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Upper bounds for the Gromov Width of Coadjoint Orbitsof Compact Lie Groups

by

Alexander Caviedes Castro

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c⃝ Copyright 2014 by Alexander Caviedes Castro

Abstract

Upper bounds for the Gromov Width of Coadjoint Orbits of Compact Lie Groups

Alexander Caviedes CastroDoctor of Philosophy

Graduate Department of MathematicsUniversity of Toronto

2014

This thesis is devoted to the problem of estimating the size of balls that can be

symplectically embedded in symplectic manifolds. This symplectic invariant is known as

the Gromov width of a symplectic manifold.

More precisely, we show that the Gromov width of a coadjoint orbit Oλ of a compact

Lie group G is bounded from above by the minimum of the nonvanishing pairings of λ

with the simple coroots. The Gromov width of a coadjoint orbit of a compact Lie group

is conjectured to be equal to this number.

The upper bound proved in this thesis extends a Theorem of Zoghi for regular coadjoint

orbits; and in type A it complements a result of Pabiniak on lower bounds for the Gromov

width of coadjoint orbits.

The upper bound for the Gromov of coadjoint orbits of compact Lie groups found in

this thesis is estimated by computing a non-vanishing Gromov-Witten invariant with one

of its constraints being a point. The approach presented here is closely related to the one

used by Gromov in his celebrated non-squeezing theorem.

ii

Dedication

To my parents,my brothersand my nephews.

iii

Acknowledgements

I would like to thank to my advisor Yael Karshon for her constant moral support andencouragement. I really enjoy our delightful meetings, her passion for mathematics andher friendly personality.

I also would like to thank to my Undergrad advisor in Colombia, Stella Huerfano, tomy Master advisor in Colombia, Bernardo Uribe, and to my Master Class advisor in theNetherlands, Gil Cavalcanti. Without their devotion and support, I would not have beenable to pursue a Ph.D. in Mathematics.

I own my love and passion for mathematics to all the people that I met at the NationalUniversity of Colombia, in special to my close friends Juan Diego Caycedo, Julio Lizarazo,Jaime Rodriguez, Mario Velasquez, Manuel Medina, Monica Ines Pinto, Diana Pulido,Nestor Forero and Miguel Angel Pachon.

I am also very grateful to the Utrecht University in the Netherlands, in particular withprofessors Marius Crainic and Jan Stienstra, organizers of the Master Class in Calabi-YauGeometry at Utrecht University, for constantly supporting Colombian students.

The year that I spent in Utrecht has been one of the most amazing experiences of mylife and it was even more enjoyable with my Master classmates Mohammad Azimi, IvanContreras, Cesar Ceballos, Alvaro Osorio and Maria Amelia Salazar. A small portion ofmy heart is still in the residence for students where I lived in the Netherlands, Parnassos.

I wish to thank to my friends Ricardo Restrepo, Carolina Benedetti and Yoe Herrera forhaving such a good time with me in Toronto, and for sharing with me all their experiencesas Ph.D. students.

A special thanks goes to my friend Irazu. Without all the coffee, I would have notbeen able to get the energy to write my thesis.

Finally, my warmest gratitude to all the people in the Math Department at the Univer-sity of Toronto, in special to Ida Bulat for her constant help and advice during my yearsin Toronto; and to Peter Crooks and Iva Halacheva for co-organizing with me and CesarCeballos the Young Tableaux Seminar at the Fields Institute.

GRACIAS TOTALES

iv

Contents

1 Introduction 1

2 J-holomorphic curves 42.1 Pseudoholomorphic theory . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Gromov width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Coadjoint Orbits of Compact Lie groups 123.1 Kostant-Kirillov-Souriau form . . . . . . . . . . . . . . . . . . . . . . . 123.2 Schubert varieties in GC/P . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Chern classes and Stable curves . . . . . . . . . . . . . . . . . . . . . . 16

4 The Gromov width of Hermitian Matrices 194.1 Geometry of Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . 204.2 Upper bounds of the Gromov width of Grassmannian manifolds . . . . . 234.3 Upper bounds for the Gromov width of Hermitian matrices . . . . . . . . 284.4 Lower bounds of the Gromov width of coadjoint orbits of type A . . . . . 31

4.4.1 Gelfand-Tsetlin Action . . . . . . . . . . . . . . . . . . . . . . . 34

5 Upper Bound for the Gromov width of Grassmannian Manifolds 365.1 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Upper Bound for the Gromov width of Grassmannian Manifolds: long root

case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Upper bounds for the Gromov width of Isotropic Grassmannians . . . . . 465.4 Upper bounds for the Gromov width of Orthogonal Grassmannians . . . 515.5 Upper bound for the Gromov width of coadjoint orbits of the exceptional

group G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

v

6 Curve Neighborhoods, Gromov Witten invariants and Gromov’s width 566.1 Curve neighborhoods of Schubert varieties . . . . . . . . . . . . . . . . 566.2 Curve neighborhoods and Gromov-Witten invariants . . . . . . . . . . . 64

7 Upper bound for the Gromov width of coadjoint orbits of compact Liegroups 697.1 Upper bounds for the Gromov width of regular coadjoint orbits . . . . . 697.2 Upper bounds for the Gromov width of coadjoint orbits . . . . . . . . . 72

8 Appendix 77

vi

Chapter 1

Introduction

The Darboux theorem in symplectic geometry states that around any point of a symplecticmanifold, there is a system of local coordinates such that the symplectic manifold lookslocally like R2n with its canonical symplectic form. A natural and fundamental problem insymplectic geometry is to know how far we can extend symplectically these coordinates inthe symplectic manifold. This is how the concept of Gromov’s width arises. The Gromovwidth of a symplectic manifold (M,ω) is defined as

Gwidth(M,ω) = sup πr2 : ∃ a symplectic embedding B2n(r) → M.

The Gromov non-squeezing theorem gives us insights of how restrictive is the Gromovwidth from above. It says that if there is a symplectic embedding of the ball B2n(r) ofradius r into a cylinder B2(λ) × R2n−2 of radius λ, then r ≤ λ. In particular,

Gwidth(B2(λ) × R2n−2) = πλ2.

Gromov’s non-squeezing theorem was proven in [22], where the connection betweenJ-holomorphic curves and sympletic geometry is established. Since then, several authorshave used Gromov’s method for bounding the Gromov width of other families of symplecticmanifolds such as Lu for symplectic toric manifolds in [40], Lu and Karshon-Tolman forcomplex Grassmannians manifolds in [39] and [32]; respectively, and Zoghi for regularcoadjoint orbits in [55] (see also McDuff-Polterovich [42], Biran [4]).

In this thesis, we want to find upper bounds for the Gromov width of arbitrary coadjointorbits of compact Lie groups with respect to the Kostant-Kirillov-Souriau form. The mainresult obtained in this thesis is the following theorem

1

Chapter 1. Introduction 2

Main Theorem. Let G be a compact connected simple Lie group with Lie algebra g.

Let T ⊂ G be a maximal torus and let R ⊂ t be the corresponding system of coroots.We identify the dual Lie algebra t∗ with the fixed points of the coadjoint action of T ong∗. Let λ ∈ t∗ ⊂ g∗, let Oλ be the coadjoint orbit passing through λ and let ωλ be theKostant-Kirillov-Souriau form defined on Oλ, then

Gwidth(Oλ, ωλ) ≤ minα∈R

⟨λ,α⟩=0

|⟨λ, α⟩|

Zoghi in his Ph.D thesis [55] has proved the same result for regular coadjoint orbits.Recall that a coadjoint orbit of a compact Lie group is regular if the stabilizer of anyelement of it under the coadjoint action is a maximal torus of the compact Lie group. Weextend Zoghi’s results to coadjoint orbits that are not necessarily regular.

On the other hand, Pabiniak has considered the problem of determining lower boundsfor the Gromov width of coadjoint orbits of compact Lie groups in [45], [46] and [47].

In [46] and [47], Pabiniak has proved that the upper bound appearing in the MainTheorem is indeed an equality for coadjoint orbits of U(n). Together with our result, thisyields the following theorem:

Theorem. Let us identify the Lie algebra of U(n) with its dual via the Ad-invariant innerproduct

(A,B) → tr(A ·B).

For (λ1, · · · , λn) ∈ Rn, let λ = i diag(λ1, · · · , λn) ∈ u(n) ∼= u(n)∗. Let Oλ be thecoadjoint orbit of U(n) passing through λ ∈ u(n)∗ and ωλ be the Kostant-Kirillov-Souriauform defined on the coadjoint orbit, then

Gwidth(Oλ, ωλ) = minλi =λj

|λi − λj|.

This thesis is organized as follows: in the second Chapter, we review the J-holomorphictools that we will use throughout the text, and then we explain how upper bounds forthe Gromov width of symplectic manifolds manifolds can be obtained by a non-vanishingGromov-Witten invariant. In the third Chapter, we give a short review about the geometryof coadjoint orbits of compact Lie groups and homogeneous spaces.

In Chapter 4 we show how to bound from above the Gromov width of standard Grass-mannian manifolds by computing certain Gromov-Witten invariants, and then we explainhow to determine upper bounds for the Gromov width of Hermitian matrices by com-

Chapter 1. Introduction 3

puting Gromov-Witten invariants on holomorphic fibrations whose fibers are isomorphicto Grassmannian manifolds. We also give an overview of Pabiniak’s results about lowerbounds for the Gromov width of Hermitian matrices.

In Chapter 5, we show the upper bound appearing in the Main Theorem for Grassman-nian manifolds coming from long simple roots. In Chapter 6, we show the upper boundsappearing in the Main Theorem for any Grassmannian manifold. In this chapter we usethe technique known as Curve neighborhoods to compute Gromov-Witten invariants andintroduced by Buch-Mihalcea in [7] and [9]. Finally, in Chapter 7 the Main Theorem isproven by computing Gromov-Witten invariants on holomorphic fibrations, in a similarway as it was done for type A.

Chapter 2

J-holomorphic curves

In this section we give a short review of pseudoholomorphic theory and Gromov-Witteninvariants, and we show how pseudoholomorphic curves are related with the Gromov widthof a symplectic manifold. Most of the material presented here is adapted from [43].

2.1 Pseudoholomorphic theory

Let (M2n, ω) be a symplectic manifold. An almost complex structure J of (M,ω) is asmooth operator J : TM → TM such that J2 = −Id. We say that an almost complexstructure J is compatible with ω if the formula

g(v, w) := ω(v, Jw)

defines a Riemannian metric. We denote the space of ω-compatible almost complexstructures by J (M,ω).

Let (CP1, j) be the Riemann sphere with its standard complex structure j. Let J ∈J (M,ω). A map u : CP1 → M is called a J-holomorphic curve of genus zero orsimply a J-holomorphic curve if

J du = du j,

or equivalently if ∂J(u) = 0 where ∂J is the operator defined by

∂Ju = (du+ J du j).

For a second homology class A ∈ H2(M,Z), we define the moduli space of simple

4

Chapter 2. J-holomorphic curves 5

J-holomorphic curves of degree A as

M∗A(M,J) = u : CP1 → M : J du = du j, u∗[CP1] = A, u is simple.

Let π1 : CP1 ×M → CP1 and π2 : CP1 ×M → M be the projections onto the firstand the second factor, respectively. Let v : π∗

1(TCP1) → π∗2(TM) be a J-antilinear map,

i.e., a map such ∂J jv = −J ∂Jv. We say that a map u : CP1 → M is a v-perturbedJ-holomorphic curve if it satisfies the equation

∂J(u)|z = v|(z,u(z))

For an antilinear map v : π∗1(TCP1) → π∗

2(TM), we denote by

MA(M,J, v) := u : CP1 → M : ∂Ju = v, u∗[CP1] = A

the moduli space of v-perturbed J-holomorphic curves of degree A.A curve u : CP1 → M is said to be multiply covered if it is the composite of a

holomorphic branched covering map (CP1, j) → (CP1, j) of degree greater than one witha J-holomorphic map CP1 → M. It is simple if it is not multiply covered.

For generic (J, v), the moduli space MA(M,J, v) is an oriented smooth manifold ofdimension equal to

dimM + 2c1(TM)(A),

where c1 denotes the first Chern class of the bundle (TM, J) (see e.g. [43, Theorem3.1.5]).

Let v : π∗1(TCP1) → π∗

2(TM) be a J-antilinear map that is equivariant with respectto the PSL(2,C) action, we denote by M∗

A,k(M,J, v) the set of equivalence classes,under the action of the reparametrization group PSL(2,C), of simple v-perturbed J-holomorphic maps

u : (CP1, z1, · · · , zk) → M

of degree A with k-marked distinct points zi ∈ CP1. For generic (J, v), the moduli spaceM∗

A,k(M,J, v) is a smooth oriented manifold of dimension equal to

dimM + 2c1(TM)(A) + 2k − 6.


We have an evaluation map

evJ = (ev1, . . . , evk) : M∗A,k(M,J, v) → Mk

defined byevJ(u, z1, · · · , zk) = (u(z1), · · · , u(zk)).

The moduli space MA,k(M,J, v) is usually not compact but it can be compactified byadding stable maps. A stable J-holomorphic map with k-marked distinct points

u : (C, z1, · · · , zk) → M

is a tree C = ∪uα of J-holomorphic maps uα : CP1 → M with at worst nodal singularities

such that if a component uα : CP1 → M is constant the number of marked and singularpoints that it contains is greater or equal to three. This implies that the automorphismgroup of u is finite. The degree of u is defined as

deg u =∑

α

deg uα ∈ H2(M,Z).

For A ∈ H2(M,Z), we denote by MA,k(M,J, v) the compactified moduli space of v-perturbed J-holomorphic stable maps of degree A with k-marked points.

As stated in Li-Tian [37], Fukaya-Ono [17], Ruan[48], Siebert [50], and more re-cently in Chen-Li-Wang [5], Fukaya-Ohta-Oh-Ono [30], Hofer-Wysocki-Zehnder [26], [27],Cieliebak-Mohnke [12] and McDuff-Wehrheim [41], the moduli space MA,k(M,J, v) car-ries a virtual fundamental class [MA,k(M,J, v)]virt ∈ H∗(MA,k(M,J, v),Q) that is usedfor defining the Gromov-Witten invariants.

2.1.1 Theorem. For generic almost complex structure J and perturbation v the modulispace MA,k(M,J, v) carries a homology class [MA,k(M,J, v)]virt ∈ H∗(MA,k(M,J, v),Q).The pushforward of [MA,k(M,J, v)]virt under evJ : MA,k(M,J, v) → Mk defines a ho-mology class

GWA,k(M) ∈ Hdim(Mk,Q)

in dimension dim = dim MA,k(M,J) = dimM + 2c1(TM)(A) + 2k − 6.The class GWA,k(M) is independent of the perturbation v and it is invariant under

smooth deformation of (ω, J) through compatible compatible structures and it is calledthe Gromov-Witten cycle of (M,ω).


For α1, · · · , αk ∈ H∗(M), the Gromov-Witten invariant is defined as

GWA,k(α1, . . . , αk) := ⟨α1 × · · · × αk,GWA,k(M)⟩ =∫

[MA,k(M,J,v)]virtev∗

1 α1 ∪ · · · ∪ ev∗k αk.

We fix geometric representatives Ai ⊂ M for the Poincaré duals of each cohomologyclass αi, and assume that

dim MA,k(M,J) = dimM + 2c1(TM)(A) + 2k − 6 =∑

i

degαi. (2.1.2)

For generic almost complex structure J and perturbation v, the Gromov-Witten invariantGWA,k(α1, . . . , αk) can be interpreted, with appropriate sign and weight, as the number ofJ-holomorphic perturbed maps of degree A with k-marked points u : (CP1, z1, · · · , zk) →M such that u(zi) ∈ Ai, i = 1, · · · , k.

2.1.3 Remark. • Gromov-Witten invariants can be defined under simple assumptions,for example, if we assume that either the symplectic manifold (M,ω) is semipositiveor the homology class A ∈ H2(M,Z) is ω-indecomposable. Examples of semipos-itive symplectic manifolds are Grassmannian manifolds which are the quotients ofcomplex Lie groups by maximal parabolic subgroups.

In these cases, for a regular almost complex structure J of (M,ω), the evaluationmap

evJ : MA,k(M,J) → Mk

represents a pseudocycle, meaning that its image can be compactified by addinga set of codimension at least two. A fundamental class can be associated to thispseudocycle; and this fundamental class can be used to define the Gromov Witteninvariant GWA,k [43, Theorem 7.1.1, Lemma 7.1.8].

• Gromov-Witten invariants are also defined in the algebraic-geometry category. Anon-trivial result due to Siebert states that when M is a complex projective manifoldits algebraic and symplectic Gromov-Witten invariants coincide [51].

• For a compact Lie group G, the complex quotient GC/P of the complexificationGC of G by a parabolic subgroup P ⊂ GC is endowed with an integrable and in-variant almost complex structure J. For A ∈ H2(GC/P,Z), the moduli space ofstable maps of degree A with k-marked point MA,k(GC/P, J) is a normal projec-tive variety and its virtual fundamental class is the same as the fundamental class


[MA,k(GC/P, J)] ∈ H∗(MA,k(GC/P, J),Q) of the moduli space MA,k(GC/P, J)(see e.g. [18]).

• The Bertini-Kleiman transversality theorem (see Theorem 4.2.1) implies that forirreducible subvarieties Γ1, . . . ,Γm ⊂ M = GC/P Poincaré dual to cohomologyclass α1, . . . , αn ∈ H∗(GC/P, /Z) such that

dim MA,k(M,J) =∑

i

degαi,

the intersection number

♯(ev−11 (g1Γ1) ∩ . . . ∩ ev−1

k (gkΓk)),

that is the number of J-holomorphic curves of degreeA passing through g1Γ1, · · · , gkΓk,

coincides with the Gromov-Witten invariant GWA,k(α1, . . . , αk), for generic g1, . . . , gk ∈GC (see e.g. [18, Lemma 14])

2.2 Gromov width

Given a symplectic manifold (M2n, ω), its Gromov’s width is defined as

Gwidth(M,ω) = sup πr2 : ∃ a symplectic embedding B2n(r) → M.

The Darboux theorem implies that the Gromov width of a symplectic manifold isalways positive. Moreover, if the symplectic manifold is compact, its Gromov width isfinite. The following statement shows the relation between pseudoholomorphic curvesand the Gromov width of symplectic manifolds:

2.2.1 Theorem. Let (M2n, ω) be a compact symplectic manifold, andA ∈ H2(M,Z)\0a second homology class. Suppose that for a dense subset of smooth ω-compatible almostcomplex structures, the evaluation map

evJ : M∗A,1(M,J) → M

has a dense image. Then for any symplectic embedding B2n(r) → M, we have

πr2 ≤ ω(A),


where ω(A) denotes the symplectic area of A. In particular,

Gwidth(M,ω) ≤ ω(A).

Proof. Suppose that there is symplectic embedding

ρ : B2n(r) → M.

Let Jst be the standard almost complex structure defined on B2n(r). Fix ϵ ∈ (0, r), letJ be a ω-compatible complex structure on M that equals ρ∗(Jst) on the open subsetρ(B2n(r − ϵ)) ⊂ M.

We claim that there exist B ∈ H2(M,Z) with ω(B) ≤ ω(A), a J-holomorphiccurve u ∈ M∗

B(M, J) and z ∈ CP1 such that ev[u, z] = u(z) = ρ(0). There existsa sequence of ω-compatible almost complex structures Jk∞

k=1 that C∞-converge to Jand Jk-holomorphic maps uk : CP1 → M of degree A and points zk ∈ CP1 such thatuk(zk) → ρ(0) ∈ ρ(B2n(r)) ⊂ M.

By Gromov’s compactness, the sequence (uk, Jk) has a subsequence (u′l, J

′l )∞

l=1 ⊂(uk, Jk) Gromov converging to a stable map

us : CP1 ⊔ · · · ⊔ CP1 → M

whose image contains ρ(0). Now, let u : CP1 → M be the restriction of us to thecomponent of the domain of us that contains the marked point. Moreover, let B =u∗([CP1]), then it satisfies

ω(B) ≤ ω(A).

Since u is J-holomorphic, its restriction to S := u−1(ρ(B2n(r − ϵ))) ⊂ CP1 gives aproper holomorphic curve u′ : S → B2n(r − ϵ) that passes through the origin. Using themonotonicity lemma in minimal surface theory (see e.g. [2, Chapter III]), the area of thisholomorphic curve is bounded from below by π(r − ϵ)2, whereas area(u′) ≤ area(u) =ω(B) ≤ ω(A), and so π(r − ϵ)2 ≤ ω(A). Since this equality is true for all ϵ > 0, weconclude that

πr2 ≤ ω(A).

2.2.2 Theorem. If ρ is a symplectic embedding of the ball B2n(r) of radius r into acylinder B2(λ) × V, where V is a compact Kähler manifold and π2(V ) = 0, then r ≤ λ.


Proof. Let V be a Kähler manifold. Let ι : B2(λ) → CP1\∞ → (CP1, λωst) be asymplectomorphism from B2(λ) onto CP1\∞. We can symplectically embed B2(λ)×Vin M = CP1 × V via ι.

Let us endow M with the symplectic structure ω = λωst ⊕ ωV and with the almostcomplex structure J = Jst ⊕JV , where ωV and JV are the symplectic and almost complexstructure of V coming from its Kähler structure.

A J-holomorphic curve u : CP1 → CP1 × V of degree A := [CP1 × p] splits in twoparts u1 : CP1 → CP1, u2 : CP1 → V. Since V is aspherical and Kähler, u2 is necessarilyconstant. On the other hand u1 is a holomorphic map of degree one, i.e, it is a Möbiustransformation. Therefore, M∗

A(M,J) = PSL(2,C) × V. Moreover, the evaluation mapevJ : M∗

A,1(M,J) → M has degree one.The homology class A = [CP1 ×p] ∈ H2(M,Z) is indecomposable, so by Gromov’s

compactness theorem for any regular almost complex structure J ′, the moduli spaceM∗

A,0(M,J ′) = M∗A(M,J ′)/PSL(2,C) is compact. Indeed, for any regular almost

complex structure J ′ there is a smooth homotopy t → Jtt connecting J with J ′ suchthat

M∗A,1(M, Jtt)

is compact and the evaluation map

evJt : M∗A,1(M, Jtt) → M

is such that

evJt |∂M∗A,1(M,Jtt) = evJ ⊔ evJ ′ : M∗

A,1(M,J) − M∗A,1(M,J ′) → M.

In particular, the evaluation maps between the moduli spaces M∗A,1(M,J),M∗

A,1(M,J ′)are compactly cobordant, so they share the same degree. Thus evJ ′ : M∗

A,1(M,J ′) → M

has degree one and in particular it is onto for any regular almost complex structure J ′,

and by the previous theorem we conclude that

πλ2 ≤ ω(A) = πr2

and we are done.

2.2.3 Theorem (Gromov’s non-squeezing Theorem). If ρ is a symplectic embeddingof the ball B2n(r) of radius r into a cylinder B2(λ) × R2n−2, then r ≤ λ.


Proof. A symplectic embedding B2n(r) → B2(λ) × R2n−2 induces a symplectic embed-ding B2n(r) → B2(λ) × V where V := R2n−2/τZ2n−2 is the (2n − 2)-torus and τ > 0is sufficiently large. Hence by the previous theorem, λ ≥ r.

2.2.4 Remark. According to Theorem 2.2.1, in order to find upper bounds for the Gromovwidth of a symplectic manifold (M,ω), we want to prove that for generic almost complexstructures J ∈ J (M,ω), the evaluation map


is onto. One way to achieve the ontoness of the evaluation map is for example byproving that a Gromov-Witten invariant with one of its constraints being a point is di-fferent from zero. More precisely, if there exist cohomology classes a1, · · · , ak such thatGWA,k(a1, . . . , ak) = 0 and a1 is Poincaré dual to the fundamental class of a point, thenfor a generic choice of almost complex structure J, the evaluation map


is onto in a dense subset of M, which, by Theorem 2.2.1, implies that

Gwidth(M,ω) ≤ ω(A).

Chapter 3

Coadjoint Orbits of Compact Liegroups

In this section we recall some general statements about homogeneous spaces GC/P,

coadjoint orbits and its geometry. Most of the material shown here can be found in theclassical literature such as [3] and [33]. Most of the material presented in Section 3.3 isadapted from [19, Chapter 2, 3].

3.1 Kostant-Kirillov-Souriau form

Let G be a connected compact Lie group, g be its Lie algebra, and g∗ be the dual ofthe Lie algebra g. The compact Lie group G acts on its Lie algebra g∗ by the coadjointaction. Let λ ∈ g∗ and Oλ be the coadjoint orbit through λ.

The coadjoint orbit Oλ carries a symplectic form defined as follows: for λ ∈ g∗ wedefine a skew bilinear form on g by

ωλ(X, Y ) = ⟨λ, [X,Y ]⟩. (3.1.1)

The kernel of ωλ is the Lie algebra gλ of the stabilizer of λ ∈ g∗ under the coadjointaction. In particular, ωλ defines a nondegenerate skew-symmetric bilinear form on g/gλ,

a vector space that can be identified with Tλ(Oλ) ⊂ g∗. Using the coadjoint action,the bilinear form ωλ induces a closed, invariant, nondegenerate 2-form on the orbit Oλ,

therefore defining a symplectic structure on Oλ. This symplectic form is known as theKostant-Kirillov-Souriau form of the coadjoint orbit Oλ.

Let T ⊂ G be a maximal torus with Lie algebra equal to t. The restricted action of

12

Chapter 3. Coadjoint Orbits 13

T ⊂ G on Oλ is Hamiltonian with momentum map

µ : Oλ → g∗ → t∗.

The Kostant Convexity Theorem states that the image of the momentum mapµ : Oλ → t∗ is the convex hull of the momentum images of the fixed points of theaction of T on Oλ [34]. This theorem is a special case of the Atiyah-Guillemin-SternbergConvexity Theorem in symplectic geometry.

The compact Lie group G admits a complexification GC. Let L = StabG λ ⊂ G

be the stabilizer of λ ∈ g∗ with respect to the coadjoint action. Fix a positive Weylchamber in t and hence a Borel subgroup B of GC, B ⊃ T, and a parabolic subgroup Pof GC, P ⊃ L, so the homogeneous spaces G/L and GC/P are diffeomorphic. One caninduce a complex structure on G/L ∼= Oλ which is homogeneous under the GC-action.Thus the coadjoint orbit Oλ get a homogeneous complex structure J. Together with theKostant-Kirillov-Souriau form, this makes the coadjoint orbit Oλ a Kähler manifold.

The homogeneous space GC/P with the torus T action is a GKM space, i.e., theclosure of every connected component of the set x ∈ GC/P : dimC (T · x) = 1 is asphere. The closure of x ∈ GC/P : dimC (T · x) = 1 is called the 1-skeleton of GC/P.

The moment graph or GKM graph of Oλ is the graph whose vertices are the T -fixedpoints and the edges are the connected components of x ∈ GC/P : dimC (T · x) = 1.

3.2 Schubert varieties in GC/P

Let R ⊂ t∗ be the root system of T in G so

gC = tC ⊕⊕α∈R

gα,

where gα := x ∈ gC : [ h , x] = α(h)x for all h ∈ tC is the root space associated withthe root α ∈ R.

Let R+ ⊂ R be a choice of positive roots with simple roots S ⊂ R+. Let W :=NG(T )/T be the Weyl group of G. For every root α ∈ R, let sα ∈ W be the reflectionassociated to it. Let B ⊂ GC be the Borel subgroup with Lie algebra

b = tC ⊕⊕

α∈R+

gα


and P ⊂ GC be a parabolic subgroup such that B ⊂ P ⊂ GC.

Let WP = NP (T )/T be the Weyl group of P and SP := α ∈ S : sα ∈ WP ⊂ S

be the set of simple roots whose corresponding reflections are in WP . The group WP isgenerated by the simple reflections sα for α ∈ SP . The parabolic subgroup P is the groupgenerated by the Borel subgroup B and NP (T ). The map

Parabolic subgroups P ⊂ GC : B ⊂ P → Subsets S ′ ⊂ S

P 7→ SP

establishes a bijection between the parabolic subgroups of GC and subsets of the set ofsimple roots S (see for instance [35, Chapter 5]).

Set RP = R ∩ ZSP and R+P = R+ ∩ ZSP , where ZSP = spanZ(SP ) is the Abelian

group spanned by SP in t∗. The Lie algebra of P is

p = b ⊕⊕

α∈R+P

g−α

For each w ∈ W, the length l(w) of w is defined as the minimum number of simplereflections sα ∈ W,α ∈ S, whose product is w.

For w′, w ∈ W, define w′ → w if there exists simple reflections s ∈ S such that

w = w′ · s

and l(w) = l(w′) + 1. Then define w′ ≤B w if there is a sequence

w′ → w1 → . . . → wm = w.

The Bruhat order on W is the partial ordering defined by the relation ≤B .

Let W P ⊂ W be the set of all minimum length representatives for cosets in W/WP .

Each element w ∈ W can be written uniquely as w = wPwP where wP ∈ W P andwP ∈ WP (see e.g. [28]). The Bruhat order on W P is the restriction to W P of theBruhat order in W. The Bruhat order on W/WP is defined by w′WP ≤B wWP if andonly if w′P ≤B wP in W P .

Let w0 be the longest element in W and let Bop = w0Bw0 ⊂ GC be the Borelsubgroup opposite to B. For w ∈ W P we define the Schubert cell

CP (w) := BwP/P ⊂ GC/P


and the opposite Schubert cell

CopP (w) := BopwP/P ⊂ GC/P.

The Schubert variety XP (w) and its opposite XopP (w) are by definition the closures of

the Schubert cells CP (w) and CopP (w), respectively. We have that for w′, w ∈ W P ,

XP (w′) ⊂ XP (w)

if and only if w′ ≤B w. Indeed,

XP (w) =⊔

w′≤Bw

CP (w′)

For w ∈ W P , the Schubert cell Cw is isomorphic to an affine space of complex dimensionequal to the length of w that is the same as the complex codimension of the oppositeSchubert cell Cop

P (w). The Schubert cells XP (w)w∈W P define a CW-complex for GC/P

with cells occurring only in even dimension. Thus, the fundamental classes [XP (w)] ofXP (w), w ∈ W P , are a free basis of H∗(GC/P,Z) as a Z-module. Likewise, the Poincarédual classes of [XP (w)], w ∈ W P , are a free basis of H∗(GC/P,Z) as a Z-module.

Let n = dimC(GC/P,Z). For an integer 1 ≤ d ≤ n, the intersection pairing is abilinear map

H2d(GC/P,Z) ⊗H2n−2d(GC/P,Z) → H2n(GC/P,Z) ∼= Z

α× β → ⟨α, β⟩

As w′, w ∈ W P varies over the permutations of length d, we have

⟨PD[XopP (w′)],PD[XP (w)]⟩ = δww′

This shows that the classes PD[XP (w)] : l(w) = d form a basis for H2n−2d(GC/P,Z)and the classes PD[Xop

P (w′)] : l(w′) = d form a dual basis for H2d(GC/P,Z).The dual application

H∗(GC/P,Z) → H∗(GC/P,Z)

[XP (w)] 7→ [XopP (w)]


is compatible with the antiautomorphism of Bruhat order

W P → W P

w 7→ w∗P := w0wwp,

where wp denotes the longest element in WP .

3.3 Chern classes and Stable curves

Let R ⊂ t∗ be the root system of a maximal torus T in G with positive roots R+

and simple roots S ⊂ R+. Let (· , ·) denote an Ad-invariant inner product defined onLie(G) = g. We identify the Lie algebra g and its dual g∗ via this inner product. Theinner product (· , ·) defines an inner product in t∗ = RR that we will denote with thesame notation (· , ·). Each root α ∈ R has a coroot α ∈ t that is identified with 2α

(α, α)via the inner product (· , ·). The coroots form the dual root system R = α : α ∈ R,with basis of simple coroots S = α : α ∈ S. For α ∈ R we let ωα ∈ t∗ denote thecorresponding fundamental weight, defined by (ωα, β) = δα,β for α ∈ R. For a parabolicsubgroup T ⊂ P ⊂ GC, we let SP := α : α ∈ SP ⊂ S.

The cohomology group H2(GC/P,Z) can be identified with the span

Zωα : α ∈ S\SP

and the homology group H2(GC/P,Z) with the quotient

ZS/ZSP .

For each α ∈ S\SP we identify the class [XP (sα)] ∈ H2(GC/P,Z) with α + ZSP ∈ZS/ZSP and we identify PD[Xop(sβ)] ∈ H2(GC/P,Z) with ωβ. The Poincaré pairingH2(GC/P,Z) ⊗H2(GC/P,Z) → Z is then given by the Ad-invariant inner product (· , ·)on t.

The following lemma, due to Bott [6], allow us to compute the first Chern classes ofline bundles over GC/P :

3.3.1 Lemma. Suppose that a torus T acts on a curve C ∼= CP1, with fixed pointsp = q, and suppose L is a T -equivariant line bundle on C. Let ηp and ηq be the weightsof T acting on the fibers Lp and Lq, and let ψp be the weight of T acting on the tangent


space to C at p. Thenηp − ηq = nψp

where n =∫

C c1(L) is the degree of L.

The collection of points wP for w ∈ W P is the set of all T -fixed points in GC/P. Foreach positive root α ∈ R+\R+

P there is a unique irreducible T -invariant curve Cα thatcontains 1 · P and sα · P. Indeed, Cα = Sl(2,C)α · P/P where Sl(2,C)α ⊂ GC is thesubgroup of GC with Lie algebra gα ⊕ g−α ⊕ [gα, g−α]. To see that Cα is unique, thereis a neighborhood of 1 ·P/P that is T -equivariantly isomorphic to gC/p. The T -invariantcurves in gC/p correspond to weight spaces g−α, for α ∈ R+\R+

P .

If λ is a weight that vanishes on all β in SP , it determines a character on P, and so aline bundle L(λ) = GC ×P C(λ) on GC/P.

The Chern class c1(L(λ)) ∈ H2(GC/P,Z) ∼= Zwα : α ∈ S \ SP is identified withthe weight λ, and we have an isomorphism

Zwα : α ∈ S \ SP → H2(GC/P,Z)

λ 7→ c1(L(λ))

Indeed, if L is any holomorphic line holomorphic line bundle on GC/P, there exists aweight λ ∈ Zwα : α ∈ S \ SP such that L = L(λ), and in particular Pic(GC/P ) ∼=Zwα : α ∈ S \ SP (see e.g. [49]).

The previous lemma implies that∫

Cα

c1(L(λ)) · (−α) = −λ− (−sα(λ)) = sα(λ) − λ

and thus(λ , α) =

∫Cα

c1(L(λ))

As a consequence, we see that

[Cα] = α + ZSP ∈ H2(GC/P,Z) ∼= ZS/ZSP

The tangent space of GC/P at the point 1 · P ∈ GC/P can be identified with

g/p =⊕

α∈R+\R+P

g−α

The weight of the line bundle ∧n T (GC/P ), n = dim(GC/P ), at the point 1 ·P ∈ GC/P


is − ∑γ∈R+\R+

Pγ, and hence

c1(TGC/P ) = c1

( n∧TGC/P

)= c1

(L

( ∑γ∈R+\R+

P

γ))

=∑

γ∈R+\R+P

γ ∈ H2(GC/P,Z) ∼= Zwα : α ∈ S\SP . (3.3.2)

3.3.3 Remark. If λ ∈ Rwα : β ∈ S\SP ⊂ t∗, the cohomology class of the Kostant-Kirillov form [ωλ] ∈ H2(Oλ,R) of the coadjoint orbit Oλ passing through λ is identifiedwith λ ∈ H2(GC/P, R) ∼= Rwα : β ∈ S\SP ⊂ t∗, and for any positive root α ∈R+\R+

P , the symplectic area

ωλ(Cα) =∫

Cα

ωλ = (λ , α)

In particular, the coadjoint orbit Oλ is prequantizable if and only if λ is integral, i.e,λ ∈ Zwα : β ∈ S\SP .

For an integral weight

λ =∑

β∈S\SP

lβwβ ∈ Z≥0wα : β ∈ S\SP ,

let Vλ be the irreducible representation of GC with highest weight λ. The Borel-Weil-Bott Theorem states that the holomorphic sections H0(GC/P, L(−λ)) of the line bundleL(−λ) is isomorphic as a GC-representation to the irreducible representation Vλ.

Let vλ be a highest weight vector of Vλ. We can embed GC/P in the projective spacePVλ by the transformation

GC/P → PVλ

[g] 7→ [g · vλ] (3.3.4)

A curve u : CP1 → GC/P of degree α = ∑β∈S\SP

mββ+ZSP ∈ H2(GC/P,Z),mβ ∈Z≥0, is push-forwarded by the embedding GC/P → PVλ to a curve of degree

∫uc1(Lλ) =

∑β∈S\SP

mβlβ ∈ H2(PVλ,Z). (3.3.5)

Chapter 4

The Gromov width of HermitianMatrices

The coadjoint orbits of a compact Lie group are endowed with a symplectic form known asthe Kostant-Kirillov-Souriau form. We wish to apply to this family of symplectic manifolds,pseudoholomorphic tools for studying the Gromov width.

In this chapter, we focus our attention on coadjoint orbits of type A that we willidentify with sets of Hermitian matrices. More precisely, for (λ1, · · · , λn) ∈ Rn, let

Hλ = A ∈ Mn(C) : A∗ = −A, spectrum = λ

and ωλ be the symplectic form defined on Hλ coming from the identification of Hλ witha coadjoint orbit of U(n) (see such identification below). The main result proven in thischapter is that

Gwidth(Hλ, ωλ) ≤ minλi =λj

|λi − λj|.

In this chapter we also describe Pabiniak’s lower bound for the Gromov widht of Hermitianmatrices. Pabiniak has shown in [46] and [47] that

Gwidth(Hλ, ωλ) ≥ minλi =λj

|λi − λj|.

Thus, the upper bound for the Gromov width of Hermitian matrices is optimal and wehave that

Gwidth(Hλ, ωλ) = minλi =λj

|λi − λj|.

19

Chapter 4. The Gromov width of Hermitian Matrices 20

4.1 Geometry of Hermitian Matrices

For the Lie group U(n), let u(n) be the Lie algebra of U(n), u(n)∗ be its dual andH := A ∈ Mn(C) : A∗ = A be the set of Hermitian matrices.

The group of unitary matrices U(n) acts by conjugation on H. The Hermitian matricesH have real eigenvalues and are diagonalizable in a unitary basis, so that the orbitsof this action correspond to sets of matrices in H with the same spectrum. Let λ =(λ1, . . . , λn) ∈ Rn and

Hλ := A ∈ Mn(C) : A∗ = A, spectrumA = λ

be the U(n)-orbit of the matrix diagonal(λ1, . . . , λn) in H. We say that the set of Her-mitian matrices Hλ is regular when all the components of λ ∈ Rn are pairwise different,otherwise we say that is non-regular.

We identify U(n)-orbits in H with adjoint orbits in u(n) by sending a matrix A ∈ Hto the matrix iA ∈ u(n). The pairing in u(n) = iH defined by

(X, Y ) = Trace(XY )

allows us to identify u(n) with u(n)∗, and adjoint orbits in u(n) with coadjoint orbits inu(n)∗. So that, the U(n)-orbits Hλ in H can be identified with the coadjoit orbits inu(n)∗.

We identify Hλ with a coadjoint orbit in u(n)∗ and define on it a symplectic form ωλ

by pulling back the Kirillov-Kostant-Souriau form defined on the coadjoint orbit. We alsoendow Hλ with a complex structure J, coming from the presentation of Hλ as a quotientof complex Lie groups Sl(n,C)/P, where P ⊂ Sl(n,C) is a parabolic subgroup of blockupper triangular matrices. The triple (Hλ, ωλ, J) is a Kähler manifold.

Let e1, . . . , en denote the standard basis of Rn. Let T = U(1)n ⊂ U(n) be thestandard maximal torus of U(n) and t ∼= Rn be its Lie algebra. We identify t∗ witht via its standard inner product so that the standard basis e1, . . . , en of t ∼= Rn isidentified with the standard basis of projections of t∗, which is also the standard basis (asa Z-module) of the weight lattice Hom(T, S1) ⊂ t∗.

The restricted action of T ⊂ U(n) on Hλ is Hamiltonian with momentum map

µ : Hλ → t∗ ≃ Rn

(aij) 7→ (a11, . . . , ann).


The image of the momentum map is the convex hull of the momentum images of thefixed points of the action of T on Hλ, i.e., the image of µ is the convex hull of all possiblepermutations of the vector (λ1, . . . , λn) ∈ t∗ ∼= Rn (see, e.g., [1, Chapter III], [24]).

The U(n)-orbit Hλ together with the torus T action is a GKM space, i.e., the closureof every connected component of the set x ∈ Hλ : dimC (T · x) = 1 is a sphere (see[54], [23]). The closure of x ∈ Hλ : dimC (T · x) = 1 is called the 1-skeleton of Hλ.

The moment graph or GKM graph of Hλ is the image of its 1-skeleton under themomentum map. The vertices and edges of this graph are in correspondence with theT -fixed points of Hλ and the closures of the connected components of the 1-skeleton ofHλ, respectively.

Two T -fixed points F, F ′ ∈ Hλ are connected by one connected component of the1-skeleton of Hλ if and only if they differ by one transposition. We denote by S2

F,F ′ ⊂ Hλ

the corresponding sphere associated to them.We now compute the symplectic area of S2

F,F ′ ⊂ Hλ with respect to ωλ in termsof λ. Let us suppose that F and F ′ differ by the transposition (i, j) ∈ Sn and the(i, i)-th component Fi ∈ λ1, . . . , λn of F is greater than its (j, j)-th component Fj ∈λ1, . . . , λn. If T ′ ⊂ T is the codimension one torus that fixes S2

F,F ′ , there exists a torusof dimension one S ⊂ T such that T ∼= T ′ × S. We will use the identification S := R/Z,

which induces an isomorphism Lie(S) ∼= R leading to Lie(S)∗ ∼= R, mapping the latticeHom(S, S1) ⊂ Lie(S)∗ isomorphically to Z ⊂ R.

The action of S on S2F,F ′ is Hamiltonian with momentum map

ι∗ µ|S2F,F ′

: S2F,F ′ → Lie(S)∗ ∼= R,

where ι : S → T is the inclusion map. The momentum image of S2F,F ′ under ι∗ µ|S2

F,F ′

is the segment line that joins ι∗(µ(F )) with ι∗(µ(F ′)). Note that the weight of T onTFS

2F,F ′ is equal to ei − ej, thus the weight of the action of S on TFS

2F,F ′ is ι∗(ei − ej),

an integer number.Let γ : [0, 1] → S2

F,F ′ → Hλ be any smooth path from F to F ′ and c : [0, 1] × S →S2

F,F ′ be the map defined by c(t, s) := s · γ(t). Then,

∫[0,1]×S

c∗(ωλ|S2F,F ′

) =∫ 1

0γ∗(ιξ

S2F,F ′

ωλ) = ι∗(µ(F )) − ι∗(µ(F ′)).

Note that the integral∫

[0,1]×S c∗ωλ is equal to the symplectic area of S2

F,F ′ times theweight ι∗(ei − ej). Since µ(F ) −µ(F ′) = (Fi −Fj)(ei − ej), and ι∗(µ(F )) − ι∗(µ(F ′)) =


(Fi − Fj)ι∗(ei − ej), we conclude that the symplectic area of S2F,F ′ is equal to Fi − Fj.

As an example, the next figure shows the moment graph of H(λ1,λ2,λ3) and some ofits edges labeled with theirs corresponding symplectic areas.

Let λb1 , λb2 , . . . , λblbe the pairwise different components of λ ∈ Rn with multiplicities

m1,m2, . . . ,ml, respectively. Let us assume without lost of generality that λ is equal to

(λb1 , · · · , λb1︸︷︷︸m1 times

, λb2 , · · · , λb2︸︷︷︸m2 times

, · · · , λbl, · · · , λbl︸︷︷︸ml times

)

Let a be the strictly increasing sequence of integers

0 = a0 < a1 < a2 < · · · < al = n

defined by aj = ∑ji=1 mi and let Fl(a;n) be the set of increasing filtrations of Cn by

complex subspaces0 = V 0 ⊂ V 1 ⊂ V 2 ⊂ · · · ⊂ V l = Cn

such that dimC Vi = ai.

Note that there is a naturally defined action of Sl(n,C) on Fl(a;n).For a flag V = (V 1, . . . , V l) ∈ Fl(a;n), denote by Pj = Pj(V ) the orthogonal

projection onto Vj. We can form the Hermitian operator

Aλ(V ) =∑

j

λbj(Pj − Pj−1).

The correspondence V 7→ Aλ(V ) defines a diffeomorphism between Fl(a;n) and Hλ.

This diffeomorphism defines by pullback a U(n)-invariant symplectic form on Fl(a;n). It


also defines an integrable almost complex structure on Fl(a;n) such that Sl(n,C) actsholomorphically on Fl(n,C), and it allows us to define a Sl(n,C) action on Hλ so themap Aλ : Fl(n,C) → Hλ is a Sl(n,C)-equivariant biholomorphism.

The (co)homology of Fl(a, n) (and hence the (co)homology of Hλ) can be computedfrom the CW-structure of Fl(a;n) coming from its Schubert cell decomposition.

Let Sn be the group of permutations of n elements. Recall that the length of apermutation is, by definition, equal to the smallest number of adjacent transpositionswhose product is the permutation. Let Wa ⊂ Sn be the subgroup generated by thesimple transpositions si = (i, i + 1) for i /∈ a1, . . . , al. Let W a ⊂ Sn be the set ofminimum length coset representatives of Sn/Wa. Let F ∈ Fl(a;n) be the partial flag

F := Ca1 ⊂ Ca2 ⊂ · · · ⊂ Can = Cn

and B be the standard Borel subgroup of Sl(n,C) of upper triangular matrices.For a permutation w ∈ W a, the Schubert cell Cw is the orbit of the induced action

of B ⊂ Sl(n,C) on Fl(a;n) through w · F. The Schubert variety Xw is by definitionthe closure of the Schubert cell Cw.

For w ∈ W a, the Schubert cell Cw is isomorphic to an affine space of complexdimension equal to the length of w. The Schubert cells Cww∈W a define a CW-complexfor Fl(a;n) with cells occurring only in even dimension. Thus, the fundamental classes[Xw] of Xw, w ∈ W a, are a free basis of H∗(Fl(a;n),Z) as a Z-module. Likewise, thePoincaré dual classes of [Xw], w ∈ W a, are a free basis of H∗(Fl(a;n),Z) as a Z-module.

The diffeomorphism Aλ : Fl(a;n) → Hλ maps the Schubert cells Cw ∈ Fl(a;n), w ∈W a, to the B-orbits of w · diagonalλ in Hλ. By abusing notation, we will denote theB-orbits of w · diagonalλ in Hλ by Cw and their closures by Xw and refer to them as theSchubert cells and Schubert varieties associated to w ∈ W a in Hλ, respectively.

4.1.1 Remark. Note that Aλ maps the Schubert varieties X(aj ,aj+1) ⊂ Fl(a;n) to thespheres S2

λ,(aj ,aj+1)·λ ⊂ Hλ. Thus, the homology group H2(Hλ,Z) is freely generated asa Z-module by the fundamental classes of S2

λ,(aj ,aj+1)·λ, 1 ≤ j ≤ l.

4.2 Upper bounds of the Gromov width of Grassman-nian manifolds

Karshon-Tolman in [32] found upper bounds for the Gromov width of Grassmannian ma-nifolds by computing a Gromov-Witten invariant. In this section, we are going to review


this idea, which would be particularly useful for considering the most general problem ofdetermining upper bounds for the Gromov width of partial flag manifolds.

We establish the convention that would be used during this section. Let G(k, n) bethe Grassmannian manifold of k-planes in Cn. Let λ ∈ Rn with k components equal to λ1,and the other n−k equal to λ2, and assume that λ1 > λ2. The set of Hermitian matricesHλ = A ∈ Mn(C) : A∗ = A, spectrumA = λ is diffeomorphic to the Grassmannianmanifold G(k, n).

Let (ωλ, J) be the Kähler structure of Hλ∼= G(k, n) defined in Section 4.1. Remark

4.1.1 implies that H2(G(k, n),Z) has one free abelian generator, let A be this generator.Let

MA(G(k, n), J) = u : CP1 → G(k, n) : u is J-holomorphic and u∗[CP1] = A

be the moduli space of J-holomorphic curves of degree A defined onG(k, n). The elementsof this moduli space would be called holomorphic lines of the Grassmannian manifoldG(k, n).

For a holomorphic curve u : CP1 → G(k, n) of degree A, we define the kernel of uas the intersection of all the subspaces V ⊂ Cn that are in the image of u. Similarly, thespan of u is the linear span of these subspaces:

ker(u) =∩

V ∈u(CP1)V, span(u) =

∑V ∈u(CP1)

V.

The kernel and span of u are of dimension k − 1 and k + 1, respectively; and theydetermine the holomorphic line up to parametrization, i.e., if there is a holomorphic linev : CP1 → G(k, n) such that ker(u) = ker(v) and span(u) = span(v), then there existsg : CP1 → CP1 ∈ PSL(2;C) such that v = u g. Moreover,

u(CP1) = V k ∈ G(k, n) : ker(u) ⊂ V k ⊂ span(u) ⊂ G(k, n),

(see for instance [10]). So MA(G(k, n), J)/PSL(2,C) ≃ Fl(k − 1, k + 1;n), whereFl(k − 1, k + 1;n) denotes the partial flag manifold of complex subspaces sequences

V k−1 ⊂ V k+1 ⊂ Cn.

For V = (V k−1, V k+1) ∈ Fl(k− 1, k+ 1;n), we will denote by uV the (unparameterized)


holomorphic line

CP1 ≃ uV = V k ∈ G(k, n) : V k−1 ⊂ V k ⊂ V k+1 ⊂ G(k, n).

Notice that MA(G(k, n), J)/PSL(2,C) ∼= Fl(k − 1, k + 1;n) is compact due to theindecomposability of A. Let us consider the evaluation map

ev2J : MA,2(G(k, n), J) → G(k, n)2.

We want to find a compact complex submanifold X ⊂ G(k, n) such that for a genericpoint p in G(k, n) the evaluation map ev2

J would be transverse to (p ×X) ⊂ G(k, n)2,

dim MA,2(G(k, n), J) + dimCX would be equal to 2 dimCG(k, n), and the number ofholomorphic curves in MA(G(k, n), J)/PSL(2,C) that pass through p and X would bedifferent to zero. If so, the Gromov-Witten invariant GWJ

A,2(PD[p] ,PD[X]) would bedifferent from zero and by Theorem 2.2.1, Remark 2.1.3 and Remark 2.2.4, we will havethat

Gwidth(Hλ, ωλ) ≤ ωλ(A) = |λ1 − λ2|.

We claim that X = V k ∈ G(k, n) : C ⊂ V k ⊂ Cn−1 ⊂ G(k, n) satisfies all theseconditions. Note that the complex submanifold X is isomorphic to the Grassmanniansubmanifold G(k − 1, n− 2).

Proving that the evaluation map ev2J is transverse to (p × X) ⊂ G(k, n)2 can be

obtained as a consequence of the Bertini-Kleiman Transversality Theorem for when theproduct Sl(n,C)2 acts on G(k, n)2 transitively:

4.2.1 Theorem. (Bertini-Kleiman [43]) Let f : U → V be a smooth map betweensmooth manifolds and letG be a Lie group that acts transitively on V. Let Z be an arbitrarysubmanifold of V and Greg be the set of elements g ∈ G for which f is transverse to gZ.Then, Greg is a set of the second category in G.

4.2.2 Remark. A similar statement holds in the algebraic geometry category for morphismsbetween complex projective varieties and group actions of complex algebraic groups (seefor instance [15, Section 5.3], [25, Theorem 10.8]).

We now prove that indeed the Gromov-Witten invariant GWJA,2(PD[p] ,PD[X]) is

different from zero.

4.2.3 Lemma. Let X = V k ∈ G(k, n) : C ⊂ V k ⊂ Cn−1 ≃ G(k − 1, n − 2) and


p ∈ G(k, n). ThenGWJ

A,2(PD[p ] ,PD[X]) = 1.

Proof. The complex dimension of X is equal to (n− k− 1)(k− 1) and as a consequenceX satisfies the dimensional constraint

dimC(MA,2(G(k, n), J)) + dimCX = dimC Fl(k − 1, k + 1;n) + 2 + dimCX

= n2 − (k − 1)2 − 22 − (n− k − 1)2

2+ 2 + (n− k − 1)(k − 1)

= 2k(n− k) = 2 dimCG(k, n).

Assume now that p = W k is a k-dimensional subspace of Cn that does not containC and transversally intersects Cn−1. We claim that (ev2

J)−1(p × X) consists of justone element, i.e., there is a unique (unparameterized) holomorphic line in G(k, n) thatintersects X and passes through W k.

Let V = (V k−1, V k+1) ∈ Fl(k − 1, k + 1;n) be such that the holomorphic line uV

passes through both X and p. So there exists V k ∈ X (that is, C ⊂ V k ⊂ Cn−1) andV k−1 ⊂ V k ⊂ V k+1. Moreover we have V k−1 ⊂ W k ⊂ V k+1 (W k is p).

Note that, we have inclusions V k−1 ⊂ Cn−1 and V k−1 ⊂ W k. Thus V k−1 ⊂ W k ∩Cn−1. But W k ∩Cn−1 is a (k − 1)-dimensional vector subspace because the intersectionis transverse. Thus V k−1 = W k ∩ Cn−1. The intersection V k−1 = W k ∩ Cn−1 does notcontain C. So there exists a unique k-dimensional vector space Uk such that V k−1 ⊂ Uk

and C ⊂ Uk ⊂ Cn−1. This vector space is Uk = V k−1 ⊕C. Thus, V k = V k−1 ⊕C. Thevector space V k+1 contains W k and V k = V k−1 ⊕ C. Observe that V k is different fromW k because V k contains C and W k does not. Therefore V k+1 = W k + V k.

In conclusion (V k−1, V k+1) = (W k ∩ Cn−1,W k + ((W k ∩ Cn−1) ⊕C)), which deter-mines a unique holomorphic line that intersects X and passes through W k.

Note that if p = W k is a k-dimensional subspace of Cn that either contains C or iscontained in Cn−1, then (ev2

J)−1(p ×X) consists of an infinite number of elements.We now prove that the evaluation map

ev2J : MA,2(G(k, n), J) → G(k, n)2

is transverse to (p × X) ⊂ G(k, n)2. The group Sl(n,C) acts transitively and holo-morphically on G(k, n) so as a consequence there exists h ∈ Sl(n,C) such that ev2

J t

(h · p × X) ⊂ G(k, n)2 and thus the preimage (ev2J)−1(h · p × X) consists of just

one point (the number of elements of the preimage (ev2J)−1(h · p × X) is either one


Figure 4.1: The vector spaces between V k−1 and V k+1 (green) corresponds to the uniqueholomorphic line in the Grassmannian manifold G(k, n) passing through the generic pointW k (blue) and X (red). The intersection of the holomorphic line with X is the k-dimensional vector spanned by V k−1 and C (black).

or infinite, but if the evaluation map is transverse to h · p ×X it has to be necessarilyone); by Proposition 7.4.5 of [43], the Gromov-Witten invariant GWJ

A,2(PD[p ] ,PD[X])is positive, so in conclusion

GWJA,2(PD[p ] ,PD[X]) = GWJ

A,2(PD[h · p] ,PD[X]) = 1

We have proved that for Grassmannian manifolds there is a non-vanishing Gromov-Witten invariant with one of its constraints being Poincaré dual to the class of a point.This would imply that the Gromov width of a Grassmannian manifolds is bounded fromabove by the symplectic area of any holomorphic line of the Grassmannian manifold. Insummary, we have the following result:

4.2.4 Theorem (Karshon-Tolman, Lu). Let

Hλ = A ∈ Mn(C) : A∗ = A, spectrumA = λ

where λ ∈ Rn is of the form

λ1 = · · · = λ1 > λ2 = · · · = · · ·λ2,


and let ωλ be the Kirillov-Kostant-Souriau form defined on Hλ. Then,

Gwidth(Hλ, ωλ) ≤ |λ1 − λ2|.

Proof. The result follows from Remark 2.2.4 and Lemma 4.2.3, and the fact that thesymplectic area of A with respect to ωλ is equal to |λ1 − λ2|.

4.3 Upper bounds for the Gromov width of Hermitianmatrices

Let λ = (λ1, . . . , λn) ∈ Rn, Hλ = A ∈ Mn(C) : A∗ = A, spectrumA = λ and (ωλ, J)be the Kähler structure of Hλ defined in Section 4.1. The following theorem appears inZoghi’s Ph.D thesis [55] as one of its main results:

4.3.1 Theorem (Zoghi). Let λ ∈ Rn be of the form λ1 > · · · > λn. Suppose that thereis an integer k such that any difference of eigenvalues λi − λj is an integer multiple ofλk+1 − λk, then

Gwidth(Hλ, ωλ) ≤ |λk − λk+1|.

In this section we show how to extend Zoghi’s result to sets of Hermitian matricesthat are not necessarily regular. But first we state the following lemma:

4.3.2 Lemma. Let B ⊂ Sl(n,C) be the Borel subgroup of upper triangular matricesand P ⊂ Sl(n,C) be any parabolic subgroup of block upper triangular matrices. Let Xbe an algebraic B-variety and π : X → Sl(n,C)/P be a B-equivariant map. If Ω is theB-stable open dense Schubert cell of Sl(n,C)/P, then π is a trivial fibration over Ω.

Proof. Let x0 ∈ Ω be any point and U ⊂ B be the unipotent radical of P. The maps : U → Ω defined by g 7→ g · x0 is an isomorphism. Let t : Ω → U be the inversefunction of s. The map

ψ : Ω × π−1(x0) → π−1(Ω)

(x, y) 7→ t(x) · y

is an isomorphism with inverse given by

ψ−1 : π−1(Ω) → Ω × π−1(x0)

m 7→ (π(m), t(π(m))−1 ·m).


The next is the main result of this chapter.

4.3.3 Theorem. For λ = (λ1, · · · , λn) ∈ Rn, let

Hλ = A ∈ Mn(C) : A∗ = A, spectrum(A) = λ

and ωλ be the symplectic form defined on Hλ coming from its identification with acoadjoint orbit of U(n) and the Kostant-Kirillov-Souriau form defined on the coadjointorbit. Then,


|λi − λj|.

Proof. The idea of the proof is to prove that a certain Gromov-Witten invariant, withone of its constraints being Poincaré dual to the fundamental class of a point, is differentfrom zero.

Let λb1 , λb2 , . . . , λblbe the pairwise different components of λ ∈ Rn with multiplicities

m1,m2, . . . ,ml, respectively. Let us assume without lost of generality that λ is equal to

(λb1 , · · · , λb1︸︷︷︸m1 times

, λb2 , · · · , λb2︸︷︷︸m2 times

, · · · , λbl, · · · , λbl︸︷︷︸ml times

)

After reordering the components of λ if necessary, we will assume that any differenceof the form |λi′ − λj′| is bigger than |λb2 − λb1 |.

We know that Hλ ≃ Fl(a;n), where a is the strictly increasing sequence of integers

0 = a0 < a1 < · · · < al = n

defined by ak = ∑kr=1 mr, for 1 ≤ k ≤ l.

We will endow Fl(a;n) with a Kähler structure coming from its identification withHλ. This Kähler structure and the one defined on Hλ would be denoted indistinguishablyby (ωλ, J).

Let a′ be the sequence of integer numbers

a2 < · · · < al = n,

and Fl(a′;n) be the corresponding partial flag manifold. Let

Wa = Sm1 × Sm2 × . . .× Sml⊂ Sn


be the subgroup generated by the simple transpositions si = (i, i+1) for i /∈ a1, . . . , al.Let W a ⊂ Sn be minimum length coset representatives of

Sn/Wa = Sn/(Sm1 × Sm2 × . . .× Sml).

Likewise, we letWa′ = Sm1+m2 × . . .× Sml

⊂ Sn

and W a′ be the minimum length coset representatives of

Sn/Wa′ = Sn/(Sm1+m2 × . . .× Sml).

Schubert varieties of Fl(a;n) and Fl(a′;n) are parametrized by elements of W a and W a′,

respectively. To avoid confusions, we will denote the Schubert varieties in Fl(a;n) byX• and the Schubert varieties in Fl(a′;n) by X ′

•. A similar thing will be done with theSchubert cells.

Let X(a1,a1+1) be the standard Schubert variety in Fl(a;n) associated to the permuta-tions (a1, a1 + 1) ∈ W a, and let A be the the fundamental class of this Schubert variety.Note that, by assumption, the symplectic area ωλ(A) = |λb2 − λb1 |.

We have a holomorphic projection

π : Fl(a;n) → Fl(a′;n)

V a1 ⊂ V a2 ⊂ · · · ⊂ V al = Cn 7→ V a2 ⊂ · · · ⊂ V al = Cn

whose fibers are isomorphic to the Grassmanian manifold G(a1, a2).The set of minimum length representatives W a′

a∼= Sm1+m2/(Sm1 × Sm2) of Wa′ on

Wa parameterizes Schubert varieties on a fiber of π. Note that (a1, a1 + 1) ∈ W a′a , so in

particular π∗(A) = 0 ∈ H∗(Fl(a′;n),Z).Let w be the permutation in W a′

a that represents in a fiber a Grassmannian manifoldisomorphic to G(a1 − 1, a2 − 2). Let w′ be the longest element in W a′

. The Schubert cellC

′w′ is open and dense in Fl(a′;n). By the previous Lemma, the restriction map

π|Xw′w: Xw′w → Fl(a′;n)

is a trivial fibration over C ′w′ with fiber isomorphic to G(a1 − 1, a2 − 2).

Now we want to count the number of holomorphic curves of degree A passing througha generic point p ∈ Fl(a;n) and Xw′w ⊂ Fl(a;n). Let u : CP1 → Fl(a;n) be one of


such holomorphic curves. The composition π u is holomorphic and (π u)∗[CP1] =π∗(A) = 0. Since Fl(a′;n) is a compact and connected Kähler manifold, the map π uis constant, which means that the image of u : CP1 → Fl(a;n) lies entirely in thefiber π−1(p) ∼= G(a1, a2) of π : Fl(a;n) → Fl(a′;n). Moreover, u : CP1 → π−1(p) ∼=G(a1, a2) ⊂ Fl(a;n) is a holomorphic line of the fiber π−1(p) ∼= G(a1, a2). If π(p) ∈ Cw′ ,

then the fiber π−1(p) intersects Xw′w in a variety isomorphic to G(a1 − 1, a2 − 2). Sincethere is just one holomorphic line passing through a generic point and G(a1 − 1, a2 − 2)in G(a1, a2) (by Lemma 4.2.3), we conclude that

GWJA,2(PD[p] ,PD[Xw′w]) = 1.

Thus, by Theorem 2.2.1 and Remark 2.2.4,

Gwidth(Hλ, ωλ) ≤ ωλ(A) = |λb2 − λb1 |.

4.4 Lower bounds of the Gromov width of coadjointorbits of type A

In [46] and [47], Pabiniak has shown that the upper bound appearing in Theorem 4.3.3is indeed the Gromov width of Hλ. In this section we give a short review of Pabiniak’sresults regarding the Gromov width of Hermitian matrices.

For a positive integer N, let N(π) = (x1, . . . , xN) ∈ RN : 0 < x1, · · · , xN < πand N(r) = 0 < y1, · · · , yN : y1 + · · · + yN < r. We endow the cartesian productN(π) × N(r) with the symplectic form

ω =n∑

i=1dxi ∧ dyi

4.4.1 Theorem. [47, Proposition 2.1] For any 0 < ρ < r, there is a symplectic embeddingof the ball B2N(√ρ) = (z1, . . . , zN) ∈ CN : |z1|2 + . . . + |zN |2 = ρ into (N(π) ×N(r), ω). In particular, if there exists a symplectic embedding of N(π) × N(r) intoa symplectic manifold (M,ω), then

Gwidth(M,ω) ≥ πr.


Proof. There is a symplectic embedding Ψ : N(π) × N(r) → B2N(√r) defined by

Ψ(x1, y1, · · · , xN , yN) = (√y1e−2ix1 , · · · ,√yNe

−2ixN )

LetSD(

√r) := B2(

√r)\x ≥ 0, y = 0 ⊂ B2(

√r) ⊂ R2

Denote by SDN(√r) := SD(

√r) × . . .× SD(

√r) ∈ R2N . We have that

Ψ(N(π) × N(r)) = B2N(√r) ∩ SDN(

√r).

For ρ < r, choose any area preserving diffeomorphism (so also preserving symplectic form)

σρ : B2(√ρ) → Im σρ ⊂ SD(√

ρ+ 1N

(r − ρ))

⊂ SD(√r)

such that if x2 + y2 ≤ a, then |σρ(x, y)2| ≤ a+ 1N

(r − ρ). Let Ψρ be the product of Nσρ’s:

Ψρ : B2(√ρ) × . . .×B2(√ρ) → SDN(√

ρ+ 1N

(r − ρ))

(x1, y1, . . . , xN , yN) 7→ (σρ(x1, y1), . . . , σρ(xN , yN))

The map Ψρ is symplectic as being the product of symplectic maps, and

Ψρ(B2N(√ρ)) ⊂ B2N(√r) ∩ SDN(

√r) = Ψ(N(π) × N(r)).

Thus, Ψ Ψρ gives symplectic embedding of B2N(√ρ) into N(π) × N(r).

Let T be a torus, and suppose that T acts on a symplectic manifold (M,ω) in a


Hamiltonian way with moment map µ : M → t∗. If the action of T on M is effective anddimT = 1

2dimM, the Hamiltonian action would be called toric. We call M a proper

Hamiltonian T -space if there exists an open and convex subset τ ⊂ t∗ containing µ(M)and such that the moment map µ : M → τ is proper as a map into τ.

Compact connected symplectic toric manifolds are classified by their momentum mapimage. Karshon-Lerman have generalized this theorem to the case of non-compact ma-nifolds with proper momentum map [31]. We have the following proposition as a conse-quence of their work:

4.4.2 Theorem. Let (M2N , ω) be a connected proper Hamiltonian TN -space with mo-mentum map µ : M → t∗. Let us suppose that there exists r > 0 and W ∈ Gl(n,Z)such that W (N(r)) ⊂ int(µ(M)), then

Gwidth(M,ω) ≥ r

Proof. Let S = W (N(r)) ⊂ int(µ(M)). The momentum map preimage µ−1(S) issymplectomorphic to TN × S (see for instance [31] or [47, Proposition 2.4]), whereTN × S is endowed with the symplectic form that it inherits as an open subset of thecotangent bundle of TN .

Note that we have a symplectic embedding

(0, 1)N × S → TN × S ∼= µ−1(S)

and symplectomorphisms

(0, π)N × N(r/π) ∼= (0, 1)N × N(r) ∼= (0, 1)N × S

By the previous Theorem,

Gwidth((0, 1)N × S) = Gwidth((0, π)N × N(r/π)) ≥ r,

and thusGwidth(M,ω) ≥ r.


4.4.1 Gelfand-Tsetlin Action

Let λ = (λ1, . . . , λn) ∈ Rn such that

λ1 ≥ . . . ≥ λn.

Let Hλ = A ∈ Mn(C) : A∗ = A, spectrum(A) = λ and N = dimC Hλ. For A ∈ Hλ,

letλ1 j(A) ≥ . . . ≥ λj j(A)

be the eigenvalues of the j × j top-left submatrix of A ordered in a non-increasing way.The min-max principle implies that

λj (l+1)(A) ≥ λj l(A) ≥ λ(j+1) (l+1)(A)

We have the following triangular array of inequalities

λ1 λ2 λ3 λn−1 λn

λ1 (n−1) λ2 (n−1) · · · λ(n−2) (n−1) λ(n−1) (n−1)

λ1 (n−2) λ(n−2) (n−2)

λ1 2 λ2 2

λ1 1

These inequalities define a polytope in Rn(n−1)/2, which we denote by P. The eigen-values λij, 1 ≤ i ≤ j < n, define a function

Λ : Hλ → Rn(n−1)/2

A 7→ (λi j(A))

whose image is the polytope P. The function Λ is called a Gelfand-Tsetlin functionand P the Gelfand-Tsetlin polytope associated with it. The number of non-constantcomponents of the Gelfand-Tsetlin function Λ : Hλ → Rn(n−1)/2 is equal to N, and wecan consider the Gelfand-Tsetlin function Λ as a function to RN .

The main properties of the Gelfand-Tsetlin function Λ are summarized in the followingproposition (see e.g. [46] for more details):

4.4.3 Proposition. The function Λ : Hλ → RN is smooth on an open-dense set U ⊂ Hλ.

Moreover, there exists a torus action TN y U ⊂ Hλ, called the Gelfand-Tsetlin action,


that is Hamiltonian on U with momentum map Λ|U : U ⊂ Hλ → RN and moment imageΛ(U) such that

int(P ) ⊂ Λ(U) ⊂ P

We want find r > 0 and W ∈ Gl(N,Z) such that W (N(r)) ⊂ int(P ). If so, we willhave that

Gwidth(Hλ, wλ) ≥ r

where ωλ is the symplectic form on Hλ defined in section 4.1.The image v := Λ(diagonal(λ1, . . . , λn)) ∈ P is a vertex in the Gelfand-Tsetlin

polytope P. Pabiniak has proved in [47, Lemma 4.1, Lemma 4.2] that there exists W ∈Gl(N,Z) such that for r = minλi =λj

|λi − λj| there is an inclusion

v +W (N(r)) ⊂ interiorP

and every edge of the simplex v+W (N(r)) is contained in an edge of the Gelfand-Tsetlinpolytope P with lattice length no smaller than minλi =λj

|λi − λj|.

By theorem 4.4.2,Gwidth(Hλ, ωλ) ≥ min

λi =λj

|λi − λj|

We summarize the results of this chapter in the following theorem

4.4.4 Theorem. For λ = (λ1, · · · , λn) ∈ Rn, let Hλ = A ∈ Mn(C) : A∗ =A, spec(A) = λ and ωλ be the Kostant-Kirillov-Souriau form defined on Hλ comingfrom its identification with a coadjoint orbit of U(n). Then,

Gwidth(Hλ, ωλ) = minλi =λj

|λi − λj|

Chapter 5

Upper Bound for the Gromov widthof Grassmannian Manifolds

Let G be a compact connected simple Lie group with Lie algebra g. Let λ ∈ t∗ ⊂ g∗, letOλ be the coadjoint orbit passing through λ and let ωλ be the Kostant-Kirillov-Souriauform defined on Oλ. Let S be a system of simple roots associated with a maximal torusT ⊂ G.

In this chapter we show that if there is a maximal parabolic subgroup P ⊂ GC

associated with a long simple root α ∈ S such that Oλ∼= GC/P, then

Gwidth(Oλ, ωλ) ≤ ⟨λ , α⟩

This upper bound would be obtained by computing a non-vanishing Gromov-Witten invari-ant with one of its constraints being Poincaré dual to a point. Having a explicit descriptionof the moduli space of holomorphic lines for Grassmannian manifolds associated with longsimple roots will allow us to compute such invariants.

The second part of this chapter is dedicated to the description of the upper bound forthe Gromov width of the Classical Grassmannian manifolds, such as Isotropic Grassman-nians and Orthogonal Grassmannians.

5.1 Fibrations

Let G be a compact simple Lie group. Let T ⊂ G be a maximal torus, B ⊂ GC be aBorel subgroup with TC ⊂ B and S be the corresponding system of simple roots. LetW = N(T )/T be the associated Weyl group. For a parabolic subgroup P ⊂ GC, B ⊂ P,

36

Chapter 5. The Gromov width of Grassmannian Manifolds 37

let WP = NP (T )/T be the Weyl group of P and SP be the subset of simple roots ofS whose corresponding reflections are in WP . Let W P ⊂ W be the set of all minimumlength representatives for cosets in W/WP . For w ∈ W P , let XP (w) ⊂ GC/P be theSchubert variety associated with w ∈ W P .

For a pair of parabolic subgroups P,Q ⊂ GC, such that B ⊂ P ⊂ Q, we have aquotient map GC/P → GC/Q. We want to study the images and preimages of Schubertvarieties under these quotient maps.

5.1.1 Lemma (Stumbo [53]). For parabolic subgroups P,Q ⊂ GC such that B ⊂ P ⊂Q define

W PQ := w ∈ WQ : l(ws) > l(w) for s ∈ SP

= minimum length representatives of elements in WQ/WP .

Given w ∈ W P , there is a unique wQ ∈ WQ and a unique wPQ ∈ W P

Q such that w =wQwP

Q. Their lengths satisfy l(w) = l(wQ) + l(wPQ).

5.1.2 Lemma. For parabolic subgroups P,Q ⊂ GC such that B ⊂ P ⊂ Q, let wpq , wp

and wq be the longest elements in W PQ ,WP and WQ, respectively. Then, wp

q = wqwp.

Proof. Let w0 be the longest element in W. The quotient map π : W → W P ∼= W/WP

is order preserving and thus the longest element in W P is π(w0). By the previous lemmaw0 = π(w0)wp, so that π(w0) = w0w

−1p . Similarly, for the quotient map π′ : W →

WQ ∼= W/WQ, we have that π′(w0) = w0w−1q is the longest element in WQ. Using again

the previous Lemma, we have that the permutation π′(w0)wpq is the longest permutation

in W P . So that π′(w0)wpq = π(w0), and thus w0w

−1q wq

p = w0w−1p or wp

q = wqw−1p . But

the longest element wp in WP satisfies w2p = e, and we are done.

5.1.3 Proposition. For parabolic subgroups P,Q ⊂ GC such that P ⊂ Q ⊂ GC, let

π : GC/P → GC/Q

be the corresponding quotient fibration. If we decompose w ∈ W P as wQwPQ, where

wQ ∈ WQ and wPQ ∈ W P

Q , then π(XP (w)) = XQ(wQ). On the other hand, if w ∈ WQ,

then π−1(XQ(w)) = XP (wwpq), where wp

q is the longest element in W PQ .

Proof. The map π : GC/P → GC/Q is B-equivariant and closed (this is a consequence offor example the closed map lemma). This implies that Schubert cells, which are B-orbits,


and Schubert varieties, which are their closures, in GC/P are mapped to Schubert cellsand Schubert varieties in GC/Q, respectively.

For w ∈ W P ⊂ W, there exist unique wQ ∈ WQ and wPQ ∈ W P

Q ⊂ WQ such thatw = wQwP

Q and l(w) = l(wQ) + l(wPQ). The Schubert cell CP (w) = BwP/P ⊂ GC/P

is mapped to the Schubert cell CQ(wQ) = BwQQ/Q ⊂ GC/Q via π, and

π(XP (w)) = π(CP (w)) = π(CP (w)) = CQ(wQ) = XQ(wQ).

On the other hand, if w ∈ WQ, then

π−1(CQ(w)) =⊔v∈W P

vQ=w

CP (v).

The maximum element, with respect to the Bruhat order defined on W P , in the setv ∈ W P : vQ = w is wwp

q , where wpq denotes the longest element in W P

Q . Since π is acontinuous map, we have that

π−1(XQ(w)) = π−1(CQ(w)) =⊔v∈W P

vQ=w

CP (v) =⊔v∈W P

v≤Bwwpq

CP (v) = XP (wwpq).

The following two technical lemmas would be needed it in the next section:

5.1.4 Lemma. Let α ∈ S be a simple root and N(α) ⊂ S be the neighbors of α inthe Dynkin diagram of G, i.e., the simple roots connected to α by an edge in the Dynkindiagram of G. Let P, P ′′ ⊂ GC be the parabolic subgroups such that SP = S\α, SP

′′ =S\(N(α) ∪ α). Then

W P′′

P · sα ⊂ W P

Proof. Let w ∈ W P′′

P . We write w = s1 · . . . · sr where s1, . . . , sr are simple reflections inSP . Suppose that there exists a simple reflection t in SP such that l(wsαt) < l(wsα). Bythe Exchange Principle (see e.g. Humphreys [28]),

wsαt = s1 · . . . · si · . . . · srsα

for some i, in particular sαtsα ∈ WP . We now consider two cases and see that this is notpossible:


1. Suppose that sαt = tsα. Thus t /∈ N(α) and

l(wsαt) = l(wtsα) = l(wt) + 1.

But t /∈ N(α) and t = sα, so t ∈ S\(N(α) ∪ sα) = SP′′ . As w ∈ W P

′′

P

l(wt) > l(w),

hencel(wsαt) = l(wtsα) = l(wt) + 1 > l(w) + 1 = l(wsα),

which contradicts our asumption of having l(wsαt) < l(wsα).

2. Suppose that sαt = tsα. If l(sαtsα) = 3, by the Deletion Principle (see e.g.Humphreys [28]) either sαtsα = sα, or sαtsα = t, which are not possible. Sol(sαtsα) = 3. Now, clearly l(sαt) = l(sαtsαsα) = 2 < l(sαtsα), so if sαtsα =s1s2s3, for some simple reflections s1, s2, s3 ∈ SP , by the Exchange Principlesαt ∈ WP which would imply that sα ∈ WP , a contradiction.

5.1.5 Lemma. Let α ∈ S be a simple root and N(α) ⊂ S be the neighbors of αin the Dynkin diagram of G. Let P, P ′ ⊂ GC be the parabolic subgroups such thatSP = S\α, SP ′ = S\N(α). Let P ′′ = P ∩ P

′, and; let π : GC/P

′′ → GC/P andπ′ : GC/P

′′ → GC/P be the natural defined quotient maps, so we have the diagram ofarrows

GC/P′′

GC/P′

GC/P

-π′

?

π


For any subset X of GC/P, we define

X := π′(π−1(X)) ⊂ GC/P′.

There exists w ∈ W P such that the fundamental class [XP (w)] ∈ H∗(GC/P′,Z) is

opposite to the fundamental class [XP (e)] ∈ H∗(GC/P′,Z), where e denotes the identity

of W.

Proof. For a permutation w ∈ W P and the corresponding Schubert variety XP (w) ⊂GC/P, the set XP (w) is a B-stable Schubert variety in GC/P

′. We will denote by w the

permutation in W P′

such thatXP (w) = XP ′ (w)

Let wp′′p , w

p′′

p′ , wp′′ , wp′ , wp and w0 be the longest elements in W P′′

P ,W P′′

P′ ,WP ′′ ,WP ′ ,WP

and W, respectively. We want to find a permutation w ∈ W P such that

[XP ′ (w)] = [XP ′ (e)]op = [XP ′ (w0ewp′)],

or equivalently a permutation w ∈ W P such that w = w0ewp′ .

Let us find first an expression for e : by the Proposition 5.1.3, we have that π−1(XP (e)) =XP ′′ (wp′′

p ), soXP

′ (e) = π′(π−1(XP (e)) = π′(XP′′ (wp′′

p )).

Note that W P′′

P = W P′ ∩P

P ⊂ W P′, in particular wp′′

p ∈ W P′, and XP ′ (e) = XP ′ (wp′′

p ),and as a consequence e = wp′′

p , or that is the same

π′(π−1(XP (e)) = XP ′(wp′′

p ) (5.1.6)

Remember that we want to find w ∈ W P such that w = w0ewp′ . If we findw ∈ W P such that wwp′′

p ∈ W P′

and wwp′′p = w0w

p′′p wp′ , then w = wwp′′

p andw = w0ewp′ . If such w exists, it should be equal to w0w

p′′p wp′(wp′′

p )−1. We have toverify that w0w

p′′p wp′(wp′′

p )−1 ∈ W P and w0wp′′p wp′ ∈ W P

′.

Letw = w0w

p′′

p wp′(wp′′

p )−1.

Notice first that W P′′

P ′ ∼= WP′/WP

′′ ∼= ⟨sα⟩ ∼= Z2, so wp′′

p′ = sα. We also know fromLemma 5.1.2 that wp′′

p = wpwp′′ and wp′′

p′ = wp′wp′′ , thus


w = w0wpwp′′wp′wp′′wp = (wpwp′′sα)∗P = (wp′′

p sα)∗P (5.1.7)

So w ∈ W P if and only if wp′′p sα ∈ W P , which we already know from the previous Lemma,

and we are done.

5.2 Upper Bound for the Gromov width of Grassman-nian Manifolds: long root case

Let G be a compact Lie group, g be its Lie algebra and g∗ be the dual of this Lie algebra.Let λ ∈ g∗ and Oλ ⊂ g∗ be the coadjoint orbit passing through λ. Let us assume thatOλ

∼= GC/P, where P ⊂ GC is a parabolic subgroup of GC. Let T ⊂ G be a maximaltorus and let B ⊂ GC be a Borel subgroup with TC ⊂ B ⊂ P. Let W = N(T )/Tbe the associated Weyl group. Let R be the corresponding set of roots and S be thecorresponding system of simple roots. Let WP be the Weyl group of P and SP be thesubset of simple roots whose corresponding reflections are in WP .

If there exists a simple root α ∈ S such that SP = S\α, we say that the parabolicsubgroup P ⊂ GC is the maximal parabolic of GC associated with the simple rootα ∈ S and we will call the corresponding homogeneous space GC/P a Grassmannianmanifold.

We will assume from now in this section that P ⊂ GC is a maximal parabolic subgroupassociated with a simple root α ∈ S and we will endow GC/P with a Kähler structurecoming from its identification with Oλ. This Kähler structure and the one defined on Oλ

would be denoted indistinguishably by (ωλ, J).The second homology group H2(GC/P,Z) of a Grassmannian manifold GC/P is cyclic

and is freely generated as a Z-module by the fundamental class of the Schubert varietyXP (sα). From now, we will denote this fundamental class by A.

Let MA(GC/P, J) be the moduli space of J-holomorphic curves of degree A. We willcall elements of the moduli space MA(GC/P, J) holomorphic lines of GC/P or justlines. Let MA, k(GC/P, J) be the moduli space of J-holomorphic maps of degree A withk-marked distinct points.

The complex group GC acts holomorphically on GC/P and trivially on H∗(GC/P,Z),as a consequence there is a group action of GC on the moduli space MA, k(GC/P, J).We will show that this action is transitive when k = 0, 1 and the simple root α is long.


First, we state the following lemma:

5.2.1 Lemma. Let β be a positive root and α be a simple root such that |α| = | β| andlet us assume that

β = α+∑

γ∈S\αnγγ

for some non-negative integers nγ. Then there exists w ∈ WP , such that wsα = sβ.

Proof. Let ζ = β − α = ∑γ∈S\α nγγ.

We use induction on the height of β. If ht β = 1, then β = α and we are done.Suppose ht β > 1. We claim that there exists γ ∈ S\α such that (β, γ) > 0. Supposethe opposite. Then (β, γ) ≤ 0 for all γ ∈ S\α. As a consequence, (β, α) > 0.Since sαβ = β − (β, α)α is a positive root, we must have (β, α) = 1, and sαβ =β − (β, α)α = ζ. We have (β, ζ) ≤ 0. But ζ, β, α have the same length, so we deducethat (ζ, α) = (α, ζ) = −1, and then (β, ζ) = 1, getting a contradiction.

Let γ ∈ S\α such that (β, γ) > 0. We have that ht sγ(β) < ht(β) and sγ(β)satisfies all the assumptions of our lemma. It follows that there exists w′ ∈ WP withw′sα = sγβ. Thus, sγw

′sα = sβ, proving the claim.

5.2.2 Theorem (Manivel-Landsberg [36], Strickland [52]). Let α ∈ S be a longsimple root and N(α) be the neighbors of α in the Dynkin diagram of G. Let P ′ ⊂ GC

be the parabolic subgroup with SP′ = S\N(α) and let P ′′ = P

′ ∩ P.

The group action of GC on the moduli spaces MA,0(GC/P, J),MA,1(GC/P, J) istransitive; and, the moduli spaces MA,0(GC/P, J),MA,1(GC/P, J) are isomorphic tothe homogeneous spaces GC/P

′, GC/P

′′, respectively.

Proof. We first claim that any element of MA,0(GC/P, J) is GC-equivalent to a T -invariant curve of GC/P. Let wα ∈ t∗ be the fundamental weight associated with thecoroot α ∈ t and let Vwα be the irreducible representation with highest weight vectorwα. We have a GC-equivariant projective embedding ι : GC/P → PVwα that maps anycurve in MA(GC/P, J) to a curve of degree one in PVwα , i.e, a projective line (seeEquation 3.3.5). Thus any J-holomorphic curve u : CP1 → GC/P is determined, up toparametrization, by two different points, i.e., if there exists another J-holomorphic curvev : CP1 → GC/P and two different points p, q ∈ GC/P such that both curves u andv pass through p and q, then there exists f ∈ PSL(2,C) such that v f = u. Thisis because ι u : CP1 → PVwα and ι u : CP1 → PVwα are lines in PVwα and thusdetermined by two points, so there exists f ∈ PSL(2,C) such that ι v f = ι u; bythe injectivity of ι, we conclude that v f = u.


Now let u : CP1 → GC/P be any J-holomorphic curve of degree A. Let p = 1 · P ∈GC/P. Since the action of GC on GC/P is transitive, there exists g ∈ GC such thatg · u : CP1 → GC/P passes through p. Take now any q ∈ g · u(CP1) ⊂ GC/P. Theelement q lies in a unique Schubert cell C(w) = BwP/P, for some w ∈ W P . There is anelement h ∈ B such that hq = wp ∈ GC/P. Thus (hg) · u := u′ : CP1 → GC/P passesthrough the two T -fixed points p and wp, and u′ ∈ MA,0(GC/P, J) is T -invariant.

In conclusion, any holomorphic line in MA,0(GC/P, J) is GC-equivalent to a T -invariant curve of GC/P.

Now we are going to prove that if α is a long root, the action of GC on MA,0(GC/P, J)is transitive. Take u to be Cα, the T -invariant J-holomorphic curve that contains p andsαp, and u′ to be Cβ, the T -invariant J-holomorphic curve that contains p and sβp,

for some β ∈ R+\R+P . We assume that u, u′ : CP1 → GC/P have the same degree in

H2(GC/P,Z). Thus they have the same degree with respect to the projective embeddingGC/P → PVwα and

⟨wα, α⟩ = 1 = ⟨wα, β⟩

We can write β = ∑γ∈S\α nγγ + nα, for some non-negative integer numbers nγ, n.

Hence,⟨wα, β⟩ = 2(wα, β)

(β, β)= n(α, α)

(β, β).

where (·, ·) is an ad-invariant inner product defined on g. If α is a long root, then n = 1.Hence, β = α + ∑

γ∈S\γ nγαγ. By the previous Lemma, there exists w ∈ W, such thatwsα = sβ. So, in conclusion, the action of GC on MA,0(GC/P, J) is transitive.

Now we are going to prove that the stabilizer of u in GC/P is the parabolic subgroupP

′ ⊂ GC such that SP ′ = S\N(α), where N(α) denotes the neighbors of α in the Dynkindiagram of G.

The curve u is a B-stable Schubert variety so that its stabilizer is a parabolic subgroupP

′ ⊂ GC, B ⊂ P′. The parabolic subgroup P ′ ⊂ GC is determined by the simple reflec-

tions sβ that fixes the curve Cα. In other words, P ′ ⊂ GC is determined by the simplereflections sβ such that sβ · p, sαp = p, sαp. Clearly sα is in the stabilizer of Cα.

We assume that β = α. This implies that sβ ∈ WP ; and, we must have sβsα = sα · P.Last equation is the same as having sαsβsα ∈ WP . As sβ ∈ WP and sα /∈ SP , we musthave sαsβ = sβsα, i.e., β is not a neighbor of α in the Dynkin diagram.

In conclusion, the parabolic subgroup P′ ⊂ GC of Cα is such that SP

′ = S\N(α),


where N(α) denotes the neighbors of α in the Dynkin diagram of G, and thus

MA,0(GC/P, J) ∼= GC/P′.

The proof of the second part of the statement follows from all these considerations.

Recall that we want to prove that there exist a cycle X ⊂ GC/P so the Gromov-Witten invariant GWA,2(PD[p] ,PD[X]) is different from zero. If so, by Remark (2.2.4)we will have that

Gwidth(GC/P, ωλ) ≤ ωλ(A)

5.2.3 Theorem. Let G be a compact connected simple Lie group with Lie algebra g. Letλ ∈ t∗ ⊂ g∗, let Oλ be the coadjoint orbit passing through λ and let ωλ be the Kostant-Kirillov-Souriau form defined on Oλ. Assume that there is a long simple root α ∈ S suchthat Oλ

∼= GC/P, thenGwidth(Oλ, ωλ) ≤ ⟨λ, α⟩

Proof. Let f : MA,1(GC/P, J) → MA,0(GC/P, J) be the forgetful map that maps a pair[u, z] to [u] and evJ : MA,1(GC/P, J) → GC/P be the evaluation map that maps a pair[u, z] to u(z). We have a diagram of arrows

MA,1(GC/P, J) MA,0(GC/P, J)

GC/P

-f

?

evJ

Let N(α) be the neighbors of α in the Dynkin diagram of G. Let P ′ ⊂ GC be the parabolicsubgroup with SP ′ = S\N(α) and let P ′′ = P

′ ∩ P. By Theorem 5.2.2, we have thatMA,0(GC/P, J) ∼= GC/P

′ and MA,0(GC/P, J) ∼= GC/P′′. The diagram of arrows shown

above can be identified with the diagram


GC/P′′

GC/P′

GC/P

-π′

?

π

where π and π′ denote the projection quotient maps (see e.g. [52, Theorem 1]).For any subset X of GC/P, we define

X := f(ev−1J (X)) = u ∈ MA,0(GC/P ) : u is incident to X ⊂ MA,0(GC/P )

The set X = f(ev−1J (X)) ⊂ MA,0(GC/P ) can be identified with the set π′(π−1(X)) ⊂

GC/P′.

By Lemma 5.1.5, there exists a permutation w ∈ W P such that the fundamental class[XP (w)] ∈ H∗(GC/P

′,Z) is dual to the fundamental class of [XP (e)] ∈ H∗(GC/P′,Z).

This implies that for generic g ∈ GC, the Schubert variety XP (w) intersects transversallygXP (e) at one point in MA,0(GC/P, J) ∼= GC/P

′. In other words, for generic g ∈ GC,

there is one holomorphic line in GC/P passing through g·P ∈ GC/P andXP (w) ⊂ GC/P.

Bertini-Kleiman’s Theorem implies that for generic g ∈ GC the evaluation map

evJ : MA,2(GC/P, J) → (GC/P )2

is transverse to g · P ×XP (w) ⊂ (GC/P )2.

Note that the Schubert variety XP (w) ⊂ GC/P satisfies the dimensional constraint

dimCXP (w) + dimC MA,2(GC/P ) = 2 dimCGC/P :

The permutation w ∈ W P is given by Equation 5.1.7, so by construction of w we havethat

dimCXP (w) = dimC(GC/P ) − l(wp′′

p sα),


where wp′′p is the longest element in W P

′′

P . Moreover,

l(wp′′

p sα) = l(wp′′

p ) + 1 = dimC(P/P ′′) + 1 = dimC(GC/P′′) − dimC(GC/P ) + 1

= dimC MA,1(GC/P ) − dimC(GC/P ) + 1

So in conclusion,

dimCXP (w) = 2 dimCGC/P − dimC MA,1(GC/P ) − 1

= 2 dimCGC/P − dimC MA,2(GC/P ).

By Proposition 7.4.5 of [43], the Gromov-Witten invariant

GWA,2(PD[XP (e)] ,PD[XP (w)])

is positive, and thus by Remark 2.2.4

Gwidth(Oλ, ωλ) ≤ ωλ(A) = ⟨λ, α⟩.

5.3 Upper bounds for the Gromov width of IsotropicGrassmannians

In this section we are going to find upper bounds for the Gromov width of Grassmannianmanifolds of type C better known as Isotropic Grassmannians.

Let (C2n,Ω) be the standard complex symplectic vector space with complex coordi-nates (z1, · · · , zn, w1, · · · , wn) and with complex bilinear skew-symmetric form

Ω =∑

dzi ∧ dwi.

Let Sp(n,C) be the complex Lie group of linear transformation on C2n that preservesΩ. Let Sp(n) = Sp(2n,C) ∩ U(2n) be the symplectic group of quaternionic unitarytransformations. The group Sp(n) is a compact form of Sp(n,C).

The set of diagonal matrices T in Sp(n) forms a maximal torus in Sp(n) with Liealgebra equals to t = i diag(λ,−λ) ∈ M2n(C) : λ ∈ Rn ∼= iRn.

Let ei : t∗ → R be the projection that maps a matrix i diag(λ,−λ) ∈ t to the i-th


component λi of λ ∈ Rn. The root system of Sp(n) with respect to the maximal torusT is the set R = ±ei ± ej (i = j), ±2ei1≤i,j≤n, with a choice of simple roots given byS = α1 = e1 − e2, α2 = e2 − e3, · · · , αn−1 = en−1 − en, αn = 2en and Dynkin diagram

Let λ ∈ Rn and

Cλ := A ∈ M2n(C) : A = QATQ,A∗ = −A, spectrumA = i(λ,−λ)

where Q is the matrix 0 −In

In 0

∈ M2n(C)

The compact group Sp(n) acts transitively on Cλ by conjugation. Indeed, the set ofmatrices Cλ corresponds to an adjoint orbit of Sp(n), and it can be identified with acoadjoint orbit of Sp(n) via an Ad-invariant product defined on the Lie algebra sp(n).Let (ωλ, J) be the Kähler structure defined on Cλ by identifying it with a coadjoint orbitof Sp(n) via an Ad-invariant inner product.

The set of matrices Cλ is isomorphic to a quotient of the form Sp(n,C)/P, whereP is a parabolic subgroup of Sp(n,C). Also, there exists a sequence of positive integersa1 ≤ a2 ≤ · · · ≤ ak ≤ n such that the homogeneous space Sp(n,C)/P is isomorphic tothe variety of isotropic flags

V a1 ⊂ V a2 ⊂ · · · ⊂ V ak ⊂ C2n : Ω|V ai = 0, 1 ≤ i ≤ k,

(see e.g. Section 23.3 of Fulton-Harris [21]).For an integer 0 < k ≤ n, let IG(k, 2n) denote the space of k-dimensional isotropic

subspaces of C2n, i.e,

IG(k, 2n) := V k ∈ G(k, 2n) : Ω|V k = 0.

When k = n, the isotropic Grassmannian IG(n, 2n) is the space of Lagrangian subspacesof C2n. The isotropic Grassmannian manifold IG(k, 2n) has dimension equal to

2k(n− k) + k(k + 1)2

and is isomorphic to the complex quotient Sp(2n,C)/Pαk, where Pαk

⊂ Sp(n,C) is the


maximal parabolic subgroup associated with the simple roots αk ∈ S.

When the components of λ = (λ1, . . . , λn) ∈ Rn are of the form

λ1 = · · · = λk > λk+1 = · · · = λn = 0,

the set of matrices Cλ is isomorphic to the isotropic Grassmannian manifold IG(k, n).LetA be the effective cyclic generator of the second homology groupH2(IG(k, 2n),Z).

The moduli space of (unparameterized) J-holomorphic curves of degree A in IG(k, 2n)is given by

MA,0(IG(k, 2n), J) ∼= (V k−1, V k+1) : V k−1 ⊂ V k+1 ⊂ (V k−1)Ω ⊂ C2n

We will call the elements of this moduli space holomorphic lines in IG(k, 2n). Theholomorphic line associated with a pair (V k−1, V k+1) ∈ MA,0(IG(k, 2n), J) consists ofall the isotropic subspaces V k ∈ IG(k, 2n) with

V k−1 ⊂ V k ⊂ V k+1

If a pair of isotropic subspaces V k1 , V

k2 ∈ IG(k, 2n) are collinear, i.e., there exists

(V k−1, V k+1) ∈ MA,0(IG(k, 2n), J) such that

V k−1 ⊂ V k1 , V

k2 ⊂ V k+1,

then V k−1 = V k1 ∩ V k

2 and V k+1 = V k1 + V k

2 (see e.g. Cohen-Cooperstein [13, Section3.4], Landsberg-Manivel [36, Remark 5.7], Strickland [52, Proposition 3] for more details).

5.3.1 Remark. The moduli space of (unparameterized) homolomorphic lines in the Grass-mannian manifold G(k, 2n) of k-dimensional vector subspaces of Cn is isomorphic to thetwo-step flag manifold

Fl(k − 1, k + 1; 2n) := (V k−1, V k+1) : V k−1 ⊂ V k+1 ⊂ C2n

We have a natural defined embedding of the isotropic Grassmannian IG(k, 2n) into thestandard Grassmannian G(k, 2n). With respect to this embedding, a holomorphic line inthe isotropic Grassmannian IG(k, 2n) is a holomorphic line in the standard GrassmannianG(k, 2n) that is totally contained in the isotropic Grassmannian IG(k, 2n). Also, a linein G(k, 2n) that contains two isotropic vector subspaces in IG(k, 2n) is totally containedin IG(k, 2n).


The dimension of MA,0(IG(k, 2n)) can be computed by considering the fibration

π : MA,0(IG(k, 2n)) → IG(k − 1, 2n)

(V k−1, V k+1) → V k−1.

This fibration has fiber isomorphic to G(2, 2n− 2k + 2), so that

dimC MA,0(IG(k, 2n)) = dimC IG(k − 1, 2n) + dimCG(2, 2n− 2k + 2)

= k(k − 1)2

+ 2(k − 1)(n− k + 1) + 2(2n− 2k)

= −3k2

2+ 2kn− k

2+ 2n− 2

= dimC IG(k, 2n) − k + 2n− 2

Just as before, we want to find a cycle X ⊂ IG(k, 2n) such that for a generic isotropicsubspace V k ∈ IG(k, 2n), the Gromov-Witten invariant GWA,2(PD[p] ,PD[X]) wouldbe different from zero; if so, we will have that

Gwidth(IG(k, 2n), ωλ) ≤ ωλ(A).

We claim that the Grassmannian submanifold

X = Σk ∈ IG(k, 2n) : C ⊂ Σk ⊂ CΩ ∼= C2n−1 ⊂ IG(k, 2n)

satisfies this condition. Note that X can be identified with the isotropic GrassmannianIG(k − 1, 2(n− 1)).

5.3.2 Theorem. Let X = Σk ∈ IG(k, 2n) : C ⊂ Σk ⊂ CΩ ∼= C2n−1 ⊂ IG(k, 2n),and p ∈ IG(k, 2n). Then

GWA,2(PD[p] ,PD[X]) = 0.

Proof. We check first that X satisfies the dimensional constraint

dimCX = 2 dimC IG(k, 2n) − dimC MA,0(IG(k, 2n), J) − 2 :


dimCX = dimC IG(k − 1, 2(n− 1))

= 2(k − 1)(n− k) + (k − 1)k2

= 2k(n− k) + k(k + 1)2

+ k − 2n

= dimC IG(k, 2n) + k − 2n

= 2 dimC IG(k, 2n) − dimC MA,0(IG(k, 2n), J) − 2

Now we prove that the Gromov-Witten invariant GWA,2(PD[p] ,PD[X]) is non-zero byconsidering the embedding of the isotropic Grassmannian IG(k, 2n) in the standard Grass-mannian G(k, 2n) :

Let Σk ∈ IG(k, 2n) → G(k, 2n). There is a unique holomorphic line in G(k, 2n)passing through Σk ∈ G(k, 2n) and

Y = V k ∈ G(k, 2n) : C ⊂ V k ⊂ CΩ ⊂ G(k, 2n),

this line intersects Y at Γk = C ⊕ (Σk ∩ CΩ) in G(k, 2n) (see proof of Lemma 4.2.3).Note that Γk is a isotropic subspace of Cn because Σk is isotropic. This implies that theline in G(k, 2n) passing through Σk and Y ⊂ G(k, 2n) is totally contained in IG(k, 2n)and intersects X = Y ∩ IG(k, 2n). Thus,

GWA,2(PD[p] ,PD[IG(k − 1, 2n− 2)]) = 0

Now we state our upper bound for the Gromov width of Isotropic Grassmannanianmanifolds:

5.3.3 Theorem. For λ = (λ1, . . . , λn) ∈ Rn such that

λ1 = · · · = λk > λk+1 = · · · = λn = 0,

letCλ = A ∈ M2n(C) : A = QATQ,A∗ = −A, spectrumA = i(λ,−λ),


In 0

∈ M2n(C)


Let ωλ be the symplectic form defined on Cλ by identifying it with a coadjoint orbit ofSp(n). Then,

Gwidth(Cλ, ωλ) ≤ λ1

Proof. We have that Cλ∼= IG(k, 2n) ∼= Sp(2n,C)/Pαk

where Pαk⊂ Sp(2n,C) is the

maximal parabolic subgroup associated with the simple root αk. Let A ∈ H2(A,Z) bethe effective cyclic generator. The symplectic area of A is equal to

ωλ(A) = ⟨λ1, αk⟩ = λ1

The result now follows from Theorem 5.3.2 and Remark 2.2.4.

5.4 Upper bounds for the Gromov width of OrthogonalGrassmannians

In this section we are going to write the statement of Theorem 5.2.3 for Grassmannianmanifolds of type B and D, better known as Orthogonal Grassmannians.

For a positive integer m, let SO(m) be the group of special orthogonal transformationson Rm which preserves the standard symmetric bilinear form defined on Rm. We will writem as 2n if m is even, and as 2n+1 if m is odd (here n is a non-negative integer number).For θ = (θ1, · · · , θn), λ = (λ1, · · · , λn) ∈ Rn, we define the matrices Rn(θ), In(λ) ∈M2n(R) as

In(λ) :=

0 λ1

−λ1 0. . .

0 λn

−λn 0

, Rn(θ) :=

cos θ1 sin θ1

− sin θ1 cos θ1. . .

cos θn sin θn

− sin θn cos θn

We make the following choice of maximal compact torus for SO(m) depending on m

being either of the form 2n+ 1 or 2n :

TSO(2n) = Rn(θ) ∈ M2n(R) : θ ∈ Rn, TSO(2n+1) =

Rn(θ) 0

0 1

∈ M2n+1(R) : θ ∈ Rn


with corresponding Lie algebras

tSO(2n) := In(λ) ∈ M2n(R) : λ ∈ Rn, tSO(2n) :=

In(λ) 0

0 0

∈ M2n+1(R) : λ ∈ Rn

These Lie algebras are identified with its corresponding duals t∗SO(2n+1), t

∗SO(2n) via an

Ad-invariant inner product. Let eini=1 be the dual basis in t∗SO(m) associated to the

standard basis of t ∼= Rn.

The root system for the group SO(2n+1), with respect to the chosen maximal torus,is the set ±ei, ±(ej ± ek) : j = k1≤i,j≤n ⊂ t∗SO(2n+1)

∼= Rn with a choice of simpleroots given by S = α1 = e1 − e2, · · · , αn−1 = en−1 − en, αn = en, and Dynkin diagram

The root system for the group SO(2n) is the set ±(ej ±ek) : j = k1≤i,j≤n with simpleroots given by S = α1 = e1 − e2, · · · , αn−1 = en−1 − en, αn = en−1 + en, in the Liealgebra t∗SO(2n)

∼= Rn, and with Dynkin diagram

Every real skew-symmetric matrix in so(m) can be diagonalized by orthogonal trans-formations to a matrix in the Lie algebra tSO(m). For λ ∈ Rn, we denote by Sλ theset of real skew-symmetric matrices of size m × m that can be diagonalized by or-thogonal transformations to the matrix In(λ) ∈ M2n(R), if m = 2n; or to the matrixIn(λ) 0

0 0

∈ M2n+1(R), if m = 2n+ 1.

If m = 2n+ 1 is odd, the matrices in Sλ can be diagonalized with special orthogonaltransformations to the corresponding matrix in tSO(m). The same is true if m = 2n is evenand at least one component of λ ∈ Rn is zero. If m = 2n is even, and all the componentsof λ are different from zero, the set of matrices Sλ has two SO(m)-orbits which consist ofthe skew symmetric matrices in Sλ with positive and negative Pfaffian. We denote thesetwo orbits by S±

λ .


LetO(m,C) be the group of complex orthogonal matrices which preserves the standardnondegenerate symmetric bilinear form defined on Cm, and let SO(m,C) be the set ofcomplex orthogonal matrices in O(m,C) with determinant one.

Let k ≤ m/2 be a positive integer. We denote by OG(k,m) the Orthogonal Grass-mannian manifold of k-dimensional isotropic subspaces in Cm with respect to the stan-dard nondegenerate symmetric bilinear form defined on Cm.

Witt’s theorem states that any isometry between two subspaces of Cm can be extendedto an isometry of the whole space (see e.g. [44]). As a consequence, given any twoisotropic subspaces in Cm of the same dimension, they can be mapped to the other by acomplex orthogonal transformation of Cm. Thus the complex orthogonal group O(m,C)acts transitively on the Orthogonal Grassmannian manifold OG(k,m).

When k = m/2, the group SO(m,C) acts transitively on OG(k,m) and the orthog-onal Grassmannannian OG(k,m) is isomorphic to a the quotient SO(m,C)/Pαk

.

When k = m/2 = n, the orthogonal Grassmannian OG(n, 2n) is the union of twoSO(2n,C)-orbits. These two SO(2n,C)-orbits are isomorphic to SO(2n,C)/Pαn−1 andSO(2n,C)/Pαn .

If the components of λ = (λ1, . . . , λn) ∈ Rn are of the form

λ1 = · · · = λk > λk+1 = · · · = λn = 0,

the set of skew-symmetric matrices Sλ is isomorphic to the orthogonal GrassmannniansOG(k, 2n) and OG(k, 2n+ 1). When the components of λ are of the form

λ1 = · · · = λn > 0,

the two SO(2n,C)-orbits of the orthogonal Grassmannnian OG(n, 2n) are isomorphicwith the two SO(2n)-orbits of Sλ, S+

λ and S−λ . Both of them are isomorphic to the

orthogonal Grassmannian OG(n, 2n+ 1).

5.4.1 Theorem. Let m be a positive integer number that we will denote by 2n if it iseven, and by 2n+ 1 if it is odd. Let λ = (λ1, . . . , λn) ∈ Rn such that its components areof the form

λ1 = · · · = λk > λk+1 = · · · = λn = 0,

where k is an integer such that 0 < k ≤ m

2. Let Sλ be the set of m×m skew-symmetric


matrices that can be diagonalized by orthogonal transformations to the matrix

0 λkIk×k

−λkIk×k 00(m−2k)×(m−2k)

∈ Mm(R)

Let ωλ be the symplectic form defined on Sλ by identifying it with a coadjoint orbit ofthe orthogonal group of matrices via an Ad-invariant inner product. Then,

Gwidth(Sλ, ωλ) ≤

λk if k < (m− 1)/2

2λk if k = (m− 1)/2

If k = m/2, the set of skew symmetric matrices Sλ has two connected S+λ and S−

λ , inthis case

Gwidth(S+λ , ωλ) = Gwidth(S−

λ , ωλ) ≤ 2λk

Proof. When k = m/2, the set of skew symmetric matrices Sλ is isomorphic withSO(m,C)/Pαk

, where αk is the simple root corresponding to the k-node in the Dynkindiagram of SO(m).

If k < (m− 1)/2, the simple root αk is long and by Theorem 5.2.3

Gwidth(Sλ, ωλ) ≤ ωλ(A) = ⟨λ, αk⟩ = λk

When k = (m − 1)/2, the set of skew symmetric matrices Sλ is isomorphic withSO(2n,C)/Pαn . Since the root αn is long, we have that

Gwidth(Sλ, ωλ) ≤ ωλ(A) = ⟨λ, αn⟩ = 2λn = 2λk

Finally, when k = m/2, the two connected components S±λ of Sλ are isomorphic with

SO(2n,C)/Pαn∼= SO(2n,C)/Pαn−1 , and as consequence

Gwidth(S+λ , ωλ) = Gwidth(S−

λ , ωλ) ≤ ⟨λ, αn⟩ = 2λn = 2λk


5.5 Upper bound for the Gromov width of coadjointorbits of the exceptional group G2

Let G = G2 and let T ⊂ G be the maximal torus whose Lie algebra t is identified withR2 and such that the set

S =α1 =

(−3

2,

√3

2), α2 = (1, 0)

⊂ t∗ ∼= R2

defines a set of simple root systems for G with Dynkin diagram

5.5.1 Theorem. For λ = (λ1, λ2) ∈ t∗, let Oλ be the G2-coadjoint orbit that passesthrough λ and let ωλ be the Kostant-Kirillov-Souriau form defined on it. Then

Gwidth(Oλ, ωλ) ≤

√

3λ2

3if λ1 = 0

2λ1 if λ2 =√

3λ1

Proof. If λ1 = 0, then Oλ∼= G2/Pα1 , where Pα1 ⊂ GC is the maximal parabolic subgroup

associated with the simple root α1 ∈ S. Since the root α1 is long, we have by Theorem5.2.3 that

Gwidth(Oλ, ωλ) ≤ ⟨λ, α1⟩ =√

3λ2

3On the other hand, if

√3λ2 = 3λ1, then Oλ

∼= GC/Pα2 , where Pα2 ⊂ GC is the maximalparabolic subgroup associated with the simple root α2 ∈ S. The homogeneous spaceGC/Pα2 can be considered as a homogenous space of type SO(7,C) : Let w1 ∈ t∗

be the fundamental weight associated with α1. Let L(w1) = GC ×P C(w1) be the linebundle defined over GC/Pα2 associated with the fundamental weight w1. The irreduciblerepresentation H0(GC/Pα2 , L(w1)) has dimension 7 (this computation can be made byusing for instance the Weyl dimensional formula). Thus, GC/Pα2 is embedded as a non-degenerate quadric in the 6 dimensional projective space P(H0(GC/Pα2 , L(w1))) ∼= CP6.

A quadric in CP6 is a complete homogeneous space for the special orthogonal groupSO(7,C). Now, if we consider our quadric as a homogeneous space of SO(7,C), then byTheorem 5.2.3,

Gwidth(Oλ, ωλ) ≤ ⟨λ, α2⟩ = 2λ1

Chapter 6

Curve Neighborhoods, GromovWitten invariants and Gromov’s width

In this chapter we introduce the concept of curve neighborhood of a variety in a coadjointorbit. Let G be a compact Lie group and P be a parabolic subgroup of GC. For a varietyZ ⊂ GC/P and a homology class A ∈ H2(GC/P,Z), the degree A neighborhood ΓA(Z)of Z ⊂ GC/P is the closure of the union of all stable curves of degree A that meet Z.

Curve neighborhoods have been defined by Buch-Mihalcea in [7], and they have beenused for computing Gromov-Witten invariants on the homogeneous space GC/P.

We will focus our attention to when P ⊂ GC is a maximal parabolic subgroup, whenthe variety Z is the point 1 · P ∈ GC/P and when A ∈ H2(GC/P,Z) is the effectivecyclic generator. We show that under these circumstances

GWA,2(PD[ΓA(Z)]op,PD[1 · P ]) = 1

This result will provide us upper bounds for the Gromov width of coadjoint orbits isomor-phic with Grassmannian manifolds of any type.

6.1 Curve neighborhoods of Schubert varieties

Let G be a compact Lie group, g be its Lie algebra and g∗ be the dual of g. Let (· , ·)denote an Ad-invariant inner product defined on g. We identify the Lie algebra g and itsdual g∗ via this inner product.

Let λ ∈ g∗ and Oλ ⊂ g∗ be the coadjoint orbit passing through λ. Let P ⊂ GC be aparabolic subgroup of GC such that Oλ

∼= GC/P. Let T ⊂ G be a maximal torus and let

56

Chapter 6. Curve Neighborhoods and Gromov’s width 57

B ⊂ GC be a Borel subgroup with TC ⊂ B ⊂ P. Let Bop be the Borel subgroup oppositeto B.

Let R ⊂ t∗ be the root system of T in G. Let R+ ⊂ R be a system of positive rootswith simple roots S ⊂ R+. Let W = NG(T )/T be the Weyl group of G. For every rootα ∈ R, let sα ∈ W be the reflection associated to it.

For the parabolic subgroup P ⊂ GC, let WP = NP (T )/T be the Weyl group of Pand SP ⊂ S be the subset of simple roots whose corresponding reflections are in WP . LetW P denote the set of minimum length representatives of W in WP . If w ∈ W, we willdenote by wP its minimum length representative in WP .

For w ∈ W P , let CP (w) = BwP/P ⊂ GC/P and CopP (w) = BopwP/P ⊂ GC/P

be the Schubert cell and the opposite Schubert cell associated with w, respectively. TheSchubert variety XP (w) and its opposite Xop

P (w) are the closure of the Schubert cellsCP (w) and Cop

P (w), respectively.Each root α ∈ R has a coroot α ∈ t. The coroot α is identified with 2α

(α, α)∈ t via the

invariant inner product (· , ·). The coroots form the dual root system R = α : α ∈ R,with basis of simple coroots S = α : α ∈ S. For α ∈ R we let ωα ∈ RR denote thefundamental weight defined by

(ωα, β) = δα,β

for α ∈ R.

The cohomology group H2(GC/P ;Z) can be identified with the span

Zωα : α ∈ S\SP

and the homology group H2(GC/P ;Z) with the quotient ZS/ZSP , as remarked in Section3.3.

Let J be the complex structure defined on GC/P coming from its presentation as aquotient of complex Lie groups. Given an effective degree A ∈ H2(GC/P,Z) = ZS/ZSP ,

let MA,k(GC/P, J) be the moduli space of k-marked stable J-holomorphic curves ofdegree A to GC/P. Recall that we have that

dim MA,k(GC/P, J) = dim(GC/P ) + 2c1(TGC/P )(A) + 2k − 6


and that Equation 3.3.2 says that

c1(TGC/P ) =∑

γ∈R+\R+P

γ ∈ H2(GC/P,Z) ∼= Zωα : α ∈ S\SP

LetevJ = (ev1, ev2) : MA,2(GC/P, J) → (GC/P )2

be the evaluation map going from the moduli space of 2-marked stable curves MA,2(GC/P, J)to (GC/P )2.

Given any subvariety Z ⊂ GC/P, define the degree A neighborhood of Z to be

ΓA(Z) := ev2(ev−11 (Z)) ⊂ GC/P,

i.e., the closure of the union of all stable J-holomorphic curves of degree A that meet Z.Using that the moduli space of stable maps MA,k(GC/P, J) is irreducible, it was

proved in [8] by Buch-Chaput-Mihalcea-Perrin that if Z is a irreducible variety of X, thenΓA(Z) is also a irreducible variety. In particular, if Z is a B-stable irreducible variety, i.e.,a B-stable Schubert variety; then so is ΓA(Z).

In this chapter, we will show that when P ⊂ GC is a maximal parabolic subgroupand when A ∈ H2(GC/P,Z) is the effective cyclic generator, the degree A neighborhoodΓA(1 · P ) of the point 1 · P ∈ GC/P is a B-stable Schubert variety. We now prove thisstatement when P ⊂ GC is a maximal parabolic subgroup associated with a long simpleroot:

6.1.1 Theorem. Let α ∈ S be a simple root, P ⊂ GC be the maximal parabolic subgroupassociated with α and A = α + ZSP ∈ H2(GC/P,Z) ∼= ZS/ZSP be the effective cyclicgenerator. Let P ′′ ⊂ GC be the parabolic subgroup with SP

′′ = S\(N(α) ∪ α), whereN(α) ⊂ S denotes the neighbors of α in the Dynkin diagram of G. If α is a long simpleroot, then the degree A neighborhood ΓA(1 ·P ) of the point 1 ·P ∈ GC/P is a B-stableSchubert variety and

ΓA(1 · P ) = XP (wpwp′′sα),

where wp and wp′′ are the longest elements in WP and WP ′′ , respectively.

Proof. Let f : MA,1(GC/P, J) → MA, 0(GC/P, J) be the forgetful map defined by[u, z] 7→ [u] and let ev : MA,1(GC/P, J) → GC/P be the evaluation map defined by[u, z] 7→ u(z). Note that the curve neighborhood ΓA(1 · P ) of the point 1 · P ∈ GC/P is


the same asΓA(1 · P ) = ev(f−1(f(ev−1(1 · P ))))

When α is a long root, we have isomorphisms MA,0(GC/P, J) ∼= GC/P′,MA,1(GC/P, J) ∼=

GC/P′′, where P

′, P

′′ ⊂ GC are the parabolic subgroups with SP′ = S\N(α) and

SP ′′ = S\(N(α) ∪ α) (see Theorem 5.2.2).Under these isomorphisms, the projection maps π : GC/P

′′ → GC/P and π′ :GC/P

′′ → GC/P′ become ev : MA,1(GC/P, J) → GC/P and f : MA,1(GC/P, J) →

MA,0(GC/P, J), respectively; and

ΓA(1 · P ) = π(π′−1(π′(π−1(1 · P )))).

From Equation 5.1.6, we know that π′(π−1(1 · P )) = XP ′(wp′′p ). As a consequence,

ΓA(1 · P ) = π(π′−1(π′(π−1(1 · P )))) = π(π′−1(XP ′ (wp′′p )) = π(XP ′′ (wp′′

p sα)),

where wp′′p denotes the longest element in the set of minimum length representatives of

WP in W P′′, W P

′′

P . We know by Lemma 5.1.4 that wp′′p sα ∈ W P , thus

ΓA(1 · P ) = π(XP ′′ (wp′′p sα)) = XP (wp′′

p sα),

and we are done.

Let us assume from now that P ⊂ GC is a maximal parabolic subgroup and A ∈H2(GC/P,Z) is the effective cyclic generator. The degree A neighborhood ΓA(1 · P ) ⊂GC/P of the point 1 · P ∈ GC/P is B-stable. A T -fixed point w · P ∈ GC/P,w ∈ W,

is in the curve neighborhood ΓA(1 · P ) if there exists a T -invariant curve of degree Apassing through 1 · P and w · P, i.e., if there exists β ∈ R+\R+

P with β = α+ ZSP suchthat w · P = sβ · P ∈ GC/P (see for instance Section 3.3). In conclusion, the set ofT -fixed points of the curve neighborhood ΓA(1 · P ) is the set

ZPA := sβ · P ∈ GC/P : β ∈ R+\R+

P , β = α + ZSP

Showing that ΓA(1 ·P ) is a B-stable Schubert variety is equivalent to showing that the setZP

A has a unique maximal element with respect to the Bruhat order defined on W/WP .

Indeed, if ZPA has a unique maximal element, say zP

A · P with zPA ∈ W P , the Schubert

variety XP (zPA) will contain ΓA(1 · P ) because it contains all its T -fixed points and since


ΓA(1 · P ) is B-stable and closed

XP (zPA) = B · zP

A ⊂ ΓA(1 · P ),

and thus XP (zPA) = ΓA(1 · P ).

Now we show that the set ZPA has a unique maximal element with respect to the

Bruhat order defined on W/WP∼= W P :

6.1.2 Theorem. Let α ∈ S be a simple root, P ⊂ GC be the maximal parabolic subgroupassociated with α and A = α + ZSP ∈ H2(GC/P,Z) ∼= ZS/ZSP be the effective cyclicgenerator. The set

ZPA = sP

β ∈ W P : β ∈ R+\R+P , β = α+ ZSP ,

has a unique maximal element with respect to the Bruhat order defined on W P . We denotethis maximum by zP

A . The degree A neighborhood ΓA(1 · P ) of the point 1 · P ∈ GC/P

is a B-stable Schubert variety, and

ΓA(1 · P ) = XP (zPA)

Proof. The statement is already proven for when α ∈ S is a long simple root, and it willbe checked case by case for when α ∈ S is a short simple root:

1. Type B: Let G be a compact Lie group of type Bn with root system

R = ±(ei ± ej) : 1 ≤ i, j ≤ n, i = j ∪ ±ek, 1 ≤ k ≤ n,

and with simple roots S = α1 = e1 −e2, α2 = e2 −e3, · · · , αn−1 = en−1 −en, αn =en, and Dynkin diagram

Let us assume that α = αn, the only short simple root.

Note that when 1 ≤ i < j ≤ n

ei − ej = αi + · · · + αj−1

ei + ej = αi + · · · αj−1 + 2αj + · · · + 2αn−1 + αn


and when 1 ≤ l ≤ n

qel = 2αl + . . .+ 2αn−1 + αn

The positive roots equal to αn mod ZSP is the set of roots ei + ej, 1 ≤ i < j ≤n ∪ el, 1 ≤ l ≤ n, and

ZPA = sP

ei+ej ∪ sP

el.

The following table indicates the minimum coset representatives of the reflectionsassociated with these roots in W P and their corresponding lengths:

Positive Root γ Minimum length representative sPγ in W P Length sP

γ

el, 1 ≤ l ≤ n sl · . . . · sn−1 · sn n− l + 1ei + ej, 1 ≤ i < j ≤ n sj · sj+1 · . . . · sn · si · si+1 · . . . sn 2n− (i+ j) + 2

The maximal element in ZPA is zP

A = sPe1+e2 , and its length is 2n − 1. The figure

below shows the set ZPA partially ordered with respect to the Bruhat order when

n = 5

2. Type C: Let G be a compact Lie group of type Cn with root system

R = ±(ei ± ej) : 1 ≤ i, j ≤ n, i = j ∪ ±2ek, 1 ≤ k ≤ n,

and with simple roots S = α1 = e1 − e2, α2 = e2 − e3, · · · , αn = en−1 − en, αn =2en, and Dynkin diagram


Note that when 1 ≤ i < j ≤ n

ei − ej = αi + · · · + αj−1

ei + ej = αi + · · · αj−1 + 2αj + · · · + 2αn−1 + 2αn

and when l ≤ n

|2el = αl + . . .+ αn−1 + αn

Suppose that α = αk for some 1 ≤ k < n. The positive roots equal to αk

mod ZSP is the set of roots

ei − ej, 1 ≤ i ≤ k < j ≤ n ∪ 2el, 1 ≤ l ≤ k ∪ ei + ej, 1 ≤ i ≤ k < j ≤ n.

The following table indicates the minimum coset representatives of the reflectionsassociated with these roots in W P and their corresponding lengths:

Positive Root γ Minimum length representative sPγ in W P Length sP

γ

ei + ej, 1 ≤ i ≤ k < j ≤ n sj · . . . sn · sn−1 · . . . · sk+1 · si · . . . · sk 2n− (i+ j) + 12el, 1 ≤ l ≤ k sl · . . . · sn−1 · sn · sn−1 · . . . sk 2n− (l + k) + 1

ei − ej, 1 ≤ i ≤ k < j ≤ n sj−1 · . . . · sk+1 · si · . . . · sk j − i

The maximal element zPA of ZP

A is sP2e1 and its length is 2n− k. As an example, the

figure below shows the set ZPA partially ordered with respect to the Bruhat order

when n = 7 and k = 4


3. Type F: Let G be a compact Lie group of type F4 with simple roots S = α1 =e2 − e3, α2 = e3 − e4, α3 = e4, α4 = (e1 − e2 − e3 − e4)/2, and Dynkin diagram

The short simple roots of S are α3 and α4.

(a) When α = α4, the maximal element of ZPA is sP

2α1+3α2+4α3+2α4 and its lengthis 10 (see Appendix).The following figure shows the Hasse diagram of the Bruhat poset ZP

A :

(b) When α = α3, the maximal element of ZPA is sP

α1+2α2+2α3+2α4 and its length is6 (see Appendix). The Hasse diagram of the Bruhat poset ZP

A is shown below

4. Type G: Let G be a compact Lie group of type G2 with simple roots S = α1, α2,where α2 is the short simple root, and Dynkin diagram

Let us suppose that α = α2. The positive roots is the set R+ = α1, α2, α1 +2α2, α1 + 3α2, 2α1 + 3α2, α1 + α2. The positive roots equal to α2 in ZSP areα2, α1 + 3α2, α1 + 3α2, α1 + α2. The maximal element in ZP

A is sPα1+3α2 with

length equal to 4.


6.2 Curve neighborhoods and Gromov-Witten invari-ants

Let us assume that P ⊂ GC is a maximal parabolic subgroup and A ∈ H2(GC/P,Z) isthe effective cyclic generator. The degree A neighborhood ΓA(1 · P ) ⊂ GC/P of thepoint 1 · P ∈ GC/P is a B-stable Schubert variety, let zP

A ∈ W P be the minimum lengthrepresentative such that

ΓA(1 · P ) = XP (zPA)

Let evJ = (ev1, ev2) : MA,2(GC/P, J) → (GC/P )2 be the evaluation map that goesfrom the moduli space of 2-marked stable curves MA,2(GC/P, J) to (GC/P )2.

Define the Gromov-Witten variety associated with the point 1 · P ∈ GC/P as

GWA = ev−12 (1 · P ) ⊂ MA,2(GC/P, J)

By definition,ev1(GWA) = ΓA(XP (e)) = XP (zP

A).

6.2.1 Theorem. Let P ⊂ GC be a maximal parabolic subgroup and A ∈ H2(GC/P,Z)be the effective cyclic generator. Then

GWA, 2([XP (zPA)]op, [1 · P ]) = 1

Proof. We identify H∗(GC/P,Z) and H∗(GC/P,Z) via Poincaré duality. Under thisidentifications, we have pullback homomorphisms

ev∗i : Hd(GC/P,Z) → Hd(MA,2(GC/P, J),Z) i ∈ 1, 2, d ∈ Z≥0

and pushforward homomorphisms

evi∗ : Hd(MA,2(GC/P, J),Z) ∼= Hdim MA,2−d(MA,2(GC/P, J),Z)

→ Hdim MA,2−d(GC/P,Z) ∼= Hdim GC/P −dim MA,2+d(GC/P,Z)


These maps are related by the Projection formula

evi∗(β ∪ ev∗i (α)) = evi∗(β) ∪ α

for α ∈ H∗(GC/P,Z) and β ∈ H∗(MA,2(GC/P, J),Z).By definition of the Gromov-Witten invariant and by the Projection Formula, we have

that

GWA,2([1 · P ] , [XP (zPA)]op) =

∫MA,2(GC/P ,Z)

ev∗1[1 · P ] ∪ ev∗

2[XP (zPA)]op

=∫

GC/Pev2∗(ev∗

1[1 · P ]) ∪ [XP (zPA)]op

We have that ev2∗(ev∗1[1 · P ]) = ev2∗[GWA], but we do not necessarily have that

ev2∗[GWA] = [ΓA(1 · P )]. It is possible to prove that ev2∗[GWA] = [ΓA(1 · P )] if andonly if dim GWA = dim ΓA(1 · P ) [9, Theorem 6.2]. Last equality is the same as having

dim GWA = dim MA,2(GC/P , J) − dimGC/P = 2c1(TGC/P )(A) − 2

= dim ΓA(1 · P ) = dimXP (zAP ) = 2l(zP

A).

If this were the case,

GWA,2([1 · P ], [XP (zPA)]op) =

∫GC/P

[XP (zPA)] ∪ [XP (zP

A)]op = 1,

and we would be done.It remains to show that l(zP

A) = c1(TGC/P )(A) − 1. If the maximal parabolic subgroupP ⊂ GC comes from a long simple root, this verification was already done in the proof ofTheorem 5.2.3, because showing that l(zP

A) = c1(TGC/P )(A) − 1 is the same as showingthat dimC MA,2 + dimXP (zP

A)op = 2 dimC(GC/P ).We will keep the same notation that we use in the proof of the previous theorem.

Now we check case by case that l(zPA) = c1(TGC/P )(A) − 1 when the maximal parabolic

subgroup P ⊂ GC comes from a short simple root. Recall that

c1(TGC/P ) =∑

γ∈R+\R+P

γ ∈ H2(GC/P,Z) ∼= Zωα : α ∈ S\SP

1. Type B: Let G be a compact Lie group of type Bn, and P ⊂ GC be the maximal


parabolic subgroup associated with the short simple root αn = en. Then

c1(TGC/P )(A) =( ∑

1≤i<j≤n

(ei +ej)+∑

1≤k≤n

ek , en

)= 2(n−1)+2 = 2n = l(zPn

A )+1

2. Type C: Let G be a compact Lie group of type Cn. Let P ⊂ GC be the maximalparabolic subgroup associated with the short simple root αk = ek−ek+1, 1 ≤ k < n.

Then

c1(TGC/P )(A) =( ∑

1≤i≤k<j≤n

(ei − ej) +∑

1≤i≤k,i<j≤n

(ei + ej) + 2∑

1≤l≤k

el ,|αk

)= n+ (n− k − 1) + 2 = 2n− k + 1 = l(zPn

A ) + 1

3. Let G be a compact Lie group of type F4

(a) Let P ⊂ GC be the maximal parabolic subgroup associated with the simpleroot α4. Then

c1(TGC/P )(A) = (c1(TGC/P ), α4) = (11(α1 + 2α2 + 3α3 + 2α4), α4)

= 11(3 · (−1) + 2 · 2) = 11 = l(zPnA ) + 1

(b) Let P ⊂ GC be the maximal parabolic subgroup associated with the simpleroot α3. Then

c1(TGC/P )(A) = (c1(TGC/P ), α3) = (7(2α1 + 4α2 + 6α3 + 3α4), α3)

= 7(4 · (−2) + 6 · 2 + 3 · (−1)) = 7 = l(zPA) + 1

4. Let G be a compact Lie group of type G2 and P ⊂ GC be the maximal parabolicsubgroup associated with the simple root α2. Then

(c1(GC/P ), α2) = (5α1 + 10α2, α2) = 5 · (−3) + 10 · 2 = 5 = l(zPA) + 1

6.2.2 Remark. Gromov-Witten invariants in the symplectic geometry category can be de-fined under simple assumptions and without using virtual fundamental classes for exampleif we assume that either the symplectic manifold (M,ω) is semipositive or the homology


class A ∈ H2(M,Z) is ω-indecomposable. That is for instance the case when M is aGrassmannian manifold (see e.g. [43, Theorem 7.1.1, Lemma 7.1.8]).

To illustrate graphically the idea of this theorem, let G be a compact Lie group oftype G2 and S = α1, α2 a system of simple roots with Dynkin diagram

Let s1 and s2 be the simple reflections in the Weyl group of G associated with the simpleroots α1 and α2, respectively. Let P be the maximal parabolic subgroup associated withthe short simple root α2. Let T be a maximal torus of G with T ⊂ P. The figure shownbelow corresponds to the moment graph of GC/P whose vertices and edges correspondto the T -fixed points and T -stable curves of GC/P, respectively.

Edges with blue color and red color in the figure correspond to T -stable curves of degreeone and two, respectively. The opposite Schubert variety XP (zP

A)op contains only theT -fixed points s1s2s1s2s1P and s2s1s2s1P. In the figure shown below, we see that thereis just one (T -stable) curve of degree one (blue) meeting the point 1 ·P and the oppositeSchubert variety XP (zP

A)op (green):


Now we apply Theorem 6.2.1 and Remark 2.2.4 to get an upper bound for the Gromovwidth of coadjoint orbits isomorphic to Grassmannian manifolds:

6.2.3 Theorem. Let G be a compact connected simple Lie group with Lie algebra g.

Let λ ∈ t∗ ⊂ g∗, let Oλ be the coadjoint orbit passing through λ and let ωλ be theKostant-Kirillov-Souriau form defined on Oλ. Assume that there is a maximal parabolicsubgroup P ⊂ GC associated with a simple root α ∈ S such that Oλ

∼= GC/P, then

Gwidth(Oλ, ωλ) ≤ ⟨λ , α⟩ = 2(λ, α)(α, α)

Chapter 7

Upper bound for the Gromov width ofcoadjoint orbits of compact Liegroups

The problem of finding upper bounds for the Gromov width of coadjoint orbits of compactLie groups has already been addressed by Masrour Zoghi in his Ph.D thesis [55] where hehas considered the problem of determining the Gromov width of regular coadjoint orbitsof compact Lie groups.

The following theorem extends Zoghi’s results to coadjoint orbits that are not neces-sarily regular and it is the main result of this thesis:

Main Theorem. Let G be a compact connected simple Lie group with Lie algebra g.

Let T ⊂ G be a maximal torus and let R be the corresponding system of coroots. Weidentify the dual Lie algebra t∗ with the fixed points of the coadjoint action of T on g∗.

Let λ ∈ t∗ ⊂ g∗, let Oλ be the coadjoint orbit passing through λ and let ωλ be theKostant-Kirillov-Souriau form defined on Oλ, then


⟨λ,α⟩=0

|⟨λ, α⟩|

7.1 Upper bounds for the Gromov width of regularcoadjoint orbits

We start this Chapter by first describing Zoghi’s upper bound for the Gromov width ofregular coadjoint orbits of compact Lie groups. We present Zoghi’s upper bound after

69

Chapter 7. The Gromov width of coadjoint orbits 70

the following theorem.

7.1.1 Theorem. Let (M,ω) be a symplectic manifold and J be a regular ω-compatiblealmost complex structure on M, and suppose that M admits a J-holomorphic CP1-fibration π : M → Y over a connected compact Kähler manifold Y, and let A ∈ H2(M,Z)denote the homology class of the fibres of π. Then, the evaluation map

evJ : MA,1(M,J) → M

is a diffeomorphism.

Sketch: Let u : CP1 → M be a J-holomorphic curve of degree A. Note that π∗u∗[CP1] =π∗(A) = 0. Since Y is a connected compact Kähler manifold, the map π u is constant.As a consequence, the image of u lies totally in a fiber of π : M → Y, let us say F ∼= CP1.

The map u : CP1 → F ∼= CP1 is is a biholomorphism, and as a consequence; the J-holomorphic curves of M of degree A, up to parametrization, are embedded curves in Mand correspond to the fibers of π : M → Y.

For a trivialization of π : M → Y over an open set V ⊂ Y, let J ′ be the restrictionon π−1(V ) ⊂ M of the almost complex structure J. Any J ′-holomorphic map u : CP1 →π−1(V ) ⊂ M of degree A ∈ H2(π−1(V ) , Z) is of the form

u : CP1 → π−1(V ) ∼= V × CP1

z 7→ (v, g · z)

for some v ∈ V ⊂ Y and some g ∈ PSL(2,C), thus MA(π−1(V ), J ′) ∼= V ×PSL(2,C).Moreover, the evaluation map

evJ ′ : MA,1(π−1(V ), J ′) → π−1(V ) ∼= V × CP1

is a diffeomorphism.Note that (π ev1

J)−1(V ) = MA,1(π−1(V ), J ′) ⊂ MA,1(M,J) and we have a com-


mutative diagram

(π evJ)−1(V ) ⊂ MA,1(M,J) π−1(V ) ⊂ M

MA,1(π−1(V ), J ′) V × CP1

-evJ

?

=

?

∼=

-evJ′

and thusevJ : MA,1(M,J) → M

is a diffeomorphism as being this a local condition.

Now we reproduce Zoghi’s upper bound for the Gromov width of regular coadjointorbits as he did in his Ph.D. thesis:

7.1.2 Theorem (Zoghi [55]). Let G be a compact simple Lie group. Let λ ∈ g∗ andlet us assume that Oλ ⊂ g∗ is a regular coadjoint orbit of G. Let B ⊂ GC be a Borelsubgroup such that Oλ

∼= GC/B and S be a system of simple roots compatible with B.If there exists α ∈ S such that for any β ∈ S, ⟨λ, β⟩ is an integer multiple of ⟨λ, α⟩; then

Gwidth(Oλ, ωλ) ≤ ⟨λ, α⟩,

where ωλ denotes the Kostant-Kirillov-Souriau form defined on Oλ.

Proof. Let J be the complex structure defined on the coadjoint orbit Oλ obtained by thepresentation of Oλ as a quotient of complex Lie groups GC/B. Let P be the minimalparabolic subgroup with B ⊂ P ⊂ GC and SP = α. We have a holomorphic CP1-fibration π : GC/B → GC/P. Let A ∈ H2(GC/B,Z) be the homology class of the fibresof π.

By the previous Theorem

evJ : MA,1(Oλ, J) → Oλ

is a diffeomorphism. If J ′ is another ωλ-compatible regular almost complex structures,we can find a smooth family of almost complex structures Jtt∈[0,1] connecting J withJ ′ such that the moduli space

M∗A,1(Oλ, Jtt) := (t, u) : u ∈ M∗

A,1(Oλ, Jt)


is a smooth oriented manifold with boundary M∗A,1(Oλ, J

′) ⊔ M∗A,1(Oλ, J), and with a

smooth evaluation mapevJt : M∗

A,1(Oλ, Jtt) → Oλ

such that

evJt |∂M∗A,1(Oλ,Jtt) = evJ ⊔ evJ ′ : M∗

A,1(Oλ, J′) − M∗

A,1(Oλ, J) → Oλ,

(see e.g. McDuff [43]). Since there exists α ∈ S such that for any β ∈ S, ⟨λ, β⟩ is aninteger multiple of ⟨λ, α⟩; the symmplectic area ωλ(A) is a cyclic generator of the image ofthe map ωλ : H2(GC/B,Z) → R, which implies that A is a ωλ-indecomposable homologyclass. Therefore, the moduli spaces MA,1(Oλ, J) and MA,1(Oλ, J

′) are compact and theevaluation maps evJ and evJ ′ are compactly cobordant (see e.g. [43]). In particular, theevaluation map

evJ ′ : MA,1(Oλ, J′) → Oλ

has degree one and hence it is onto, which by Theorem 2.2.1 implies that


7.2 Upper bounds for the Gromov width of coadjointorbits

The Gromov width of arbitrary coadjoint orbits of compact Lie group would be estimatedby computing Gromov-Witten invariants on holomorphic fibrations whose fibers are isomor-phic to Grassmannian manifolds. The following result found in Li-Ruan [38, Proposition2.10], whose proof we reproduce here, is the key point of this argument:

7.2.1 Theorem. Let (M,ω) be a symplectic manifold and J be a regular ω-compatiblealmost complex structure on M, and suppose that M admits a J-holomorphic fibrationπ : M → Y. Let ι : π−1(y) → M be the inclusion map for a generic fiber over y ∈ Y.

Then, for A ∈ H2(π−1(y),Z) and α2, · · · , αk ∈ H∗(M,R)

GWπ−1(y)A,k (PD[pt], ι∗α2, . . . , ι

∗αk) = GWMA,k(PD[pt], α2, . . . , αk).


Proof. Choose an almost complex structures J ′ on Y such that π is an almost complexfibration. Suppose that u : CP1 → M is a J-holomorphic map of degree A. Then, π fis holomorphic with zero homology class. Therefore, im(π f) is a point. Namely, im(f)is contained in a fiber. Choose a point pt ∈ π−1(b). Then we have the identification ofthe moduli spaces of k marked genus zero stable curves with the first marked point goingto pt

MA, k(M, pt, J) = MA, k(π−1(b), pt, J |π−1(b))

furthermore, they have the same virtual fundamental cycles. As π is almost complex wehave the splitting u∗(TX) = u∗(Tπ−1(b)) ⊕ Cl, where l is the codimension of a fiberand Cl is the trivial complex bundle of dimension l. As CP1 has genus zero, we haveH1(CP1,Cl) = 0. It implies that

[ MA, k(M, pt, J) ]virt = [ MA, k(π−1(b), pt, J |π−1(b)) ]virt

By integrating α2, · · · , αk against the virtual fundamental cycles, we obtain

GWπ−1(y)A,k (PD[pt], ι∗α2, . . . , ι

∗αk) = GWMA,k(PD[pt], α2, . . . , αk)

7.2.2 Remark. The last part of the proof of the previous theorem can be omitted whenM = GC/P and Y = GC/Q, where G is a compact Lie group and P,Q are parabolicsubgroups of GC. In this case, the virtual fundamental classes of the moduli spaces ofcurves in M and Y are the same as the fundamental classes of those moduli spaces (seee.g. [18]).

The next statement is the main result of this thesis:

7.2.3 Theorem. Let G be a compact connected simple Lie group with Lie algebra g.

Let T ⊂ G be a maximal torus and let R ⊂ t be the corresponding system of coroots.We identify the dual Lie algebra t∗ with the fixed points of the coadjoint action of Ton g∗. Let λ ∈ t∗ ⊂ g∗, Oλ be the coadjoint orbit passing through λ and ωλ be theKostant-Kirillov-Souriau form defined on Oλ, then


⟨λ,α⟩=0

|⟨λ, α⟩|

Proof. For the coadjoint orbit Oλ there exists a parabolic subgroup P ⊂ GC such that


Oλ∼= GC/P. For each α ∈ S\SP , we have a parabolic subgroup Q ⊂ P with SQ =

SP ⊔ α, and a holomorphic fibration

πα : GC/P → GC/Q.

The fiber Q/P can be identified with the quotient of a simple Lie group and a maximalparabolic subgroup. Let A ∈ H2(Q/P,Z) be the effective cyclic generator of the secondhomology group of the Grassmannian Q/P. The Schubert variety XP (sα) = BsαP/P istotally contained in Q/P and its fundamental class is the same as the second homologyclass A ∈ H2(Q/P,Z). The symplectic area of A with respect to ωλ is equal to ⟨λ, α⟩.

By Theorem 6.2.1, there exists a Schubert variety X ⊂ Q/P such that

GWQ/PA,2 (PD[pt] ,PD[X]) = 0.

The inclusion map ι : Q/P → GC/P is cohomologically surjective, and as a consequencethere exists β ∈ H∗(GC/P,Z) such that ι∗β = PD[X]. Thus, by Theorem 7.2.1

GWQ/PA, 2 (PD[pt] ,PD[X]) = GWGC/P

A, 2 (PD[ pt] , β) = 0,

so that we would have by Remark 2.2.4 that


The above inequality holds for any α ∈ S\SP , and as consequence for any α ∈ R+\R+P ,

and we are done.

7.2.4 Remark. In the previous proof, if we assume that for any β ∈ R the pair ⟨λ, β⟩is an integer multiple of ⟨λ, α⟩, the effective cyclic generator A of the second homologygroup of the fiber H2(Q/P,Z) would be ωλ-indecomposable in H2(GC/P,Z). This simpleassumption will imply that the evaluation map

evJ : MA,2(Oλ, J) → O2λ

is a pseudocycle (see [43, Lemma 7.1.8]) and as a consequence the Gromov-Witten in-variant

GWGC/PA, 2 (PD[ pt] , β)

can be computed using evJ : MA,2(Oλ, J) → O2λ without having to define virtual funda-


mental classes on it.

7.2.5 Remark. Cieliebak-Mohnke have provided an alternative definition of the Gromov-Witten invariant in [12] with the assumption that [ωλ] ∈ H2(Oλ,Z). This definition makesuse of Donaldson’s divisors as auxiliary data to define the Gromov-Witten invariant. Thisdefinition of the Gromov-Witten invariant is one of the most widely accepted by thesymplectic community.

7.2.6 Corollary. Let λ = (λ1, · · · , λn) ∈ Rn, and

Hλ = A ∈ Mn(C) : A∗ = A, spectrumA = λ.

Let ωλ be the symplectic form defined on Hλ obtained by identifying Hλ with acoadjoint orbit of U(n) via an Ad-invariant inner product. Then


|λi − λj|.

7.2.7 Corollary. Let m be a positive integer number that we will denote by 2n if it iseven, and by 2n + 1 if it is odd. Let λ = (λ1, . . . , λn) ∈ Rn

≥0 and let λi1 , . . . , λir bethe non-zero components of λ counted with multiplicity. Let Sλ be the set of real skew-symmetric matrix of size m×m that can be diagonalized by orthogonal transformationsto a matrix of the form

0 λi1

−λi1 0. . .

0 λir

−λir 00

Let ωλ be the Kirillov-Kostant-Souriau form defined on Sλ by identifying it with a coadjointorbit of the special orthogonal group SO(n,R). Then,

Gwidth(Sλ, ωλ) ≤

min λk =0,

λi =±λj

|2λk|, |λi ± λj| if m = 2n+ 1

minλi =±λj|λi ± λj| if m = 2n

.

7.2.8 Corollary. Let λ ∈ Rn and

Cλ = A ∈ M2n(C) : A = QATQ,A∗ = −A, spectrumA = i(λ,−λ)



In 0

.

Let ωλ be the symplectic form defined on Cλ by identifying it with a coadjoint orbit ofSp(n) via an Ad-invariant inner product. Then

Gwidth(Cλ, ωλ) ≤ minλk =0,λi =±λj

|λk|, |λi ± λj|

Chapter 8

Appendix

Dynkin diagram of F4

Positive roots α of F4 Positive coroots α of F4 sα

α1 + 2α2 + 3α3 + 2α4 2α1 + 4α2 + 3α3 + 2α4 s4s3s2s3s1s2s3s4s3s2s3s1s2s3s4

α2 + 2α3 + 2α4 α2 + α3 + α4 s4s3s2s3s4

α1 + α2 + 2α3 + 2α4 α1 + α2 + α3 + α4 s4s3s1s2s3s4s1

2α1 + 3α2 + 4α3 + 2α4 2α1 + 3α2 + 2α3 + α4 s1s2s3s4s2s3s1s2s3s4s1s2s3s2s1

α3 + α4 α3 + α4 s3s4s3

α1 + 2α2 + 2α3 + α4 2α1 + 4α2 + 2α3 + α4 s2s3s1s2s3s4s3s2s3s1s2

α2 + 2α3 α2 + α3 s3s2s3

α1 + 2α2 + 3α3 + α4 2α1 + 4α2 + 3α3 + α4 s3s2s3s1s2s3s4s3s2s3s1s2s3

α1 + α2 α1 + α2 s1s2s1

α1 + 2α2 + 2α3 α1 + 2α2 + α3 s2s3s1s2s3s1s2

α1 + 2α2 + 4α3 + 2α4 α1 + 2α2 + 2α3 + α4 s3s4s2s3s1s2s3s4s1s2s3

α1 + α2 + 2α3 α1 + α2 + α3 s3s1s2s3s1

α1 + 2α2 + 2α3 + 2α4 α1 + 2α2 + α3 + α4 s4s2s3s1s2s3s4s1s2

α1 + α2 + 2α3 + α4 2α1 + 2α2 + 2α3 + α4 s3s1s2s3s4s3s2s3s1

α1 + α2 + α3 + α4 2α1 + 2α2 + α3 + α4 s1s2s3s4s3s2s1

α1 + 3α2 + 4α3 + 2α4 α1 + 3α2 + 2α3 + α4 s2s3s4s2s3s1s2s3s4s1s2s3s2

α2 + α3 2α2 + α3 s2s3s2

α2 + 2α3 + α4 2α2 + 2α3 + α4 s3s2s3s4s3s2s3

α2 + α3 + α4 2α2 + α3 + α4 s2s3s4s3s2

α1 + α2 + α3 2α1 + 2α2 + α3 s1s2s3s2s1

77

Chapter 8. Appendix 78

Positive roots α of F4 with α = α4 + ZSP where P ⊂ F4 is the maximal parabolicsubgroup associated with the simple root α4 :

Positive Root α Minimum length representative sPα in W P Length

α3 + α4 s3s4 2α2 + α3 + α4 s2s3s4 3

α1 + α2 + α3 + α4 s1s2s3s4 4α2 + 2α3 + α4 s3s2s3s4 4

α1 + α2 + 2α3 + α4 s3s1s2s3s4 5α2 + 2α3 + 2α4 s4s3s2s3s4 5

α1 + 2α2 + 2α3 + α4 s2s3s1s2s3s4 6α1 + α2 + 2α3 + 2α4 s4s3s1s2s3s4 6α1 + 2α2 + 3α3 + α4 s3s2s3s1s2s3s4 7α1 + 2α2 + 2α3 + 2α4 s4s2s3s1s2s3s4 7α1 + 2α2 + 4α3 + 2α4 s3s4s2s3s1s2s3s4 8α1 + 3α2 + 4α3 + 2α4 s2s3s4s2s3s1s2s3s4 92α1 + 3α2 + 4α3 + 2α4 s1s2s3s4s2s3s1s2s3s4 10

Positive roots α of F4 with α = α3 + ZSP , where P ⊂ F4 is the maximal parabolicsubgroup associated with the simple root α3 :

Positive Root α Minimum length representative sPα in W P Length

α2 + α3 s2s3 2α3 + α4 s4s3 2α2 + 2α3 s3s2s3 3

α1 + α2 + α3 s1s2s3 3α2 + α3 + α4 s4s2s3 3α1 + α2 + 2α3 s3s1s2s3 4α2 + 2α3 + 2α4 s4s3s2s3 4α1 + α2 + α3 + α4 s4s1s2s3 4α1 + 2α2 + 2α3 s2s3s1s2s3 5

α1 + α2 + 2α3 + 2α4 s4s3s1s2s3 5α1 + 2α2 + 2α3 + 2α4 s4s2s3s1s2s3 6

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