Lower bounds for Gromov width of

coadjoint orbits of U(n) and SO(n).

Milena Pabiniak

ETH, January 11th 2012

Problem suggested by

prof Yael Karshon,

extension of PhD work of her student

Masrour Zoghi

Key points:

• Hamiltonian torus action ⇒ symplectic embeddings of balls

• Action of the Gelfand-Tsetlin torus (Cetlin, Zetlin).

1

Let (M,ω) be a 2N-dimensional symplectic

manifold.

Gromov Non-Squeezing Theorem ⇒being symplectomorphism is much more

restrictive then just being volume preserv-

ing.

The Gromov width of M is the supremum of the set of a’s such

that a ball of capacity a

B2Na =

{z ∈ CN

∣∣∣∣ π N∑i=1

|zi|2 < a

},

can be symplectically embedded in (M,ω).

2

G- compact connected Lie group, G 3 g : G→ G, g(h) = ghg−1.

Derivative at e gives the adjoint action: Adg : TeG = g→ g.

⇒ the coadjoint action, Gy g∗,

Ad∗gξ(X) = ξ(Adg−1X), ξ ∈ g∗, X ∈ g.

For matrix groups, coadjoint action is by conjugation.

T ⊂ G choice of maximal torus

(t∗)+ choice of positive Weyl chamber

coadjoint points in positive Weyl chamber

orbits1−1↔ (t∗)+

Fact: For any λ ∈ (t∗)+, the coadjoint orbit through λ, Oλ, is

a symplectic manifold with Kostant-Kirillov symplectic form ω.

3

Example: G = U(n)

u(n)∗ ∼= u(n) ∼= n× n Hermitian matrices,

coadjoint action is conjugation

T =

eit1

eit2

. . .eitn

, t∗+ =

a1

a2. . .

an

; a1 ≥ a2 ≥ . . . ≥ an

Coadjoint orbits ∼= Hermitian matrices with the same eigenvalues

4

Example: G = SO(2n+ 1), coadjoint action is conjugationso(2n+ 1)∗ ∼= (2n+ 1)× (2n+ 1) skew symmetric matricesLet

R(α) =

(cos(α) − sin(α)sin(α) cos(α)

), L(a) =

(0 −aa 0

)Then

TSO(2n+1) =

R(α1)R(α2)

. . .R(αn)

1

; αj ∈ S1

(tSO(2n+1))∗+ =

L(λ1)L(λ2)

. . .L(λn)

0

; λj ∈ R, λ1 ≥ . . . ≥ λn ≥ 0

Coadjoint orbits ∼= matrices with char. pol. t

∏nj=1(t2 + λ2

j ).

5

Example: G = SO(2n), coadjoint action is conjugationso(2n)∗ ∼= (2n)× (2n) skew symmetric matrices

TSO(2n) =

R(α1)R(α2)

. . .R(αn)

; αj ∈ S1

(tSO(2n))

∗+ =

L(λ1)L(λ2)

. . .L(λn)

; λj ∈ R, λ1 ≥ . . . ≥ λn−1 ≥ |λn|

6

Theorem 1. Let G = U(n), SO(2n+ 1), or SO(2n) and

λ = (λ1 > . . . > λn) ∈ int t∗+,

a point in the interior of the positive Weyl chamber (λ regular),

M := Oλ - G-coadjoint orbit through λ.

The Gromov width of M is at least the minimum

min{∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot}.

Method:

- construct a proper, centered, Hamiltonian T -space,

- use it to construct explicit embeddings of symplectic balls;

7

The root system of U(n) consists of vectors in Rn

±(ej − ek), j, k = 1, . . . , n, j < k, of squared length 2,

Note that

⟨(ej − ek)∨, λ

⟩= 2

⟨ej − ek, λ

⟩⟨ej − ek, ej − ek

⟩ = (λj − λk).

Therefore for λ in our chosen positive Weyl chamber

min{∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot} = min{λ1 − λ2, . . . , λn−1 − λn}.

8

The root system of SO(2n+ 1) consists of vectors in Rn±ej, j = 1, . . . , n, of squared length 1,±(ej ± ek), j < k, of squared length 2.

Therefore this root system for SO(2n+ 1) is non-simply laced.Note that ⟨

(±ej)∨, λ⟩

= ±2

⟨ej, λ

⟩⟨ej, ej

⟩ = ±2λj,

and ⟨(ej ± ek)∨, λ

⟩= 2

⟨ej ± ek, λ

⟩⟨ej ± ek, ej ± ek

⟩ = λj ± λk

Therefore for λ in our chosen positive Weyl chamber

min{∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot} = min{λ1 − λ2, . . . , λn−1 − λn, 2λn}.

9

The root system of SO(2n) consists of vectors in Rn

±(ej ± ek), j < k, of squared length 2.

This root system is simply laced.

Note that

⟨(ej ± ek)∨, λ

⟩= 2

⟨ej ± ek, λ

⟩⟨ej ± ek, ej ± ek

⟩ = λj ± λk

Therefore for λ in our chosen positive Weyl chamber

min{∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot} = min{λ1−λ2, . . . , λn−1−λn, λn−1+λn}.

10

Why do we care about such lower bound?

1. (Zoghi) For regular, indecomposable (i.e. with some inte-grality conditions) U(n) coadjoint orbits their Gromov width isgiven by min{

∣∣⟨α∨, λ⟩∣∣ ;α∨ a coroot}.2. For regular (Z., P.) and for a class of not regular U(n) coad-joint orbits (P.) the above formula is a lower bound of theirGromov width. The same formula gives the Gromov width ofcomplex Grassmannians (Karshon,Tolman).3. (Zoghi) For any compact connected Lie group G, an up-per bound of the Gromov width of a regular, indecomposablecoadjoint G orbit is given by the above formula.

Corollary 2. (P., Zoghi) The Gromov width of regular, indecom-posable SO(n) coadjoint orbits is min{

∣∣⟨α∨, λ⟩∣∣ ;α∨ a coroot}.

4. (Caviedes) is working on the upper bounds for non-regularmonotone U(n) orbits.

11

Action is Hamiltonian if there exists a T -invariant momentum

map Φ: M → t∗, such that

ι(ξM)ω = d 〈Φ, ξ〉 ∀ ξ ∈ t,

where ξM is the vector field on M corresponding to ξ ∈ t.

This sign convention ⇒ for p ∈ MT , the isotropy weights of

T y TpM are pointing out of the momentum map image.

(S1)2 y C2 gives

NOT

12

Let T ⊂ t∗ be an open convex set which contains Φ(M).

The quadruple (M,ω,Φ, T ) is a proper Hamiltonian T-manifold

if Φ is proper as a map to T .

We will identify Lie(S1) with R using the convention that the

exponential map exp : R ∼=Lie(S1) → S1 is given by t → e2πit,

that is S1 ∼= R/Z.

13

A proper Hamiltonian T -manifold (M,ω,Φ, T ) is centered about

a point α ∈ T if

∀K⊂T ∀ctd X⊂MK , α ∈ Φ(X).

Not centered:

α

Centered:

αα

14

Hamiltonian T action on M is called toric if dimT = 12 dimM.

Example 3.M - compact symplectic toric manifold

Φ: M → t∗ - moment map

Then:

∆ := Φ(M) is a convex polytope,

and for any α ∈∆, ⋃F face of ∆

α∈F

Φ−1(rel-int F )

is the largest subset of M that is centered about α.

15

Proposition 4. (Karshon, Tolman) Let:

(M2n, ω,Φ, T ) - a proper Hamiltonian T -manifold,

centered about α ∈ T and

Φ−1({α}) = {p} a single fixed point.

Then

M is equivariantly symplectomorphic to{z ∈ Cn | α+ π

∑|zj|2ηj ∈ T

},

where −η1, . . . ,−ηn are the isotropy weights at p.

16

Example: Isotropy weights at α: −η1,−η2

α

η2

η1 5η1

2η2

−η1

−η2

Φ−1(shaded region T ) is equivariantly symplectomorphic to

W := {z ∈ C2|α+ π(|z1|2η1 + |z2|2η2) ∈ T }Notice that

z ∈ B2 = {z ∈ C2∣∣∣∣π(|z1|2+|z2|2) < 2} ⇒ α+π(|z1|2η1+|z2|2η2) ∈ T

⇒ B2 ↪→W ∼= Φ−1(T ) ⊂M embeds symplectically

17

The Gelfand-Tsetlin functions for a group G.

Consider a sequence of subgroups

G = Gk ⊃ Gk−1 ⊃ . . . ⊃ G1,

with maximal tori T = TGk ⊃ TGk−1⊃ . . . ⊃ TG1

.

Inclusion Gj ↪→ G ⇒ an action of Gj on the G-coadjoint orbit Oλ.

This action is Hamiltonian with momentum map

Φj : Oλ → g∗j

18

Every Gj orbit intersects the (chosen) positive Weyl chamber

(tGj)∗+ exactly once.

This defines a continuous (but not everywhere smooth) map

sj : g∗j → (tGj)∗+.

Let Λ(j) denote the composition sj ◦Φj:

Oλ Φj//

Λ(j) ##HHHH

HHHH

HHg∗jsj

��

(tGj)∗+

The functions {Λ(j)}, j = 1, . . . , k−1, form the Gelfand-Tsetlin

system denoted by Λ : Oλ → RN .

19

Example: G = U(n) ⊃ U(n− 1) ⊃ . . . ⊃ U(1), B 7→(B 00 I

).

- maximal tori: diagonal matrices,- t∗: diagonal Hermitian matrices,- positive Weyl chambers: eigenvalues in non-increasing order.

Then for a Hermitian matrix A,

Φj(A) is its j × j top left submatrix and

Λ(j)(A) = (λ(j)1 (A) ≥ . . . ≥ λ(j)

1 (A)) ∈ Rj

is a sequence of eigenvalues of Φj(A) ordered in a non-increasingway.

Due to this ordering, the function Λ(j) is not smooth on thewhole orbit. The singularities may occur when eigenvalues coin-cide.

20

Gelfand-Tsetlin system for SO(n).

SO(n) ⊃ SO(n− 1) ⊃ . . . ⊃ SO(2).

⇒ SO(k) also acts on Oλ by a subaction of a coadjoint action.

This action is Hamiltonian with a momentum map

Φk : Oλ → so(k)∗,

Φk(A) − k × k top left submatirx of A.

21

Then

λ(k)1 (A) ≥ λ(k)

2 (A) ≥ . . . ≥ λ(k)

bk2c(A)

are such that

Φk(A) ∼SO(k)

L(λ(k)

1 (A)). . .

L(λ(k)

bk2c(A))

0

if k odd

or

Φk(A) ∼SO(k)

L(λ(k)

1 (A)). . .

L(λ(k)

bk2c(A))

if k even .

22

Why not smooth everywhere?

Due to ordering. The singularities may occur when generalized

eigenvalues coincide.

Proposition 5. The functions Λ(k) are smooth at the preimage

of the interior of the positive Weyl chamber,

UGk := (Λ(k))−1(int (tGk)∗+).

For λ regular, UGk is open and dense subset of Oλ.

23

Torus action induced by the Gelfand-Tsetlin system

On UGk, Λ(k) is inducing a smooth action of TGk.

For t ∈ TGk and A ∈ Oλ this new action is

t ∗A =

(B−1 tB

I

)A

(B−1 tB

I

)−1

where B ∈ Gk is such that

BΦk(A)B−1 ∈ (tGk)∗+.

Proposition 6. Λ(k) is a momentum map for the Hamiltonian

action of the torus TGk on UGk.

24

Putting together actions of all TGk, we obtain the action of the

Gelfand-Tsetlin torus TGT

TGT = TGn−1⊕ TGn−2

⊕ . . .⊕ TG1

on the set

U :=n−1⋂k=1

UGk.

Momentum map for this action is

Λ = (Λ(n−1),Λ(n−2), . . . ,Λ(1)) : Oλ → t∗GT∼= RN .

25

Case 1: regular U(n) orbit.

Λ(j)(A) eigenvalues of j × j top left minor of A (non-increasing)

The classical min max principle implies that for any A ∈ Oλ

λ1 ≥ λ(n−1)1 (A) ≥ λ2 ≥ λ

(n−1)2 (A) ≥ λ3 ≥ . . . ≥ λ

(n−1)n−1 (A) ≥ λn

and more generally

λ(l+1)j (A) ≥ λ(l)

j (A) ≥ λ(l+1)j+1 (A).

P -the polytope in RN cut out by the above inequalities,

N = (n− 1) + . . .+ 1 = 12n(n− 1).

Proposition 7. For any λ, regular or not, the dimension of the

polytope P is half of the dimension of Oλ.

26

The Gelfand-Tsetlin polytope

Min-max principle inequalities ⇒ Λ(Oλ) ⊂ P.

Proposition 8. The image Λ(Oλ) is exactly P.

To prove Proposition 8 succesively apply:Lemma 9. For any a1 ≥ b1 ≥ a2 ≥ . . . ≥ ak ≥ bk ≥ ak+1 ∈ R

∃ x1, . . . , xk ∈ C, xk+1 ∈ R

such that the Hermitian matrix

A :=

b1 0 x1

. . . ...0 bk xkx1 . . . xk xk+1

,has eigenvalues a1, . . . , ak+1

.

27

The Gelfand-Tsetlin action is smooth on U .

λ is generic ⇒ λ ∈ U ,

because λ is diagonal matrix with all diagonal entries distinct.

Moreover, for any face S 3 Λ(λ) have Λ−1( rel-intS) ⊂ U.

The region

W :=⋃

S; Λ(λ)∈SΛ−1(rel-int S) ⊂ U

is centered around Λ(λ) and Λ−1( Λ(λ) ) = {λ}.

28

Need:

-weights of T action on TλW = TλOλ,-lattice lengths of edges of P from Λ(λ).

For U(n) the standard action of maximal torus is a subaction of

the Gelfand-Tsetlin action.

⇒ ∃ projection map pr : RN → Rn

OλΦ &&NNNNNNNNNNNN

Λ // P ⊂ RNpr

��

Q ⊂ Rn

For any edge e ∈ P starting from Λ(λ) there is an edge e′ in the

1-skeleton of Q, such that pr(e) ⊂ e′.

29

Qλ = µ(Oλ) ∈ R2 Pλ = Λ(Oλ) ∈ R3

The ”x-ray” for standard action and momentum polytope for the

Gelfand-Tsetlin action for generic U(3) coadjoint orbit, Oλ.

30

Stadnard action of maximal torus of U(n):

vertices of Q ↔ Sn := permutations on n elements

edges of Q ↔ transpositions

direction of edges = - weights = ±(ej − ek)

lattice lengths = (λj − λk), j < k

31

Example:transposition (i, k) ∈ Sn ↔ µ({Fa | a ∈ CP1})- edge in Q from λ

Fa =

. . . ... 0 ... 0

. . .(λi+|a|2λk)

A . . . a(λi−λk)A . . .

0 ... . . . ... 0

. . . a(λi−λk)A . . .

(λk+|a|2λi)A . . .

0 ... 0 ... . . .

where A =

√1 + |a|2.

As |a| goes 0→∞, then (λi+|a|2λk)A goes from λi to λk

Λ({Fa | a ∈ CP1,(λi + |a|2λk)

A≤ λi+1 })− edge in P

of lattice length λi − λi+1 (w.r.t. weight lattice).

32

Proposition 10. Any edge in P starting from Λ(λ) has the latticelength at least

min{λi − λj |λi > λj}.

There is an edge with length exactly the min{λi − λj |λi > λj}.Moreover, the same is true for all vertices of P of the form

Λ( fixed point of the standard action ).

This fact + Proposition 4 prove that we can embed a ball ofcapacity min{λi − λj |λi > λj}, so the Gromov width of Oλ is atleast

min{λi − λj |λi > λj}.

For λ regular this was already proved by Zoghi (using the cen-tered region for the standard action of maximal torus).

33

Note:We used the fact that λ is regular only to show that λ ∈ U .

λ not regular ⇒ Φn−1(Oλ) ⊂ wall of the positive Weyl chamber.Gelfand-Tsetlin functions are smooth on a bigger set then U :

Lemma 11. If λ is not regular but there is only one eigenvalue

that is repeated

⇒ ∃ Tn−fixed point F , with neighborhood equipped with a smooth

Gelfand-Tsetlin action. Moreover the region

⋃S; Λ(F )∈S

Λ−1(rel-int S) ⊂ U.

is centered about Λ(F ).

34

Consider example with λ non generic:

λ : (5,4,4,4,3,1).

Here is the Tn fixed point and its Gelfand-Tsetlin functions

(bolded ones are constant on the whole orbit)

F =

15

34

44

,

5 4 4 3 15 4 3 1

5 3 15 1

1

35

Theorem 12. Let Oλ be the orbit of the U(n) coadjoint action

through λ=diag(λ1, . . . , λn), where

λ1 > λ2 > . . . > λl = λl+1 = . . . = λl+s > λl+s+1 > . . . > λn.

The Gromov width of Oλ is at least min{λi − λj |λi > λj }.

36

Case: SO(2n+ 1)

T -maximal torus of SO(2n+ 1), dimT = n

λ ∈ t∗+ regular, Oλ-coadjoint orbit through λ, dimOλ = 2n2

T y Oλ coadjoint (conjugation).

Centered region for this action is “too small”:

For the SO(5) orbit through λ = (6,1) =

L(6)L(1)

0

min{

∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot} = min{5,7,12,2} = 2,

while the centered region is

37

λ

σβ(λ)

σα(p)

E1

E2

2

2

1

5

α = e1 + e2

e1

β = e2

σβ(α) 38

⇒ Instead use the Gelfand-Tsetlin action.

Let {x(k)j |1 ≤ k ≤ 2n, 1 ≤ j ≤ bk2c} be basis of Rn2

.

Proposition 13. The image of the Gelfand-Tsetlin functions

Λ : Oλ → Rn2is the polytope, which we will denote by P, defined

by the following set of inequalities x(2k)1 ≥ x(2k−1)

1 ≥ x(2k)2 ≥ x(2k−1)

2 ≥ . . . ≥ x(2k)k−1 ≥ x

(2k−1)k−1 ≥ |x(2k)

k |,x

(2k+1)1 ≥ x(2k)

1 ≥ x(2k+1)2 ≥ x(2k)

2 ≥ . . . ≥ x(2k+1)k ≥ |x(2k)

k |,

for all k = 1, . . . , n, where x(2n+1)j = λj.

Moreover, the dimension of the polytope P is n2, what is half of

the dimension of Oλ.

39

Graphically,

. . .

. . .. . .

. . .

. . .

λ1 λ2 λn

x(2n)1 x

(2n)2 x

(2n)n−1 |x(2n)n |

λn−1

x(2n−1)1 x

(2n−1)2 x

(2n−1)n−1

Every coordinate is between its top right and top left neighbors.

40

Where do these inequalities come from?

Any A ∈ Oλ can be SO(2n+ 1) conjugated toL(λ1)

. . .L(λn)

0

and U(2n+ 1) conjugated to

iλ1. . .

iλn0−iλn

. . .−iλ1

41

Therefore the Hermitian matrix 1iA can be U(2n+1) conjugated

to

diag(λ1, . . . , λn,0,−λn, . . . ,−λ1)

and similarly Φ2n(1iA) can be U(2n) conjugated to

diag(λ(2n)1 , . . . , |λ(2n)

n |,−|λ(2n)n |, . . . ,−λ(2n)

1 ).

Min-max principle ⇒ intertwining inequalities on the eigenvalues:

λj ≥ λ(2n)j ≥ λj+1, λn ≥ |λ(2n)

n | ≥ 0

42

Note:

- λ is in U as all generalized eigenvalues for λ are distinct.

- λ is fixed under the action of Gelfand-Tsetlin torus

- Λ(λ) is a vertex of P as all Gelfand-Tsetlin functions are equal

to their upper bounds.

Moreover:

∀ ( face S of P, Λ(λ) ∈ S) Λ−1( rel-intS) ⊂ U.

Therefore the region

T :=⋃

S; Λ(λ)∈SΛ−1(rel-int S) ⊂ U.

is centered around Λ(λ).

43

Need: weights of the Gelfand-Tsetlin action on TλOλ and edgesof P from Λ(λ).

Warning: For SO(2n+ 1) the standard action of maximal torusis NOT a subaction of the Gelfand-Tsetlin action.

First: identify the edges of P starting from Λ(λ):

• pick one of these inequalities defining P that is an equationon Λ(λ)

• consider the set of points in P satisfying all the equationsthat Λ(λ) does except possibly this chosen one. Call this setE.

44

Back to example of SO(5) orbit through λ = (6,1):

At Λ(λ) = (6,1,6,6) all Gelfand-Tsetlin functions are equal totheir upper bounds.

Can choose the equation λ(4)1 = λ

(3)1 .

Then the set E case consists of points satisfying

6 1= =

x(4)1 x

(4)2

≥x

(3)1

=

x(2)1

45

That is,

(x(4)1 , x

(4)2 , x

(3)1 , x

(2)1 ) = (6,1, s, s) ∈ R4

where

s ∈ [1,6].

Note that the edge E connects Λ(λ) = (6,1,6,6) and (6,1,1,1),

so

~E = (0,0,−5,−5).

46

The preimage Λ−1(E) consists of matrices A of the form

A :=

0 −s 0 −a 0s 0 0 −b 00 0 0 −c 0a b c 0 00 0 0 0 0

,

where a, b, c ∈ R are such that

(A)4 := Φ4(A) ∼SO(4)

0 −66 0

0 −11 0

47

Calculating weights of the action of the Gelfand Tsetlin torus

TGT = TSO(4) ⊕ TSO(3) ⊕ TSO(2).

First of TSO(4). Let

R =

(R(α1)

R(α2)

)∈ TSO(4).

Then

R ∗A =

(B−1RB

1

) ((A)4 0

0 0

) (B−1R−1B

1

)

=

(B−1R (B(A)4B

−1) R−1B 00 0

)=

((A)4

0

)= A

The action is trivial.

48

Now let

R =

(R(α1)

1

)∈ TSO(3), W =

(a b c0 0 0

).

Then

R∗A =

(R

I2

) ((A)3 −WT

W 0

) (R−1

I2

)=

((A)3 −RWT

W R−1 0

).

As

−RWT =

R(α1)

(−a−b

)0

−c 0

The action has weight 1.

Similarly TSO(2) acts with weight 1.

49

Therefore the Gelfand-Tsetlin torus acts on TλOλ with weight

−η = (0,0,1,1).

Recall that ~E = (0,0,−5,−5), so

~E = (6− 1)η = 〈(e1 − e2)∨, (6,1)〉η = 2〈e1 − e2, λ〉

〈e1 − e2, e1 − e2〉η.

50

Lemma 14. Every edge E of P starting from Λ(λ) is at least

r := min{∣∣∣⟨α∨, λ⟩∣∣∣ ;α∨ a coroot}

multiple of ηE where (−ηE) is the weight of the action along this

edge.

51

Denote the weights by −η1, . . . ,−ηn2.

The centered region T is equivariantly symplectomorphic to

W :={z ∈ Cn | λ+ π

∑|zj|2ηj ∈ T

}.

Due to Lemma 14, for any z in a ball of capacity r,

Br = {z ∈ Cn2∣∣∣∣ π∑n2

i=1 |zi|2 < r}, have

λ+ πn2∑i=1

|zi|2ηi ∈ T .

Thus:

⇒ Br ⊂ W⇒ Br symplectically embeds into T ⊂ Oλ.⇒ r is the lower bound for Gromov width of Oλ.

52