Jordan triples in analysis on bounded symmetric...

Jordan triples in analysis on bounded symmetricdomains

Guy Roos

St [email protected]

Hsinchu, Taiwan 2010/04/08-12

Guy Roos (St Petersburg) Part II. Jordan triples and symmetric domains Hsinchu, Taiwan 2010/04/08-12 1 / 46

Part II. Hermitian Jordan triples and bounded symmetricdomains

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 domain Ω ⊂ V ' Cn is called symmetric if for each x ∈ Ωthere is an involutive holomorphic automorphism sx (s2

x = idΩ) such that xis an isolated fixed point of sx .Bounded symmetric domains are homogeneous (under the group Aut Ω ofholomorphic automorphisms).Any bounded symmetric domain Ω is biholomorphic to a bounded circledhomogeneous domains, which is unique up to linear isomorphisms and iscalled the circled realization of Ω.We will always consider bounded symmetric domains in their circledrealization.A bounded symmetric domain is called irreducible if it is not equivalent tothe direct product of two bounded symmetric domains.


Classification of bounded symmetric domains

Type Im,n (1 ≤ m ≤ n). V =Mm,n(C) (space of m× n matrices withcomplex entries).

Ω =

x ∈ V | Im − x tx 0

.

Type IIn (n ≥ 2) V = An(C) (space of n× n alternating matrices).

Ω = x ∈ V | In + xx 0 .

Type IIIn (n ≥ 1). V = Sn(C) (space of n× n symmetric matrices).

Ω = x ∈ V | In − xx 0 .

Type IVn (n > 2). V = Cn, σ(x) = ∑ x2i , ρ(x) = ∑ |xi |2. The domain Ω is

defined by1− 2ρ(x) + |σ(x)|2 > 0, 1− ρ(x) > 0.

Type V . V =M2,1(OC) ' C16, exceptional type.Type VI. V = H3(OC) ' C27, exceptional type.


References

Cartan, Elie. Sur les domaines bornes homogenes de l’espace de nvariables complexes. Abh. Math. Sem. Univ. Hamburg, 11 (1935),1–114.

Helgason, Sigurdur. Differential geometry, Lie groups, and symmetricspaces. 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, IwanamiShoten 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. Koranyi, Q.k. Lu, G. Roos, Analysis and geometry oncomplex homogeneous domains, Progress in Mathematics, vol.185,Birkhauser, Boston, 2000.

Roos, Guy. Exceptional symmetric domains. Symmetries in complexanalysis, 157–189, Contemp. Math., 468, Amer. Math. Soc.,Providence, RI, 2008. MR2484095 (2009m:32041).





3 Spectral theory




7 Compactification


Automorphism group

Let Ω be a bounded circled homogeneous domain in a complex vectorspace V .Then the automorphism group Aut Ω is a real Lie group. Denote byG = (Aut Ω)0 its identity component and byK = G0 = g ∈ G | g(0) = 0.Then K is a Lie group of linear automorphisms of Ω.


Automorphism group

Let ω be a volume form on V , invariant by K and by translations. Let K(z)be the Bergman kernel of Ω with respect to ω and let

hz(u, v) = ∂u∂v logK(z)

be its Bergman metric at z ∈ Ω. The Bergman metric is invariant by theautomorphisms of Ω: for g ∈ G,

hz(u, v) = hg(z)(g′(z) · u, g′(z) · v) (u, v ∈ V , z ∈ Ω).

For g ∈ K , this implies

h0(u, v) = h0(gu, gv) (u, v ∈ V).

So K is a compact subgroup of the unitary group U(V , h0).


Infinitesimal automorphisms

A vector field on Ω is a map ξ : Ω→ V . The Lie bracket of two suchvector fields is defined by

[ξ, η](z) = ξ ′(z) · η(z)− η′(z) · ξ(z).

A one parameter subgroup (gt)t∈R of G will be identified with theholomorphic vector field

ξ(z) =dd t

(gt(z)∣∣∣∣t=0

.

These vector fields form the Lie algebra g of G.Theorem. The elements ξ of g are the vector fieds

ξ(z) = Uz + v −Q(z)v,

where U ∈ Lie K, v ∈ V and Q : V → EndC V is a quadratic map withvalues in C-linear endomorphisms of V.Proof. See [Loos1977], §2.


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) the C-linear operator defined by

D(x, y)z = xyz (z ∈ V).

For v ∈ V , denote by ξv the vector field

ξv(z) = v −Q(z)v (z ∈ V).


Jordan triple associated to a bounded symmetric domain

Proposition. The following identities hold:

[ξu, ξv ] = D(u, v)− D(v, u),[[ξu, ξv ], ξw ] = ξu,v,w−v,u,w,

[D(u, v), D(x, y)] = D(uvx, y)− D(x, vuy,h0(uvw, z) = h0(w, vuz).


Hermitian Jordan triples

The triple product (x, y, z) 7→ xyz is complex bilinear and symmetricwith respect to (x, z), complex antilinear with respect to y. It satisfies theJordan identity

xyuvw − uvxyw = xyuvw − uvxyw. (J)

Definition. The space V endowed with the triple product xyz verifyingthe identity (J) is called an Hermitian Jordan triple.


Bergman operator

Let (V , , , ) be an Hermitian Jordan triple. The mapQ : V −→ End R(V) defined by Q(x)y = 1

2xyx is called.quadraticrepresentation . 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 manyidentities; the most important of these are

Q(Q(x)y) = Q(x)Q(y)Q(x),Q(B(x, y)z) = B(x, y)Q(z)B(y, 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 Ω is

K(z) =1

vol Ω1

det B(z, z).

The Bergman metric at 0 is h0(u, v) = tr D(u, v).The Bergman metric at z ∈ Ω is

hz (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∂t logK(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 Hermitianpositive.


Examples

Type Im,n (1 ≤ m ≤ n). V = Mm,n(C) (space of m× n matrices withcomplex entries).

Ω =

x ∈ V | Im − x tx 0

,

D(x, y)z = x, y, z = xy∗z + zy∗x,

Q(x)y = xy∗x,

B(x, y)z = (Im − xy∗)z(In − y∗x).

det B(x, x) = (det(Im − xx∗))m+n .


Examples

Type IIn (n ≥ 2) V = An(C) (space of n× n alternating matrices).

Ω = x ∈ V | In + xx 0 .

Same triple product and same operators D, Q and B as type Im,n.

det B(x, x) = (det(In + xx))n−1 .


Ω = x ∈ V | In − xx 0 .

Same triple product and same operators D, Q and B as type Im,n.

det B(x, x) = (det(In − xx))n+1 .


Examples

Type IVn (n > 2).V = Cn, σ(x) = ∑ x2

i , ρ(x) = ∑ |xi |2, (x | y) = ∑ xiyi , (x : y) = ∑ xiyi .

Ω =

1− 2ρ(x) + |σ(x)|2 > 0, 1− ρ(x) > 0

,

D(x, y)z = x, y, z = 2 ((x | y)z + (z | y)x − (x : z)y)Q(x)y = 2(x | y)x − σ(x)y,

det B(x, x) =(

1− 2ρ(x) + |σ(x)|2)n

.





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7 Compactification


Tripotents

Let (V , , , ) be a Jordan triple. An element c ∈ V is called tripotent if

ccc = 2c.

If c is a tripotent, the operator D(c, c) annihilates the polynomialT(T − 1)(T − 2).The decomposition

V = V0(c)⊕ V1(c)⊕ V2(c),

where Vj(c) is the eigenspace

Vj(c) = x ∈ V | D(c, c)x = jx ,

is called the Peirce decomposition of V (with respect to the tripotent c).


Orthogonality of tripotents

Two tripotents c1 and c2 are called orthogonal if D(c1, c2) = 0.If c1 and c2 are orthogonal tripotents, then D(c1, c1) and D(c2, c2)commute and c1 + c2 is also a tripotent.A non zero tripotent c is called primitive if it is not the sum of non zeroorthogonal tripotents.A tripotent c is called maximal if there is no non zero tripotent orthogonalto c.


Frames

A frame of V is a maximal sequence (c1, . . . , cr) of pairwise orthogonalprimitive tripotents.Let V be a positive Jordan triple. Then there exist frames for V . All frameshave the same number of elements, which is the rank r of V .Let c = (c1, . . . , cr) be a frame. For 0 ≤ i ≤ j ≤ r , let

Vij(c) =

x ∈ V | D(ck , ck )x = (δki + δk

j )x, 1 ≤ k ≤ r

.

In particular,

V00(c) = 0,

Vjj(c) = V2(cj) (0 < j),Vij(c) = V1(ci) ∩ V1(cj) (0 < i < j),

V0j(c) = V1(cj) ∩⋂i,j

V0(ci) (0 < j).

The decomposition V =⊕

0≤i≤j≤r Vij(c) is called the simultaneous Peircedecomposition with respect to the frame c.


Let (pij) be the family of orthogonal projectors onto the subspaces Vij(c).Then, for x = λ1c1 + λ2c2 + · · ·+ λrcr , λi ∈ R (1 ≤ i ≤ r) and λ0 = 0,

D(x, x) = ∑0≤i≤j≤r

(λ2

i + λ2j)

pij,

Q(x)2 = ∑0≤i≤j≤r

λ2i λ2

j pij,

B(x, x) = ∑0≤i≤j≤r

(1− λ2

i) (

1− λ2j)

pij.


Spectral decomposition

Let V be a positive Jordan triple. Then any x ∈ V can be written in aunique way

x = λ1c1 + λ2c2 + · · ·+ λpcp ,

where λ1 > λ2 > · · · > λp > 0 and c1, c2 . . . , cp are pairwise orthogonaltripotents.The element x is regular iff p = r (the rank of V ); then (c1, c2, . . . , cr) is aframe of V .The decomposition x = λ1c1 + λ2c2 + · · ·+ λpcp is called the spectraldecomposition of x.The map x 7→ λ1, where x = λ1c1 + λ2c2 + · · ·+ λpcp is the spectraldecomposition of x (λ1 > λ2 > · · · > λp > 0) is a norm on V , called thespectral norm. We will denote the spectral norm by ‖x‖Ω. It satisfies

‖x‖2Ω = ‖Q(x)‖ =

12‖D(x, x)‖ ,

where ‖u‖ denotes the operator norm of an R-linear operator u ∈ EndR Vwith respect to the Hermitian norm ‖x‖2 = tr D(x, x).


Symmetric domain associated to a Jordan triple

Let V be a positive Jordan triple. Then V is associated to a uniquebounded circled symmetric domain Ω.The bounded symmetric domain Ω is the unit ball of V for the spectralnorm.It can also be defined as

Ω = x ∈ V | B(x, x) 0 .





3 Spectral theory




7 Compactification


Minimal polynomial

Let V be a positive Jordan triple of rank r . For k ≥ 0, let

x(k+1,y) = Q(x)k y (x, y ∈ V)

be the (k + 1)-th power of x in the Jordan algebra V (y).There exist polynomials m1, . . . , mr on V × V, homogeneous of respectivebidegrees (1, 1), . . . , (r , r), such that for each x ∈ V,

x(r+1,y) −m1(x, y)x(r,y) + · · ·+ (−1)rmr(x, y)x = 0.

The polynomial

m(T , x, y) = T r −m1(x, y)T r−1 + · · ·+ (−1)rmr(x, y)

is called the generic minimal polynomial of V (at (x, y)).The (inhomogeneous) polynomial N : V × V → C defined by

N(x, y) = m(1, x, y)

is called the generic norm.Guy Roos (St Petersburg) Part II. Jordan triples and symmetric domains Hsinchu, Taiwan 2010/04/08-12 27 / 46

Minimal polynomial

If x = ∑ λjcj , where c = (c1, . . . , cr) is a frame and λj ≥ 0, then

m(T , x, x) =r

∏i=1

(T − λ2i ),

N(x, x) =r

∏i=1

(1− λ2i ).

The domain Ω associated to V is also characterized by the set ofpolynomial inequalities

∂j

∂T j m(T , x, x)∣∣∣∣T=1

> 0, 0 ≤ j ≤ r − 1.


Quasi-inverse

Let V be a positive Jordan triple. For z, u ∈ V such thatB(z, u) = idV −D(z, u) + Q(x)Q(u) is invertible, the quasi-inverse zu isdefined by

zu = B(z, u)−1 (z −Q(z)u) .

Main properties.

zu+v = (zu)v ,

B(z, u)B(zu, v) = B(z, u + v).

The derivative of the rational map τu defined by

τu(z) = zu

isτ′u(z) = B(z, u)−1.


Automorphisms

For u ∈ Ω, the operator B(u, u) is positive (with respect to the Hermitianscalar product tr D(x, y)), so that B(u, u)t is well defined for t ∈ R. Letu ∈ Ω and consider a spectral decomposition

u = λ1e1 + · · ·+ λrer ,

1 > λ1 ≥ · · · ≥ λr ≥ 0,

where e = (e1, . . . , er) is a frame. Then B(u, u)t is given by

B(u, u)t = ∑0≤i≤j≤r

(1− λ2

i)t (

1− λ2j)t

pij,

where (pij) is the family of orthogonal projectors onto the subspaces of thesimultaneous Peirce decomposition with respect to the framee = (e1, . . . , er) .


Automorphisms

For u ∈ Ω, denote by τu the translation z 7→ z + u and by τu the rationalmap

τu(z) = zu.

The mapψu = τu B(u, u)1/2 τ−u

is an automorphism of Ω which sends 0 to u.The derivative of ψu at u is

ψ′u(u) = B(u, u)1/2.





3 Spectral theory




7 Compactification


Simple Jordan triples

Let V be a positive Jordan triple.An ideal I of V is a vector subspace of V such that

VVI ⊂ I, VIV ⊂ I.

A Jordan triple V is called simple if each ideal of V is 0 or V .Each positive Jordan triple V is semi-simple, that is, V is a (unique) directsum of simple ideals.A positive Jordan triple V is simple if and only if the associated boundedsymmetric domain Ω is irreducible.


Classification of simple positive Jordan triples

Type Im,n (1 ≤ m ≤ n). V = Mm,n(C) (space of m× n matrices withcomplex entries).

x, y, z = xy∗z + zy∗x.

Type IIn (n ≥ 2) V = An(C) (space of n× n alternating matrices). Sametriple product as type Im,n.Type IIIn (n ≥ 1). V = Sn(C) Same triple product as type Im,n.Type IVn (n > 2).V = Cn, σ(x) = ∑ x2

i , ρ(x) = ∑ |xi |2, (x | y) = ∑ xiyi , (x : y) = ∑ xiyi .

x, y, z = 2 ((x | y)z + (z | y)x − (x : z)y) .

Type V . V =M2,1(OC) ' C16, exceptional type.Type VI. V = H3(OC) ' C27, exceptional type.


Numerical invariants

Let V be a simple positive Jordan triple.For any frame c of V , the subspaces Vij = Vij(c) of the simultaneousPeirce decomposition have the following properties: V00 = 0 ; Vii = Cei

(0 < i); all Vij ’s (0 < i < j) have the same dimension a; all V0i ’s (0 < i)have the same dimension b.The numerical invariants of V are the rank r and the two integers

a = dim Vij (0 < i < j),b = dim V0i (0 < i).

The genus of V is the number g defined by

g = 2 + a(r − 1) + b.

The positive Jordan triple V is said to be of tube type if b = 0.


Numerical invariants by type

Type r a b g Tube type

Im,n 1 ≤ m ≤ n m 2 n−m m + n If m = n

IIn n = 2p n2 = p 4 0 2 (n− 1) Yes

IIn n = 2p + 1[ n

2

]= p 4 2 2(n− 1) No

IIIn n ≥ 1 n 1 0 n + 1 Yes

IVn n > 2 2 n− 2 0 n Yes

V 2 6 4 12 No

VI 3 8 0 18 Yes


Minimal polynomial

If V is a simple Hermitian Jordan triple of genus g,

h0(u, v) = tr D(u, v) = gm1(u, v),det B(x, y) = N(x, y)g.


Generic norm by type

Type Im,n (1 ≤ m ≤ n). V =Mm,n(C) (space of m× n matrices withcomplex entries).

N(x, y) = Det(Im − x ty).

Type IIn (n ≥ 2). V = An(C) (space of n× n alternating matrices).

N(x, y)2 = Det(In + xy).


N(x, y) = Det(In − xy).

Type IVn (n > 2). V = Cn. N(x, y) = 1− 2(x|y) + σ(x)σ(y).Type V . V =M2,1(OC). N(x, y) = 1− (x|y) + (x]|y]).Type VI. V = H3(OC). N(x, y) = 1− (x|y) + (x]|y])− det x det y.





3 Spectral theory




7 Compactification


Boundary structure

Let V be a simple positive Jordan triple of rank r and let Ω be theassociated irreducible homogeneous circled domain.Let M be the manifold of tripotents of V . Then M is the disjoint union

M =r

äk=1

Mk ,

where Mk is the manifold of tripotents of rank k : e ∈ Mk iff e is the sum ofk pairwise orthogonal minimal tripotents. The manifolds Mk are theconnected components of M (with different dimensions). The tangentspace TeMk is

i V+2 (e)⊕ V1(e),

where V1(e) = v ∈ V | D(e, e)x = x,

V2(e) = v ∈ V | D(e, e)x = 2x = V+2 (e)⊕ i V+

2 (e),V+

2 (e) = v ∈ V | Q(e)x = x ,

V−2 (e) = i V+2 (e) = v ∈ V | Q(e)x = −x .


Boundary structure

Theorem.The boundary of Ω is the disjoint union

∂Ω =r

äk=1

∂k Ω

of locally closed submanifolds ∂k Ω, with ∂k+1Ω ⊂ ∂k Ω.For e ∈ Mk , let Ωk (e) be the bounded symmetric domain associatedto the positive Jordan triple subsystem V0(e). Thene + Ωk (e) ⊂ ∂k Ω and

∂Ω ∩ (e + V0(e)) = e + Ωk (e).

The submanifold ∂k Ω is the disjoint union

∂k Ω = äe∈Mk

(e + Ωk (e)) .

The tangent space Tx ∂k Ω at x ∈ e + Ωk (e) is

i V+2 (e)⊕ V1(e)⊕ V0(e).


Example

Type IVn (n > 2). V = Cn, σ(x) = ∑ x2i , ρ(x) = ∑ |xi |2, (x | y) = ∑ xiyi ,

(x : y) = ∑ xiyi .

x, y, z = 2 ((x | y)z + (z | y)x − (x : z)y) ,

M1 =

e ∈ V | σ(e) = 0, ρ(e) =12

,

M2 = e ∈ V | ρ(e) = |σ(e)| = 1 .

For e ∈ M1, the spaces of the Peirce decomposition are V2(e) = Ce,V0(e) = Ce, V1(e) = 〈e, e〉⊥ and Ω0(e) = ∆e. For e ∈ M2,V0(e) = V1(e) = 0.

dimR M1 = 2n− 3, dimR M2 = n,

∂1Ω = äe∈Mk1

(e + ∆e) , ∂2Ω = M2.

In this example, the manifold M2 = ∂2Ω is totally real (domain of tubetype).





3 Spectral theory




7 Compactification


Compactification

Let V be a positive Jordan triple of rank r . Denote by N(x, y) its genericnorm:

N(x, y) = 1−m1(x, y) + · · ·+ (−1)k mk (x, y) + · · ·+ (−1)rmr(x, y).

The Hermitian scalar product m1 induces an isomorphism V∗ → V andisomorphisms jk : Pk (V)→ k V , where Pk (V) is the space ofk -homogeneous polynomials on V and k V the k -th tensor symmetricpower of V .For x ∈ V , mk (x, ) ∈ Pk (V); let σk (x) = jk (mk (x, )). Let Wk denote thelinear span of the range of σk : V → k V . For k = 0, take W0 = C andσ0(x) = 1. Note that σ1 = idV .


Compactification

Consider W = W0 ⊕W1 ⊕ · · · ⊕Wr and its complex projective spacePC(W). The canonical compactification map

χ : V → PC(W)

is then defined by

χ(x) = [1, x, σ2(x), . . . , σr(x)] .

Theorem. The image of χ is an open dense subset of a (compact)algebraic submanifold X of PC(W).The complex manifold X (or the embedding χ : V → X ) is called thecanonical compactification of the Jordan triple V .


Example

Type IVn (n > 2). V = Cn.

x, y, z = 2 ((x | y)z + (z | y)x − (x : z)y) ,

N(x, y) = 1− 2(x|y) + σ(x)σ(y).

Here Wo = C, W1 = V , W2 = Cq, where q = j2σ. ThenW = C⊕ V ⊕Cq and the compactification map is

χ(x) = [1, x, σ(x)q] .

The compactification X is the complex quadric

X = [λ, v, µq] ∈ PC(W) | λµ− σ(v) = 0 .


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