v2) with equality if and only if ηa = 0 for alla ∈ E.

Let B1 be the diagonal matrix 1rank(Q,v1)

tr β1 · id ∈ u(Vect(Q,v1)) and let B2 be the diag-onal matrix 1

rank(Q,v2)tr β2 · id ∈ u(Vect(Q,v2)). From [1, Section 12], we have

(4.28) p(Φ(A)− α) ≥ p

(B1 00 B2

)for any convex invariant function p : h → R. On the other hand, by definition, p

(B1 00 B2

)=

p(ν), where ν = (ν1, . . . , νrank(Q,v)) and

(4.29) νk =

−i 1

rank(Q,v1)tr β1 1 ≤ k ≤ rank(Q,v1)

−i 1rank(Q,v2)

tr β2 rank(Q,v1) + 1 ≤ k ≤ rank(Q,v)

Since −i tr β1 ≥ degα(Q,v1) and −i tr β2 ≤ degα(Q,v2) then ν ≥ ν by definition of thepartial ordering on H-N types, with equality ν = ν if and only if ηa = 0 for all a ∈ E.

In summary, we have proved the inequalities

(4.30) P (A) = p(Φ(A)− α) ≥ p(ν) ≥ p(ν)


for all convex invariant functions p, where the last inequality follows from [1, 8.23]. More-over, if ηa 6= 0 for some a ∈ E then ν > ν, and by Lemma 4.24 there exists a convexinvariant function p such that P (A) ≥ p(ν) > p(ν).

Now suppose that the Harder-Narasimhan filtration of arbitrary length L. As in the2-step case we may write

(4.31) A =

A1

a η1,2a η1,3

a · · · η1,La

0 A2a η2,3

a · · · η2,La

... . . . . . . . . . ...

... . . . . . . . . . ηL−1,La

0 · · · · · · 0 ALa

.

The moment map Φ(A) − α can then be expressed in a block diagonal form generalisingthe two-step case, with diagonal terms

(4.32) βj = i∑a∈E

([Aj

a, (Aja)∗] +

∑k>j

ηj,ka (ηj,k

a )∗ −∑k<j

(ηk,ja )∗ηk,j

a

)− αj.

Therefore we have

(4.33) −i tr βj =∑a∈E

(∑k>j

tr ηj,ka (ηj,k

a )∗ −∑k<j

tr(ηk,ja )∗ηk,j

a

)+ degα(Q,vj).

Taking a sum over j from 1 to `, a computation shows that for all ` ≤ n we have

(4.34) −i∑j=1

tr βj ≥∑j=1

degα(Q,vj).

As above, equality for all ` occurs if and only if ηj,ka = 0 for all a ∈ E and j < k. Let Bj =

1rank(Q,vj)

tr βj · id ∈ u(Vect(Q,vj)). Let ν = (ν1, . . . , νrank(Q,v)), where νk = −i 1rank(Q,vj)

tr βj if∑j−1`=1 rank(Q,v`) < k ≤

∑j`=1 rank(Q,v`). Then ν ≥ ν since−i

∑`j=1 tr βj ≥

∑`j=1 degα(Q,vj),

and the same argument as in the 2-step case allows us to conclude

(4.35) P (A) = p(Φ(A)− α) ≥ p(ν) ≥ p(ν).

Moreover, if ηj,ka 6= 0 for some j, k and a ∈ E, then there exists a convex invariant function

p such that p(ν) > p(ν), and so P (A) > p(ν). �

The following lemma says that we can use the complex group action to force P (A) tobe close to a critical value.

Lemma 4.26. Let A be a representation of Harder-Narasimhan type ν, and let {pj : g → R} be afinite collection of convex invariant functions. Then for any ε > 0 there exists g ∈ GC such that|pj(Φ(g · A)− α)− pj(ν)| < ε for all j.

Proof. Firstly note that since {pj} is a finite collection of continuous functions then it issufficient to show that for any δ > 0 there exists g ∈ GC such that ‖Φ(g · A)− Λν‖2 < δ.

Lemma 4.3 shows that the result holds if A is semistable, and we will prove the generalcase by induction on the length of the H-N filtration as follows (cf [5, Theorem 3.10]). Let


A1 be the maximal semistable sub-representation of A ∈ Rep(Q,v). Then we can write Aas the extension of A1 by the quotient A2 = A/A1,

(4.36) 0 → A1 → A→ A2 → 0.

With respect to this extension we can write A =

(A1 A21

0 A2

). Using the inductive hypoth-

esis, apply an element of GC of the form g =

(g1 00 g2

)such that

(4.37) ‖Φ(g1 · A1)− Λν1‖2 + ‖Φ(g2 · A2)− Λν2‖2 <1

2δ.

We may apply a GC-transformation of the form h =

(t 00 t−1

)to get

(4.38)(A1 A21

0 A2

)7→(A1 t2A21

0 A2

)to scale the extension class A21 such that ‖Φ(h · g · A)− Λν‖2 < δ. �

The next result states that for any convex invariant function p : h → R, the infimumof P = p ◦ ι ◦ (Φ − α) on a GC-orbit of a representation of Harder-Narasimhan type ν isexactly p(ν).

Proposition 4.27. Let GC =∏

`∈I GL(V`) denote the complexification of G. Suppose p : h → R

is a convex invariant function, and let P = p◦ ι◦(Φ−α) be the associated function on Rep(Q,v).Let A ∈ Rep(Q,v) be a representation with Harder-Narasimhan type ν. Then

p(ν) = infg∈GC

P (g · A).

Proof. This follows from Proposition 4.25 and Lemma 4.26. �

On the other hand, if A ∈ Rep(Q,v) is a critical point of P = p ◦ ι ◦ (Φ− α) for a convexinvariant function p : h → R, then in fact A achieves the infimum of P on the GC-orbit ofA.

Proposition 4.28. In the setting of Proposition 4.27, suppose also that the Hessian of the convexinvariant function p is strictly positive definite. If A ∈ Rep(Q,v) is a critical point of P , then

P (A) = infg∈GC

P (g · A).

The proof of Proposition 4.28 relies on the following lemmas. The basic idea is to showthat if A is a critical point of P , then A is a local minimum of P on the orbit GC ·A. This isdone by examining the Hessian of P on the image of the complexified infinitesmal action;Lemma 4.29 and Corollary 4.30 are preliminary computations to this end. The connectionbetween criticality for P on Rep(Q,v) and criticality for P restricted to a GC-orbit is madein Lemma 4.31. From here, a simple connectivity argument allows us to conclude that Ais a global minimum.

Recall that ρC denotes the complexified infinitesmal group action of GC, i.e. ρC : gC ×Rep(Q,v) → T Rep(Q,v). For the following, all notation follows that in the statement ofProposition 4.27.


Lemma 4.29. If A ∈ Rep(Q,v) is a critical point of P , then HP (A) : Rep(Q,v) → Rep(Q,v)preserves im ρCA.

Proof. Since the Hessian of a function is self-adjoint, it suffices to show that HP (A) pre-serves the orthogonal complement (im ρCA)⊥ = ker(ρ∗A)∩ker(ρ∗AI). Let δA ∈ ker ρ∗A∩ker ρ∗AI .Then

ρ∗AHP (A)(δA) = −ρ∗AIδρA (πg grad p(Φ(A)− α)) (δA)− ρ∗AIρA

(πgHp(Φ(A)−α)(dΦA(δA))

)by (4.14)

= −ρ∗AIδρA(πg grad p(Φ(A)− α))(δA)− ρ∗AIρA

(πgHp(Φ(A)−α)(ρ

∗AIδA)

)by Lemma 2.1

= −ρ∗AIδρA(πg grad p(Φ(A)− α))(δA) since δA ∈ ker ρ∗AI

= −[πg grad p(Φ(A)− α), ρ∗AIδA] + (δρ(δA))∗(IρA(πg grad p(Φ(A)− α)))

by Lemma 2.3= −[πg grad p(Φ(A)− α), ρ∗AIδA] by (4.13), since A is critical for P= 0, since δA ∈ ker ρ∗AI.

(4.39)

A similar calculation shows thatρ∗AIHP (A)(δA) = ρ∗AIIδρA(πg grad p(Φ(A)− α))(δA)

= ρ∗AIδρA(πg grad p(Φ(A)− α))(IδA) by I-linearity of δρA

= [πg grad p(Φ(A)− α),−ρ∗AδA]

= 0,

(4.40)

and so HP (A) ∈ ker(ρ∗A) ∩ ker(ρ∗AI), as desired. �

Corollary 4.30. The space im ρCA ⊆ TA Rep(Q,v) splits into eigenspaces of HP .

Proof. The operator HP (A) is self-adjoint and preserves im ρCA. �

Lemma 4.31. The point A ∈ Rep(Q,v) is critical for P if and only if dPA(ρCA(u)) = 0 for allu ∈ gC.

Proof. If A is a critical point of P , then by definition we have dPA(ρCA(u)) = 0 for anyu ∈ gC. To see the converse, note that by (4.13) we have

0 = dPA(ρCA(u)) = gRep(Q,v)

(−IρA(πg grad p(Φ(A)− α)), ρCA(u)

)for any u ∈ gC. In particular, for u = −iπg grad p(Φ(A)− α), this equation becomes

(4.41) 0 = gRep(Q,v) (−IρA(πg grad p(Φ(A)− α)),−IρA(πg grad p(Φ(A)− α))) .

Therefore 0 = −IρA(πg grad p(Φ(A)−α)) = gradP (A), and soA is a critical point of P . �

Now we can complete the proof of Proposition 4.28.

Proof of Proposition 4.28. The idea of the proof is to show that if A is critical for P , then Ais a local minimum for P on the orbit GC ·A. Infinitesimally, this reduces to showing thatthe Hessian is non-negative definite on im ρCA. Once we have proved this, then Lemma4.31 shows that A is critical for P on Rep(Q,v) if and only if A is critical for P on GC · A.


Since GC · A is connected and P is differentiable, the fact that all the critical points of Pare local minima implies that A must be a global minimum for P on GC · A.

To show that the Hessian of P is non-negative definite on im ρCA, by Corollary 4.30 it issufficient to show that there are no negative eigenvalues. Suppose first that we have asolution δA ∈ TA Rep(Q,v) to the following eigenvalue equation with λ < 0. Then

λδA = HP (A)(δA)

= −IρA(πgHp(Φ(A)−α)(ρ∗AIδA))− IδρA(πg grad p(Φ(A)− α))(δA), by (4.14)

= −IρAHι∗p(Φ(A)−α)(ρ∗AIδA)− IδρA(grad ι∗p(Φ(A)− α))(δA),

(4.42)

where ι : g → h is the linear inclusion. Applying ρ∗A to both sides of the equation gives

λρ∗AδA = −ρ∗AIρA

(Hι∗p(Φ(A)−α)(ρ

∗AIδA)

)− ρ∗AIδρA(grad ι∗p(Φ(A)− α))(δA)

= −[Hι∗p(Φ(A)−α)(ρ∗AIδA),Φ(A)− α]− [grad ι∗p(Φ(A)− α), ρ∗AIδA]

+ (δρ(δA))∗(IρA(grad ι∗p(Φ(A)− α))) by Lemmas 2.2 and 2.3= −[Hι∗p(Φ(A)−α)(ρ

∗AIδA),Φ(A)− α]− [grad ι∗p(Φ(A)− α), ρ∗AIδA] since A critical for P

= −Hι∗p(Φ(A)−α)([ρ∗AIδA,Φ(A)− α])−Hι∗p(Φ(A)−α)([Φ(A)− α, ρ∗AIδA])

= 0,

(4.43)

where the second to last equality follows from Lemma 4.20 (2) and (3). Since λ < 0, thisimplies ρ∗AδA = 0. Similarly we apply ρ∗I to (4.42) to get

λρ∗AIδA = −ρ∗AIIρA(Hι∗p(Φ(A)−α)(ρ∗AIδA))− ρ∗AIIδρA(grad ι∗p(Φ(A)− α))(δA)

= ρ∗AρA(Hι∗p(Φ(A)−α)(ρ∗AIδA))− ρ∗AIδρA(grad ι∗p(Φ(A)− α))(IδA) by I-linearity of δρA

= ρ∗AρA(Hι∗p(Φ(A)−α)(ρ∗AIδA)− [grad ι∗p(Φ(A)− α),−ρ∗AδA]

by criticality of A and Lemma 2.3= ρ∗AρA(Hι∗p(Φ(A)−α)(ρ

∗AIδA)) since ρ∗AδA = 0.

(4.44)

Since λ < 0, in order to show that ρ∗AIδA = 0 it would suffice to show that the quantity

λ‖ρ∗AIδA‖2 = 〈λρ∗AIδA, ρ∗AIδA〉g= 〈ρ∗AρAHι∗p(Φ(A)−α)(ρ

∗AIδA), ρ∗AIδA〉g

= gRep(Q,v)(ρAHι∗p(Φ(A)−α)(ρ∗AIδA), ρAρ

∗AIδA)

(4.45)

is non-negative. Using the definition of the infinitesmal action, we compute the righthand side as

gRep(Q,v)(ρAHι∗p(Φ(A)−α)(ρ∗AIδA), ρAρ

∗AIδA) = gRep(Q,v)

(⊕a∈E

[Hι∗p(Φ(A)−α)(ρ∗AIδA), Aa],

⊕a∈E

[ρ∗AIδA,Aa]

)=∑a∈E

ga([Hι∗p(Φ(A)−α)(ρ∗AIδA), Aa], [ρ

∗AIδA,Aa]).

(4.46)

Since πg is linear and grad ι∗p(u) = πg grad p(u) for u ∈ g, πgHp(u)(v) = Hι∗p(u)(v) for allu, v ∈ g ⊆ h. Hence

[Hι∗p(Φ(A)−α)(ρ∗AIδA), Aa] = πa[Hp(Φ(A)−α)(ρ

∗AIδA), Aa],


using the same arguments as in the proof of Lemmas 4.22. By using the same orthogo-nality argument as in the proof of Lemma 4.23, we may conclude that the last expressionin (4.46) is equal to∑

a∈E

ggl(Vect(Q,v))([ρ∗AIδA,Aa], [Hp(Φ(A)−α)(ρ

∗AIδA), Aa])

where now the entries are considered as elements in hC = gl(Vect(Q,v)). Fix a ∈ E. Thecorresponding term in the sum above is

ggl(Vect(Q,v))([ρ∗AIδA,Aa], HpC(Φ(A)−α+i(Φ(A)−α))([ρ

∗AIδA,Aa])

+ggl(Vect(Q,v))([ρ∗AIδA,Aa],

d

dt

∣∣∣∣t=0

HpC(Φ(A)−α+i(Φ(A)−α)+t(ρ∗AIδA+iρ∗AIδA))([Φ(A)− α,Aa])).

(4.47)

The first term in this sum is ≥ 0 for reasons argued previously, and Lemma 4.22 showsthat the second term is zero, since ρA(Φ(A) − α) = 0 at a critical point of P when theHessian of p is strictly positive definite.

Equation (4.45) becomes

(4.48) λ‖ρ∗AIδA‖2 = ggl(Vect(Q,v))([ρ∗AIδA,Aa], HpC(Φ(A)−α+i(Φ(A)−α))([ρ

∗AIδA,Aa]) ≥ 0,

which can only be true if ρAρ∗AIδA = 0 ⇔ ρ∗AIδA = 0. Therefore, any solution δA to the

eigenvalue equation (4.42) with λ < 0 must satisfy δA ∈ ker(ρ∗A) ∩ ker(ρ∗AI) = (im ρCA)⊥.Therefore HP (A) is non-negative definite on im ρCA, which completes the proof. �

The following corollary immediately follows from Proposition 4.28.

Corollary 4.32. If A ∈ Rep(Q,v) is not semistable, then GC · A ∩ Φ−1(α) is empty.

As a consequence, we may conclude that the Kahler quotient Xα(Q,v) may be identi-fied with the corresponding GIT quotient.

Lemma 4.33. Let Q = (I, E), v ∈ ZI≥0, G =∏

`∈I U(V`), and Φ : Rep(Q,v) → g∗ ∼= g be as inLemma 3.10. Let α ∈ Z(g) be an integral central parameter. Then the Kahler quotient

Xα(Q,v) := Φ−1(α)/G

is homeomorphic to the geometric invariant theory quotient

Rep(Q,v)αR−ss//αGC.

Now we use the result of Proposition 4.28 to show that the Harder-Narasimhan type ata critical point is given by the value of the moment map Φ(A)− α. The following lemmaessentially says that if the filtration changes H-N type in the limit then it must happenabove the critical set with the same H-N type as the initial conditions.

Lemma 4.34. Let ν and µ be Harder-Narasimhan types. If Bν ∩ Sµ 6= ∅ then µ ≥ ν with respectto the partial ordering on H-N type.

Proof. If A0 ∈ Bν then Proposition 4.27 shows that P (A0) ≥ p(ν) for any convex invariantp. In particular, this is true for all points along the gradient flow γ(A0, t). If A0 ∈ Sµ thenγ(A0, t) ∈ Sµ for all t, and limt→∞ γ(A0, t) ∈ Cµ. Therefore p(µ) ≥ p(ν) for every convexinvariant p, and the result follows from [1, (8.23)]. �

Lemma 4.35. If µ ≥ ν and ‖µ‖2 = ‖ν‖2, then µ = ν.


Proof. Using the notation in Section 12 of [1], we see that µ ≥ ν implies that

(4.49) Σnν ⊆ Σnµ,

where Σnµ denotes the orbit of µ ∈ Rn under the permutation group Σ, and C denotes theconvex hull of a finite set C ⊂ Rn.

Now suppose that µ > ν. Then ν ∈ Σnµ \ Σnµ, since if ν ∈ Σnµ then µ = ν as Harder-Narasimhan types. Since Σnµ \ Σnµ is Σn-invariant, then we can extend this to

(4.50) Σnν ⊂ Σnµ \ Σnµ

All of the points in Σnµ are the same distance from the origin in Rn, and so no three pointsare collinear (they all lie on a sphere). Since Σnµ is the minimal convex set containing Σnµ,which is contained in a sphere, then all points x ∈ Σnµ \ Σnµ satisfy |x| < |µ|, and so wecan extend (4.50) to the convex hull of Σnν.

(4.51) Σnν ⊂ Σnµ \ Σnµ

In particular, the above shows that ‖ν‖2 < ‖µ‖2, where ‖ · ‖ denotes the distance fromthe origin in Rn. Therefore we have shown that µ > ν implies that ‖µ‖2 > ‖ν‖2, andconversely, if µ ≥ ν and ‖µ‖2 = ‖ν‖2, then µ = ν. �

Proposition 4.36. If A ∈ Cµ then A has Harder-Narasimhan type µ, and A splits according tothe Harder-Narasimhan type.

Proof. If A ∈ Cµ then Proposition 3.11 and Lemma 4.22 show that P (A) = p(µ), and thatA is critical for every convex invariant function p. Suppose that A has H-N type ν. ThenLemma 4.34 shows that ν ≤ µ, and so by Lemma 4.35 the problem reduces to showingthat ‖ν‖2 = ‖µ‖2. Proposition 4.28 shows that when the Hessian of p is strictly positivedefinite then P (A) = infg∈GC

P (g · A), and combining this with Proposition 4.27 gives us

(4.52) p(µ) = P (A) = infg∈GC

P (g · A) = p(ν).

This is true for every convex invariant function p for which the Hessian is strictly positivedefinite, in particular it is true for the function ‖ ·‖2, and so Lemma 4.35 shows that µ = ν.

Proposition 3.11 shows that a critical representation A splits into subrepresentations,A = ⊕λAλ, and each subrepresentation has degree-rank ratio given by the values alongthe diagonal of Φ(A) − α. By definition these are the same as the values of µ, which wehave shown above to be the type of the Harder-Narasimhan filtration. Since the Harder-Narasimhan filtration is canonical, then the subrepresentations Aλ are the same as thosein the Harder-Narasimhan filtration of A. �

Corollary 4.37. For each Harder-Narasimhan type ν, A ∈ Cν implies that P (A) = p(Φ(A) −α) = p(ν).

Equation (3.2) shows that the finite-time gradient flow is generated by the action ofGC, and so by the result of Lemma 4.11, the type of the Harder-Narasimhan filtration ispreserved by the finite-time gradient flow. However, in order to relate the Morse strati-fication {Sν} by gradient flow to the algebraic stratification {Bν} by Harder-Narasimhantype, we also need to show that the Harder-Narasimhan type is preserved in the limit.This is the content of Theorem 4.15, for which we first need some technical results.


In the following we use p0 to denote the convex invariant function p0(u) = truu∗, andwe write P0(A) = p0(Φ(A) − α) = ‖Φ(A) − α‖2. In general, lowercase letters are used forconvex invariant functions on h, and uppercase letters are used for real-valued functionson the space Rep(Q,v), where Pj(A) = pj(Φ(A)− α).

We can also find a neighbourhood of each critical set such that within this neighbour-hood, the algebraic stratum lies inside the analytic stratum.

Lemma 4.38. For each Harder-Narasimhan type ν there exists a neighbourhood Vν of Cν suchthat Bν ∩ Vν ⊆ Sν .

Proof. There are only finitely many possible H-N types, and hence a finitely many con-nected critical sets of P0(A) = ‖Φ(A)−α‖2. In particular, there are only a finite number ofH-N types µ such that p0(µ) = p0(ν). Let U ε

ν = P−10 (p0(ν)− ε, p0(ν) + ε), and choose ε > 0

such that the only critical sets intersecting U εν are those for which the Harder-Narasimhan

type µ satisfies p0(µ) = p0(ν). For notation, let {µi} be the set of such H-N types.The previous lemma shows that we can restrict attention to those critical sets Cµi

forwhich µi ≥ ν. For each µ ∈ {µi}µi>ν choose a convex invariant functional pµ such thatpµ(µ) > pµ(ν), and let eµ = 1

2(pµ(µ)− pµ(ν)). Define Wµ = U ε

ν ∩P−1µ (pµ(ν)− eµ, pµ(ν) + eµ)

and note that Cν ⊂ Wµ for each µ ∈ {µi}µi>ν .To complete the proof we need to show that if ε is small enough, then Wµ ∩ Bν ∩ Sµ

is empty for all µ ∈ {µi}µi>ν . Suppose that A ∈ Wµ ∩ Bν ∩ Sµ and let A∞ denote thelimit of the gradient flow equation (3.1) with initial condition A. Lemma 4.23 shows thatPµ(A∞) ≤ Pµ(A), and since A ∈ Sµ ∩Wµ we have

(4.53) pµ(µ) = Pµ(A∞) ≤ Pµ(A) < pµ(ν) + eµ.

We also have

(4.54) pµ(ν) + eµ = pµ(ν) +1

2(pµ(µ)− pµ(ν)) =

1

2(pµ(ν) + pµ(µ)) < pµ(µ),

which leads to a contradiction. Therefore Wµ ∩Bν ∩Sµ = ∅ for µ ∈ {µi}µi>ν , and thereforethe set

(4.55) Vν =⋂

µ∈{µi}µi>ν

Wµ

has the required properties. �

The final step is to show that if a GC orbit intersects Sν then the entire orbit lies insideSν .

Proposition 4.39. Let A0 ∈ Sν ∩ Bν , and let {gk} ⊆ GC be a sequence that converges to someg∞ ∈ GC, and for which Ak = gk · A0 ∈ Sν ∩ Bν . Then A∞ = g∞ · A0 ∈ Sν .

Proof. Let Ak(t) be the gradient flow of ‖Φ(A) − α‖2 at time t with initial condition Ak.Suppose for contradiction that A∞ ∈ Sµ for some µ 6= ν. Then µ > ν by Lemma 4.34,and there exists a convex invariant function f such that f(µ) > f(ν). Since A∞ ∈ Sµ,then f(µ) ≤ f(Φ(A∞(t)) − α) for all t, it suffices to show that there exists T such thatf(Φ(A∞(T )) − α) < f(µ). More specifically, since f is continuous it suffices to show thatthere exists φ ∈ g such that f(ν) = f(φ), and for every ε > 0 there exists T such that|Φ(A∞(T ))− α− φ| < ε.


Let ε′ = 12ε, and choose δ as in Proposition 3.23. Since gk → g∞, there exists k such that

gk = g∞ ·g−1k satisfies σ(gk

−1(gk∗)−1) < δ. Theorem 3.19, Proposition 3.23, and convergence

of the gradient flow imply ‖Φ(A∞(t))− Φ(gu(t) · Ak(t))‖ < 12ε for all t. Convergence of the

gradient flow shows that Ak(t) → Ak(∞), with f(Φ(Ak(∞)) − α) = f(ν) (since Ak ∈ Sν).Since G is compact, and Φ is continuous and G-equivariant, then there exists φ ∈ g suchthat φ = limt→∞ gu(t)·(Φ(Ak(t))−α)·gu(t)

−1. Since f isG-invariant and continuous, f(φ) =f(ν). Therefore there exists T such that ‖Φ(gu(T ) · Ak(T ))− α− φ‖ < 1

2ε. Combined with

the previous statement, this shows that |Φ(A∞(T ))− α− φ| < ε, as required.�

Lemma 4.40. If A ∈ Bν , and there exists g0 ∈ GC such that g0 · A ∈ Sν then g · A ∈ Sν for allg ∈ GC.

Proof. Let G be the subset of GC for which g · A ∈ Sν . Since GC preserves Bν , Lemma4.38 shows that G is open in GC. Proposition 4.39 shows that G is closed in GC, and soG = GC. �

We may now prove the main theorem.

Proof of Theorem 4.15. Let A0 ∈ Bν . Using Lemma 4.26 with the finite collection of convexinvariant functions in the proof of Lemma 4.38 shows that we can find g ∈ GC such thatg · A0 ∈ Bν ∩ Vν . Lemma 4.38 then says that g · A0 ∈ Sν , and Lemma 4.40 implies thatA0 ∈ Sν . Therefore Bν ⊆ Sν for each Harder-Narasimhan type ν. Since {Bν} and {Sν} areboth partitions of the same space Rep(Q,v), we must have Bν = Sν for all ν. �

We next show that the sub-representations in the Harder-Narasimhan filtration con-verge along the gradient flow of ‖Φ(A)− α‖2. Given a representation A ∈ Rep(Q,v) withH-N filtration

0 ⊂ A1 ⊂ · · · ⊂ AL = A,

define π(i) to be the orthogonal projection onto the subspace of Vect(Q,v) associated toAi ∈ Rep(Q,vi). The induced representation on the image of π(i) is then Ai = A ◦ π(i).Using these projections we may also denote the H-N filtration of A by {π(i)}`

i=1. For asolution γ(A0, t) = g(t) · A0 to the gradient flow equation (3.1), we let the correspondingprojection be π(i)

t , i.e. the orthogonal projection onto the vector space g(t)π(i)0 Vect(Q,v).

Proposition 4.41. Let {π(i)t } be the H-N filtration of a solutionA(t) to the gradient flow equations

(3.1), and let {π(i)∞ } be the H-N filtration of the limit A∞. Then there exists a subsequence tj such

that π(i)tj → π

(i)∞ for all i.

Proof. For each i, π(i)t (being projections) are uniformly bounded operators, so there exists

a subsequence {tj} such that limj→∞ π(i)tj → π

(i)∞ for some π(i)

∞ . Hence the goal is to showthat π(i)

∞ = π(i)∞ for all i.

Firstly note that on each vector space Vj the projections π(i)t and π(i)

∞ have the same rank(since π(i)

t is the orthogonal projection onto the space g(t)π(i)0 Vect(Q,v), and π

(i)tj → π

(i)∞

as projections). Theorem 4.15 shows that the type of the Harder-Narasimhan filtration ispreserved in the limit of the gradient flow, and so the ranks of the maximal semistablesub-representations are also preserved. Therefore rank(π

(i)∞ ) = rank(π

(i)tj ) = rank(π

(i)∞ ) on

each vector space Vj . Since the ranks are the same on each vector space in Vect(Q,v),


by the definition of α-degree, degα(π(i)∞ ) = degα(π

(i)∞ ). Thus the degree-rank ratios of the

sub-representations corresponding to the projections π(i)∞ and π

(i)∞ are the same.

For the case i = 1, the fact that the maximal α-semistable sub-representation is unique(Proposition 4.7) implies that π(1)

∞ = π(1)∞ . Now we roceed by induction: Fix k and assume

that π(i)∞ = π

(i)∞ for all i < k. Let A(i)

∞ be the sub-representation of A corresponding tothe projection π

(i)∞ , and let A(i)

∞ be the sub-representation of A corresponding to the pro-jection π

(i)∞ . Then A

(k)∞ /A

(k−1)∞ has the same α-degree and rank as A(k)

∞ /A(k−1)∞ , which is

the maximal semistable sub-representation of A∞/A(k−1)∞ . Again, uniqueness of the max-

imal semistable sub-representation implies A(k)∞ /A

(k−1)∞ = A

(k)∞ /A

(k−1)∞ . Together with the

inductive hypothesis this gives us π(i)∞ = π

(i)∞ for all i ≤ k. �

5. AN ALGEBRAIC DESCRIPTION OF THE LIMIT OF THE FLOW

The results of Sections 3 and 4 show that the negative gradient flow of f = ‖Φ−α‖2 withinitial condition A ∈ Rep(Q,v) converges to a critical point A∞ of f , and that the Harder-Narasimhan type of this limit point A∞ is the same as the Harder-Narasimhan type of theinitial condition A. In this section we provide a more precise description of the limit ofthe flow in terms of the Harder-Narasimhan-Jordan-Holder filtration (defined below) ofthe initial condition A ∈ Rep(Q,v). Our main result, Theorem 5.4, should be viewed asa quiver analogue of the of Daskalopoulos and Wentworth for the case of holomorphicbundles over Kahler surfaces [5].

We first recall the definition of the Harder-Narasimhan-Jordan-Holder filtration, whichis a refinement of the Harder-Narasimhan filtration using stable subrepresentations. Asbefore, let Q = (I, E) be a finite quiver, v ∈ ZI≥0 a choice of dimension vector, and α ∈(iR)I a stability parameter.

Definition 5.1. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0, andlet α be a stability parameter. Let A ∈ Rep(Q,v)α−ss be an α-semistable representation. Afiltration of A with induced subrepresentations of A

(5.1) 0 = A0 ⊂ A1 ⊂ · · · ⊂ Am = A,

where Aj ∈ Rep(Q,vj), is called a Jordan-Holder filtration if• for each k, 1 ≤ k ≤ m, the induced quotient representationAk/Ak−1 ∈ Rep (Q,vk − vk−1)

is α-stable, and• µα (Q,vk − vk−1) = µα(Q,v) for each k.

Given a Jordan-Holder filtration of A as above, we define the associated graded object ofthe filtration as the direct sum

GrJH(A;Q,v) :=m⊕

k=1

Ak/Ak−1,

which by construction is also a representation of Q with dimension vector v. The nextproposition states that Jordan-Holder filtrations always exist for α-semistable representa-tions A ∈ Rep(Q,v)α−ss and that their associated graded objects are uniquely determinedup to isomorphism in Rep(Q,v) by the isomorphism type of A.


Proposition 5.2. (cf. [15, Chapter 5, Theorem 7.18]) Let Q = (I, E) be a (finite) quiver,v ∈ ZI≥0 a dimension vector, and α ∈ (iR)I a stability parameter. Let A ∈ Rep(Q,v)α−ss. Thenthere exists a Jordan-Holder filtration of A. Moreover, the associated graded object of the Jordan-Holder filtration is uniquely determined up to isomorphism in Rep(Q,v) by the isomorphism classin Rep(Q,v) of A.

GivenA ∈ Rep(Q,v) any representation, we may now combine the Harder-Narasimhanfiltration of A by α-semistable subrepresentations with a Jordan-Holder filtration of eachα-semistable piece in the H-N filtration, thus obtaining a double filtration called a Harder-Narasimhan-Jordan-Holder (H-N-J-H) filtration of A. This is the quiver analogue of theHarder-Narasimhan-Seshadri filtration for holomorphic bundles (see e.g. [5, Proposition2.6]). The proof is a straightforward application of Proposition 5.2 and is hence omitted.

We first set some notation for double filtrations. We will say a collection {Aj,k} is adouble filtration of A when

(5.2) 0 = A0,0 ⊆ A1,0 ⊆ · · · ⊆ AL,0 = A

is a filtration of A by subrepresentations, and furthermore, for each j with 1 ≤ j ≤ L, wehave

(5.3) Aj,0 ⊆ Aj,1 ⊆ Aj,2 ⊆ · · · ⊆ Aj,mj= Aj+1,0

a sequence of intermediate subrepresentations. Notating by Aj,k the quotient Aj,k/Aj,0,the sequence (5.3) then immediately gives rise to an induced filtration (again by subrep-resentations)

(5.4) 0 = Aj,0 ⊆ Aj,1 ⊆ · · · ⊆ Aj+1,0

of the quotient representation Aj,mj= Aj,mj

/Aj,0 = Aj+1/Aj,0.

Proposition 5.3. (cf. [5, Proposition 2.6]) Let Q = (I, E) be a quiver with specified dimentionvector v ∈ ZI≥0, and α ∈ (iR)I a stability parameter. Let A ∈ Rep(Q,v). Then there existsa double filtration {Aj,k} of A such that the filtration (5.2) is the Harder-Narasimhan filtrationof A, and for each j, 1 ≤ j ≤ L, the filtration (5.3) is a Jordan-Holder filtration of the quotientAj+1,0 ∈ Rep(Q,vj+1,0 − vj,0). Moreover, the isomorphism class in Rep(Q,v) of the associatedgraded object

(5.5) GrHNJH(A,Q,v) :=L⊕

j=1

mj⊕k=1

Aj,k/Aj,k−1

is uniquely determined by the isomorphism class of A ∈ Rep(Q,v).

We may now state the main theorem of this section. The point is that it is preciselythe graded object of the H-N-J-H filtration of the initial condition which determines the(isomorphism type of the) limit under the negative gradient flow.

Theorem 5.4. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0, Rep(Q,v)its associated representation space, and Φ : Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I U(V`) a moment map

for the standard Hamiltonian action of G =∏

`∈I U(V`) on Rep(Q,v). Let A0 ∈ Rep(Q,v), andlet A∞ = limt→∞ γ(A0, t) be its limit under the negative gradient flow of ‖Φ − α‖2. Then A∞is isomorphic, as representations of the quiver Q, to the associated graded object of the H-N-J-Hfiltration of the initial condition A0.


Firstly, recall that Proposition 4.41 already shows that the vector spaces that define theHarder-Narasimhan filtration converge in the limit of the gradient flow. To prove thatthe gradient flow converges to the graded object of the H-N-J-H filtration we need toshow that the corresponding sub-representations also converge. The next propositionis a key step in the argument; it shows that α-stability and α-semistability conditionson two representations, together with knowledge of the relationship between their α-slopes, can place strong restrictions on homomorphisms between them. Again, this is aquiver analogue of a similar statement for holomorphic bundles ([15, Proposition V.7.11]),a well-known corollary of which is that a semistable bundle with negative degree has noholomorphic sections.

Proposition 5.5. Let Q = (I, E) be a finite quiver, v1,v2 ∈ ZI≥0 be dimension vectors, and α ∈(iR)I a stability parameter. Suppose A1 ∈ Rep(Q,v1) with associated vector spaces {V`}, A2 ∈Rep(Q,v2) with associated vector spaces {V ′

` }, and f = {f` : V` → V ′` }`∈I is a homomorphism

of quivers from A to A′.• If µα(Q,v1) = µα(Q,v2), A1 is α-stable, and A2 is α-semistable, then f is either zero or

injective.• If µα(Q,v1) = µα(Q,v2), A1 is α-semistable, and A2 is α-stable, then f is either zero or

surjective.• If µα(Q,v1) = µα(Q,v2) and both A1 and A2 are α-stable, then f is either zero or an

isomorphism.• If µα(Q,v1) > µα(Q,v2) and both A1 and A2 are α-semistable, then f is zero.

Proof. LetK = ker(f) andL = im(f) (where by ker(f) we mean the direct sum⊕`∈I ker(f`),and similar for im(f). If L = {0} then there is nothing to prove, so we assume that L 6= {0}and hence also K 6= Vect(Q,v1). Since f is a homomorphism of representations of quiv-ers, it is straightforward to see that K is a subrepresentation of A1, and L is a subrep-resentation of A2. Let A1|K and A2|L denote the restrictions of A1 and A2 to K and Lrespectively, with associated dimension vectors vK ,vL. Then we have an exact sequenceof representations of quivers

(5.6) 0 → A1|K → A1 → A2|L → 0

where the first map is by inclusion and the second induced by f . Now assume µα(Q,v1) =µα(Q,v2), A1 is α-stable, and A2 is α-semistable. Since by assumption A1|K is not equal toA1, then ifK 6= 0 it is a proper subrepresentation ofA1 and α-stability implies µα(Q,vK) <µα(Q,v1). From this we get

µα(Q,v1) < µα(Q,vL) by Lemma 4.6 applied to (5.6)≤ µα(Q,v2) by α-semistability of A2

= µα(Q,v1) by assumption,(5.7)

which is a contradiction. Hence K = {0} and f is injective. This proves the first claim.A similar argument shows that if µα(Q,v1) = µα(Q,v2), A1 is α-semistable, and A2 is

α-stable, then f is either 0 or surjective. This proves the second claim, and hence also thethird.

Finally suppose that µα(Q,v1) > µα(Q,v2) and both A1, A2 are α-semistable. Sup-pose for a contradiction that L 6= {0} and K 6= {0}. Then by α-semistability we haveµα(Q,vL) ≤ µα(Q,v2) and µα(Q,vK) ≤ µα(Q,v1), so µα(Q,v1) ≤ µα(Q,vL) by Lemma 4.6


applied to (5.6), which gives a contradiction. On the other hand if K = {0} then A1 isisomorphic to A2|L and µ(Q,v1) = µ(Q,vL) ≤ µ(Q,v2), again a contradiction. HenceL = {0} and the last claim is proved. �

Define γ(A0, t)a to be the component of γ(A0, t) ∈ Rep(Q,v) along the edge a ∈ E.Since the finite-time gradient flow lies in a GC orbit then for t, T ∈ R we can define theisomorphism gt,T : Rep(Q,v) → Rep(Q,v) such that gt,T γ(A0, t)ag

−1t,T = γ(A0, T )a for each

a ∈ E. Then we have

(5.8) gt,T γ(A0, t)ag−1t,T = γ(A0, T )a ∀a ∈ E ⇔ gt,T γ(A0, t)a = γ(A0, T )agt,T

Similarly, if At is a sub-representation of γ(A0, t) with associated projection πt, then theinduced isomorphism ft,T = gt,T ◦ πt also satisfies

(5.9) ft,T γ(A0, t)a πt = γ(A0, T )aft,T πt

Lemma 5.6. For some sub-representation A0 ⊂ A0, let f0,t = f0,t/‖f0,t‖, where f0,t is the map de-fined above for the gradient flow with initial conditionA0. Then there exists a sequence tj such thatlimj→∞ f0,tj = f0,∞ for some non-zero map f0,∞ satisfying f0,∞ γ(A0, 0) π0 = γ(A0,∞) f0,∞ π0.

Proof. Since ‖f0,t‖ = 1 for all t then there exists a sequence tj and a non-zero map f0,∞ suchthat f0,tj → f0,∞, and so it only remains to show that f0,∞γ(A0, 0)π0 = γ(A0,∞)f0,∞π0. Toshow this, first note that

(5.10) γ(A0,∞)f0,tπ0 − f0,tγ(A0, 0)π0

= γ(A0, t)f0,tπ0 − f0,tγ(A0, 0)π0 + (γ(A0,∞)− γ(A0, t)) f0,tπ0.

Equation (5.9) and Theorem 3.1 show that the right-hand side converges to zero, and sowe have

(5.11) limt→∞

(γ(A0,∞)f0,tπ0 − f0,tγ(A0, 0)π0

)= 0.

Now consider the equation

γ(A0,∞)f0,∞π0 − f0,∞γ(A0, 0)π0

= γ(A0,∞)(f0,∞ − f0,tj)π0 − (f0,∞ − f0,tj)γ(A0, 0)π0 + γ(A0,∞)f0,tj π0 − f0,tjγ(A0, 0)π0.

The convergence f0,tj → f0,∞, together with equation (5.11), shows that all of the termson the right-hand side converge to zero, and so

(5.12) γ(A0,∞)f0,∞π0 = f0,∞γ(A0, 0)π0

as required. �

Proof of Theorem 5.4. Let A0 ∈ Rep(Q,v) be the initial condition for the gradient flow of‖Φ(A) − α‖2, and consider the sub-representation A0 ∈ Rep(Q, v) corresponding to thefirst term in the H-N-J-H filtration of A0 ∈ Rep(Q,v). Lemma 5.6 shows that there is anon-zero map f0,∞ : Vect(Q, v) → Vect(Q, v) such that A∞ ◦ f0,∞ = f0,∞ ◦ A0, where A∞ =

limt→∞ A0 and f0,∞ satisfies the conditions of Proposition 5.5. The proof of Proposition4.41 then shows that degα(Q,v∞)

rank(Q,v∞)= degα(Q,v0)

rank(Q,v0).

Applying Proposition 5.5 shows that f0,∞ is injective (since it is non-zero), and there-fore an isomorphism onto the image f0,∞ ◦ Vect(Q, v). Now repeat the process on the


quotient representation A0/A0. Doing this for each term in the H-N-J-H filtration of A0

shows that (along a subsequence) the flow converges to the graded object of the H-N-J-Hfiltration of A0. Theorem 3.1 shows that the limit exists along the flow independently ofthe subsequence chosen, which completes the proof of Theorem 5.4. �

6. FIBRE BUNDLE STRUCTURE OF STRATA

The results in Section 4 and 5 establish the tight relationship between the negative gra-dient flow on Rep(Q,v) and the Harder-Narasimhan stratification of Rep(Q,v). In par-ticular, since Theorem 4.15 shows that the analytic and algebraic stratifications are equiv-alent, we may without ambiguity refer to “the” stratum Rep(Q,v)µ = Sµ associated to aH-N type µ ∈ Rrank(Q,v).

In this section, we turn our attention back to the analytical description of the stratumRep(Q,v)µ, and in particular construct an explicit system of local coordinates around eachpoint in Rep(Q,v)µ. The key idea in our construction is to first build a coordinate systemnear a critical point A ∈ Cµ ⊆ Rep(Q,v)µ, since the criticality of A gives us a conve-nient way of parametrizing a neighborhood of A. These local coordinates near Cµ maythen be translated by GC to the other points in Rep(Q,v)µ, since (as seen in Lemma 4.26)any point in Rep(Q,v)µ can be brought arbitrarily close to a point in Cµ via the action ofGC. The main results in this direction are Propositions 6.2 and 6.6. Moreover, as a conse-quence of these explicit local descriptions, we also obtain a formula for the codimensionof Harder-Narasimhan strata and an explicit description of the stratum as a fiber bundlein Proposition 6.8. They also lead to an identification of a tubular neighborhood Uµ ofRep(Q,v)µ with the (disk bundle of the) normal bundle to Rep(Q,v)µ.

We note that Reineke also provides a description of each Harder-Narasimhan stratumas a fiber bundle [24, Proposition 3.4]. However, with our analytical approach we areable to do somewhat more, i.e. we obtain explicit local coordinates near all points inthe stratum. We expect these descriptions to be useful in the Morse-theoretic analysisof the hyperkahler Kirwan map for Nakajima quiver varieties. Indeed, in Section 7 wetake a step in this direction by using these local coordinates to give a description of thesingularities of ΦC

−1(0), the zero level set of the holomorphic moment map, which is animportant intermediate space used in the construction of the hyperkahler quotient.

We begin by using the complex group action to provide specific local coordinates aroundany point in Rep(Q,v). Let µ ∈ Rrank(Q,v) be a H-N type, and let Rep(Q,v)∗,µ denote thesubset of the H-N stratum Rep(Q,v)µ which preserves a fixed filtration, denoted ∗, oftype µ. Equip gC with the direct sum metric gC ∼= g ⊕ ig, and let ρ∗

C: TA Rep(Q,v) → gC

denote the adjoint of the infinitesmal action of gC.

Lemma 6.1. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let µ ∈Rrank(Q,v) be a H-N type and Rep(Q,v)∗,µ as above. SupposeA ∈ Rep(Q,v)∗,µ. Then the function

ψA : (ker ρCA)⊥ × ker(ρCA)∗ // Rep(Q,v)

(u, δA) // exp(u) · (A+ δA)

is a local diffeomorphism at the point (0, 0) ∈ ker(ρ∗AC)⊥ × ker(ρCA)∗.


Proof. The derivative of ψ at (u, δA) = (0, 0) is given by

(6.1) (dψA)(0,0)(δu, δa) = ρCA(δu) + δa.

By the definition of the domain of ψA, it is straightforward to see that (dψA)(0,0) is injective.Moreover, since TA Rep(Q,v) ∼= im ρCA ⊕ ker(ρCA)∗, then (dψA)(0,0) is also surjective. Theresult follows. �

With respect to the filtration ∗, the space ker(ρCA)∗ ⊂ TA Rep(Q,v) ∼= Rep(Q,v) splitsinto two subspaces

(6.2) ker(ρCA)∗ ∼=(ker(ρCA)∗ ∩ TA Rep(Q,v)∗,µ

)⊕(ker(ρCA)∗ ∩ TA(Rep(Q,v)∗,µ)⊥

)Therefore, the previous claim shows that there exists ε1 such that any representation B ∈Rep(Q,v) satisfying ‖B − A‖ < ε1 can be written uniquely as

(6.3) B = exp(u) · (A+ δa+ σ)

where δa ∈ TA Rep(Q,v)∗,µ ∩ ker(ρCA)∗ and σ ∈ TA(Rep(Q,v)∗,µ)⊥ ∩ ker(ρCA)∗. The nextproposition shows that if A is critical for f = ‖Φ− α‖2 and B ∈ Rep(Q,v)µ, then σ = 0. Itis a key step toward describing local coordinates at points on the stratum Rep(Q,v)µ.

Proposition 6.2. Let Q = (I, E), v ∈ ZI≥0, µ, and Rep(Q,v)∗,µ be as in Lemma 6.1. Let µ bea non-minimal Harder-Narasimhan type of Rep(Q,v). Suppose A ∈ Rep(Q,v)∗,µ is a criticalpoint of ‖Φ − α‖2, and suppose B ∈ Rep(Q,v)µ satisfies ‖B − A‖ < ε1 as above. Then thereexist unique elements u ∈ (ker ρCA)⊥ and δa ∈ ker(ρCA)∗ ∩ TA Rep(Q,v)∗,µ such that

B = exp(u) · (A+ δa).

As in the case of the Yang-Mills functional [4], the proof of Proposition 6.2 is by contra-diction. Namely, we show that if σ 6= 0 then there exists g ∈ GC such that ‖Φ(g ·B)−α‖2 <‖Φ(A) − α‖2, which implies that g · B has a different Harder-Narasimhan type to A; thisis because a critical point is an infimum of ‖Φ(A)− α‖2 on each stratum (by Propositions4.27 and 4.28). We have the following.

Lemma 6.3. Let Q = (I, E), v ∈ ZI≥0, µ, and Rep(Q,v)∗,µ be as in Lemma 6.1. SupposeA ∈ Rep(Q,v)∗,µ is a critical point of ‖Φ − α‖2. Then there exists ε2 > 0 and η > 0 such thatif σ ∈ TA(Rep(Q,v)∗,µ)⊥ ∩ ker(ρCA)∗ and ‖σ‖ = ε2, then ‖Φ(A + σ) − α‖ < ‖Φ(A) − α‖ − η.Moreover, the constant η can be chosen uniformly over the set Cµ ∩ Rep(Q,v)∗,µ.

Proof. If A is critical, then

(6.4) ‖Φ(A+ σ)− α‖2 = ‖Φ(A)− α‖2 +⟨H‖Φ(A)−α‖2(σ), σ

⟩+O(σ3).

Since σ ∈ ker(ρCA)∗, then a calculation shows that

(6.5) H‖Φ(A)−α‖2(σ) = −2IδρA(Φ(A)− α)(σ) = 2∑a∈E

[−i(Φ(A)− α), σa] .

Combining Lemma 3.10 and Proposition 4.36 shows that written with respect to thefixed filtration ∗, the element −i(Φ(A) − α) is diagonal with entries exactly the Harder-Narasimhan type of A, in descending order. Since σ is lower-triangular with respect tothe Harder-Narasimhan filtration of A, a calculation shows that⟨

H‖Φ(A)−α‖2(σ), σ⟩< 0.


Hence (6.4) shows that there exists η > 0, ε2 > 0 such that ‖Φ(A+σ)−α‖2 < ‖Φ(A)−α‖2−η.Since the operator −IδρA(Φ(A) − α) : TA Rep(Q,v) → TA Rep(Q,v) only depends on thevalue of Φ(A)− α, this estimate is uniform over the set Cµ ∩ Rep(Q,v)∗,µ.

�

We may now prove the Proposition.

Proof of Proposition 6.2. Let gt = exp(−tΛµ), where Λµ is the diagonal (with respect to thesplitting given by the critical representation A) matrix the entries of which are given bythe Harder-Narasimhan type µ ofA. Since gt is constant on each summand of the splittingof Vect(Q,v) defined by A, we have gt ·A = A. Hence for σ ∈ ker(ρCA)∗∩TA(Rep(Q,v)∗,µ)⊥

we have gt · (A+σ) = A+ gt ·σ. If ‖σ‖ < 12ε2, then since σ is lower-triangular with respect

to the filtration ∗ and since µ is non-minimal (hence has filtration length at least 2), wehave ‖gt · (A + σ) − A‖ = ‖gt · σ‖ → ∞ as t → ∞. Therefore there exists t > 0 such that‖gt ·(A+σ)‖ = ε2; for such a t, by Lemma 6.3 we have ‖Φ(gt ·(A+σ))−α‖ < ‖Φ(A)−α‖−η.

Now note that since Φ : Rep(Q,v) → g∗ is continuous, given the value of η in Lemma6.3, there exists ε3 > 0 such that if ‖δa‖ < ε3 then ‖Φ(A+δa+σ)−Φ(A+σ)‖ < 1

2η. As above,

for any t ≥ 0 we have ‖gt · δa‖ ≤ ‖δa‖, so if ‖δa‖ < ε3, then ‖gt · (A+ δa)−A‖ ≤ ‖δa‖ < ε3,which implies ‖Φ(gt · (A+ δa+ σ))− Φ(gt · (A+ σ))‖ < 1

2η. By the triangle inequality we

conclude

(6.6) ‖Φ(gt · (A+ δa+ σ))− α‖ < ‖Φ(A)− α‖ − 1

2η,

which contradicts Proposition 4.28. Therefore for any σ ∈ ker(ρCA)∗ ∩ TA Rep(Q,v)∗,µ with0 < ‖σ‖ < 1

2ε2, the element exp(u) · (A + δa + σ) /∈ Rep(Q,v)µ. The result now follows

from Lemma 6.1. �

We have just seen that near a critical point A of ‖Φ − α‖2 of H-N type µ, the stratumRep(Q,v)µ has a local manifold structure with dimension dim(ker ρCA)⊥ + dim(ker(ρCA)∗ ∩TA Rep(Q,v)∗,µ). In particular, locally near A the dimension of the normal bundle toRep(Q,v)µ is dim(ker(ρ∗AC)∗ ∩ TA Rep(Q,v)⊥∗,µ). Our next series of computations showsthat in fact these dimensions are independent of A, which implies in particular that theH-N strata are locally closed manifolds with well-defined (constant-rank) normal bun-dles in Rep(Q,v). Along the way, we also compare our formula for the dimension of thenormal bundle with that given in Kirwan’s manuscript [14].

Let Rep(Q,v)LT∗ denote the subspace of representations in Rep(Q,v) that are lower-

triangular with respect to a given fixed filtration ∗, and let (gC)LT∗ denote the subspace

of gC for which the associated endomorphisms of Vect(Q,v) are lower-triangular withrespect to the same filtration. We begin by observing that the subspace (gC)

LT∗ injects via

the infinitesmal action to TA Rep(Q,v) for any representation which is split with respectto that same filtration.

Lemma 6.4. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let ∗ denotea fixed filtration of Vect(Q,v) of H-N type µ, and let A ∈ Rep(Q,v) be a representation of H-Ntype µ which is split with respect to the fixed filtration. Then

dim ker(ρCA |(gC)LT∗

) = 0.

Proof. Let Vect(Q,v) ∼= Vect(Q,v1)⊕ · · · ⊕Vect(Q,vL) be the given splitting of the vectorspaces, ordered by decreasing α-slope. Then an element u ∈ (gC)

LT∗ consists of a sum of


maps ujk : Vect(Q,vj) → Vect(Q,vk) where j < k, so the α-slope of the domain is alwaysgreater than the α-slope of the image. If u ∈ ker ρCA then [u,Aa] = 0 for all edges a ∈ E,and so each component ujk of u is a map between semistable representations where theα-slope is strictly decreasing. Therefore Proposition 5.5 shows that ujk = 0 for all pairsj < k, and hence dim ker ρCA

∣∣gLTC

= 0. �

The next result is a formula for the dimension of the normal bundle to the stratumlocally near a critical point A, which in particular shows that it is independent of thechoice of A. This formula is also contained in a different form in Proposition 3.4 of [24]and Lemma 4.20 of [14]; however, one of the advantages of our formula is that it canalso be used in the hyperkahler case, and in particular on the singular space ΦC

−1(0) (seeSection 7).

Proposition 6.5. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let µ bea H-N type of Rep(Q,v) and ∗ a fixed filtration of type µ. Suppose A ∈ Rep(Q,v)∗,µ is a criticalpoint of ‖Φ − α‖2. The (complex) codimension of the H-N stratum Rep(Q,v)µ in Rep(Q,v),locally near A, is

(6.7) d(Q,v, µ) := dim Rep(Q,v)LT∗ − dim(gC)

LT∗ .

In particular, the codimension is independent of the choice of A in Cµ.

Proof. From Lemma 6.1 and Proposition 6.2 we see that at a critical representation A inRep(Q,v)µ, the dimension of the subspace in TA Rep(Q,v) orthogonal to the stratum isdim(ker(ρC

A)∗ ∩ (TA Rep(Q,v)∗,µ)⊥), where ∗ denotes the H-N filtration associated to A. Itis straightforward from the definition that

Rep(Q,v)LT∗ = (TA Rep(Q,v)∗,µ)⊥

under the standard identification of TA Rep(Q,v) with Rep(Q,v), so the dimension of thenormal space is dim(ker(ρCA)∗ ∩ Rep(Q,v)LT

∗ . We may compute

dim(ker(ρCA)∗ ∩ Rep(Q,v)LT

∗)

= dim Rep(Q,v)LT∗ − dim im(ρCA)∗

∣∣Rep(Q,v)LT

∗

= dim Rep(Q,v)LT∗ − dim(gC)

LT∗ + dim ker ρCA

∣∣(gC)LT

∗

= dim Rep(Q,v)LT∗ − dim(gC)

LT∗ ,

where the second equality uses that (ρCA)∗(Rep(Q,v)LT∗ ⊆ (gC)

LT∗ and the last equality uses

Lemma 6.4. Finally, we observe that the last quantity dim Rep(Q,v)LT∗ − dim(gC)

LT∗ is the

same for any choice of filtration ∗ of H-N type µ, so d(Q,v, µ) is indeed independent ofthe choice of critical point A. �

In Theorem 3.1 we saw that the gradient flow of ‖Φ−α‖2 converges. As a consequence,Kirwan’s Morse theory in [14] applies to Rep(Q,v) and so [14, Lemma 4.20] (which com-putes the dimension of the negative normal bundle at the critical sets of ‖Φ−α‖2) shouldalso give the same answer as Proposition 6.5 above. To see the translation between thetwo formulae, we recall that Kirwan’s formula is equivalent to

(6.8) d(Q,v, µ) = m− dimG+ dim stab(β),

where m = dim Rep(Q,v)LT∗ , G is our group, β = Φ(A) − α at a critical point A ∈

Rep(Q,v)µ and the stabilizer is with respect to the adjoint action of G on g. To make theidentification between the two formulæ, it remains to show that dimG − dim stab(β) =


dim(gC)LT∗ . This follows from observing that stab(β) is the block-diagonal part of g (with

respect to the H-N filtration ∗ associated to A), dim(gC)LT∗ = dim(gC)

UT∗ where (gC)

UT∗ de-

notes the upper-triangular part of gC with respect to ∗, and that dim(gC)LT∗ +2 dim stab(β)+

dim(gC)UT∗ = dim gC = 2 dimG.

By using the complex group action to get close to Cµ, we may use Proposition 6.2 toalso give local coordinates near any point in Rep(Q,v)∗,µ, not just those which are critical.

Proposition 6.6. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let µbe a non-minimal Harder-Narasimhan type of Rep(Q,v). Suppose A ∈ Rep(Q,v)∗,µ and letA∞ denote the limit point of the negative gradient flow with respect to ‖Φ − α‖2 with initialcondition A. Then there exists ε > 0 and g ∈ GC such that for any B ∈ Rep(Q,v)µ with‖B − A‖ < ε, there exist unique elements u ∈ (ker ρCA∞)⊥ and δa ∈ ker(ρCA∞)∗ ∩ Rep(Q,v)∗such that B = g−1 exp(u) · (A∞ + δa).

Proof. By Proposition 6.2, there exists an ε1 > 0 such that the open set {B ∈ Rep(Q,v) |‖B − A∞‖ < ε1} in Rep(Q,v)µ is a local coordinate chart for Rep(Q,v) centered at A∞.Since the finite-time negative gradient flow of ‖Φ − α‖2 is contained in a GC-orbit, fromresults of Section 3 there exists g ∈ GC such that ‖g ·A−A∞‖ < 1

2ε1. Then by the choice of

ε1 above, the open set {B ∈ Rep(Q,v)µ | ‖B − g ·A‖ < 12ε1} in Rep(Q,v)µ is a coordinate

neighborhood centered around g ·A, with local coordinates given by Proposition 6.2. Theelement g−1 induces a diffeomorphism which translates this coordinate neighborhood toan open neighborhood around B. In particular, there exists some ε > 0 such that theconditions in the Proposition are satisfied.

�

The next result shows that the H-N strata have a fibre bundle structure. We first needthe following.

Lemma 6.7. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let µ be aHarder-Narasimhan type. Then Rep(Q,v)µ = GC · Rep(Q,v)∗,µ.

Proof. Since the action of GC preserves Harder-Narasimhan types, GC · Rep(Q,v)∗,µ ⊆Rep(Q,v)µ. Conversely, any filtration of type µ is equivalent to the given filtration ∗ bysome element of GC, and therefore any representation of type µ can be mapped by GC toa representation that preserves the filtration ∗. Hence Rep(Q,v)µ ⊆ GC · Rep(Q,v)∗,µ. �

From the lemma above we can conclude that each Harder-Narasimhan stratum fibersover a homogeneous space of G. Let GC,∗, G

split∗ denote the subgroups of GC and G, re-

spectively, which preserve the given fixed filtration ∗.

Proposition 6.8. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Let µ bea H-N type. Then

Rep(Q,v)µ∼= GC ×G∗,C

Rep(Q,v)∗,µ ∼= G×Gsplit∗

Rep(Q,v)∗,µ.

Proof. Given Lemma 6.7, to prove the first equality it only remains to show that the mapϕ : GC ×G∗,C

Rep(Q,v)∗,µ → Rep(Q,v)µ given by [g, A] 7→ g · A is a local diffeomorphism.If A is a critical point, Proposition 6.2 shows that dϕ[id,A] is surjective. Since G∗,C is themaximal subgroup of GC that preserves Rep(Q,v)∗,µ, this means dϕ[id,A] is injective also.Therefore ϕ is a local diffeomorphism at [id, A]. By Lemma 6.7, the action of GC trans-lates this local diffeomorphism over the whole stratum Rep(Q,v)µ. The second equalityfollows from the isomorphism of groups GC = G×Gsplit

∗G∗,C [4, Theorem 2.16].


�

We conclude with a description of an open neighborhood of each H-N stratum as a diskbundle.

Proposition 6.9. Let Q = (I, E) be a quiver with specified dimension vector v ∈ ZI≥0. Letµ be a H-N type for Rep(Q,v). Then there exists an open neighbourhood Uµ of the H-N stra-tum Rep(Q,v)µ in Rep(Q,v) that is homeomorphic to a disk bundle of rank d(Q,v, µ) overRep(Q,v)µ.

Proof. Proposition 6.2 shows that near each A ∈ Rep(Q,v)∗,µ which is critical for ‖Φ−α‖2,the stratum Rep(Q,v)µ is locally a manifold of codimension d(Q,v, µ). This codimensionis independent of choice of critical representation Cµ∩Rep(Q,v)∗,µ. Propostion 6.6 showsthat Rep(Q,v)µ is locally a manifold of the same codimension d(Q,v, µ) at any point inRep(Q,v)∗,µ. By Lemma 6.7, local coordinates near points in Rep(Q,v)∗,µ can be translatedby the GC-action to any point in Rep(Q,v)µ; hence Rep(Q,v)µ is a manifold of Rep(Q,v)of constant codimension d(Q,v, µ). In particular, there exists a tubular neighborhood Uµ

of Rep(Q,v)µ in Rep(Q,v) satisfying the conditions of the theorem.�

7. APPLICATIONS

7.1. Hyperkahler quotients. As mentioned throughout, one of the main applications wehave in mind for the tools developed in this manuscript is the study of the topologyof Nakajima quiver varieties. In the construction of Nakajima quiver varieties as hy-perkahler quotients (as recounted in Section 2), it is necessary to take a level set not onlyof the real moment map Φ = ΦR but also of the holomorphic moment map ΦC as in (2.14)before taking the quotient by G. Hence in order to analyze the topology of the Nakajimaquiver variety via equivariant Morse theory, we take the approach of first restricting tothe level set ΦC

−1(0), and then using the Morse theory of ‖ΦR−α‖2 on ΦC−1(0) to analyze

the G-equivariant topology of ΦR−1(α) ∩ ΦC

−1(0). The first few steps of such an analysisfollow by a straightforward application of our results in Sections 3 and 4. Indeed, in Sec-tion 3 we observed that the negative gradient flow of ‖ΦR −α‖2 preserves ΦC

−1(0), so theMorse stratification on T ∗ Rep(Q,v) restricts to one on ΦC

−1(0). In Section 4, we addition-ally saw that the Harder-Narasimhan stratification on T ∗ Rep(Q,v) naturally restricts to awell-behaved stratification on ΦC

−1(0). Nevertheless, completing such a Morse-theoreticanalysis of ΦC

−1(0) is made non-trivial by the fact that ΦC−1(0) is usually a singular space.

In particular, the analogue of Proposition 6.9 is no longer true on ΦC−1(0), i.e. open neigh-

borhoods of the strata ΦC−1(0)µ := ΦC

−1(0) ∩ T ∗ Rep(Q,v)µ in ΦC−1(0) no longer have

a description as a (constant-rank) disk bundle over the strata, because now the normaldirections jump in rank. We discuss this situation in more detail below.

We follow the notation of Sections 2 and 4. Let T ∗ Rep(Q,v) = Rep(Q,v) be the originalhyperhamiltonian G-space, i.e. the cotangent bundle to the space of representations ofa quiver Q = (I, E). Let α = (iR)rank(Q,v) be a choice of parameter such that (Q,v, α) is2-filtered. Let µ ∈ Rrank(Q,v) be a H-N type of T ∗ Rep(Q,v) with respect to α, and considerthe corresponding H-N stratum ΦC

−1(0)µ of ΦC−1(0). In order to understand the topology

of the H-N stratification of ΦC−1(0), we must analyze the open neighborhoodsUµ∩ΦC

−1(0)of ΦC

−1(0)µ, where the Uµ are the open sets in Rep(Q,v) constructed in Proposition 6.9.


Let πµ : Uµ → Sµ denote the orthogonal projection of the tubular neighborhood to thestratum Sµ. The main result of this section is that although ΦC

−1(0) is singular, if (Q,v, α)is 2-filtered, then at any critical A ∈ Cµ ∩ ΦC

−1(0), the inverse image π−1µ (A) ∩ ΦC

−1(0)

is still a vector space. In other words, the restriction of πµ to Uµ ∩ ΦC−1(0) has fibers

which are vector spaces, although the dimension of these fibers may jump in rank alongCµ ∩ ΦC

−1(0).

Theorem 7.1. Let Q = (I, E) be a quiver and v ∈ ZI≥0 a choice of dimension vector. Supposeα ∈ (iR)I is a choice of stability parameter such that (Q,v, α) is 2-filtered. Let f = ‖ΦR − α‖2

be the norm-square of the real moment map on T ∗ Rep(Q,v) = Rep(Q,v), and let (A,B) ∈Crit(f) ∩ ΦC

−1(0)µ ⊆ T ∗ Rep(Q,v) for a H-N type µ. Then locally near (A,B),

(7.1) ΦC−1(0) ∩ π−1

µ (A,B) = ker(dΦC)(A,B) ∩ π−1µ (A,B),

where we consider both π−1µ (A,B) and ker(dΦC)(A,B) as affine subspaces of T ∗ Rep(Q,v) going

through (A,B).

Proof. Let (A + δA,B + δB) ∈ T ∗ Rep(Q,v) where (δA, δB) ∈ T(A,B)(T∗ Rep(Q,v)) ∼=

T ∗ Rep(Q,v). Then from the formula for ΦC we see that (A + δA,B + δB) is containedin ker(dΦC)(A,B) exactly if for all ` ∈ I,∑

in(a)=`

(AaδBa + δAaBa)−∑

out(a′)=`

(δBa′Aa′ +Ba′δAa′) = 0.

On the other hand, (A+ δA,B + δB) is in ΦC−1(0) if and only if for all ` ∈ I∑

in(a)=`

(AaδBa + δAaBa + δAaδBa)−∑

out(a′)=`

(Ba′δAa′ + δBa′Aa′ + δBa′δAa′) = 0,

where we have used that ΦC(A,B) = 0. Hence to prove (7.1) near (A,B), it suffices toshow that

(7.2)∑

in(a)=`

δAaδBa −∑

out(a)=`

δBa′δAa′ = 0

for all ` if (δA, δB) ∈ π−1µ (A,B).

Let µ denote the H-N type and ∗ denote the H-N filtration of (A,B). By assumption, theHarder-Narasimhan filtration of A has H-N α-length at most 2. If the H-N α-length is 1then the representation is α-semistable, and there is nothing to prove. If the H-N α-lengthis 2, let V` = (V`)1 ⊕ (V`)2 denote the splitting of A corresponding to the H-N filtration forall ` ∈ I. Then Rep(Q,v)LT

∗ , written with respect to a basis compatible with ∗, consists ofhomomorphisms in Hom((Vt(a))1, (Vh(a))2) for each a ∈ E. In particular, since this holds foreach δAa, δBa, we may conclude that each summand in (7.2) is separately equal to zero,so the sum is also equal to zero.

�

7.2. Kirwan surjectivity for representation varieties in cohomology and K-theory. TheMorse theory on the spaces of representations of quivers Rep(Q,v) developed in the pre-vious sections allows us to immediately conclude surjectivity results for both rationalcohomology and integralK-theory rings of the quotient moduli spaces of representationsRep(Q,v)//αG. We explain this briefly in this section.


We refer the reader to [14], [8, Section 3] for a more detailed account of what follows.The original work of Kirwan proves surjectivity in rational cohomology in the follow-ing general setting. For G a compact connected Lie group, suppose (M,ω) is a compactHamiltonian G-space with a moment map Φ : M → g∗. Assuming 0 ∈ g∗ is a regularvalue of Φ, Kirwan gives a Morse-theoretic argument using the negative gradient flow ofthe norm-square ‖Φ‖2 in order to prove that the G-equivariant inclusion ι : Φ−1(0) ↪→ Minduces a ring homomorphism κ : H∗

G(M ;Q) → H∗G(Φ−1(0);Q) ∼= H∗(M//G;Q) which is

a surjection. This argument is inductive on the Morse strata Sβ ⊆ M, defined as the setof points which limits to a component Cβ of the critical set Crit(‖Φ‖2) under the negativegradient flow of f = ‖Φ‖2. Here, the limit point always exists for any initial conditiondue to the compactness of the original space M .

The surjectivity of the inclusion map in the base case follows from the definition off , since the minimal critical set C0 = f−1(0) is precisely the level set Φ−1(0). The induc-tive step uses the long exact sequence of the pair (M≤β,M<β) for M≤β := tγ≤βSγ,M<β :=tγ<βSγ for an appropriate partial ordering on the indexing set of components of Crit(‖Φ‖2 ={Cβ} (see e.g. [8, Section 3]). The long exact sequence splits into short exact sequences,and hence yields surjectivity at each step, by an analysis of the G-equivariant negativenormal bundles of M<β in M≤β and an application of the Atiyah-Bott lemma [1, Propo-sition 13.4] in rational cohomology. Moreover, this Morse-theoretic argument of Kirwan,together with an integral topological K-theoretic version of the Atiyah-Bott lemma [8,Lemma 2.3], also implies that the Kirwan surjectivity statement (in the same setting asabove) holds also in integral K-theory [8, Theorem 3.1].

As mentioned above, in Kirwan’s original manuscript [14] she always assumes that theoriginal symplectic manifold (M,ω) of which we take the quotient is compact. In [8] thisis slightly generalized to the setting of Hamiltonian G-spaces with proper moment mapΦ. However, neither of these assumptions necessarily hold in our situation, namely, inthe Kahler quotient construction of the moduli spaces of quiver representations, since theoriginal space Rep(Q,v) is the affine space of quiver representations, and its moment mapΦ is not necessarily proper. On the other hand, Kirwan also comments in [14, Section 9]that her results and proofs generalize immediately to the situation of any Hamiltonian ac-tion (M,ω,Φ : M → g∗) provided that one can prove explicitly, in the given case at hand,that the negative gradient flow with respect to the norm-square ‖Φ − α‖2 does indeedconverge. Since this is precisely what we proved in Section 3, we have the following as acorollary.

Theorem 7.2. Let Q = (I, E) be a quiver, and v ∈ ZI≥0 a dimension vector. Let Rep(Q,v) be itsassociated associated representation space, Φ : Rep(Q,v) → g∗ ∼= g ∼=

∏`∈I u(V`) the moment

map for the standard Hamiltonian action of G =∏

`∈I U(V`) on Rep(Q,v), and α ∈ (iR)I ∼=Z(g) a choice of parameter such that G acts freely on Φ−1(0). Then the G-equivariant inclusionι : Φ−1(α) ↪→ Rep(Q,v) induces a ring homomorphism in G-equivariant rational cohomology

κ : H∗G(Rep(Q,v);Q) → H∗

G(Φ−1(α);Q) ∼= H∗(Rep(Q,v)//αG;Q)

which is surjective.

In the case of topological integral K-theory, we must restrict to the case of quivers Q =(I, E) such that the components of the critical sets of ‖Φ‖2 are compact; this is because theK-theoretic Atiyah-Bott lemma of [8] requires a compact base for its equivariant bundles.


It is known that this condition is satisfied if, for instance, Q = (I, E) has no orientedcycles [10]. Hence we have the following corollary.

Theorem 7.3. Let Q = (I, E),v ∈ ZI≥0,Rep(Q,v), G =∏

`∈I U(V`),Φ : Rep(Q,v) → g∗,

and α ∈ (iR)I be as in Theorem 7.2. Assume further that Q = (I, E) has no oriented cycles.Then the G-equivariant inclusion ι : Φ−1(α) ↪→ Rep(Q,v) induces a ring homomorphism inG-equivariant integral topological K-theory

κ : K∗G(Rep(Q,v);Q) → K∗

G(Φ−1(α);Q) ∼= K∗(Rep(Q,v)//αG;Q)

which is surjective.

7.3. Equivariant Morse theory and equivariant Kirwan surjectivity. In the study of thetopology of quotients via the Morse theory of the moment map, it is often possible tomake Kirwan surjectivity statements onto the equivariant cohomology of the quotientwith respect to some residual group action, not just the ordinary (non-equivariant) co-homology of the quotient. To prove such an equivariant version of Kirwan surjectivity,however, it is of course necessary to check that the relevant Morse-theoretic argumentsmay all be made equivariant with respect to the appropriate extra symmetry. This is thegoal of this section, for the arguments presented in Sections 3 to 6.

First, we begin with a brief recollection of the precise statement of equivariant Kirwansurjectivity. Suppose (M,ω) is a symplectic manifold, and further suppose that two com-pact connected Lie groups G and K act Hamiltonianly on (M,ω) with moment maps ΦG

and ΦK respectively. Assume that the actions of G and K commute, and that ΦG andΦK are K-invariant and G-invariant, respectively. Then there is a residual (Hamiltonian)K-action on the G-symplectic quotient M//G, and we may ask the following question: isthe ring map induced by the natural G×K-equivariant inclusion Φ−1

G (0) ↪→M ,

(7.3) κK : H∗G×K(M ;Q) → H∗

G×K(Φ−1G (0);Q)

surjective? Note that if G acts freely on Φ−1G (0), then as usual, the target of (7.3) is iso-

morphic to H∗K(M//G;Q). Hence κK is the K-equivariant version of the usual Kirwan

surjectivity question.

Remark 7.4. In the case of Nakajima quiver varieties, there is a well-studied extra S1-actioncommuting with the usualG =

∏`∈I U(V`)-action on T ∗ Rep(Q,v) which acts by spinning

only the fiber directions of the cotangent bundle with weight 1. It is straightforward tocheck that this S1-action and the givenG-action satisfy all the hypotheses required for thequestion given in (7.3) to make sense, so this is a specific example of the situation underdiscussion.

Theorem 7.5. . Let K ⊆ U(Rep(Q,v)) be a Lie subgroup such that G =∏

`∈I U(V`) ↪→U(Rep(Q,v)) and K commute, let ΦG denote the usual induced G-moment map on Rep(Q,v),and let α ∈ Z(g). Then the inclusion ι : Φ−1

G (α) ↪→ Rep(Q,v) induces a ring homomorphism

ι∗ : H∗G×K(Rep(Q,v;Q) → H∗

G×K(Φ−1G (α);Q)

which is a surjection. In particular, if G acts freely on Φ−1G (α), the composition of ι∗ with the

isomorphism H∗G×K(Φ−1

G (α);Q) ∼= H∗K(Rep(Q,v)//αG;Q) induces a surjection of rings

(7.4) κK : H∗G×K(Rep(Q,v;Q) → H∗

K(Rep(Q,v)//αG;Q).


Moreover, if the quiver Q has no oriented cycles, the inclusion ι induces a surjection

ι∗ : K∗G×K(Rep(Q,v) → K∗

G×K(Φ−1G (0))

and if G acts freely on Φ−1G (0) then the composition of ι∗ with the isomorphism K∗

G×K(Φ−1G (0)) ∼=

K∗K(Rep(Q,v)//αG) induces a surjection of rings

(7.5) κK : K∗G×K(Rep(Q,v) → K∗

K(Rep(Q,v)//αG).

Proof. From the arguments given in Section 7.2, it is evident that it suffices to show thatall the steps in the Morse theory of Section 7.2 may be made K-equivariant. We begin byshowing that ΦG is K-invariant, i.e. for all k ∈ K, u ∈ g, A ∈ Rep(Q,v), we have

〈ΦG(k · A), u〉g = 〈ΦG(A), u〉g.

By definition 〈ΦG(A), u〉g = 〈Φ(A), ιg(u)〉u(Rep(Q,v) where ιg : g ↪→ u(Rep(Q,v)) is the natu-ral inclusion, so we may compute

〈ΦG(k · A), u〉g = 〈Adk Φ(A), ιg(u)〉u(Rep(Q,v))

= 〈Φ(A),Adk−1 ιg(u)〉u(Rep(Q,v))

= 〈Φ(A), ιg(u)〉u(Rep(Q,v))

= 〈ΦG(A), u〉g,

(7.6)

for all k ∈ K, u ∈ g, A ∈ Rep(Q,v), as desired, where in the second-to-last equality weuse that G and K commute. In particular, the norm-square ‖ΦG − α‖2 for any α ∈ Z(g)is also K-invariant, since the metric on g is induced from that on u(Rep(Q,v)). Moreover,sinceK ⊆ U(Rep(Q,v)), by definition it preserves the metric on Rep(Q,v), so the negativegradient vector field of the function ‖ΦG−α‖2 isK-invariant, implying that the associatedflow is K-equivariant and the associated Morse strata are K-invariant.

To complete the argument, it suffices to note that the G × K-invariance of the metricimplies that the negative normal bundles at the critical sets areG×K-equivariant bundles,and that if there exists an S1 ⊆ G satisfying the hypotheses of the G-equivariant Atiyah-Bott lemmas, then the same S1 ⊆ G ⊆ G × K satisfies the hypotheses of the G × K-equivariant Atiyah-Bott lemma. This is true in either rational cohomology [1, Proposition13.4] or in integral topological K-theory [8, Lemma 2.3]. The result follows. �

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