Some geometric and topological features

of symmetric R-spaces

Geodesics and moment maps

in collaboration with Jurgen Berndt and Anna Fino

Index numbers: Riem. invariants for flag manifolds

Index number=∑

(Z2-Betti numbers)

(Takeuchi, Sanchez)

• Aim:

- interpretation within symplectic topology and Morsetheory

- sketch of an alternative simple proof

- interplay with submanifold geometry

1

A complex flag manifold can be realized as a coadjointorbit of a connected, compact, semisimple real Liegroup

=⇒ carries a natural symplectic structure

Moreover a real flag manifold =

connected component of the fixed point set of an anti-symplectic involution on “its complexification” (whichis a complex flag manifold).

• Main tools:

- moment mappings of Hamiltonian torus actions

- convexity theorems :

by Atiyah, Guillemin-Sternberg and Kostant

(for Hamiltonian torus actions on cpt symplectic mflds)

by Duistermaat

(for fixed point sets of antisymplectic involutions)

2

COMPLEX FLAG MANIFOLD =

orbit of the adjoint representation of a compact Liegroup K

Example:

K = SU(3)

k = su(3) = {A ∈ gl(3, C) | A = −tA, trA = 0}

Cartan subalgebra t ={diag (λ1, λ2, λ3) | λ1+λ2+λ3 = 0}

• H=diag(λ1, λ2, λ3), λ1 6= λ2 6= λ3

=⇒ Ad(SU(3))·H=“full flag manifold”

Manifold F1,2,3 of full flags V1 ⊂ V2 ⊂ V3 in C3

• H=diag(λ1, λ2, λ3), λ1 = λ2 6= λ3

=⇒ Ad(SU(3))·H=“partial flag manifold”

Manifold of partial flags V1 = V2 ⊂ V3 in C3 ∼= CP 2

3

Geometry of complex flag manifoldsas submanifolds of Euclidean space

M = Ad(K)X ∼= K/KX ↪→ (k,−B( , ))

standard immersion of a flag manifold

TXM = imad(X) kX = ker ad(X)

reductive decomposition (⊥): k = kX ⊕ TXM

−→ kX = ker ad(X) = νXM

X regular ⇐⇒ kX as minimal dim

principal orbit (full flag manifold)

kX =: t Cartan subalgebra

4

Example: K = SU(3)

t = {diag (λ1, λ2, λ3) | λ1 + λ2 + λ3 = 0}

General fact (Ad(SU(3))X) ∩ t =

Weyl group orbit

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Geometric property:

t meets all orbits orthogonally

polar action

“There is a linear subspace which meets all orbits orthogonally”

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One can choose X ∈ t

• the orbits are parallel and equidistant

• If M = Ad(K)H is a principal orbit,

ξ ∈ t = νXM induces a ∇⊥-parallel normal field −→νM is flat

6

Spit k into real root spaces (w.r. to t):

[T, Xα] = α(T )Yα , [T, Yα] = −α(T )Xα , T ∈ t

complex (Kahler) structure on complex flag manifoldJ : Xα 7→ Yα , Yα 7→ −Xα

On a principal orbit M = Ad(K)H

THM = span {Xα, Yα}

AξXα = − α(ξ)α(H)Xα AξYα = − α(ξ)

α(H)Yα

−→ M isoparametric submanifold

7

M isoparametric submanifold of Rn: (Terng)

• νM flat

• the eigenvalues of A relative to parallel normal fields

are constant.

−→ {Aξ}ξ∈νM simult. diagonalisable

common eigenvalues: λi(ξ) =: 〈ni, ξ〉, i = 1, ..., g

ni: curvature normals

common eigendistributions: Ei, i = 1, ..., g

are autoparallel

−→ integrable with totally geodesic leaves

8

Si(p) leaf through p ∈M= sphere of dimension dimEi

−→ M is foliated by spheres

A full flag manifold is isoparametric

Thorbergsson: codimM ≥ 3,

M isoparametric full and irreducible

=⇒ full flag manifold

9

Endpoint map:

tξ : M → Rn

x 7→ x + ξ(x) = exp(ξ(x))

focal point in direction ξ = critical value of tξ .

x + ξ(x) focal point in dir. of ξ ⇐⇒ ker(id−Aξ(x)) isnon trivial

10

In general: if ξ ∈ νM parallel normal isoparametricsection (Aξ has constant eigenvalues)

⇒ the image of tξ , Mξ = {x + ξ(x) | x ∈M}

is a submanifold of Rn

dimMξ = dimM − dim(ker(id−Aξ(x)))

parallel focal manifold

TpMξ = sum of the eigendistributions Ej “non focalised”( 6∈ ker(id−Aξ))

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→ one has submersions

π : M →Mξ : x 7→ x + ξ(x),

π−1(p) isoparametric in νpMξ

(consequence of Olmos’ normal holonomy theorem)

12

Aξ =

〈n1, ξ〉 . . . 0

. . .

..

.0 . . . 〈ng, ξ〉

.

the focal points belong to

`i(p) := {1− 〈ni(p), ξ(p)〉 = 0} focal hyperplanes

Coxeter group (of a complete isoparam. subm.)

Terng= finite group generated by reflections σi w.r. to `i

(= Weyl group for flag manifold)

Fact: σi(p)= antipodal point of p in Si(p)

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Height functions and topology of cx flag manifolds

M = Ad(K) ·H (with H ∈ t)

X ∈ t regular element −→

Critical set = (Ad(K) ·H) ∩ t

all critical points of fX are non degenerate ≡ fX Morsefunction

Index of a critical point = 2 × number of hyperplanescrossed by a line joining X to the critical point

Example for K = SU(3)

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H regular (F1,2,3) H singular (CP 2)

14

Morse theory=⇒

bi(M) = number of critical points of index i

=⇒

fX is a perfect Morse function

fX determines a cell decomposition ≡ Bruhat

decomposition

15

k-Symmetric structure:

∀ complex flag manifold M ∃ k0 ≥ 2 :

∀k ≥ k0 ∃ k-symmetric struct. {θ(k)x | x ∈ M} on M

(Jimenez)

k-symmetry θ(k)x = isometry of M of order k

for which x is an isolated fixed point.

REAL FLAG MANIFOLD =

orbit of the isotropy representation of a symmetricspace G/K

A complex flag manifold is a real flag manifold

[take G/K = G(= G×G/∆)]

16

BUT

Important fact:

we can regard a real flag manifold as a real form of acomplex flag manifold

=⇒

∃ a natural immersion of a real flag manifold M into

a complex flag manifold MC “its complexification”

M ↪→MC

17

Index numbers

• 2-number of a Riem. mfld M Chen, Nagano

“#2(M)”

#2(M)=maximalf cardinality of subsets A ⊂M :∀ pair of points x, y ∈ A ∃ a closed geodesic γ in Mon which x and y are antipodal

i.e. γ : [0,2b] → M is a closed geodesic of length 2b: x = γ(0), y = γ(b).

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sγ∨

y = γ(b)

x = γ(0) = γ(2b)

Takeuchi: M symmetric R-space

(=particular real flag mfld which is also symmetric)

18

=⇒

#2(M) =∑

bZ2

i (= dimH∗(M, Z2))

bZ2

i : Z2-Betti numbers M endowed with the normalmetric

Note: For a symmetric space M ,

#2(M) = maximal possible cardinality of the 2-setsA2 ⊂M :

∀x ∈ A2 the symmetry sx of M fixes every point ofA2.

symmetric space ≡ 2-symmetric space =⇒

19

• k-number of a k-symmetric space M Sanchez

“#k(M)”

#k(M)=maximal cardinality of the k-sets Ak ⊂M :

∀x ∈ Ak the symmetry θ(k)x fixes every point of Ak .

Sanchez: M complex flag manifold

=⇒

#k(M) =∑

bi (= dimH∗(M, Z))

bi : Z-Betti numbers

M endowed with the normal metric

20

• Index number of a flag manifold M Sanchez

“#I(M)”

#I(M) = maximal cardinality of p-sets Ap ⊂M :

∀x ∈ Ap the symmetry θ(p)x in the p-symmetric struc-

ture on MC fixes every point of Ap.

Sanchez: M real flag manifold

=⇒

#I(M) =∑

bZ2

i (= dimH∗(M, Z2))

21

Symplectic structure on complex flag manifolds

Adjoint orbits ←→ Coadjoint orbits

Ad : k ∈ K 7→ Ad(k) : k→ k Ad∗ : k ∈ K 7→ Ad∗(k) : k∗ → k∗

(Ad∗(K) · η)X := η(Ad(k)X)

Ad(K) ·H ←→ Ad∗(K) · η (η = 〈H, 〉)

Symplectic structure on M = Ad∗(K) · η:

ωη(ad∗Xη, ad∗Y η) := η([X, Y ])

22

T := a maximal torus of K

Important fact: The natural K -action on M restrictsto a Hamiltonian action of T on M = Ad∗(K) · η

Hamiltonian action of a group G on a symplectic man-ifold N

⇐⇒

∃ Lie alg. hom (g, [ , ])→ (C∞(M), { , } : X 7→ fX :

∀X ∈ g the Killing vector field X∗ induced by theaction =

Hamiltonian vector field XfXassociated with fX

{fX}X∈g

f : M → g∗

momentum map

If G = T and M = Ad∗(K) · η

=⇒ momentum map ≡ {the height functions }

fX : M → R : p 7→ 〈p, X〉

23

Convexity Theorems

Guillermin - Sternberg, Atiyah

(N, ω) compact symplectic manifold

T torus acting on N in a Hamiltonian way

f : N → t∗ momentum mapping of the HamiltonianT -action

=⇒

• f(F ) is finite F = fixed point set of the T -action• f(N) = convex hull of f(F );• dimH∗(N, Z) = dimH∗(CX, Z) ∀X ∈ t,(CX = critical point set of fX )

24

For a complex flag manifold (=coadj. orbit η ↔ H)

take T =maximal torus

f : Ad∗(K) · η ↪→ k∗proj−→ t∗ =⇒

f(F ) = (Ad∗(K) · η) ∩ t = Weyl group orbit

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25

Duistermaat

(N, ω) compact symplectic manifoldT torus acting on N in a Hamiltonian wayτ antisymplectic involution on N Q= fixed pointset 6= ∅ :fX ◦ τ = fX ∀X ∈ t

=⇒

• f(Q) = f(N) is a convex polytope• dimH∗(Q, Z2) = dimH∗(CX, Z2) ∀X ∈ t,CX = set of critical points of fX|Q

We can replace Q by any connected component Q0

of Q

26

Proof and geom. interpretation of Sanchez Thm forcomplex flag manifolds

Basic algebraic fact: k-set Ak ⊂ Ad(K) ·H =⇒

∃ Cartan subalgebra t : Ak = t ∩ (Ad(K) ·H)

convexity=⇒

#k(Ad(K) ·H) = cardinality of A =

= cardinality of t ∩ (Ad(K) ·H) == n.of vertices of convexpolytope =

=∑

bi =

= n.of cells inBruhat decomposition == minimal number of cells needed to

have aCW − complex structure

27

M := Ad (K)p symmetric irreducible R-space(G/K symmetric space of compact type)

(M endowed with normal metric =induced metric - by irred.)

−→ M is a focal mfld of a principal orbit M of theisotropy repr. of a symmetric space G/K .

M is isoparametric

π : M →M projection π(p) = p + ξ(p),ξ parallel normal field

TM = ⊕Ei

Ei eigendistributions of the shape operator of M

TM = ⊕ non focalized eigendistributions Ei

28

NOTE: A non focalized eigendistribution Ei a leafSi(q) homothetically sent to M .

[π∗|Ei= (id−Aξ)|Ei

is a multiple of the identity] −→

• M is foliated by totally geodesic spheres

• σi ∈W , (reflections w.r. to `i)q ∈M 7→ antipodal point in Si(q).

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Si(q)

q

σi(q)

maximal 2-set A2 = a ∩M = νpM ∩M = W · q

MOMENTUM MAP ≡ projection ⊥ M → νpM = a

29

=⇒ The half closed geodesics connecting

points in a maximal 2-set A2 = W · q which

correspond one each other by reflections σi

with respect to the walls `i of the Weyl cham-

bers are mapped by the moment map µ into

lines in a = νpM .

30

K = SU(3) (CP 2)

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