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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,) 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 as such

that a ball of capacity a

B2Na ={z CN

Ni=1

|zi|2 < a},

can be symplectically embedded in (M,).

2

G- compact connected Lie group, G 3 g : G G, g(h) = ghg1.Derivative at e gives the adjoint action: Adg : TeG = g g. the coadjoint action, Gy g,

Adg(X) = (Adg1X), 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

orbits11 (t)+

Fact: For any (t)+, the coadjoint orbit through , O, isa symplectic manifold with Kostant-Kirillov symplectic form .

3

Example: G = U(n)

u(n) = u(n) = n n Hermitian matrices,coadjoint action is conjugation

T =

eit1

eit2. . .

eitn

, t+ =

a1a2

. . .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() =

(cos() sin()sin() cos()

), L(a) =

(0 aa 0

)Then

TSO(2n+1) =

R(1)R(2)

. . .R(n)

1

; j S1

(tSO(2n+1))+ =

L(1)L(2)

. . .L(n)

0

; j R, 1 . . . n 0

Coadjoint orbits = matrices with char. pol. tnj=1(t

2 + 2j ).

5

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

TSO(2n) =

R(1)R(2)

. . .R(n)

; j S1

(tSO(2n))+ =

L(1)L(2)

. . .L(n)

; j R, 1 . . . n1 |n|

6

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

= (1 > . . . > n) int t+,

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

M := O - G-coadjoint orbit through .The Gromov width of M is at least the minimum

min{, ; 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),

= 2

ej ek,

ej ek, ej ek

= (j k).Therefore for in our chosen positive Weyl chamber

min{, ; a coroot} = min{1 2, . . . , n1 n}.

8

The root system of SO(2n+ 1) consists of vectors in Rnej, 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),

= 2

ej,

ej, ej

= 2j,and

(ej ek),

= 2

ej ek,

ej ek, ej ek

= j kTherefore for in our chosen positive Weyl chamber

min{, ; a coroot} = min{1 2, . . . , n1 n, 2n}.

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),

= 2

ej ek,

ej ek, ej ek

= j kTherefore for in our chosen positive Weyl chamber

min{, ; a coroot} = min{12, . . . , n1n, n1+n}.

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{

, ; 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{

, ; 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 : M t, such that

(M) = d , t,

where M is the vector field on M corresponding to t.

This sign convention for p MT , the isotropy weights ofT 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 (M).The quadruple (M,,, T ) is a proper Hamiltonian T-manifoldif is proper as a map to T .

We will identify Lie(S1) with R using the convention that theexponential map exp : R =Lie(S1) S1 is given by t e2it,that is S1 = R/Z.

13

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

a point T if

KT ctd XMK , (X).

Not centered:

Centered:

14

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

Example 3.M - compact symplectic toric manifold

: M t - moment mapThen:

:= (M) is a convex polytope,

and for any , F face of

F

1(rel-int F )

is the largest subset of M that is centered about .

15

Proposition 4. (Karshon, Tolman) Let:

(M2n, ,, T ) - a proper Hamiltonian T -manifold,centered about T and

1({}) = {p} a single fixed point.

Then

M is equivariantly symplectomorphic to{z Cn | +

|zj|2j T

},

where 1, . . . ,n are the isotropy weights at p.

16

Example: Isotropy weights at : 1,2

2

1 51

22

1

2

1(shaded region T ) is equivariantly symplectomorphic to

W := {z C2|+ (|z1|21 + |z2|22) T }Notice that

z B2 = {z C2(|z1|2+|z2|2) < 2} +(|z1|21+|z2|22) T

B2 W = 1(T ) M embeds symplectically17

The Gelfand-Tsetlin functions for a group G.

Consider a sequence of subgroups

G = Gk Gk1 . . . G1,

with maximal tori T = TGk TGk1 . . . TG1.

Inclusion Gj G an action of Gj on the G-coadjoint orbit O.

This action is Hamiltonian with momentum map

j : O gj

18

Every Gj orbit intersects the (chosen) positive Weyl chamber

(tGj)+ exactly once.

This defines a continuous (but not everywhere smooth) map

sj : gj (tGj)

+.

Let (j) denote the composition sj j:

O j

//

(j) ##HH

HHHH

HHHH

gjsj

(tGj)+

The functions {(j)}, j = 1, . . . , k1, form the Gelfand-Tsetlinsystem denoted by : O 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,

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

(j)(A) = ((j)1 (A) . . . (j)1 (A)) R

j

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

Due to this ordering, the function (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 O by a subaction of a coadjoint action.

This action is Hamiltonian with a momentum map

k : O so(k),

k(A) k k top left submatirx of A.

21

Then

(k)1 (A)

(k)2 (A) . . .

(k)

bk2c(A)

are such that

k(A) SO(k)

L((k)1 (A)) . . .

L((k)bk2c

(A))

0

if k oddor

k(A) SO(k)

L((k)1 (A)) . . .

L((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 (k) are smooth at the preimage

of the interior of the positive Weyl chamber,

UGk := ((k))1(int (tGk)

+).

For regular, UGk is open and dense subset of O.

23

Torus action induced by the Gelfand-Tsetlin system

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

For t TGk and A O this new action is

t A =(B1 tB

I

)A

(B1 tB

I

)1

where B Gk is such that

Bk(A)B1 (tGk)+.

Proposition 6. (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 = TGn1 TGn2 . . . TG1on the set

U :=n1k=1

UGk.

Momentum map for this action is

= ((n1),(n2), . . . ,(1)) : O tGT = RN .

25

Case 1: regular U(n) orbit.

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

The classical min max principle implies that for any A O

1 (n1)1 (A) 2

(n1)2 (A) 3 . . .

(n1)n1 (A) n

and more generally

(l+1)j (A)

(l)j (A)

(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 , regular or not, the dimension of the

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