NCTS Colloquium Jordan Theory and Analysis:Jordan triples in analysis on bounded symmetric domains

Jordan triples in analysis on bounded symmetric

domains

Guy Roos St Petersburg [email protected] Hsinchu, Taiwan 2010/04/08-12

Part I. Examples of bounded symmetric domains

1 The unit disc ofC

2 The Hermitian ball ofCn

3 The Lie ball

Riemann’s Mapping Theorem

Unit disc ofC:

∆

=

{

z

∈

_C

| |

z

| <

1

}

Theorem (Riemann’s Mapping Theorem). Every simply connected

Automorphisms of

∆

G

=

Aut∆, the group of biholomorphic automorphisms of∆.

For a

∈

∆, letφa :∆

→

∆ φa

(

z

) =

z

+

a 1

+

az

(

z

∈

∆

)

. Thenφa

∈

Aut∆, φ−_a1

=

φ−a, φa

(

0

) =

a.

The group Aut∆acts transitively on∆.

Symmetries:

s₀

(

z

) = −

z

(

z

∈

∆

)

,

sa

=

φa

◦

s0

◦

φ−a.

Then sa

∈

Aut∆,

Automorphisms of

∆

Let K

=

G0

= {

u

∈

G

|

u

(

0

) =

0

}

.Then

K

=

U

(

1

) =

{

h_λ

| |

λ

| =

1

}

, hλ

(

z

) =

λz.

Eachφ

∈

G may be uniquely written

φ

(

z

) =

λ z

+

a

1

+

az

(

z

∈

∆

)

,

Bergman space of

∆

Flat symplectic formβ0and flat volume formω0for∆:

β0

(

z

) =

ω0

(

z

) =

i 2πd z

∧

d z

=

1 π d x

∧

d y. Bergman space of∆: A2

(

∆

) =

Hol∆

∩

L2

(

∆, ω0

)

,

with the scalar product

(

f

|

g

) = (

f

|

g

)

_A2_(∆)

=

Z

∆f gω0

(

f,g

∈

A 2

₍

_∆

₎₎

and the norm

k

f

k

2

=

k

f

k

2_A2_(∆)

=

Z

∆

|

f

|

2

Bergman kernel of

∆

The Bergman space is a Hilbert space of holomorphic functions, that is,

A2

(

∆

) →

R

f

7→

sup

{|

f

(

z

)| |

z

∈

C

}

is continuous for every compact subset C

⊂

_∆.

For z

∈

∆, let Kz

∈

A2

(

∆

)

be the Riesz representative of the

continuous linear formεz :A2

(

∆

) →

C,εz

(

f

) =

f

(

z

)

: εz

(

f

) = (

f

|

Kz

)

(

f

∈

A2

(

∆

))

.

The Bergman kernel of ∆is

Properties of the Bergman kernel

K

(

z,t

) =

K

(

t,z

)

, K

(

z,z

) =

k

Kz

k

2

>

0,

(

z,t

) 7→

K

(

z,t

)

is

holomorphic w.r. to z and antiholomorphic w.r. to t. Reproducing property:

f

(

z

) =

Z

∆K

(

z,t

)

f

(

t

)

ω0

(

t

)

(

f

∈

A

2

₍

_∆

₎

_{, z}

_∈

_∆

₎

_.

Transformation by automorphisms: forφ

∈

Aut∆,

K

(

z,t

) =

K

(

φ

(

z

)

, φ

(

t

))

φ0

(

z

)

φ0

(

t

)

(

z,t

∈

∆

)

, K

(

z,z

) =

K

(

φ

(

z

)

, φ

(

z

))

φ0

(

z

)

  2

(

z

∈

∆

)

.

Bergman kernel of

∆

The Bergman kernel of ∆is equal to

K

(

z,t

) =

1

(

1

−

zt

)

2.

Proof

Apply the transformation formula toφa: K

(

0,0

) =

K

(

a,a

)

|

φ0_a

(

0

)|

2.

Asφa0

(

0

) =

1

−

aa, we have K

(

a,a

) =

K(0,0)

(1−aa)2.Hence the Bergman

kernel of∆is

K

(

z,t

) =

K

(

0,0

)

(

1

−

zt

)

2.

K

(

0,t

) =

K

(

0,0

)

for all t

∈

_∆and the reproducing formula implies

1

=

Z ∆K

(

0,t

)

ω0

(

t

) =

K

(

0,0

)

Z ∆ω0. As

R

_∆ω0

=

1, we have K

(

0,0

) =

1.

Bergman metric of

∆

The Bergman metric of ∆is defined by

h_z

(

u,v

) =

∂u∂vlog K

(

z,z

) =

∂2 ∂z∂z

log K

(

z,z

)

uv.

Computation of the Bergman metric of ∆

h_z

(

u,v

) =

2

(

1

−

zz

)

2uv.

The Bergman metric is an Hermitian metric on∆, invariant by Aut∆:

Forφ

∈

Aut∆such thatφ

(

z

) =

0,

Bergman metric of

∆

The hyperbolic symplectic form βassociated to the Bergman metric is

β

(

z

) =

i

2∂∂log K

(

z,z

)

. It is related to the Bergman metric by

β

(

z

)(

u,v

) = −

Im hz

(

u,v

)

.

Computation of the hyperbolic symplectic form

β

(

z

) =

i d z

∧

d z

(

1

−

zz

)

2

=

2

β0

(

z

)

Bergman operator of

∆

The Bergman operator B

(

z,z

)

(z

∈

∆) is the complex linear operator

B

(

z,z

)

:C

→

_C, such that

hz

(

u,v

) =

h0

(

B

(

z,z

)

−1u,v

)

(

z

∈

∆, u,v

∈

C

)

.

The Bergman operator for∆is

Part I. Examples of bounded symmetric domains

1 The unit disc ofC

2 The Hermitian ball ofCn

3 The Lie ball

The Hermitian ball of

C

n

V

=

Cn_{, Hermitian vector space of dimension n, with Hermitian}

scalar product

(

z

|

t

) =

n

∑

j=1 zjtj

and Hermitian norm

k

z

k

2

= (

z

|

z

)

.

Unit Hermitian ball of V

=

Cn_:

Ω

=

Bn

=

{

z

∈

Cn

| k

z

k <

1

}

.

G

=

AutΩ, the group of biholomorphic automorphisms ofΩ.

Automorphisms of B

n Let M

=

  A B C d  

∈

U

(

n,1

)

,

with A

∈

End_CV , B

∈

V , C

∈

V0, d

∈

C. Then the mapφM defined

by

φM

(

z

) = (

Az

+

B

)(

Cz

+

d

)

−1

is a biholomorphic automorphism of Bn.

The condition M

∈

U

(

n,1

)

writes as MJn,1M∗

=

Jn,1, where

J_n,1

=

diag

(

I_n,

−

1

)

. So M

∈

U

(

n,1

)

if and only if AA∗

−

BB∗

=

In,

AC∗

−

dB

=

0,

Automorphisms of B

n

Theorem. The group Aut Bn acts transitively on Bn. Proof. Let a

=

λe

∈

Bn, e

∈

∂Bn, 0

≤

λ

<

1.

Let

B

=

da, C

=

B∗, d

=

√

1

1

−

λ2

and let A

∈

End_CV be defined by

Ae

=

de, Az

=

z if

(

u

|

e

) =

0. Thenφa defined by φa

(

z

) = (

Az

+

B

)(

Cz

+

d

)

−1

=

Az

+

B

(

z

|

B

) +

d

(

z

∈

Bn

)

Derivative of

φ

a Lemma. φ0_a

(

0

) ·

e

=

 1

− k

a

k

2e, φ_a0

(

0

) ·

u

=

 1

− k

a

k

21/2u if

(

u

|

e

) =

0, Jφa

(

0

) =

 1

− k

a

k

2 n+1 2 . Proof. For z

∈

B_n, u

∈

V φ_a0

(

z

) ·

u

=

Au

(

z

|

aB

) +

d

−

(

u

|

B

)(

Az

+

B

)

((

z

|

B

) +

d

)

2 , φ_a0

(

0

) ·

u

=

Au d

−

(

u

|

B

)

B d2 . Then φ0_a

(

0

) ·

e

=

1

−

λ2
e, φ0_a

(

0

) ·

u

=

√

1

−

λ2u if

(

u

|

e

) =

0.

Linear automorphisms of B

n

Lemma (H. Cartan). Let u

∈

Aut B_nsuch that u

(

0

) =

0 and u0

(

0

) =

idV. Then u

=

idBn.

Proposition. K _U

(

n

)

: u

∈

Aut Bn and u

(

0

) =

0 if and only if u is

the restriction to B_n of a unitary linear endomorphism of V .

Each automorphismφ

∈

Aut Bn may be written

φ

=

u

◦

φa,

where a

=

φ

(

0

)

and u

∈

K .

Symmetry around 0:

s₀

(

z

) = −

z

(

z

∈

B_n

)

.

Symmetry around a

∈

Bn: takeφ

∈

Aut Bnsuch thatφ

(

0

) =

a and

s_a

=

φ

◦

s0

◦

φ−1. Then sa

∈

Aut Bn,

s_a2

=

idBn, sa

(

a

) =

a, s

0

Bergman space of B

n

Flat K ¨ahler form for Bn: β0

(

z

) =

i 2π∂∂

(

z

|

z

) =

i 2π n

∑

j=1 d zj

∧

d zj.

Flat volume form for B_n:

ω0

=

βn₀

=

n! n ^ j=1  i 2π d z_j

∧

d z_j  . Bergman space of Bn: A2

(

B_n

) =

Hol B_n

∩

L2

(

B_n, ω0

)

,

with scalar product and norm

(

f

|

g

) = (

f

|

g

)

_A2_(B n)

=

Z Bn f gω0

(

f,g

∈

A2

(

Bn

))

,

k

f

k

2

=

k

f

k

2_A2_(B n)

=

Z Bn

|

f

|

2ω0

(

f

∈

A2

(

Bn

))

.

Bergman kernel of B

n

The Bergman space A2

(

Bn

)

is a Hilbert space of holomorphic

functions.

The Bergman kernel of Bnis

K

(

z,t

) =

Kz

(

t

)

(

z,t

∈

Bn

)

,

where Kz

∈

A2

(

Bn

)

is characterized by

Properties of the Bergman kernel

K

(

z,t

) =

K

(

t,z

)

, K

(

z,z

) =

k

K_z

k

2

>

0,

(

z,t

) 7→

K

(

z,t

)

is holomorphic w.r. to z and antiholomorphic w.r. to t.

Reproducing property: f

(

z

) =

Z

Bn

K

(

z,t

)

f

(

t

)

ω0

(

t

)

(

f

∈

A2

(

Bn

)

, z

∈

Bn

)

.

Transformation by automorphisms: forφ

∈

Aut Bn,

K

(

z,t

) =

K

(

φ

(

z

)

, φ

(

t

))

Jφ

(

z

)

Jφ

(

t

)

(

z,t

∈

Bn

)

,

K

(

z,z

) =

K

(

φ

(

z

)

, φ

(

z

))

|

Jφ

(

z

)|

2

(

z

∈

Bn

)

,

where

Jφ

(

z

) =

det_Cφ0

(

z

)

Bergman kernel of B

n

Theorem. The Bergman kernel of B_n is equal to

K

(

z,t

) =

1

(

1

− (

z

|

t

))

n+1.

Proof. Applying the transformation formula to h_λ :B_n

→

B_n, h_λ

(

z

) =

λz,

where

|

λ

| =

1, gives K

(

0,a

) =

K

(

0, λa

)

for allλ(

|

λ

| =

1), hence

K

(

0,a

) =

K

(

0,0

)

for all a

∈

B_n. By the reproducing formula for the function 1, K

(

0,0

) =

Z Bn ω0 −1

=

1.

Letφ

∈

Aut Bn such thatφ

(

0

) =

a. Then

|

Jφ

(

0

)|

2

=



1

− k

a

k

2n+1. By the transformation formula,

K

(

a,a

) =

1

|

Jφ

(

0

)|

2

=

1  1

− k

a

k

2n+1 .

Bergman metric of B

n

The Bergman metric of Bnis defined by

hz

(

u,v

) =

∂u∂vlog K

(

z,z

)

.

The hyperbolic symplectic form associated to the Bergman metric is

β

(

z

) =

i

2∂∂log K

(

z,z

)

. It is related to the Hermitian metric by

β

(

z

)(

u,v

) = −

Im hz

(

u,v

)

.

The Bergman metric is an Hermitian metric on Bn, invariant by Aut Bn:

forφ

∈

Aut Bn such thatφ

(

z

) =

0,

Computation of the Bergman metric of B

n K

(

z,z

) =

1

(

1

− (

z

|

z

))

n+1, ∂vlog K

(

z,z

) = (

n

+

1

)

(

z

|

v

)

1

− (

z

|

z

)

, hz

(

u,v

) =

∂u∂vlog K

(

z,z

)

= (

n

+

1

)

(

u

|

v

)

1

− (

z

|

z

)

+

(

u

|

z

)(

z

|

v

)

(

1

− (

z

|

z

))

2 ! .

The Bergman metric at z

∈

Bn is

h_z

(

u,v

) = (

n

+

1

)

(

1

− (

z

|

z

)) (

u

|

v

) + (

u

|

z

)(

z

|

v

)

(

1

− (

z

|

z

))

2 .

Bergman metric of B

n

h0

(

u,v

) = (

n

+

1

)(

u

|

v

)

Let a

=

λe

∈

Bn, e

∈

∂Bn, 0

≤

λ

<

1. Define

V2

(

e

) =

Ce, V1

(

e

) =

{

z

∈

V

| (

z

|

e

) =

0

}

.

Then the Bergman metric at a is h_a

(

e,e

) = (

n

+

1

)

1

(1−λ2)2,

h_a

(

u,v

) = (

n

+

1

)

1

1−λ2

(

u

|

v

)

(

u,v

∈

V1

(

e

))

,

Bergman operator of B

n

The Bergman operator B

(

z,z

)

(z

∈

B_n) is the complex linear

operator, such that

h_z

(

u,v

) =

h₀

(

B

(

z,z

)

−1u,v

)

(

z

∈

B_n, u,v

∈

V

=

Cn

)

.

Let z

=

λe

∈

Bn, e

∈

∂Bn, 0

≤

λ

<

1. The Bergman operator for Bn

is B

(

z,z

)

u

=

   1

−

λ2
2u

(

u

∈

V2

(

e

)

, 1

−

λ2
u

(

u

∈

V1

(

e

)

.

Part I. Examples of bounded symmetric domains

1 The unit disc ofC

2 The Hermitian ball ofCn

3 The Lie ball

Notations

V0, real Euclidean vector space of dimension n

>

1, with Euclidean

scalar product x

·

y and norm

|

x

|

2

=

x

·

x.

V

=

C

⊗

_RV0

=

V0

⊕

i V0, complexified vector space of V0.

Conjugation on V w.r. to V0:

z

=

x

+

i y

7→

z

=

x

−

i y

(

x,y

∈

V0

)

.

Hermitian scalar product on V :

(

x

+

i y

|

u

+

i v

) =

x

·

u

+

y

·

v

+

i y

·

u

−

i x

·

v

(

x,y,u,v

∈

V0

)

.

The associated Hermitian norm is denoted by

|

z

|

. Let

ρ

(

z

) =

|

z

|

2

= (

z

|

z

)

(

z

∈

V

)

.

Complex bilinear form on V and associated quadratic form:

(

z:t

) =

z

|

t

(

z,t

∈

V

)

,

Notations

V0, real Euclidean vector space of dimension n

>

1, with Euclidean

scalar product x

·

y and norm

|

x

|

2

=

x

·

x.

V

=

C

⊗

_RV0

=

V0

⊕

i V0, complexified vector space of V0.

Conjugation on V w.r. to V0:

z

=

x

+

i y

7→

z

=

x

−

i y

(

x,y

∈

V0

)

.

Hermitian scalar product on V :

(

x

+

i y

|

u

+

i v

) =

x

·

u

+

y

·

v

+

i y

·

u

−

i x

·

v

(

x,y,u,v

∈

V0

)

.

The associated Hermitian norm is denoted by

|

z

|

. Let

ρ

(

z

) =

|

z

|

2

= (

z

|

z

)

(

z

∈

V

)

.

Complex bilinear form on V and associated quadratic form:

(

z:t

) =

z

|

t

(

z,t

∈

V

)

,

Notations

V0, real Euclidean vector space of dimension n

>

1, with Euclidean

scalar product x

·

y and norm

|

x

|

2

=

x

·

x.

V

=

C

⊗

_RV0

=

V0

⊕

i V0, complexified vector space of V0.

Conjugation on V w.r. to V0:

z

=

x

+

i y

7→

z

=

x

−

i y

(

x,y

∈

V0

)

.

Hermitian scalar product on V :

(

x

+

i y

|

u

+

i v

) =

x

·

u

+

y

·

v

+

i y

·

u

−

i x

·

v

(

x,y,u,v

∈

V0

)

.

The associated Hermitian norm is denoted by

|

z

|

. Let

ρ

(

z

) =

|

z

|

2

= (

z

|

z

)

(

z

∈

V

)

. Complex bilinear form on V and associated quadratic form:

(

z:t

) =

z

|

t

(

z,t

∈

V

)

,

Notations

V0, real Euclidean vector space of dimension n

>

1, with Euclidean

scalar product x

·

y and norm

|

x

|

2

=

x

·

x.

V

=

C

⊗

_RV0

=

V0

⊕

i V0, complexified vector space of V0.

Conjugation on V w.r. to V0:

z

=

x

+

i y

7→

z

=

x

−

i y

(

x,y

∈

V0

)

.

Hermitian scalar product on V :

(

x

+

i y

|

u

+

i v

) =

x

·

u

+

y

·

v

+

i y

·

u

−

i x

·

v

(

x,y,u,v

∈

V0

)

.

The associated Hermitian norm is denoted by

|

z

|

. Let

ρ

(

z

) =

|

z

|

2

= (

z

|

z

)

(

z

∈

V

)

.

Complex bilinear form on V and associated quadratic form:

(

z:t

) =

z

|

t

(

z,t

∈

V

)

,

The Lie norm

Definition. The Lie norm (or maximal norm) on V is the greatest norm

k k

on the complex vector space V , whose restriction to V0is the Euclidean

norm of V0.

k

z

k =

inf (

∑

j∈J

|

λj

| |

xj

| |

z

=

∑

j∈J λjxj, Jfinite, λj

∈

C, xj

∈

V0 ) .

Proposition (L. Dru´zkowski, 1974). Let ν

(

z

) =

k

z

k

2. Then ν

(

z

) =

ρ

(

z

) +

q

ρ2

(

z

) −

|

σ

(

z

)|

2.

For z

∈

V ,ν

(

z

)

is the greatest root of the polynomial

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2.

Definition. The Lie ball Ln is the unit ball of the Lie norm.

L_n

=

nz

∈

V

|

ρ

(

z

) <

1, 2ρ

(

z

) −

|

σ

(

z

)|

2

<

1

o .

The Lie norm

Definition. The Lie norm (or maximal norm) on V is the greatest norm

k k

on the complex vector space V , whose restriction to V0is the Euclidean

norm of V0.

k

z

k =

inf (

∑

j∈J

|

λj

| |

xj

| |

z

=

∑

j∈J λjxj, Jfinite, λj

∈

C, xj

∈

V0 ) .

Proposition (L. Dru´zkowski, 1974). Let ν

(

z

) =

k

z

k

2. Then ν

(

z

) =

ρ

(

z

) +

q

ρ2

(

z

) −

|

σ

(

z

)|

2.

For z

∈

V ,ν

(

z

)

is the greatest root of the polynomial

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2.

Definition. The Lie ball Ln is the unit ball of the Lie norm.

L_n

=

nz

∈

V

|

ρ

(

z

) <

1, 2ρ

(

z

) −

|

σ

(

z

)|

2

<

1

o .

The Lie norm

Definition. The Lie norm (or maximal norm) on V is the greatest norm

k k

on the complex vector space V , whose restriction to V0is the Euclidean

norm of V0.

k

z

k =

inf (

∑

j∈J

|

λj

| |

xj

| |

z

=

∑

j∈J λjxj, Jfinite, λj

∈

C, xj

∈

V0 ) .

Proposition (L. Dru´zkowski, 1974). Let ν

(

z

) =

k

z

k

2. Then ν

(

z

) =

ρ

(

z

) +

q

ρ2

(

z

) −

|

σ

(

z

)|

2.

For z

∈

V ,ν

(

z

)

is the greatest root of the polynomial

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2. Definition. The Lie ball Ln is the unit ball of the Lie norm.

L_n

=

nz

∈

V

|

ρ

(

z

) <

1, 2ρ

(

z

) −

|

σ

(

z

)|

2

<

1

o .

The Lie norm

Definition. The Lie norm (or maximal norm) on V is the greatest norm

k k

on the complex vector space V , whose restriction to V0is the Euclidean

norm of V0.

k

z

k =

inf (

∑

j∈J

|

λj

| |

xj

| |

z

=

∑

j∈J λjxj, Jfinite, λj

∈

C, xj

∈

V0 ) .

Proposition (L. Dru´zkowski, 1974). Let ν

(

z

) =

k

z

k

2. Then ν

(

z

) =

ρ

(

z

) +

q

ρ2

(

z

) −

|

σ

(

z

)|

2.

For z

∈

V ,ν

(

z

)

is the greatest root of the polynomial

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2. Definition. The Lie ball Ln is the unit ball of the Lie norm.

L_n

=

nz

∈

V

|

ρ

(

z

) <

1, 2ρ

(

z

) −

|

σ

(

z

)|

2

<

1

o .

Case n

=

2

Let n

=

dim_RV0

=

dim_CV

=

2.

If

(

e₁,e₂

)

is an orthonormal basis for V₀(and V ), the Lie norm of z

=

z1e1

+

z2e2is

k

z

k =

max z1

+

i z2  ,  z1

−

i z2  
.

The Lie ball L₂is linearly isomorphic to the bidisc ∆

×

∆.

Boundary of the Lie ball

Proposition. The boundary ∂Ln of the Lie ball is the disjoint union of a

real analytic hypersurface ∂1Lnand of a compact submanifold ∂2Ln:

∂1Ln

=

n z

∈

V

|

2ρ

(

z

) −

|

σ

(

z

)|

2

=

1, ρ

(

z

) <

1 o , ∂2Ln

=

{

z

∈

V

|

ρ

(

z

) =

|

σ

(

z

)| =

1

}

=

{

z

∈

V

|

z

=

λx, λ

∈

C, x

∈

V0,

|

λ

| = |

x

| =

1

}

.

∂2Ln 

(

S1

×

Sn−1

)

/

{±

I

}

;∂2Ln is orientable if n is even, non

orientable if n is odd.

∂2Ln

⊂

∂1Ln

=

∂Ln

Boundary of the Lie ball

Proposition. The boundary ∂Ln of the Lie ball is the disjoint union of a

real analytic hypersurface ∂1Lnand of a compact submanifold ∂2Ln:

∂1Ln

=

n z

∈

V

|

2ρ

(

z

) −

|

σ

(

z

)|

2

=

1, ρ

(

z

) <

1 o , ∂2Ln

=

{

z

∈

V

|

ρ

(

z

) =

|

σ

(

z

)| =

1

}

=

{

z

∈

V

|

z

=

λx, λ

∈

C, x

∈

V0,

|

λ

| = |

x

| =

1

}

.

∂2Ln 

(

S1

×

Sn−1

)

/

{±

I

}

;∂2Ln is orientable if n is even, non

orientable if n is odd.

∂2Ln

⊂

∂1Ln

=

∂Ln

Boundary of the Lie ball

If z

∈

∂Ln, thenρ

(

z

) =

1₂

(

1

+

|

σ

(

z

)|

2

) ∈ [

1₂,1

]

. An element e

∈

∂Lnwill

be called minimal ifρ

(

e

) =

1₂, which is equivalent toσ

(

e

) =

0. The set of

minimal elements M1

=

 e

∈

V

|

ρ

(

e

) =

1 2, σ

(

e

) =

0 

is a real-analytic compact submanifold of V , and dim_RM1

=

2n

−

3.

Proposition. The map

Φ :M1

×

∆

→

∂1Ln

(

e, λ

) 7→

e

+

λe

is a real-analytic diffeomorphism.

Boundary of the Lie ball

If z

∈

∂Ln, thenρ

(

z

) =

1₂

(

1

+

|

σ

(

z

)|

2

) ∈ [

1₂,1

]

. An element e

∈

∂Lnwill

be called minimal ifρ

(

e

) =

1₂, which is equivalent toσ

(

e

) =

0. The set of

minimal elements M1

=

 e

∈

V

|

ρ

(

e

) =

1 2, σ

(

e

) =

0 

is a real-analytic compact submanifold of V , and dim_RM1

=

2n

−

3.

Proposition. The map

Φ :M1

×

∆

→

∂1Ln

(

e, λ

) 7→

e

+

λe

is a real-analytic diffeomorphism.

Polar decomposition

Two minimal elements e1,e2are called strongly∂s orthogonal if

e₂

=

θe1 with

|

θ

| =

1; this is equivalent to

|(

e₁ :e₂

)| =

1

2.

A pair

(

e₁,e₂

)

of strongly orthogonal minimal elements is called a

frame of V . A frame is a maximal sequence of pairwise strongly orthogonal minimal elements.

The manifold of frames (also called F ¨urstenberg-Satake boundary of L_n)

F =



(

e1,e2

) ∈

M1

×

M1

| |(

e1:e2

)| =

1 2 

Polar decomposition

Two minimal elements e1,e2are called strongly∂s orthogonal if

e₂

=

θe1 with

|

θ

| =

1; this is equivalent to

|(

e₁ :e₂

)| =

1

2.

A pair

(

e₁,e₂

)

of strongly orthogonal minimal elements is called a

frame of V . A frame is a maximal sequence of pairwise strongly orthogonal minimal elements.

The manifold of frames (also called F ¨urstenberg-Satake boundary of L_n)

F =



(

e1,e2

) ∈

M1

×

M1

| |(

e1:e2

)| =

1 2 

Polar decomposition

Theorem. Each z

∈

V has a polar decomposition z

=

λ1e1

+

λ2e2,

where

(

e1,e2

)

is a frame and λ1

≥

λ2

≥

0.

Proof. Use the decomposition of the boundary∂Ln.

If z

=

λ1e1

+

λ2e2is a polar decomposition for z, then

ρ

(

z

) =

1 2 λ 2 1

+

λ22
,

|

σ

(

z

)|

2

₌

λ2₁λ2₂

andλ2₁,λ2₂are the roots of

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2.

The realsλ1,λ2are called singular values of z; in particular,λ1is the Lie

Polar decomposition

Theorem. Each z

∈

V has a polar decomposition z

=

λ1e1

+

λ2e2,

where

(

e1,e2

)

is a frame and λ1

≥

λ2

≥

0.

Proof. Use the decomposition of the boundary∂Ln.

If z

=

λ1e1

+

λ2e2is a polar decomposition for z, then ρ

(

z

) =

1 2 λ 2 1

+

λ22
,

|

σ

(

z

)|

2

₌

λ2₁λ2₂

andλ2₁,λ2₂are the roots of

m

(

X;z

) =

X2

−

2ρ

(

z

)

X

+

|

σ

(

z

)|

2.

The realsλ1,λ2are called singular values of z; in particular,λ1is the Lie

Polar coordinates

An element z

∈

V is called regular if its singular valuesλ1,λ2 satisfy

λ1

>

λ2

>

0.

The polar decomposition of a regular element is unique.

The set V_regof regular elements

V_reg

=

{

z

∈

V

|

0

<

|

σ

(

z

)| <

ρ

(

z

)}

is an open dense subset of V .

Let

C₂

=

λ

= (

λ1, λ2

) ∈

R2

|

λ1

>

λ2

>

0 ,

Σ2

=



λ

= (

λ1, λ2

) ∈

R2

|

1

>

λ1

>

λ2

>

0 .

The polar coordinates mapΨ :C₂

× F →

V_regdefined by Ψ

(

λ1, λ2;e1,e2

) =

λ1e1

+

λ2e2is a diffeomorphism.

Its restrictionΨ : Σ₂

× F →

L_n

∩

V_regis a diffeomorphism onto the open dense subset of regular points in Ln.

Polar coordinates

An element z

∈

V is called regular if its singular valuesλ1,λ2 satisfy

λ1

>

λ2

>

0.

The polar decomposition of a regular element is unique.

The set V_regof regular elements

V_reg

=

{

z

∈

V

|

0

<

|

σ

(

z

)| <

ρ

(

z

)}

is an open dense subset of V . Let

C₂

=

λ

= (

λ1, λ2

) ∈

R2

|

λ1

>

λ2

>

0 ,

Σ2

=



λ

= (

λ1, λ2

) ∈

R2

|

1

>

λ1

>

λ2

>

0 .

The polar coordinates mapΨ :C₂

× F →

V_regdefined by

Ψ

(

λ1, λ2;e1,e2

) =

λ1e1

+

λ2e2is a diffeomorphism.

Its restrictionΨ : Σ₂

× F →

L_n

∩

V_regis a diffeomorphism onto the

Linear automorphisms of L

n

G

=

Aut Ln, the group of biholomorphic automorphisms of Ln.

K

=

G0

= {

u

∈

G

|

u

(

0

) =

0

}

.

Proposition. The elements of K are the linear endomorphisms

u

∈

End_CV which preserve the Lie norm; u

∈

K if and only if

u

(

z

) =

αev

(

z

)

,

where αis a complex number,

|

α

| =

1, andev is the C-linear extension to

Compactification of V

Notations. W

=

C2

_⊕

_{V, W} 0

=

R2

⊕

V0. An element w

∈

W is written w

=

w1ε1

+

w2ε2

+

w0, w1,w2

∈

C, w0

∈

V . Quadratic form on W : S

(

w

) =

w1
2

+

w2
2

+

σ

(

w0

)

(

w

∈

W

)

.

Indefinite Hermitian form with signature

(

2,n

)

on W :

h

w,t

i =

w1t1

₊

_w2_t2

_{− (}

_w0

_|

_t0

₎

₍

_w,t

_∈

_W

₎

_, R

(

w

) =

h

w,w

i =

 w1   2

+

w2   2

−

ρ

(

w0

)

(

w

∈

W

)

. Complex quadric: Q_n

=

{[

w

] ∈

P_C

(

W

) |

q

(

w

) =

0

}

.

Compactification of V

Letξe:V

→

W be the real-analytic map

e

ξ

(

z

) = (

1

+

σ

(

z

))

ε1

+

i

(

1

−

σ

(

z

))

ε2

+

2 i z

(

z

∈

V

)

. Compactification of V : ξ :V

→

Qn

ξ

(

z

) = [

ξe

(

z

)]

(

z

∈

V

)

.

Proposition. The map ξis a biholomorphism of V onto the open dense

subset ξ

(

V

)

of Qn

ξ

(

V

) =



[

w

] ∈

Qn

|

w2

+

i w1 ,0 . The inverse map is given by

ξ−1

([

w

]) =

w

0

Compactification of V

Letξe:V

→

W be the real-analytic map

e

ξ

(

z

) = (

1

+

σ

(

z

))

ε1

+

i

(

1

−

σ

(

z

))

ε2

+

2 i z

(

z

∈

V

)

. Compactification of V : ξ :V

→

Qn

ξ

(

z

) = [

ξe

(

z

)]

(

z

∈

V

)

.

Proposition. The map ξis a biholomorphism of V onto the open dense

subset ξ

(

V

)

of Qn

ξ

(

V

) =



[

w

] ∈

Qn

|

w2

+

i w1 ,0 .

The inverse map is given by

ξ−1

([

w

]) =

w 0

Compactification of V

Let z

∈

V . Then R  e ξ

(

z

)



=

2  1

−

2ρ

(

z

) +

|

σ

(

z

)|

2  . Proposition. Let U+

⊂

Qn be defined by

U

=

{[

w

] ∈

Qn

|

R

(

w

) >

0

}

.

Then U+has two connected components U₀+, U+₁ such that ξ

(

Ln

) =

U+0.

If z

∈

V has the polar decomposition z

=

λ1e1

+

λ2e2, then

Rξe

(

z

)



=

2

(

1

−

λ2₁

)(

1

−

λ2₂

)

.

Compactification of V

Let z

∈

V . Then R  e ξ

(

z

)



=

2  1

−

2ρ

(

z

) +

|

σ

(

z

)|

2  .

Proposition. Let U+

⊂

Qn be defined by

U

=

{[

w

] ∈

Qn

|

R

(

w

) >

0

}

.

Then U+has two connected components U₀+, U+₁ such that ξ

(

Ln

) =

U+0.

If z

∈

V has the polar decomposition z

=

λ1e1

+

λ2e2, then

Rξe

(

z

)



=

2

(

1

−

λ2₁

)(

1

−

λ2₂

)

.

Compactification of V

Let z

∈

V . Then R  e ξ

(

z

)



=

2  1

−

2ρ

(

z

) +

|

σ

(

z

)|

2  .

Proposition. Let U+

⊂

Qn be defined by

U

=

{[

w

] ∈

Qn

|

R

(

w

) >

0

}

.

Then U+has two connected components U₀+, U+₁ such that ξ

(

Ln

) =

U+0.

If z

∈

V has the polar decomposition z

=

λ1e1

+

λ2e2, then

Rξe

(

z

)



=

2

(

1

−

λ2₁

)(

1

−

λ2₂

)

.

Automorphisms of L

n

Let M:W

→

W be an invertible endomorphism of W . The map

ΦM :PC

(

W

) →

PC

(

W

)

defined by

ΦM

([

w

]) = [

Mw

]

(

w

∈

W

)

is then a biholomorphic automorphism ofP_C

(

W

)

.

If M

∈

O

(

W,S

)

(the complex orthogonal group of S), thenΦM

restricts to a biholomorphism of the quadric Q_n.

If moreover M

∈

U

(

W,R

)

(the unitary group of the indefinite Hermitian form R), thenΦ_M

(

U+

) =

U+, andΦ_M U₀+

=

U+₀ or ΦM U0+

=

U₁+. Let w₀

=

ξ

(

0

)

; ifΦM

(([

w0

]) ∈

U0+, thenΦM U0+

=

U₀+and ΨM

=

ξ−1

◦

ΦM

◦

ξ

∈

Aut Ln.

O

(

W,S

) ∩

U

(

W,R

) '

O

(

W0,R0

)

(the real orthogonal group of the

Automorphisms of L

n

Let M:W

→

W be an invertible endomorphism of W . The map

ΦM :PC

(

W

) →

PC

(

W

)

defined by

ΦM

([

w

]) = [

Mw

]

(

w

∈

W

)

is then a biholomorphic automorphism ofP_C

(

W

)

.

If M

∈

O

(

W,S

)

(the complex orthogonal group of S), thenΦM

restricts to a biholomorphism of the quadric Q_n.

If moreover M

∈

U

(

W,R

)

(the unitary group of the indefinite Hermitian form R), thenΦ_M

(

U+

) =

U+, andΦ_M U₀+

=

U+₀ or ΦM U0+

=

U₁+. Let w₀

=

ξ

(

0

)

; ifΦM

(([

w0

]) ∈

U0+, thenΦM U0+

=

U₀+and ΨM

=

ξ−1

◦

ΦM

◦

ξ

∈

Aut Ln.

O

(

W,S

) ∩

U

(

W,R

) '

O

(

W0,R0

)

(the real orthogonal group of the

Automorphisms of L

n

Let M:W

→

W be an invertible endomorphism of W . The map

ΦM :PC

(

W

) →

PC

(

W

)

defined by

ΦM

([

w

]) = [

Mw

]

(

w

∈

W

)

is then a biholomorphic automorphism ofP_C

(

W

)

.

If M

∈

O

(

W,S

)

(the complex orthogonal group of S), thenΦM

restricts to a biholomorphism of the quadric Q_n.

If moreover M

∈

U

(

W,R

)

(the unitary group of the indefinite

Hermitian form R), thenΦ_M

(

U+

) =

U+, andΦ_M U₀+

=

U+₀ or

ΦM U0+

=

U₁+. Let w₀

=

ξ

(

0

)

; ifΦM

(([

w0

]) ∈

U0+, thenΦM U0+

=

U₀+and ΨM

=

ξ−1

◦

ΦM

◦

ξ

∈

Aut Ln.

O

(

W,S

) ∩

U

(

W,R

) '

O

(

W0,R0

)

(the real orthogonal group of the

Automorphisms of L

n

Let M:W

→

W be an invertible endomorphism of W . The map

ΦM :PC

(

W

) →

PC

(

W

)

defined by

ΦM

([

w

]) = [

Mw

]

(

w

∈

W

)

is then a biholomorphic automorphism ofP_C

(

W

)

.

If M

∈

O

(

W,S

)

(the complex orthogonal group of S), thenΦM

restricts to a biholomorphism of the quadric Q_n.

If moreover M

∈

U

(

W,R

)

(the unitary group of the indefinite

Hermitian form R), thenΦ_M

(

U+

) =

U+, andΦ_M U₀+

=

U+₀ or

ΦM U0+

=

U₁+. Let w₀

=

ξ

(

0

)

; ifΦM

(([

w0

]) ∈

U0+, thenΦM U0+

=

U₀+and ΨM

=

ξ−1

◦

ΦM

◦

ξ

∈

Aut Ln.

O

(

W,S

) ∩

U

(

W,R

) '

O

(

W0,R0

)

(the real orthogonal group of the

Automorphisms of L

n

Let M:W

→

W be an invertible endomorphism of W . The map

ΦM :PC

(

W

) →

PC

(

W

)

defined by

ΦM

([

w

]) = [

Mw

]

(

w

∈

W

)

is then a biholomorphic automorphism ofP_C

(

W

)

.

If M

∈

O

(

W,S

)

(the complex orthogonal group of S), thenΦM

restricts to a biholomorphism of the quadric Q_n.

If moreover M

∈

U

(

W,R

)

(the unitary group of the indefinite

Hermitian form R), thenΦ_M

(

U+

) =

U+, andΦ_M U₀+

=

U+₀ or

ΦM U0+

=

U₁+. Let w₀

=

ξ

(

0

)

; ifΦM

(([

w0

]) ∈

U0+, thenΦM U0+

=

U₀+and ΨM

=

ξ−1

◦

ΦM

◦

ξ

∈

Aut Ln.

O

(

W,S

) ∩

U

(

W,R

) '

O

(

W0,R0

)

(the real orthogonal group of the

Automorphisms of L

n

Let

(

f₁,. . . ,f_n

)

be an orthonormal basis for the Euclidean space V₀. Then

(

ε1, ε2,f1,. . . ,fn

)

is anR-basis for W0and aC-basis for W . We

will identify an endomorphism of W0 or W with its matrix in this basis.

The matrix of the quadratic form R0in this basis is

I_2,n

=

diag

(

I₂,

−

I_n

)

. So M

∈

O

(

W₀,R₀

)

iff M is real and satisfies MI2,nM0

=

I2,n. Write M

=

  A B C D  , with A

∈ M

2,2

(

R

)

, B

∈ M

2,n

(

R

)

, C

∈ M

n,2

(

R

)

, D

∈ M

n,n

(

R

)

. Then M

∈

O

(

W0,R0

)

iff AA0

−

BB0

=

I2, CA0

−

DB0

=

0, CC0

−

DD0

= −

In.

Automorphisms of L

n

Let M as above. For z

∈

L_n,

ΨM

(

z

) =

w0 i w1

+

w2 , with   w₁ w2  

=

A   1

+

σ

(

z

)

i

(

1

−

σ

(

z

))

 

+

2 i Bz, w0

=

C   1

+

σ

(

z

)

i

(

1

−

σ

(

z

))

 

+

2 i Dz.

Automorphisms of L

n In particular, ΨM

(

0

) =

 

(

i,1

)

A   1 i     −1 C   1 i  . If we write A

=

  α β γ δ 

and C

= (

C₁,C₂

)

, this is equivalent to

ΨM

(

0

) =

C₁

+

i C₂ iα

−

β

+

γ

−

iδ.

Automorphisms of L

n

Bergman kernel of L

n

Notations

β0

(

z

) =

_2πi ∂∂

(

z

|

z

)

,ω0

=

βn₀

A2

(

L_n

) =

Hol L_n

∩

L2

(

L_n, ω0

)

, Bergman space of Ln , with Hilbert

product

(

f

|

g

)

_A2_(L n)

=

R

Lnf gω0

K

(

z,t

)

, Bergman kernel of Ln, reproducing kernel for A2

(

Ln

)

:

f

(

z

) =

R

LnK

(

z,t

)

f

(

t

)

ω0

(

t

)

(f

∈

A

2

₍

_L n

)

) Transformation law: forφ

∈

Aut Ln,

Bergman kernel of L

n

Let z

∈

Ln andφ

∈

Aut Ln such thatφ

(

0

) =

z. By the transformation

law, K

(

z,z

) =

K

(

0,0

)

|

Jφ

(

0

)|

2 . As L_n is circled, K

(

0,t

) =

K

(

0,0

) =

Z Ln ω0 −1 .

Bergman kernel of L

n

Proposition. Let φ

∈

Aut Lnsuch that φ

(

0

) =

z. Then

|

Jφ

(

0

)|

2

=



1

−

2ρ

(

z

) +

|

σ

(

z

)|

2

n .

Theorem. The Bergman kernel of L_n is equal to

K

(

z,t

) =

_ K

(

0,0

)

1

−

2

(

z

|

t

) +

σ

(

z

)

σ

(

t

)

Bergman metric of L

n

The Bergman metric of Ln is defined by

hz

(

u,v

) =

∂u∂vlog K

(

z,z

)

.

The hyperbolic symplectic form associated to the Bergman metric is

β

(

z

) =

i

2∂∂log K

(

z,z

)

. It is related to the Hermitian metric by

β

(

z

)(

u,v

) = −

Im hz

(

u,v

)

.

The Bergman metric is an Hermitian metric on Ln, invariant by Aut Ln:

forφ

∈

Aut Ln such thatφ

(

z

) =

0,

Computation of the Bergman metric of L

n K

(

z,z

) =

_ K

(

0,0

)

1

−

2

(

z

|

z

) +

σ

(

z

)

σ

(

z

)

n ∂vlog K

(

z,z

) =

2n

(

z

|

v

) −

σ

(

z

)(

z:v

)

1

−

2

(

z

|

z

) +

σ

(

z

)

σ

(

z

)

h_z

(

u,v

) =

2n

(

u

|

v

) −

2

(

z:u

)(

z:v

)

1

−

2

(

z

|

z

) +

σ

(

z

)

σ

(

z

)

+

4n 

(

z

|

v

) −

σ

(

z

)(

z :v

)

 

(

u

|

z

) − (

z:u

)

σ

(

z

)

  1

−

2

(

z

|

z

) +

σ

(

z

)

σ

(

z

)

2 .

Bergman metric of L

n

The Bergman metric at 0 is

h0

(

u,v

) =

2n

(

u

|

v

)

. Let z

=

λ1e1

+

λ2e2

∈

Ln. Let V11

=

Ce1, V22

=

Ce2, V12

= (

V11

⊕

V22

)

⊥. For u

=

u₁

+

u₂

+

u0, u₁

∈

Ce₁, u₂

∈

Ce₂, u0

∈

V₁₂, hz

(

u,u

) =

2n 

(

u1

|

u1

)

(

1

−

λ2₁

)

2

+

(

u2

|

u2

)

(

1

−

λ2₂

)

2

+

(

u 0

_|

_u0

₎

(

1

−

λ2₁

)(

1

−

λ2₂

)

 .

Bergman operator of L

n

The Bergman operator B

(

z,z

)

(z

∈

L_n) is the complex linear

operator, such that

hz

(

u,v

) =

h0

(

B

(

z,z

)

−1u,v

)

(

z

∈

Ln, u,v

∈

V

=

Cn

)

.

The Bergman operator for L_n is

B

(

z,z

)

u

=

          

(

1

−

λ2₁

)

2u ifu

∈

Ce1,

(

1

−

λ2₂

)

2u ifu

∈

Ce2,

(

1

−

λ2₁

)(

1

−

λ2₂

)

u ifu

∈

V12.

Part I. Examples of bounded symmetric domains

1 The unit disc ofC

2 The Hermitian ball ofCn

3 The Lie ball

Type I

m,n

Rectangular matrices

Domains of type Im,n

Let 1

≤

m

≤

n, V

= M

m,n

(

C

)

(space of m

×

n matrices with complex

entries) and

Ω

=

{

Z

∈

V

|

Im

−

ZZ∗



0

}

.

Lemma. Let Z

∈

V . There exist λ1

≥ · · · ≥

λm

≥

0 and unitary matrices

P

∈

U

(

m

)

, Q

∈

U

(

n

)

such that Z

=

P          λ1 0 0 0 0 0 . .. . .. 0 0 λm 0 0          Q−1.

Type I

m,n

Rectangular matrices

Then Z

∈

_Ωif and only ifλ1

<

1, that is, if.

dk d Tk det

(

TIm

−

ZZ ∗

₎

    T=1

>

0

(

0

≤

k

≤

m

−

1

)

. Boundary ofΩ: ∂Ω

=

m [ j=1 ∂jΩ, where Z

∈

∂jΩifλ1

= · · · =

λj

=

1

>

λj+1.

Type I

m,n

Rectangular matrices

Automorphisms. Let M

=

  A B C D  be an

(

m

+

n

) × (

m

+

n

)

matrix, with A m

×

m, B m

×

n, . . . If MJm,nM∗

=

Jm,n(where

J_m,n

=

diag

(

I_m,

−

I_n

)

), thenΦ_M defined by

ΦM

(

Z

) = (

AZ

+

B

)(

CZ

+

D

)

−1

provides an automorphism ofΩ.

The automorphism group acts transitively onΩ.

Bergman kernel. The Bergman kernel ofΩis

K(

Z,W

) =

K(

0,0

)

det

(

Im

−

ZZ∗

)

m+n

Type III

n

Symmetric matrices

For n

≥

1, V

= S

_n

(

C

)

(space of n

×

n symmetric matrices),

Ω

=



Z

∈

V

|

In

−

Z Z



0 .

Bergman kernel. The Bergman kernel ofΩis

K(

Z,W

) =

K(

0,0

)

det

(

Im

−

Z Z

)

n+1

Type II

n

Alternating matrices

For n

≥

2, V

= A

_n

(

C

)

(space of n

×

n antisymmetric matrices),

Ω

=



Z

∈

V

|

In

+

Z Z



0 .

Bergman kernel. The Bergman kernel ofΩis

K(

Z,W

) =

K(

0,0

)

det

(

Im

+

Z Z

)

n−1

Jordan triples in analysis on bounded symmetric

domains

Guy Roos St Petersburg [email protected] Hsinchu, Taiwan 2010/04/08-12

Part II. Hermitian Jordan triples and bounded symmetric

domains

1 Complex bounded symmetric domains

2 Jordan triple associated to a bounded symmetric domain 3 Spectral theory

4 Minimal polynomial and quasi-inverse 5 Simple Jordan triples

6 Boundary structure 7 Compactification

Bounded symmetric domains

A bounded domainΩ

⊂

V

'

Cn_{is called symmetric if for each x}

_∈

_Ω

there is an involutive holomorphic automorphism sx(sx2

=

idΩ) such that x

is an isolated fixed point of sx.

Bounded symmetric domains are homogeneous (under the group AutΩof holomorphic automorphisms).

Any bounded symmetric domainΩis biholomorphic to a bounded circled homogeneous domains, which is unique up to linear isomorphisms and is called the circled realization ofΩ.

We will always consider bounded symmetric domains in their circled realization.

A bounded symmetric domain is called irreducible if it is not equivalent to the direct product of two bounded symmetric domains.

Bounded symmetric domains

A bounded domainΩ

⊂

V

'

Cn_{is called symmetric if for each x}

_∈

_Ω

there is an involutive holomorphic automorphism sx(sx2

=

idΩ) such that x

is an isolated fixed point of sx.

Bounded symmetric domains are homogeneous (under the group AutΩof

holomorphic automorphisms).

Any bounded symmetric domainΩis biholomorphic to a bounded circled homogeneous domains, which is unique up to linear isomorphisms and is called the circled realization ofΩ.

We will always consider bounded symmetric domains in their circled realization.

A bounded symmetric domain is called irreducible if it is not equivalent to the direct product of two bounded symmetric domains.

Bounded symmetric domains

A bounded domainΩ

⊂

V

'

Cn_{is called symmetric if for each x}

_∈

_Ω

there is an involutive holomorphic automorphism sx(sx2

=

idΩ) such that x

is an isolated fixed point of sx.

Bounded symmetric domains are homogeneous (under the group AutΩof

holomorphic automorphisms).

Any bounded symmetric domainΩis biholomorphic to a bounded circled

homogeneous domains, which is unique up to linear isomorphisms and is

called the circled realization ofΩ.

We will always consider bounded symmetric domains in their circled realization.

A bounded symmetric domain is called irreducible if it is not equivalent to the direct product of two bounded symmetric domains.

Bounded symmetric domains

A bounded domainΩ

⊂

V

'

Cn_{is called symmetric if for each x}

_∈

_Ω

there is an involutive holomorphic automorphism sx(sx2

=

idΩ) such that x

is an isolated fixed point of sx.

Bounded symmetric domains are homogeneous (under the group AutΩof

holomorphic automorphisms).

Any bounded symmetric domainΩis biholomorphic to a bounded circled

homogeneous domains, which is unique up to linear isomorphisms and is

called the circled realization ofΩ.

We will always consider bounded symmetric domains in their circled realization.

A bounded symmetric domain is called irreducible if it is not equivalent to the direct product of two bounded symmetric domains.

Classification of bounded symmetric domains

Type Im,n

(

1

≤

m

≤

n

)

. V

= M

m,n

(

C

)

(space of m

×

n matrices with

complex entries).

Ω

=



x

∈

V

|

I_m

−

xtx



0 .

Type II_n

(

n

≥

2

)

V

= A

n

(

C

)

(space of n

×

n alternating matrices).

Ω

=

{

x

∈

V

|

I_n

+

xx



0

}

.

Type IIIn

(

n

≥

1

)

. V

= S

n

(

C

)

(space of n

×

n symmetric matrices).

Ω

=

{

x

∈

V

|

In

−

xx



0

}

.

Type IVn

(

n

>

2

)

. V

=

Cn,σ

(

x

) =

∑x_i2, ρ

(

x

) =

∑

|

xi

|

2. The domainΩis

defined by

1

−

2ρ

(

x

) +

|

σ

(

x

)|

2

>

0, 1

−

ρ

(

x

) >

0.

Type V . V

= M

_2,1

(

O_C

) '

C16_{, exceptional type.}

References

Cartan, ´Elie. Sur les domaines born ´es homog `enes de l’espace de n variables complexes. Abh. Math. Sem. Univ. Hamburg, 11 (1935), 1–114.

Helgason, Sigurdur. Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. xv+628 pp. ISBN: 0-12-338460-5 MR0514561 (80k:53081)

Satake I. Algebraic Structures of Symmetric Domains, Iwanami Shoten and Princeton University Press, 1980.

References

Loos, Ottmar. Bounded symmetric domains and Jordan pairs, Math. Lectures, Univ. of California, Irvine, 1977.

Roos, Guy. Jordan triple systems, pp. 425–534, in J. Faraut,

S. Kaneyuki, A. Kor ´anyi, Q.k. Lu, G. Roos, Analysis and geometry on complex homogeneous domains, Progress in Mathematics, vol.185, Birkh ¨auser, Boston, 2000.

Roos, Guy. Exceptional symmetric domains. Symmetries in complex analysis, 157–189, Contemp. Math., 468, Amer. Math. Soc.,

Part II. Hermitian Jordan triples and bounded symmetric

domains

1 Complex bounded symmetric domains

2 Jordan triple associated to a bounded symmetric domain 3 Spectral theory

4 Minimal polynomial and quasi-inverse 5 Simple Jordan triples

6 Boundary structure 7 Compactification

Automorphism group

LetΩbe a bounded circled homogeneous domain in a complex vector

space V .

Then the automorphism group AutΩis a real Lie group. Denote by

G

= (

AutΩ

)

0its identity component and by

K

=

G0

=

{

g

∈

G

|

g

(

0

) =

0

}

.

Automorphism group

Letω be a volume form on V , invariant by K and by translations. Let

K(

z

)

be the Bergman kernel ofΩwith respect toωand let

hz

(

u,v

) =

∂u∂vlog

K(

z

)

be its Bergman metric at z

∈

Ω. The Bergman metric is invariant by the

automorphisms ofΩ: for g

∈

G,

hz

(

u,v

) =

hg(z)

(

g0

(

z

) ·

u,g0

(

z

) ·

v

)

(

u,v

∈

V, z

∈

Ω

)

.

For g

∈

K , this implies

h₀

(

u,v

) =

h₀

(

gu,gv

)

(

u,v

∈

V

)

.

Infinitesimal automorphisms

A vector field onΩis a mapξ :Ω

→

V . The Lie bracket of two such

vector fields is defined by

[

ξ, η

](

z

) =

ξ0

(

z

) ·

η

(

z

) −

η0

(

z

) ·

ξ

(

z

)

.

A one parameter subgroup

(

g_t

)

_t∈Rof G will be identified with the

holomorphic vector field

ξ

(

z

) =

d d t

(

gt

(

z

)

    t=0 .

These vector fields form the Lie algebragof G.

Theorem. The elements ξ of gare the vector fieds

ξ

(

z

) =

Uz

+

v

−

Q

(

z

)

v,

where U

∈

Lie K , v

∈

V and Q:V

→

End_CV is a quadratic map with

values inC-linear endomorphisms of V .

Jordan triple associated to a bounded symmetric domain

Let Q

(

x,z

) =

Q

(

x

+

z

) −

Q

(

x

) −

Q

(

z

)

and define the triple product

{

x,y,z

} =

Q

(

x,y

)

z

(

x,y,z

∈

V

)

.

For x,y

∈

V , denote by D

(

x,y

)

theC-linear operator defined by

D

(

x,y

)

z

= {

xyz

}

(

z

∈

V

)

.

For v

∈

V , denote byξvthe vector field

Jordan triple associated to a bounded symmetric domain

Proposition. The following identities hold:

[

ξu, ξv

] =

D

(

u,v

) −

D

(

v,u

)

,

[[

ξu, ξv

]

, ξw

] =

ξ{u,v,w}−{v,u,w},

[

D

(

u,v

)

,D

(

x,y

)] =

D

({

uvx

}

,y

) −

D

(

x,

{

vuy

}

,

Hermitian Jordan triples

The triple product

(

x,y,z

) 7→ {

xyz

}

is complex bilinear and symmetric

with respect to

(

x,z

)

, complex antilinear with respect to y. It satisfies the Jordan identity

{

xy

{

uvw

}} − {

uv

{

xyw

}} = {{

xyu

}

vw

} − {

u

{

vxy

}

w

}

. (J)

Definition. The space V endowed with the triple product

{

xyz

}

verifying the identity (J) is called an Hermitian Jordan triple.

Bergman operator

Let

(

V,

{

, ,

})

be an Hermitian Jordan triple. The map

Q :V

−→

End_R

(

V

)

defined by Q

(

x

)

y

=

1₂

{

xyx

}

is called.quadratic

representation . The Bergman operator B is defined by

B

(

x,y

) =

idV

−

D

(

x,y

) +

Q

(

x

)

Q

(

y

)

(

x,y

∈

V

)

.

The quadratic representation and the Bergman operator satisfy to many identities; the most important of these are

Q

(

Q

(

x

)

y

) =

Q

(

x

)

Q

(

y

)

Q

(

x

)

,

Bergman operator

Theorem. Let Ω

⊂

V be a bounded circled homogeneous domain and let

(

V,

{

, ,

})

be the associated Jordan triple. Then

The Bergman kernel of Ωis

K(

z

) =

1

volΩ

1 det B

(

z,z

)

.

The Bergman metric at 0 is h₀

(

u,v

) =

tr D

(

u,v

)

.

The Bergman metric at z

∈

Ωis

h_z

(

u,v

) =

h₀

(

B

(

z,z

)

−1u,v

)

(

z

∈

Ω; u,v

∈

V

)

.

The Jordan triple product on V is characterized by h0

({

uvw

}

,t

) =

∂u∂v∂w∂tlog

K(

z

)

|

z=0 . Definition. A Jordan triple system is called Hermitian positive if

(

u

|

v

) =

tr D

(

u,v

)

is positive definite.

As the Bergman metric of a bounded domain is always definite positive, the Jordan triple associated to a bounded symmetric domain is Hermitian positive.