III The The Shilov Shilov boundary boundary III V Positive - - PowerPoint PPT Presentation

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III III The The Shilov Shilov boundary boundary

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V V Positive Positive Hermitian Hermitian Jordan Triple System Jordan Triple System {x,y,z} {x,y,z} C C-

• linear

linear in x in x and and z, z, conjugate conjugate linear linear in y in y symmetric symmetric in (x,z) in (x,z) a a quintic quintic identity identity τ τ(x,y) = tr (z (x,y) = tr (z→ → {x,y,z}) positive {x,y,z}) positive definite definite hermitian hermitian form form c tripotent {c,c,c} = 2c c tripotent {c,c,c} = 2c lxl lxl spectral spectral norm norm of x

• f x

D unit D unit ball ball for for the the spectral spectral norm norm G = G = Hol Hol(D) (D)0

0, K

, K stabilizer stabilizer of 0 in G.

• f 0 in G.

Example Example. . V = Mat (p,q, V = Mat (p,q, C C) (p ) (p≤ ≤q, n=p+q) q, n=p+q) {x,y,z} = {x,y,z} = xy xy*z+ *z+zy zy*x, *x, lxl lxl = = llxll llxllop

• p

D = {x in V, D = {x in V, lxl lxl<1}, G = PU(p,q), K = P(U(p) <1}, G = PU(p,q), K = P(U(p)xU xU(q)) (q)) k(z,w)= k(z,w)= det det(1 (1p

p-

• zw*)

zw*)-

• n

n

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III.1 III.1 Tripotents

Tripotents and and Peirce Peirce frames frames

Let V Let V be be a PHJTS, a PHJTS, with with triple triple product product {.,.,.}. Assume, for {.,.,.}. Assume, for convenience convenience, , that that V V is is simple (i.e. simple (i.e. cannot cannot be be written written as a as a sum sum of

• f two

two PHJTS). PHJTS). Recall Recall that that a tripotent a tripotent is is an an element element c c which which satisfies satisfies {c,c,c} = 2c. {c,c,c} = 2c. There There is is a (partial) a (partial) order

• rder on
• n tripotents

tripotents : if c : if c and and d are d are two two tripotents tripotents, , then then say say c c < d if < d if there there exists exists a tripotent f≠0, orthogonal to c a tripotent f≠0, orthogonal to c and and such such that that d=c+f. d=c+f. A tripotent c A tripotent c is is primitive primitive (or minimal) if c (or minimal) if c can can not not be be written written as a as a sum sum of

• f

two two (non (non zero zero) ) tripotents tripotents. . Any Any tripotent tripotent can can be be written written as a as a sum sum of

• f

primitive orthogonal primitive orthogonal tripotents tripotents. . Any Any two two minimal minimal tripotents tripotents are are conjugate conjugate under under an an automorphism automorphism of V.

• f V.

A A Peirce Peirce frame frame is is a maximal set of a maximal set of orthognal

• rthognal primitive

primitive tripotents tripotents. . Any Any two two Peirce Peirce frames frames are are conjugate conjugate under under an an automorphism automorphism of V. In

• f V. In

particular particular, , the the number number of

• f elements

elements is is the the same same for all for all frames frames (call (call it it the the rank rank of V).

• f V).

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Let c Let c be be a tripotent. a tripotent. Then Then TFAE TFAE (i) c= c (i) c= c1

1+…+

+…+c cr

r,

, where where (c (c1

1,…,

,…, c cr

r)

) is is a Peirce a Peirce frame frame ( (ii ii) c ) c is is a maximal tripotent a maximal tripotent ( (iii iii) V = V ) V = V2

2 (c) + V

(c) + V1

1(c) (i.e. V

(c) (i.e. V0

0(c) = 0)

(c) = 0) ( (iv iv) ) R R c c1

1+

+R R c c2

2+… +

+… +R R c cr

r is

is a maximal flat a maximal flat space space in V. in V.

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III.2 III.2 The The Shilov Shilov boundary boundary

Let D Let D be be a a domain domain in in some some complex complex vector vector space space. . The The Shilov Shilov boundary boundary S of D S of D is is the the smallest smallest closed closed subset subset of

• f the

the boundary boundary of D, for

• f D, for which

which the the maximum maximum principle principle for for the the modulus modulus

• f
• f holomorphic

holomorphic function function applies applies. . The The Shilov Shilov boundary boundary may may be be much much smaller smaller than than the the topological topological boundary boundary. . Example Example 1

• 1. Let D

. Let D be be the the product product of

• f two

two copies of copies of the the complex complex unit unit disc.

• disc. Then

Then the the Shilov Shilov boundary boundary of D is

• f D is the

the product product of

• f two

two copies copies

• f
• f the

the unit unit circle circle, as , as can can be be seen seen by by applying applying twice twice the the maximum maximum principle principle w.r.t. w.r.t. each each variable. variable. Example Example 2

• 2. Let D

. Let D be be the the unit unit ball ball in Mat in Mat (p,q) (p,q) with with p ≤ q, p ≤ q, then then

• x

x is is in in the the topological topological boundary boundary of D

• f D iff

iff 1 1 is is an an eigenvalue eigenvalue of

• f xx

xx* *

• x

x is is in in the the Shilov Shilov boundary boundary iff iff xx xx*= *= Id Idp

p

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Example Example 3.

• 3. The

The Siegel disc Siegel disc and and the the Lagrangian Lagrangian manifold manifold Let V = Let V = Symm Symm(r, (r, C C) ) be be the the PHJTS, PHJTS, with with product product {x,y,z} = {x,y,z} = xy xy*z+ *z+zy zy*x . *x . D = {x in V ; 1 D = {x in V ; 1-

• xx*>>0}

xx*>>0} The The group G group G is is Sp Sp(2r, (2r, R R)( )( mod mod{±1}), {±1}), and and K K is is isomorphic isomorphic to U(r) to U(r) acting on V by (u,X) acting on V by (u,X)→ →uXu uXut

t .

. D D is is called called the the Siegel disc Siegel disc. . Its Its Shilov Shilov boundary boundary is is S = { S = { σ σ in V ; in V ; σ σ σ σ* = 1}. * = 1}. S S is is isomorphic isomorphic to to the the Lagrangian Lagrangian manifold ( manifold (also also to U(r)/O(r)). to U(r)/O(r)). Recall

• Recall. (E,

. (E, ω) ω) a real a real symplectic symplectic vector vector space space , of dimension 2r. , of dimension 2r. A A Lagrangian Lagrangian L L is is a maximal a maximal totally totally isotropic isotropic vector vector subspace subspace of E

• f E

( (hence hence of dimension r).

• f dimension r). The

The Lagrangian Lagrangian manifold manifold is is the the set of all set of all lagrangians lagrangians. . It It sits sits in in the the Grassmanian Grassmanian of

• f r

r-

• subspaces

subspaces in E. in E.

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The The Shilov Shilov boundary boundary S of S of the the open unit

• pen unit ball

ball D in V D in V can can be be described described in in the the following following equivalent equivalent ways ways : :

i) i)

S S is is the the set of maximal set of maximal tripotents tripotents

ii) ii)

S S is is the the set of set of extremal extremal points of points of the the closed closed open

• pen ball

ball (as a (as a convex convex set) set)

iii) iii)

S S is is the the set of points in set of points in the the closed closed unit unit ball ball which which are are at at maximal distance of maximal distance of the the origin

• rigin for

for the the distance distance associated associated to to Hermitian Hermitian form form τ τ. .

SLIDE 8

III.3 Action of G on S III.3 Action of G on S and and S S x

x S

S

The The action of a action of a holomorphic holomorphic diffeomorphism diffeomorphism of D

• f D always

always extend extend to to some some neigh’d neigh’d of

• f the

the closure closure of D.

• f D. Hence

Hence the the action of G action of G extends extends to to the the closure closure of D. In

• f D. In particular

particular, G , G acts acts on S.

• n S.

Proposition 1 Proposition 1 S S is is a a connected connected compact manifold. G compact manifold. G acts acts transitively transitively on S,

• n S, and

and S S is is the the unique unique closed closed G G-

• orbit
• rbit in

in the the boundary boundary of D. K (a maximal compact

• f D. K (a maximal compact subgroup

subgroup of G)

• f G) acts

acts already already transitively transitively on S.

• n S. The

The stabilizer stabilizer of a point in S

• f a point in S is

is a a (maximal) (maximal) parabolic parabolic subgroup subgroup of G.

• f G.

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III.3 Action of G on S III.3 Action of G on S and and S S x

x S

S

The The action of a action of a holomorphic holomorphic diffeomorphism diffeomorphism of D

• f D always

always extend extend to to some some neigh’d neigh’d of

• f the

the closure closure of D.

• f D. Hence

Hence the the action of G action of G extends extends to to the the closure closure of D. In

• f D. In particular

particular, G , G acts acts on S.

• n S.

Proposition 1 Proposition 1 S S is is a a connected connected compact manifold. G compact manifold. G acts acts transitively transitively on S,

• n S, and

and S S is is the the unique unique closed closed G G-

• orbit
• rbit in

in the the boundary boundary of D. K (a maximal compact

• f D. K (a maximal compact subgroup

subgroup of G)

• f G) acts

acts already already transitively transitively on S.

• n S. The

The stabilizer stabilizer of a point in S

• f a point in S is

is a a (maximal) (maximal) parabolic parabolic subgroup subgroup of G.

• f G.

Proposition 2 Proposition 2 G has a (unique) open G has a (unique) open orbit

• rbit in S

in S x x S. S. A pair ( A pair (σ,τ) σ,τ) in in S S x

x S in

S in the the open

• pen orbit
• rbit is

is said said to to be be transverse. transverse. Example Example Let S Let S be be the the Lagrangian Lagrangian manifold. A pair of

• manifold. A pair of Lagrangians

Lagrangians (L (L1

1 , L

, L2

2)

) is is tranverse tranverse iff iff L L1

1 ∩

∩ L L2

2 = {0}.

= {0}. The The symplectic symplectic group group is is transitive on pairs of transverse transitive on pairs of transverse Lagrangians Lagrangians (Darboux). (Darboux).

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A pair ( A pair (σ,τ σ,τin S in S x

x S is transverse

S is transverse

• iff there exists

iff there exists a a geodesic geodesic γ γ(t) in D (t) in D such that such that γ γ(+ (+∞ ∞ ) = ) = σ σ, , γ γ( (-

• ∞

∞ ) = ) = τ. τ.

• iff

iff the the Bergman Bergman kernel kernel extends extends by by continuity continuity to to ( (σ,τ) ( σ,τ) (i.e. i.e. k( k(σ,τ) σ,τ) is is defined defined). ).

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III.4 III.4 Euclidean Euclidean Jordan Jordan algebra algebra

A A Euclidean Euclidean Jordan Jordan algebra algebra is is a a Euclidean Euclidean vector vector space space (W, < , >) (W, < , >) with with a a bilinear bilinear product product x.y x.y and and a unit e a unit e such such that that

i) i)

x.y = y.x x.y = y.x

ii) ii)

x x2

2(x.y) = x.(x

(x.y) = x.(x2

2.y) (

.y) (weak weak associativity associativity) )

iii) iii)

e.x = x.e = x e.x = x.e = x

iv) iv)

<x.y,z> = <x,y.z> <x.y,z> = <x,y.z> Example Example W = W = Symm Symm(r, (r,R R), x.y = 1/2( ), x.y = 1/2(xy xy+ +yx yx), e = id, <x.y> = tr( ), e = id, <x.y> = tr(xy xy). ).

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III.4 III.4 Euclidean Euclidean Jordan Jordan algebra algebra

A A Euclidean Euclidean Jordan Jordan algebra algebra is is a a Euclidean Euclidean vector vector space space (W, < , >) (W, < , >) with with a a bilinear bilinear product product x.y x.y and and a unit e a unit e such such that that

i) i)

x.y = y.x x.y = y.x

ii) ii)

x x2

2(x.y) = x.(x

(x.y) = x.(x2

2.y) (

.y) (weak weak associativity associativity) )

iii) iii)

e.x = x.e = x e.x = x.e = x

iv) iv)

<x.y,z> = <x,y.z> <x.y,z> = <x,y.z> Example Example W = W = Symm Symm(r, (r,R R), x.y = 1/2( ), x.y = 1/2(xy xy+ +yx yx), e = id, <x.y> =t r( ), e = id, <x.y> =t r(xy xy). ). Hermitification Hermitification of a

• f a Euclidean

Euclidean Jordan Jordan algebra algebra W. W. Let Let W W be be the the complexification of W. complexification of W. Extend Extend the the Jordan Jordan product product in a in a C C-

• linear

linear way

• way. On

. On W W define define {x , y , z} = x . (y {x , y , z} = x . (y-

• . z) + z . (y

. z) + z . (y-

• . x)

. x) -

• y

y-

• . (x . z) .

. (x . z) . Then Then W W is is a PHJTS ( a PHJTS (called called the the Hermitification Hermitification of W).

• f W).

Example Example. . W W = = Symm Symm(r, (r, C C), {x,y,z} = ( ), {x,y,z} = (xy xy*z+ *z+zy zy*x). *x).

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III.5 Tube III.5 Tube-

• type

type domains domains

D D is is said said to to be be of

• f tube type

tube type if S if S is is totally totally real ( real (equivalently equivalently if if dim dimR

R S =

S = dim dimC

C V).

V). Propostion Propostion 1

• 1. Let V

. Let V be be a PHJTS, a PHJTS, and and let D let D be be its its unit unit ball ball for for the the spectral spectral norm norm. . Then Then D D is is of tube type if

• f tube type if and

and only

• nly if V

if V is is the the hermitification hermitification of

• f some

some Euclidean Euclidean Jordan Jordan algebra algebra. . Let W Let W be be a a Euclidean Euclidean Jordan Jordan algebra algebra, , W W its its hermitification hermitification. . Then Then D D is is holomorphically holomorphically equivalent equivalent to to the the tube V+i tube V+iΩ Ω through through the the Cayley Cayley transform transform, , where where Ω Ω is is the the ( (interior interior of)

• f) the

the cone cone of

• f

squares {x squares {x2

2, x in V}.

, x in V}. The The Cayley Cayley transform transform extends extends to a dense to a dense

• pen of S
• pen of S and

and maps maps it it to V+i0. to V+i0.

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Example Example 1

• 1. V=

. V= Symm Symm(r, (r,R R) ) The The cone cone Ω Ω is is the the set of positive set of positive definite definite matrices. matrices. The The Siegel disc {z in Siegel disc {z in Symm Symm(r, (r,C C), 1 ), 1-

• zz*>>0}

zz*>>0} is is of tube type,

• f tube type,

holomorphically holomorphically equivalent equivalent to to the the Siegel Siegel upper upper half half plane plane {x+ {x+iy iy, x, y in , x, y in Symm Symm(r, (r,R R), y >> 0} ), y >> 0} through through the the Cayley Cayley transform transform c(z) = i (1+z) (1 c(z) = i (1+z) (1-

• z)

z) -

• 1

1.

. The The Shilov Shilov boundary boundary is is S = {z in S = {z in Symm Symm(r,C), z*=z (r,C), z*=z -

• 1

1}. S

}. S is is isomorphic isomorphic to to the the Lagrangian Lagrangian manifold. manifold.

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Example Example 2 2 The The matrix matrix unit unit ball ball I Ip

p,q ,q is

is of tube

• f tube-
• type

type iff iff p=q. p=q. The The associated associated PHJTS PHJTS is is Mat (p,q, Mat (p,q, C C). ). If p=q, If p=q, then then the the PHJTS PHJTS is is the the Hermitification Hermitification of

• f the

the Euclidean Euclidean Jordan Jordan algebra algebra Herm Herm(p, (p,C C) ) with with x.y = 1/2 ( x.y = 1/2 (xy xy+ +yx yx) ) and and <x,y>= <x,y>= Re Re(tr (tr xy xy) ) The The Shilov Shilov boundary boundary is is S = U(p). S = U(p). Proposition 2 Proposition 2 If D If D is is of tube

• f tube-
• type,

type, then then S S is is a compact a compact Riemannian Riemannian symmetric symmetric space space K/L. K/L.

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III.4 Action of G on S III.4 Action of G on S x

x S

S x

x S

S

Proposition 4 Proposition 4 Let D Let D be be a a bounded bounded symmetric symmetric domain domain of tube type.

• f tube type.

The The action of G on S action of G on S x x S S x x S has a S has a finite finite number number of

• f orbits
• rbits,

, and and in in particular particular (r+1) open (r+1) open orbits

• rbits.

. Let V Let V be be the the Euclidean Euclidean Jordan Jordan algebra algebra,to ,to which which D D is is associated associated. . Let (c Let (c1

1,c

,c2

2,…,

,…,c cr

r)

) be be a Peirce a Peirce frame frame such such that that e = c e = c1

1+c

+c2

2,…+

,…+c cr

r . For

. For 0 ≤ k ≤ r, let 0 ≤ k ≤ r, let e ek

k = c

= c1

1 + …+

+ …+c ck

k -

• c

ck

k+1 +1 -

• …

…-

• c

cr

r .

.Then Then (e, (e, -

• e,

e, ie iek

k) (

) (0 ≤ k ≤ r) 0 ≤ k ≤ r) is is a set of a set of representatives representatives of

• f the

the open

• pen orbits
• rbits.

. N.B. If D N.B. If D is is not of tube not of tube-

• type, S

type, S is is not a not a symmetric symmetric space space of U,

• f U,

and and there there are are infinitely infinitely many many G G-

• orbits
• rbits and

and no open no open G G-

• orbit
• rbit in

in S S x

x S

S x

x S

S . .

SLIDE 17

In In the the tube tube-

• type case, S

type case, S is is a a Riem Riem. . Symmetric Symmetric space

• space. A maximal flat

. A maximal flat torus of S torus of S is is of

• f the

the form form T = { T = { ξ ξ1

1c

c1

1+

+ ξ ξ2

2c

c2

2+

+… …+ + ξ ξr

rc

cr

r, l

, lξ ξj

jl

l=1, 1 =1, 1≤ ≤j j≤ ≤r } r } where where (c (c1

1, c

, c2

2,

, … …, , c cr

r)

) is is Peirce Peirce frame frame. . Proposition 5 Proposition 5 (KH (KH Neeb Neeb, JLC ‘07) Let D , JLC ‘07) Let D be be a a bounded bounded symmetric symmetric domain domain

• f tube type.
• f tube type. Fix

Fix a maximal torus T in S. Let a maximal torus T in S. Let σ σ1

1,

, σ σ2

2,

, σ σ3

3 be

be in S. in S. Then Then there there exists exists g in G g in G such such that that g( g(σ σ1

1), g(

), g(σ σ2

2), g(

), g(σ σ3

3)

) belong belong to T. to T. Example Example. . Normal Normal form form of a triplet of

• f a triplet of Lagrangians

Lagrangians . . Let L Let L(1)

(1), L

, L(2)

(2), L

, L(3)

(3) be

be three three arbitray arbitray Lagrangians Lagrangians in E. in E. Then Then there there exists exists a a symplectic symplectic basis ( basis (e ej

j,

,f fj

j) of E

) of E such such that that

L L(k)

(k) =

= span span { cos { cos θ θ1

1(k) (k) e

e1

1+sin

+sin θ θ1

1(k) (k) f

f1

1, cos

, cos θ θ2

2(k) (k) e

e2

2+sin

+sin θ θ2

2(k) (k) f

f2

2, …, cos

, …, cos θ θr

r(k) (k) e

er

r+sin

+sin r

r(k) (k)f

fr

r}

} for k=1,2,3. for k=1,2,3.

SLIDE 18

List of simple List of simple Euclidean Euclidean Jordan Jordan algebras algebras, , bounded bounded domains domains of tube

• f tube

type type and and their their Shilov Shilov boundaries boundaries V V D D S S Symm Symm(r, (r,R R) ) unit unit ball ball in in Symm Symm(r, (r,C C) ) Lagrangian Lagrangian manifold manifold Herm Herm(r, (r,C C) ) unit unit ball ball in Mat(r, in Mat(r, C C) ) U(r) U(r) Herm Herm(r, (r,H H) ) unit unit ball ball in in Skew Skew(2r, (2r,C C) ) U(2r)/SU(m, U(2r)/SU(m,H H) ) R R x

x R

Rd

d-

• 1

1 (*) (*)

Lie Lie ball ball in in C Cd

d

(U(1) (U(1)x

xS

Sd

d-

• 1

1) /

) / Z Z2

2

Herm Herm(r, (r,O O) ) E E7(

7(-

• 25)

25)/U(1)E

/U(1)E6

6

U(1)E U(1)E6

6 / F

/ F4

4

(*) (*) ( (λ λ, x) . ( , x) . (μ μ,y) = ( ,y) = (λ λ + + μ μ + <x,y>, + <x,y>, μ μ x + x + λ λ y) y)