THE BISYMPLECTOMORPHISM GROUP OF A BOUNDED SYMMETRIC DOMAIN

ANTONIO J. DI SCALA, ANDREA LOI, AND GUY ROOS

Abstract. We determine the group of diffeomorphisms of a bounded symmetric domain, which preserve simultaneously the hyperbolic and the flat symplectic form.

Contents

Introduction 1 1. Hermitian positive Jordan triples 2 2. Symplectic forms in polar coordinates 7 3. Symplectic duality 10 4. The bisymplectomorphism group 11 References 19

Introduction

Let Ω be an Hermitian bounded symmetric domain in a complex vector space V ; we always assume that Ω is given in its circled realization. The domain Ω is endowed with two natural symplectic forms: the flat form ω_{0} and the hyperbolic form ω_{−}. In a similar way, the ambient vector space V is also endowed with two natural symplectic forms: the Fubini-Study form ω_{+} and the flat form ω_{0} (see Section 1 for the definition of ω_{−}, ω_{0}, ω_{+}). It has been shown in [1] that there exists a diffeomorphism F : Ω → V such that

F^{∗}ω_{0} = ω_{−}, F^{∗}ω_{+} = ω_{0}. (0.1)

This map is the same that was used in [4], with the property

F^{∗}ω_{+}^{n} = ω_{0}^{n}

(n = dim_{ℂ} V), to show that the flat volume of a bounded symmetric domain, with some natural normalization, is equal to the degree of a canonical projective embedding of its compact dual. In the one-dimensional case, where V = ℂ and Ω is the unit disc Δ, this map is simply f : Δ → ℂ given by

f(z) = \frac{z}{\sqrt{1 − |z|^{2}}} . (0.2)

Date: July 11, 2007.

2000 Mathematics Subject Classification. Primary 53D05, 58F06; Secondary 32M15, 17C10.

Key words and phrases. Kähler metrics; bounded symmetric domains; symplectic duality; Jordan triple systems; Bergman operator.

Research partially supported by GNSAGA (INdAM) and MIUR of Italy.

Even in this case, it does not seem that the property (0.1) had been noticed before. In the general case, the map F may be defined by

F (z) = B(z, z)^{−1/4}z, (0.3)

where B(z, z) denotes the Bergman operator of the Jordan triple structure on V associated to Ω; it may also be defined by functional calculus in Hermitian positive Jordan triples. In view of the property (0.1), the map F is called map of (bi)symplectic duality. The part F^{∗}ω_{0} = ω_{−} tells that F is a realization of the isomorphism of Mc Duff [3] for the bounded symmetric domain Ω; but the property (0.1), which involves two pairs of symplectic forms, is much stronger. In order to determine all diffeomorphisms F : Ω → V verifying (0.1), we determine the group of bisymplectomorphims of Ω, that is, diffeomorphisms ϕ : Ω → Ω such that

ϕ^{∗}ω_{0} = ω_{0}, ϕ^{∗}ω_{−} = ω_{−}. (0.4)

This group is infinite-dimensional, but is the semi-direct product of the compact group K of linear automorphisms of Ω with an infinite-dimensional Abelian group of “radial circular diffeomorphisms” (Theorem 4); this is the main result of this article and may be considered as a kind of Schwarz lemma.

The plan of this article is as follows. In Section 1, we recall known facts about Jordan triple systems associated to bounded complex symmetric domains (see mainly [2]); the only result we could not find in the literature is Proposition 1, which describes the tangent space of the manifold of frames (the “Fürstenberg-Satake boundary” of Ω) in terms of Peirce decomposition in Jordan triples. In Section 2, we compute the symplectic forms ω_{0} and ω_{−} using spectral decomposition in Jordan triples, which is the appropriate generalization of polar coordinates. From this, we derive in Section 3 a simple proof of the property (0.1), different from the proof given in [1]. Section 4 is devoted to the study and characterization of bisymplectomorphisms.

1. Hermitian positive Jordan triples

Let Ω be a bounded symmetric domain in a finite dimensional complex vector space V. We will always consider such a domain in its (unique up to linear isomorphism) circled realization. Consider the associated Jordan triple system (V, { , , }). For basic facts about Hermitian positive Jordan triples and their correspondence with complex symmetric domains, see [2], [4]. We recall hereunder those which will be used here.

1.1. Definitions and notations. Consider the operators on the Jordan triple V defined by

D(x, y)z = {x, y, z} ,  (1.1)
Q(x, z)y = {x, y, z} ,  (1.2)
Q(x, x) = 2Q(x),  (1.3)
B(x, y) = id_{V} −D(x, y) + Q(x)Q(y).  (1.4)

The operators D(x, y) and B(x, y) are ℂ-linear, the operator Q(x) is ℂ-antilinear. The hermitian form

(u | v) = tr D(u, v) (1.5) is a Hermitian scalar product on V; with respect to this product, D(x, x) and B(x, x) are self-adjoint.

For z ∈ V, the odd powers z^{(2p+1)} of z in the Jordan triple system V are defined by

z^{(1)} = z,  z^{(2p+1)} = Q(z)z^{(2p−1)}.  (1.6)

An element e ∈ V is called tripotent if e ≠ 0 and e^{(3)} = e. Two tripotents e_{1}, e_{2} are called (strongly) orthogonal if D (e_{1}, e_{2}) = 0. A tripotent element is called minimal, or primitive, if it is not the sum of two orthogonal tripotents. A tripotent element e is called maximal if there is no tripotent orthogonal to e.

1.2. Spectral decomposition. Each element z ∈ V has a unique spectral decomposition

z = λ_{1}e_{1} + ⋯ + λ_{s}e_{s}  (λ_{1} > ⋯ > λ_{s} > 0),  (1.7)

where (e_{1}, …, e_{s}) is a sequence of pairwise orthogonal tripotents. The integer s = rk z is called the rank of z. Let V_{z}^{+} be the ℝ-subspace of V generated by the odd powers z, …, z^{(2p+1)}, … and V_{z} = V_{z}^{+} ⊕ i V_{z}^{+} the ℂ-subspace generated by the odd powers of z. Then rk z = dim_{ℝ} V_{z}^{+} and (e_{1}, …, e_{s}) is an ℝ-basis of V_{z}^{+}. The rank of V is r = rk V = max {rk z | z ∈ V }; elements z such that rk z = rk V are called regular. If z ∈ V is regular, with spectral decomposition

z = λ_{1}e_{1} + ⋯ + λ_{r}e_{r}  (λ_{1} > ⋯ > λ_{r} > 0),  (1.8)

then (e_{1}, …, e_{r}) is a (Jordan) frame of V, that is, a maximal sequence of pairwise orthogonal minimal tripotents.

1.3. Peirce decompositions. Let e ∈ V be a tripotent. Then the eigenvalues of D(e, e) are contained in {0, 1, 2}. Define the Peirce subspaces of e as

V_{i}(e) = {z ∈ V | D(e, e)z = iz}  (i ∈ {0, 1, 2}).  (1.9)

The decomposition

V = V_{0}(e) ⊕ V_{1}(e) ⊕ V_{2}(e) (1.10)

is called the Peirce decomposition of V w.r. to e. A tripotent e is maximal if V_{0}(e) = 0, minimal if V_{2}(e) = ℂe. The Peirce subspaces compose according to the law

{V_{i}(e), V_{j}(e), V_{k}(e)} ⊂ V_{i−j+k}(e), (1.11)

where V_{m}(e) = 0 if m ∉ {0, 1, 2}; in particular, Peirce subspaces are Jordan subsystems of V. The ℂ-antilinear operator Q(e) is 0 on V_{0}(e) ⊕ V_{1}(e); its restriction to V_{2}(e) is involutive. Let

V_{2}^{+}(e) = {v ∈ V | D(e, e)v = 2v, Q(e)v = v} . (1.12)

Then the decomposition of V (e) into (real) eigenspaces of Q(e) is

V_{2}(e) = V_{2}^{+}(e) ⊕ i V_{2}^{+}(e). (1.13)

Let e = (e_{1}, …, e_{s}) be a sequence of pairwise orthogonal tripotents. Then the operators D(e_{j}, e_{j}), 1 ≤ j ≤ s commute and have the common eigenspaces

V_{jj}(e) = V_{2}(e_{j})  (1 ≤ j ≤ s),
V_{jk}(e) = V_{1}(e_{j}) ∩ V_{1}(e_{k})  (1 ≤ j < k ≤ s),

V_{0j}(e) = V_{1}(e_{j}) ∩ ⋂_{k≠j}V_{0}(e_{k})  (1 ≤ j ≤ s),  (1.14)
V_{00}(e) = ⋂_{k}V_{0}(e_{k})

(some of these subspaces may be 0). The decomposition

V = ⨁_{0≤j≤k≤s}V_{jk}(e) (1.15)

is called the simultaneous Peirce decomposition of V w.r. to e. If

z = λ_{1}e_{1} + ⋯ + λ_{s}e_{s}  (λ_{j} ∈ ℂ),
e = e_{1} + ⋯ + e_{s}

and v ∈ V_{jk}(e), then

D(z, z)v = (|λ_{j}|^{2} + |λ_{k}|^{2}) v  (1.16)
Q(z)v = λ_{j}λ_{k}Q(e)v,  (1.17)
Q(z)Q(z) = |λ_{j}λ_{k}|^{2} v,  (1.18)
B(z, z)v = (1 − |λ_{j}|^{2}) (1 − |λ_{k}|^{2}) v,  (1.19)
B(z, −z)v = (1 + |λ_{j}|^{2}) (1 + |λ_{k}|^{2}) v,  (1.20)

where λ_{0} = 0. So the V_{jk}(e)’s are eigenspaces for all the operators D(z, z), B(z, z), B(z, −z), z = λ_{1}e_{1} + ⋯ + λ_{s}e_{s}.

1.4. Hermitian metrics and symplectic forms. Let V be a Hermitian positive Jordan triple and let Ω be the associated Hermitian bounded symmetric domain. Let

h_{0}(z)(u, v) = (u | v) = tr D(u, v) (1.21)

be the flat Hermitian metric and let

ω_{0}(z) = \frac{i}{2}∂\overline{∂} (z | z) ,
ω_{0}(z)(u, v) = \frac{i}{2} ((u | v) − (v | u))

be the associated flat symplectic form. If Ω is endowed with the volume form ω_{0}^{n} (n = dim_{ℂ} V), the Bergman kernel of Ω is

K(x, y) = \frac{C}{det B(x, y)},

with C = (∫_{Ω} ω_{0}^{n})^{−1}. The Bergman metric of Ω is

h_{−}(z)(u, v) = ∂_{u}\overline{∂}_{v} ln K(z, z) = −∂_{u}\overline{∂}_{v} ln det B(z, z).

It satisfies the relation

h_{−}(z)(u, v) = h_{0} (B(z, z)^{−1}u, v) . (1.22)

In view of this relation, B(z, z) is called the Bergman operator at z ∈ Ω.

The hyperbolic symplectic form of Ω, associated to the Bergman metric, is defined by

ω_{−}(z) = −\frac{i}{2}∂\overline{∂} ln det B(z, z). (1.23)

From (1.22), it results that the forms ω_{0} and ω_{−} are related with the Bergman operator by

ω_{−}(z)(u, v) = ω_{0}(B(z, z)^{−1}u, v), (1.24)

for z ∈ Ω and u, v ∈ T_{z}Ω.

The (generalized) Fubini-Study metric on V is defined by

h_{+}(z)(u, v) = ∂_{u}\overline{∂}_{v} ln det B(z, −z).

The associated Kähler form is

ω_{+}(z) = \frac{i}{2}∂\overline{∂} ln det B(z, −z).

It is related to the flat form by

ω_{+}(z)(u, v) = ω_{0}(B(z, −z)^{−1}u, v). (1.25)

1.5. Polar coordinates. Let M be the set ot tripotents elements of the positive Jordan triple V. Then M is a compact submanifold of V (with connected components of different dimensions). At e ∈ M, the tangent space T_{e}M and the normal space N_{e}M to M are

T_{e}M = i V_{2}^{+}(e) ⊕ V_{1}(e),  (1.26)
N_{e}M = V_{0}(e) ⊕ V_{2}^{+}(e)  (1.27)

(see [2], Theorem 5.6).

The height k of a tripotent element e is the maximal length of a decomposition e = e_{1} + ⋯ + e_{k} into a sum of pairwise orthogonal (minimal) tripotents. Minimal tripotents have height 1, maximal tripotents have height r = rk V. Denote by M_{k} the set of tripotents of height k. If V is simple (that is, if Ω is irreducible), the submanifolds M_{k} are the connected components of M.

The set F of frames (also called Fürstenberg-Satake boundary of Ω):

F = {(e_{1}, …, e_{r}) | e_{j} ∈ M_{1}, e_{j} ⊥ e_{k} (1 ≤ j < k ≤ r)} , (1.28)

(where e_{j} ⊥ e_{k} means orthogonality of tripotents: D(e_{j}, e_{k}) = 0 or equivalently {e_{j}, e_{j}, e_{k}} = 0) is a submanifold of V^{r}. The following proposition provides a description of the tangent space of F.

Proposition 1. Let e = (e_{1}, …, e_{r}) ∈ F ⊂ V^{r} and e = e_{1} + ⋯ + e_{r}. Then (v_{1}, …, v_{r}) ∈ T_{e}F if and only if

v_{j} = i α_{j}e_{j} + v_{j0} + ∑_{\begin{matrix}1≤k≤r \\ k≠j \end{matrix}} v_{jk}  (1 ≤ j ≤ r),  (1.29)

where α_{j} ∈ ℝ, v_{j0} ∈ V_{0j}(e), v_{jk} ∈ V_{jk}(e) = V_{kj}(e) and

Q(e)v_{jk} = −v_{kj}  (1 ≤ j < k ≤ r).  (1.30)

Proof. Let (v_{1}, …, v_{r}) ∈ T_{e}F. As e_{j} are minimal tripotents, we have

v_{j} = T_{e_{j}}M_{1} = i ℝe_{j} ⊕ V_{1}(e_{j}) = i ℝe_{j} ⊕ V_{0j}(e) ⊕ ⨁_{\begin{matrix}1≤k≤r \\ k≠j \end{matrix}}V_{jk}(e),

which shows that v_{j} has the form (1.29).

The orthogonality conditions in a frame are

{e_{j}, e_{j}, e_{k}} = 0 (1 ≤ j < k ≤ r).

Differentiating these conditions yields

{v_{j}, e_{j}, e_{k}} + {e_{j}, v_{j}, e_{k}} + {e_{j}, e_{j}, v_{k}} = 0 (1 ≤ j < k ≤ r).

As D(e_{j}, e_{k}) = 0, this condition is reduced to

Q(e_{j}, e_{k})v_{j} + D(e_{j}, e_{j})v_{k} = 0  (1 ≤ j < k ≤ r).  (1.31)

Let

v_{j} = i α_{j}e_{j} + v_{j0} + ∑_{\begin{matrix}1≤m≤r \\ m≠j \end{matrix}} v_{jm} (1 ≤ j ≤ r).

Then

D(e_{j}, e_{j})v_{k} = v_{kj}, Q(e_{j}, e_{k})e_{j} = 0,
Q(e_{j}, e_{k}) = Q(e_{j} + e_{k}) − Q(e_{j}) − Q(e_{k}),

and we get from (1.17)

Q(e_{j} + e_{k})v_{jm} = δ_{k}^{m}Q(e)v_{jm}, Q(e_{j})v_{jm} = 0, Q(e_{k})v_{jm} = 0,
Q(e_{j}, e_{k})v_{j} = Q(e)v_{jk}.

This shows that the conditions (1.31) are equivalent to (1.30). □

Comparing the description of T_{e}F with the simultaneous Peirce decomposition of V w.r. to e, it is easily checked that T_{e}F is a real vector space of dimension 2n − r, where n = dim_{ℂ} V. This implies that the map

{λ_{1} > ⋯ > λ_{r} > 0} × F → V_{reg}
((λ_{1}, …, λ_{r}) , (e_{1}, …, e_{r})) ↦ ∑ λ_{j}e_{j}  (1.32)

is a diffeomorphism onto the set V_{reg} of regular elements of V; its restriction

{1 > λ_{1} > ⋯ > λ_{r} > 0} × F → Ω_{reg}

is a diffeomorphism onto the set Ω_{reg} of regular elements of Ω. This map plays the same role as polar coordinates in rank one.

1.6. Functional calculus. Using the spectral decomposition, it is possible to associate to an odd function f : (−1, 1) → ℂ (resp. f : ℝ → ℂ) a “radial” map F : Ω → V (resp. F : V → V) in the following way. Let z ∈ V and let

z = λ_{1}e_{1} + ⋯ + λ_{k}e_{k}, λ_{1} > ⋯ > λ_{k} > 0

be the spectral decomposition of z. Define the map F = f̃ associated to f by

F (z) = f(λ_{1})e_{1} + ⋯ + f(λ_{k})e_{k}. (1.33)

Since f is odd, it follows from properties of tripotents that, for

z = λ_{1}e_{1} + ⋯ + λ_{r}e_{r},

THE BISYMPLECTOMORPHISM GROUP OF A BOUNDED SYMMETRIC DOMAIN 7

where e = (e_{1}, …, e_{r}) is a frame and λ_{1}, …, λ_{r} ∈ ℝ, we have

F (z) = f(λ_{1})e_{1} + ⋯ + f(λ_{r})e_{r}. (1.34) If f is continuous, then F is continuous.

If

f(t) = ∑_{k=0}^{N} a_{k}t^{2k+1}

is a polynomial, then F is the map defined by

F (z) = ∑_{k=0}^{N} a_{k}z^{(2k+1)}  (z ∈ V).  (1.35)

If f is analytic, then F is real-analytic; if f is given near 0 by

f(t) = ∑_{k=0}^{∞} a_{k}t^{2k+1},

then F has the Taylor expansion near 0 ∈ V:

F (z) = ∑_{k=0}^{∞} a_{k}z^{(2k+1)}. (1.36)

If f is C^{∞}, then F is also C^{∞}.

2. Symplectic forms in polar coordinates

2.1. We compute the flat symplectic form ω_{0} in the “polar coordinates”

{λ_{1} > ⋯ > λ_{r} > 0} × F → V_{reg}
((λ_{1}, …, λ_{r}) , (e_{1}, …, e_{r})) ↦ ∑ λ_{j}e_{j}.

If (z_{1}, …, z_{n}) (n = dim V) are orthonormal coordinates for the Hermitian product (u | v), then

ω_{0} = i ∑_{m=1}^{n} d z_{m} ∧ d \overline{z}_{m}.

From

z = ∑_{j=1}^{r} λ_{j}e_{j},

we have

d z_{m} ∧ d \overline{z}_{m} = ∑_{j,k=1}^{r} λ_{j}λ_{k} d e_{jm} ∧ d \overline{e}_{km} + ∑_{j,k=1}^{r} e_{jm}\overline{e}_{km} d λ_{j} ∧ d λ_{k}

+ ∑_{j,k=1}^{r} λ_{k} d λ_{j} ∧ (e_{jm} d \overline{e}_{km} − \overline{e}_{km} d e_{jm}) .

Using (e_{j} | e_{k}) = δ_{jk}, we get

ω_{0} = ∑_{j,k=1}^{r} λ_{j}λ_{k}ω_{jk} + 2 ∑_{j,k=1}^{r} λ_{k} d λ_{j} ∧ η_{jk}, (2.1)

where

η_{jk} = \frac{i}{2} ∑_{m=1}^{n} (e_{jm} d \overline{e}_{km} − \overline{e}_{km} d e_{jm})|_{F} = i ∑_{m=1}^{n} e_{jm} d \overline{e}_{km}|_{F} ,  (2.2)

ω_{jk} = i ∑_{m=1}^{n} d e_{jm} ∧ d \overline{e}_{km}|_{F} = d η_{jk}.  (2.3)

We compute η_{jk} and ω_{jk} using the description of T_{e}F given in Proposition 1. Let v, w ∈ T_{e}F with v = (v_{1}, …, v_{r}), w = (w_{1}, …, w_{r}),

v_{j} = i α_{j}e_{j} + v_{j0} + ∑_{\begin{matrix}1≤m≤r \\ m≠j \end{matrix}} v_{jm}  (1 ≤ j ≤ r),

w_{j} = i β_{j}e_{j} + w_{j0} + ∑_{\begin{matrix}1≤m≤r \\ m≠j \end{matrix}} w_{jm}  (1 ≤ j ≤ r),

α_{j} ∈ ℝ, v_{j0} ∈ V_{0j}(e), v_{jk} ∈ V_{jk}(e),
β_{j} ∈ ℝ, w_{j0} ∈ V_{0j}(e), w_{jk} ∈ V_{jk}(e),
Q(e)v_{jk} = −v_{kj}, Q(e)w_{jk} = −w_{kj}  (1 ≤ j < k ≤ r).

Then, as the Peirce subspaces are orthogonal w.r. to ( | ), we deduce from (2.2)-(2.3)

η_{jj}(e)(v) = i (e_{j} | v_{j}) = α_{j},  (2.4)
η_{jk}(e)(v) = i (e_{j} | v_{k}) = 0  (j ≠ k),
ω_{jk} = d η_{jk} = 0  (j ≠ k)

and

ω_{jj}(e)(v, w) = \frac{i}{2} ((v_{j} | w_{j}) − (w_{j} | v_{j}))
= ⟨v_{j0} | w_{j0}⟩ + ∑_{\begin{matrix}1≤m≤r \\ m≠j \end{matrix}} ⟨v_{jm} | w_{jm}⟩ ,  (2.5)

where ⟨ | ⟩ denotes the symplectic product

⟨x | y⟩ = \frac{i}{2} ((x | y) − (y | x)) . (2.6)

Finally, we have

ω_{0} = ∑_{j=1}^{r} λ_{j}^{2}ω_{jj} + 2 ∑_{j=1}^{r} λ_{j} d λ_{j} ∧ η_{jj}, (2.7)

with ω_{jj} and η_{jj} given by (2.5), (2.4). The expression (2.5) shows that the ω_{jj} (1 ≤ j ≤ r) are linearly independent at each point e ∈ F.

As

2 (Q(z)x | y) = (D(z, x)z | y) = (z | D(x, z)y)
= (z | D(y, z)x) = (D(z, y)z | x) = 2 (Q(z)y | x) ,

the Q operator satisfies

(Q(z)x | y) = (Q(z)y | x) ,  (2.8)
⟨Q(z)x | y⟩ = − ⟨x | Q(z)y⟩  (2.9)

for all z, x, y ∈ V.

For v, w ∈ T_{e}F, using

Q(e)v_{jk} = −v_{kj}, Q(e)w_{jk} = −w_{kj} (1 ≤ j ≠ k ≤ r),

we have then

⟨v_{jk} | w_{jk}⟩ = ⟨Q(e)v_{kj} | Q(e)w_{kj}⟩ = − ⟨v_{kj} | Q(e)^{2}w_{kj}⟩ ,

that is,

⟨v_{jk} | w_{jk}⟩ = − ⟨v_{kj} | w_{kj}⟩ , (2.10)

as w_{kj} ∈ V_{2}(e) and Q(e) is involutive on V_{2}(e). In view of (2.10), the flat symplectic form ω_{0} in polar coordinates may be rewritten

ω_{0} = ∑_{j=1}^{r} λ_{j}^{2}θ_{j0} + ∑_{\begin{matrix}j,k \\ 1≤j<k≤r \end{matrix}} (λ_{j}^{2} − λ_{k}^{2}) θ_{jk} + 2 ∑_{j=1}^{r} λ_{j} d λ_{j} ∧ η_{jj}, (2.11)

where the η_{jj}’s are defined by (2.4), and the θ_{j0}’s and θ_{jk}’s by

θ_{j0}(e)(v, w) = ⟨v_{j0} | w_{j0}⟩ ,  (1 ≤ j ≤ r),  (2.12)
θ_{jk}(e)(v, w) = ⟨v_{jk} | w_{jk}⟩  (1 ≤ j < k ≤ r).  (2.13)

for v, w ∈ T_{e}F. Note that these forms are (pull-backs of) forms on the manifold of frames F and that the θ_{j0}’s are 0 when the domain is of tube type.

2.2. We compute now the hyperbolic form ω_{−} and the Fubini-Study form ω_{+} in polar coordinates.

Using

B(z, z)^{−1}v_{j0} = (1 − λ_{j}^{2})^{−1} v_{j0}  (1 ≤ j ≤ r),
B(z, z)^{−1}v_{jk} = (1 − λ_{j}^{2})^{−1} (1 − λ_{k}^{2})^{−1} v_{jk}  (1 ≤ j < k ≤ r),
B(z, z)^{−1}e_{j} = (1 − λ_{j}^{2})^{−2} e_{j}  (1 ≤ j ≤ r)

in (2.11), (2.12), (2.13), we obtain

η_{jj}(e)(B(z, z)^{−1}v) = (1 − λ_{j}^{2})^{−2} η_{jj}(e)(v),  (2.14)
θ_{j0}(e)(B(z, z)^{−1}v, w) = (1 − λ_{j}^{2})^{−1} θ_{j0}(e)(v, w),  (2.15)
θ_{jk}(e)(B(z, z)^{−1}v, w) = (1 − λ_{j}^{2})^{−1} (1 − λ_{k}^{2})^{−1} θ_{jk}(e)(v, w),  (2.16)

for z = λ_{1}e_{1} + ⋯ + λ_{r}e_{r} and v, w ∈ T_{e}F. From (1.24):

ω_{−}(z)(v, w) = ω_{0}(B(z, z)^{−1}v, w)

and the expression of ω_{0} in polar coordinates, we have

ω_{−} = ∑_{j=1}^{r} \frac{λ_{j}^{2}}{1 − λ_{j}^{2}} θ_{j0} + ∑_{\begin{matrix}j,k \\ 1≤j<k≤r \end{matrix}} \frac{λ_{j}^{2} − λ_{k}^{2}}{(1 − λ_{j}^{2}) (1 − λ_{k}^{2})}θ_{jk} + 2 ∑_{j=1}^{r} \frac{λ_{j} d λ_{j}}{(1 − λ_{j}^{2})^{2}} ∧ η_{jj}.

(2.17)

In the same way, the Fubini-Study symplectic form on V is

ω_{+} = ∑_{j=1}^{r} \frac{λ_{j}^{2}}{1 + λ_{j}^{2}} θ_{j0} + ∑_{\begin{matrix}j,k \\ 1≤j<k≤r \end{matrix}} \frac{λ_{j}^{2} + λ_{k}^{2}}{(1 + λ_{j}^{2}) (1 + λ_{k}^{2})}θ_{jk} + 2 ∑_{j=1}^{r} \frac{λ_{j} d λ_{j}}{(1 + λ_{j}^{2})^{2}} ∧ η_{jj}.

(2.18)

3. Symplectic duality

Consider the real analytic maps f =] − 1, 1[→ ℝ and g : ℝ →] − 1, 1[, inverse of each other, defined by

f(t) = \frac{t}{\sqrt{1 − t^{2}}}  (−1 < t < 1),  (3.1)
g(t) = \frac{t}{\sqrt{1 + t^{2}}}  (t ∈ ℝ).  (3.2)

By the functional calculus described in Subsection 1.6, we associate to these maps the real analytic diffeomorphisms, also inverse of each other

F = f̂: Ω → V,
G = ĝ: V → Ω,

where Ω is the bounded symmetric domain associated to the Jordan triple V. If e = (e_{1}, …, e_{r}) is a frame and z = ∑_{j=1}^{r} λ_{j}e_{j}, then

F (z) = ∑_{j=1}^{r} \frac{λ_{j}}{\sqrt{1 − λ_{j}^{2}}} e_{j}  (z ∈ Ω),  (3.3)

G(z) = ∑_{j=1}^{r} \frac{λ_{j}}{\sqrt{1 + λ_{j}^{2}}} e_{j}  (z ∈ V).  (3.4)

Using (1.16)-(1.20), the maps F and G may also be defined by

F (z) = B(z, z)^{−1/4}z = (id_{V} −\frac{1}{2}D(z, z))^{−1/2} z  (z ∈ Ω),  (3.5)

G(z) = B(z, −z)^{−1/4}z = (id_{V} −\frac{1}{2}D(z, −z))^{−1/2} z  (z ∈ V).  (3.6)

The following theorem is the main result of [1]. We give here a different and simpler proof, using the expression of the symplectic forms ω_{0}, ω_{−}, ω_{+} in generalized polar coordinates.

Theorem 1. (Symplectic duality)

F^{∗}ω_{0} = ω_{−}, F^{∗}ω_{+} = ω_{0},  (3.7)
G^{∗}ω_{0} = ω_{+}, G^{∗}ω_{−} = ω_{0}.  (3.8)

Proof. In polar coordinates, the map F is written

((λ_{1}, …, λ_{r}) , e) ↦ ((μ_{1}, …, μ_{r}) , e)

with

μ_{j} = \frac{λ_{j}}{\sqrt{1 − λ_{j}^{2}}} . (3.9)

As, by (2.11),

ω_{0} = ∑_{j=1}^{r} λ_{j}^{2}θ_{j0} + ∑_{\begin{matrix}j,k \\ 1≤j<k≤r \end{matrix}} (λ_{j}^{2} − λ_{k}^{2}) θ_{jk} + 2 ∑_{j=1}^{r} λ_{j} d λ_{j} ∧ η_{jj},

we obtain, using (3.9),

F^{∗}ω_{0} = ∑_{j=1}^{r} \frac{λ_{j}^{2}}{1 − λ_{j}^{2}} θ_{j0} + ∑_{\begin{matrix}j,k \\ 1≤j<k≤r \end{matrix}} (\frac{λ_{j}^{2}}{1 − λ_{j}^{2}} − \frac{λ_{k}^{2}}{1 − λ_{k}^{2}} ) θ_{jk}+

+ 2 ∑_{j=1}^{r} \frac{λ_{j} d λ_{j}}{(1 − λ_{j}^{2})^{2}} ∧ η_{jj},

which, compared to (2.17), gives F^{∗}ω_{0} = ω_{−} on the open dense subset Ω_{reg} of regular elements, and by continuity on all of Ω.

The relation G^{∗}ω_{0} = ω_{+} is proved along the same lines. The relations F^{∗}ω_{+} = ω_{0} and G^{∗}ω_{−} = ω_{0} follow, as F and G are inverse of each other. □

In view of this theorem, the map F (or the map G = F^{−1}) is called the duality map.

Example 1. (Type I_{1,1}) Here V = ℂ, Ω is the unit disc,

ω_{0} = \frac{i}{2} d z ∧ d \overline{z}, ω_{−} = \frac{i}{2} \frac{d z ∧ d \overline{z}}{(1 − z\overline{z})^{2}} , ω_{+} = \frac{i}{2} \frac{d z ∧ d \overline{z}}{(1 + z\overline{z})^{2}} .

The duality map is

F (z) = \frac{z}{\sqrt{1 − z\overline{z}}} .

Example 2. (Type I_{1,n}) Here V = ℂ^{n} with the Hermitian norm ∥z∥^{2} = ∑ z_{j}\overline{z}_{j}, Ω is the unit Hermitian ball,

ω_{0} = \frac{i}{2} ∑ d z_{j} ∧ d \overline{z}_{j}, ω_{−} = \frac{ω_{0}}{(1 − ∥z∥^{2})^{n+1}} , ω_{+} = \frac{ω_{0}}{(1 + ∥z∥^{2})^{n+1}} .

The duality map is

F (z) = \frac{z}{\sqrt{1 − ∥z∥^{2}}} .

4. The bisymplectomorphism group

4.1. In this section we study to which extent a diffeomorphism F : Ω → V satisfying the property (3.7) is unique. If two diffeomorphisms F_{1}, F_{2} : Ω → V satisfy (3.7), then F_{2} = F_{1} ∘ f, where f : Ω → Ω preserves ω_{0} and ω_{−}. This leads us to the following definition.

Definition 1. A bisymplectomorphism of Ω is a diffeomorphism f : Ω → Ω which satisfies

f^{∗}ω_{0} = ω_{0},  (4.1)
f^{∗}ω_{−} = ω_{−}.  (4.2)

Clearly, bisymplectomorphisms of Ω form a group, which will be denoted by B(Ω) and called the bisymplectomorphism group of Ω.

From (1.24), we derive a characterization of B(Ω) in terms of the Bergman operator:

Proposition 2. Let Ω be a bounded symmetric domain. Then a diffeomorphism f ∈ Diff(Ω) is a bisymplectomorphism if and only if it satisfies

f^{∗}ω_{0} = ω_{0},  (4.3)
B (f(z), f(z)) ∘ d f(z) = d f(z) ∘ B(z, z) (z ∈ Ω).  (4.4)

Note that the second condition implies that the tangent map d f(z) maps invariant subspaces of B_{z} = B(z, z) to invariant subspaces of B_{f(z)}, and that B_{f(z)} has the same eigenvalues as B_{z}.

Proof. Let z ∈ Ω, u, v ∈ T_{z}Ω. We have from (1.24) and (4.1)

ω_{−}(z)(u, v) = ω_{0}(B(z, z)^{−1}u, sv)
= ω_{0}(d f(z)B(z, z)^{−1}u, d f(z)v),
(f^{∗}ω_{−}) (z)(u, v) = ω_{−}(f(z))(d f(z)u, d f(z)v)
= ω_{0} (B(f(z), f(z))^{−1} d f(z)u, d f(z)v) ,

so that, assuming (4.1), the condition (4.2) is equivalent to

ω_{0}(d f(z)B(z, z)^{−1}u, d f(z)v) = ω_{0} (B(f(z), f(z))^{−1} d f(z)u, d f(z)v)

for all u, v. As ω_{0} is non singular and d f(z) is bijective, this is equivalent to

d f(z) ∘ B(z, z)^{−1} = B(f(z), f(z))^{−1} ∘ d f(z),

that is, to (4.4). □

4.2. Here we study diffeomorphisms of Ω satisfying the condition (4.4). Recall that B_{z} = B(z, z) : V → V is a ℂ-linear operator, self-adjoint w.r. to the Hermitian metric h_{0}, positive if z ∈ Ω. Let r denote the rank of Ω and V.

For z ∈ Ω, consider the spectral decomposition

z = λ_{1}e_{1} + λ_{2}e_{2} + ⋯ + λ_{s}e_{s}, 1 > λ_{1} > λ_{2} > ⋯ > λ_{s} > 0, (4.5)

where s = rk z ≤ r = rk V. An element is called regular if rk z = rk V; for regular elements, which form an open dense subset of Ω, the decomposition (4.5) is the decomposition using generalized polar coordinates.

Let

V = ⨁_{0≤i≤j≤s}V_{ij}

be the simultaneous Peirce decomposition relative to (e_{1}, …, e_{s}). Note that some subspaces V_{ij} may be 0. The operator B(z, z) may only have the eigenvalues

(1 − λ_{i}^{2})^{2} (1 ≤ i ≤ s), (1 − λ_{i}^{2})(1 − λ_{j}^{2}) (1 ≤ i < j ≤ s),  (4.6)

(1 − λ_{i}^{2}) (1 ≤ i ≤ s), 1,

which occur respectively on the subspaces

V_{ii}, V_{ij}, V_{0i}, V_{00}. (4.7)

The relation (4.4) then implies that B(z, z) and B(f(z), f(z)) have the same eigenvalues with the same multiplicities. Moreover, if all eigenvalues in the list (4.6) occurring for non-zero V_{ij} are different, the non-zero subspaces of the list (4.7) are the eigenspaces of B(z, z) and are mapped by d f(z) to the corresponding eigenspaces of B(f(z), f(z)).

Definition 2. Let Ω be an irreducible bounded symmetric domain and denote by (V, { , , }) be the corresponding Jordan triple. An element z ∈ V is called superregular if z is regular and if all eigenvalue in the list (4.6), occurring on non-zero V_{ij}, are different.

If V is of tube type or of rank 1, any regular element z ∈ V is super-regular. If V is not of tube type, an element z ∈ V is super-regular if it is regular and if its spectral values satisfy

1 − λ_{i}^{2} ≠ (1 − λ_{j}^{2})^{2} (4.8)

for all (i, j), i < j. Clearly, super-regular elements form an open dense subset of V. Let z ∈ Ω be an element of rank one. Then the spectral decomposition of z is

z = λ_{1}e_{1}, 0 < λ_{1} < 1,

where e_{1} is a tripotent; the associated Peirce decomposition is

V_{11} = V_{2}(e_{1}), V_{01} = V_{1}(e_{1}), V_{00} = V_{0}(e_{1})

and the eigenvalues of B(z, z) on these subspaces are respectively

(1 − λ_{1}^{2})^{2} , 1 − λ_{1}^{2}, 1.

It then follows that

f(z) = λ_{1}ε_{1},

where ε_{1} is a tripotent such that V_{2}(ε_{1}) = d f(z)V_{2}(e_{1}), which means that e_{1} and ε_{1} have the same height dim V_{2}(e_{1}) = dim V_{2}(ε_{1}). In particular, if e_{1} is minimal (resp. maximal), then ε_{1} is minimal (resp. maximal).

Let V_{z} be the ℂ-subspace generated by the odd powers z, …, z^{(2p+1)}, … Then dim_{ℂ} V_{z} = rk z ≤ r, and dim_{ℂ} V_{z} = r if and only if z is a regular element. Denote

P_{z} = V_{z} ∩ Ω.

For z ∈ Ω, z ≠ 0 with spectral decomposition z = ∑_{j=1}^{k} α_{j}e_{j} (α_{1} > ⋯ > α_{k} > 0; k = rk_{V} x), P_{z} is the k-polydisc

P_{z} = {u = ∑ u_{j}e_{j} | |u_{j}| < 1} .

Note that if u ∈ P_{z} and rk_{V} u = rk_{V} z, then V_{u} = V_{z} and P_{u} = P_{z}.

Lemma 3. Let f : Ω → Ω be a diffeomorphism of Ω such that

B (f(z), f(z)) ∘ d f(z) = d f(z) ∘ B(z, z)

for all z ∈ Ω. Then for all z ∈ Ω,

d f(z)V_{z} = V_{f(z)}. (4.9)

Proof. Let z be a super-regular element and let

z = α_{1}e_{1} + ⋯ + α_{r}e_{r}

be the spectral decomposition of z in V; then V_{z} := ℂe_{1} ⊕ ⋯ ⊕ ℂe_{r} is the sum of the eigenspaces of B(z, z) relative to the eigenvalues

(1 − α_{j}^{2})^{2} (1 ≤ j ≤ r),

so that it follows from (4.4) that

d f(z)V_{z} = V_{f(z)}.

By continuity, (4.9) holds for all z ∈ Ω. □

Proposition 4. Let f : Ω → Ω be a diffeomorphism of Ω such that

B (f(z), f(z)) ∘ d f(z) = d f(z) ∘ B(z, z)

for all z ∈ Ω. Then for any element z ∈ Ω, z ≠ 0, we have

f (P_{z}) = P_{f(z)}.

Proof. Let s = rk_{V} z. We already know that d f(z)V_{z} = V_{f(z)}, which implies that rk_{V} f(z) = z. So there exist continuous functions β_{j} such that

d f(z)u = ∑_{j=1}^{s} β_{j}(u, z) (f(z))^{(2j+1)} (z ∈ Ω, u ∈ P_{z}).

Consider a C^{1} path η : [0, 1] → P_{z} from η(0) = z to η(1) = w; let g(t) = f(η(t)). Then g satisfies the differential equation

g′(t) = ∑_{j=1}^{s} β_{j}(η′(t), η(t)) (g(t))^{(2j+1)} ,  (4.10)
g(0) = f(z).

Let

f(z) = ∑_{k=1}^{s} α_{k}ε_{k}

be the spectral decomposition of f(z). Let h : [0, 1] → ℂ^{s} be the solution of the differential system

h′_{k}(t) = ∑_{j=1}^{s} β_{j}(η′(t), η(t)) (h_{k}(t))^{2j+1)} (1 ≤ k ≤ s),
h_{k}(0) = α_{k}.

Then the solution of (4.10) is

g(t) = ∑_{k=1}^{s} h_{k}(t)ε_{k},

which shows that g(1) = f(w) belongs to P_{f(z)}. □

Let K denote the group of linear automorphisms of Ω.

Proposition 5. Let f : Ω → Ω be a diffeomorphism of Ω such that

B (f(z), f(z)) ∘ d f(z) = d f(z) ∘ B(z, z)

for all z ∈ Ω. Then any K-orbit is globally invariant by f.

Proof. Assume first that z = λ_{1}e_{1} + ⋯ + λ_{r}e_{r} is super regular. Then f(z) = μ_{1}ε_{1} +⋯+μ_{r}ε_{r}, and equality between the eigenvalues of B(z, z) and B(f(z), f(z)) implies λ_{j} = μ_{j}, hence f(z) ∈ Kz and f(Kz) = Kz. By continuity, this also holds for any z ∈ Ω. □

4.3. We now go back to bisysmplectomorphisms of Ω.

Proposition 6. Let Ω be a bounded circled symmetric domain and denote by K the group of linear automorphisms of Ω. For each bisymplectomorphism f ∈ B(Ω), we have d f(0) ∈ K.

As K ⊂ B(Ω), it will be sufficient to study the subgroup

B_{0}(Ω) = {f ∈ B(Ω) | d f(0) = id_{V} } . (4.11)

Proof. Let (e_{1}, …, e_{r}) be a frame of V and let V = ⨁V_{ij} be the associated simultaneous Peirce decomposition. Consider a regular element

z = α_{1}e_{1} + ⋯ + α_{r}e_{r},

1 > α_{1} > ⋯ > α_{r} > 0. For f ∈ B(Ω) and z super-regular, we have f(z) = α_{1}ε_{1} + ⋯ + α_{r}ε_{r}, where (ε_{1}, …, ε_{r}) is a frame of V, which may depend on (α_{1}, …, α_{r}). As V_{jj} = ℂe_{j} is the eigenspace of B(z, z) for the eigenvalue (1 − α_{j}^{2})^{2}, we deduce from (4.4) that

d f(z) (e_{j}) ∈ ℂε_{j}.

By continuity, there exists a frame (e′_{1}, …, e′_{r}) such that

d f(0) (e_{j}) ∈ ℂe′_{j} (1 ≤ j ≤ r).

Multiplying e′_{j} by a suitable complex of modulus 1, we may assume that

d f(0) (e_{j}) = λ′_{j}e′_{j}, λ′_{j} > 0 (1 ≤ j ≤ r). (4.12)

We have also

d f(0)(e_{j}) = lim_{t→0+} \frac{f(te_{j})}{t}

and f(te_{j}) is a multiple of a minimal tripotent, with spectral norm ∥f(te_{j})∥ ≤ t ∥e_{j}∥, hence ∥d f(0)(e_{j})∥ ≤ ∥e_{j}∥ and λ′_{j} ≤ 1. The same argument applied to f^{−1} gives λ′_{j} = 1. So the image of a frame (e_{1}, …, e_{r}) under d f(0) is a frame (e′_{1}, …, e′_{r}) and the Peirce spaces V_{jk} relative to (e_{1}, …, e_{r}) are mapped by d f(0) onto the corresponding Peirce spaces V′_{jk} relative to (e′_{1}, …, e′_{r}). In particular, d f(0) is an ℝ-linear map from ℂe_{j} onto ℂe′_{j}. We have d f(0)(i e_{j}) = βe′_{j}, with |β| = 1. As

ω_{0}(u, v) = \frac{i}{2} (tr D(u, v) − tr D(v, u)) ,

we have

ω_{0}(e_{j}, i e_{j}) = tr D(e_{j}, e_{j}),
ω_{0} (d f(0) (e_{j}) , d f(0) (i e_{j})) = Im β tr D(e′_{j}, e′_{j}).

From tr D(e′_{j}, e′_{j}) = tr D(e_{j}, e_{j}) and f^{∗}ω_{0} = ω_{0}, we get Im β = 1, β = i, which means that d f(0) is ℂ-linear on ℂe_{1} ⊕ ⋯ ⊕ ℂe_{r}. Finally, d f(0) is ℂ-linear on V, maps minimal tripotents to minimal tripotents and Ω to Ω. □

4.4. The unit disc. Let Δ be the unit disc of ℂ. The associated triple product is \{u, v, w\} = 2u\overline{v}w. The Bergman operator is B(z, z)w = (1−|z|^{2})^{2}w. The Bergman metric is

h_{z}(u, v) = \frac{2u\overline{v}}{(1 − |z|^{2})^{2}} .

The two symplectic forms are

ω_{0} = i d z ∧ d \overline{z},  (4.13)
ω_{−} = \frac{ω_{0}}{(1 − |z|^{2})^{2}} .  (4.14)

Denote by S^{1} the unit circle in ℂ and consider the “polar coordinates” diffeomorphism

Θ : (0, 1) × S^{1} → Δ \ {0},
(r, ζ) ↦ rζ.

Then we have

Θ^{∗}ω_{0} = 2r d r ∧ \frac{d ζ}{i ζ} .

The following theorem characterizes the elements of B(Δ).

Theorem 2. The elements f ∈ B(Δ) are the maps defined by

f(z) = u (|z|^{2}) z (z ∈ Δ) ,

where u is a smooth function u : [0, 1) → S^{1} ≃ U(1).

In other words, the restriction of f to a circle of radius r (0 < r < 1) is the rotation u (r^{2}).

Proof. From (4.14), we see that a diffeomorphism f : Δ → Δ is a bisymplectomorphism if and only if f preserves ω_{0} and |f(z)| = |z| for all z ∈ Δ.

If |f(z)| = |z| for all z ∈ Δ, the map F = Θ^{−1} ∘ f ∘ Θ may be written

(r, ζ) →^{F} (r, Z(r, ζ))

for some smooth function Z : (0, 1) × S^{1} → S^{1}. We have

F^{∗} (Θ^{∗}ω_{0}) = 2r d r ∧ \frac{d Z}{i Z} = 2r d r ∧ \frac{d_{ζ} Z}{i Z} .

If f preserves ω_{0}, then F preserves Θ^{∗}ω_{0}, which implies

\frac{d_{ζ} Z}{Z} = \frac{d ζ}{ζ} . (4.15)

For r fixed, let u_{r}(ζ) = Z(r, ζ) ; the condition (4.15) is then equivalent to

\frac{d u_{r}}{u_{r}} = \frac{d ζ}{ζ} (4.16)

for all r ∈ (0, 1). The condition (4.16) is in turn equivalent to

u_{r}(ζ) = v(r)ζ, (4.17) with v(r) ∈ S^{1}. The function v, which is given by

v(r) = \frac{Z(r, ζ)}{ζ},

is smooth on (0, 1) and

f(rζ) = rv(r)ζ,

or

f(z) = v (|z|) z (4.18)

for z ∈ Δ, z ≠ 0; on the other hand, f(0) = 0. Let g be the restriction of f to (−1, 1); from (4.18), we see that g is odd. For x ≠ 0, we have

g(x) = ∫_{0}^{1} \frac{d}{d s}g(sx) d s = x ∫_{0}^{1} g′(sx) d s,

which shows that

v(r) = ∫_{0}^{1} g′(sr) d s

extends to a smooth even function v : (−1, 1) → S^{1}. Let u : [0, 1) → S^{1} be defined by

u(r) = v (\sqrt{r}) .

It follows then from Whitney’s theorem [5] that u is smooth at 0.

Conversely, if

f(z) = u(|z|^{2})z,

with u : [0, 1) → S^{1} smooth, then f satisfies |f(z)| = |z|, f is a diffeomorphism with inverse f^{−1}(w) = u(|w|^{2})^{−1}w; also, f preserves ω_{0} on Δ \ {0}, hence on Δ by continuity. This implies that f ∈ B(Δ). □

4.5. The polydisc. Let Δ^{r} ⊂ ℂ^{r} be the product of r unit discs. The Jordan triple product on V = ℂ^{r} is just the component-wise product

\{x, y, z\} = 2 (x_{1}\overline{y_{1}}z_{1}, …, x_{r}\overline{y_{r}}z_{r}) .

Let (e_{1}, …, e_{r}) denote the canonical basis of ℂ^{r}. The minimal tripotents of V are the elements λ_{j}e_{j}, 1 ≤ j ≤ r, |λ_{j}| = 1. Any frame (maximal ordered set of mutually orthogonal tripotents) has the form

(λ_{j}e_{σ(j)})_{1≤j≤r} ,

where |λ_{j}| = 1 and σ ∈ 𝔖_{r} is a permutation of {1, ⋯, r}. The corresponding Peirce decomposition is V = ℂe_{σ(1)} ⊕ ⋯ ⊕ ℂe_{σ(r)}.

Theorem 3. A diffeomorphism f : Δ^{r} → Δ^{r} belongs to B_{0} (Δ^{r}) if and only if there exist smooth functions u_{j} : [0, 1) → S^{1} such that u_{j}(0) = 1 and

f (z_{1}, …, z_{r}) = ∑_{j=1}^{r} u_{j} (|z_{j}|^{2}) z_{j}e_{j}  (z_{j} ∈ Δ) .  (4.19)

Proof. Let f ∈ B_{0}(Δ^{r}) be a bisymplectomorphism with d f(0) = id_{V}. Consider a regular element z ∈ Δ^{r}, that is,

z = z_{1}e_{1} + ⋯ + z_{r}e_{r},

with all |z_{j}| different. The spaces ℂe_{j} of the corresponding Peirce decomposition are mapped by d f(z) to the spaces ℂe_{σ(j)} of another Peirce decomposition, for some permutation σ ∈ 𝔖_{r}. This means that [d f(z) (e_{j})] = [e_{σ(j)}] for all regular z ∈ Δ^{r}, where [ ] denotes the class in ℙ(ℂ^{r}); by continuity, this is true for all z ∈ Δ^{r}. As {[e_{1}] , …, [e_{r}]} is discrete in ℙ(ℂ^{r}) and d f(0) = id, we have [d f(z) (e_{j})] = [e_{j}] and

d f(z) (ℂe_{j}) = ℂe_{j} (4.20)

for all z ∈ Δ^{r}. This shows that f (z_{1}, …, z_{r}) = ∑_{j=1}^{r} f_{j}(z)e_{j}, with |f_{j}(z)| = |z_{j}|. From (4.20), we deduce that f_{j} depends only of z_{j}. Each f_{j} is then a bisymplectomorphism of the unit disc. According to Theorem 2, there exists a smooth function u_{j} : [0, 1) → S^{1} ≃ U(1) such that

f_{j}(z_{j}) = u_{j} (|z_{j}|^{2}) z_{j} (z_{j} ∈ Δ) .

Finally, any f ∈ B_{0}(Δ^{r}) has the form (4.19).

Conversely, each f of the form (4.19) is easily seen to be a bisymplectomorphism. □

4.6. The general case. We assume now that the domain Ω is irreducible, that is, not a product of two bounded symmetric domains. In Theorem 4, we will characterize the bisymplectomorphisms of Ω. This theorem may be considered as a kind of Schwarz lemma.

Let f ∈ B_{0}(Ω). If z = ∑ λ_{j}e_{j} is a regular element, then P_{z} = (ℂe_{1} ⊕ ⋯ ⊕ ℂe_{r})∩ Ω and f (P_{z}) = P_{z}. For a frame e = (e_{1}, …, e_{r}), let P (e) = (ℂe_{1} ⊕ ⋯ ⊕ ℂe_{r}) ∩ Ω. By the same argument as in the proof of Theorem 3, the restriction of f to P (e) has the form

∑_{j=1}^{r} λ_{j}e_{j} ↦ ∑ λ_{j}u_{j} (|λ_{j}|^{2}) e_{j},

where the u_{j}’s are smooth functions u_{j} : [0, 1) → S^{1} ≃ U(1). In the “polar coordinates” ((λ_{1}, …, λ_{r}) , e = (e_{1}, …, e_{r})), the map f is then represented by

((λ_{1}, …, λ_{r}) , e) ↦ ((λ_{1}, …, λ_{r}) , (u_{1} (λ_{1}^{2}, e) e_{1}, …, u_{r} (λ_{r}^{2}, e) e_{r})) . (4.21)

Let w_{j} (λ_{j}, e) = u_{j} (λ_{j}^{2}, e). We obtain

f^{∗}η_{jj} = i f^{∗} ∑_{m=1}^{n} e_{jm} d \overline{e}_{jm} = i ∑_{m=1}^{n} w_{j}e_{jm} (\overline{w}_{j} d \overline{e}_{jm} + \overline{e}_{jm} d \overline{w}_{j})

= η_{jj} − i (e_{j} | e_{j}) \frac{d w_{j}}{w_{j}} ,

f^{∗}ω_{jj} = ω_{jj}.

As Ω is irreducible, (e_{j} | e_{j}) has the same value g for all minimal tripotents. Finally

f^{∗}ω_{0} = ∑_{j=1}^{r} λ_{j}^{2}ω_{jj} + 2 ∑_{j=1}^{r} λ_{j} d λ_{j} ∧ (η_{jj} − i g \frac{d w_{j}}{w_{j}} )

= ω_{0} − 2 i g ∑_{j=1}^{r} λ_{j} d λ_{j} ∧ \frac{d w_{j}}{w_{j}} .

Then f^{∗}ω_{0} = ω_{0} implies that f ∈ B_{0}(Ω), written in the form (4.21), satisfies

∑_{j=1}^{r} λ_{j} d λ_{j} ∧ \frac{d w_{j}}{w_{j}} = 0. (4.22)

As w_{j} depends only on λ_{j} and e, this implies that d_{e} w_{j} = 0. As the manifold of frames is connected when the domain Ω is irreducible, w_{j} does not depend on e ∈ F. As a permutation of a frame is again a frame, we have w_{1} = ⋯ = w_{r} and u_{1} = ⋯ = u_{r}.

Finally, an element f ∈ B_{0}(Ω) is written in polar coordinates

((λ_{1}, …, λ_{r}) , (e_{1}, …, e_{r})) ↦ ((λ_{1}, …, λ_{r}) , (u (λ_{1}^{2}) e_{1}, …, u (λ_{r}^{2}) e_{r})) ,

(4.23)

where u is a smooth function u : [0, 1) → S^{1} ≃ U(1).

Theorem 4. Let Ω be an irreducible Hermitian bounded circled symmetric domain and let K be the isotropy group of 0. The analytic (resp. C^{∞}) bisymplectomorphisms of Ω are the maps ϕ = f ∘ g, where g = d ϕ(0) ∈ K and f is associated to v(t) = tu(t^{2}), with u : [0, 1) → S^{1} ≃ U(1) analytic (resp. C^{∞}) and u(0) = 1.

References

[1] Di Scala, A.J. and Loi, A., Symplectic duality of symmetric spaces, arXiv:math.DG/0603141.

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

[3] McDuff, Dusa, The symplectic structure of Kähler manifolds of non-positive curvature, J. Differ. Geom. 28, No.3 (1988), 467-475.

[4] Roos, Guy, Jordan triple systems, pp. 425–534, in J. Faraut, S. Kaneyuki, A. Korányi, Q.k. Lu, G. Roos, Analysis and geometry on complex homogeneous domains, Progress in Mathematics, vol.185, Birkhäuser, Boston, 2000.

[5] Whitney, H., Differentiable even functions, Duke Math. J., 10 (1943), 159–160.

A.D.: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

   E-mail address: antonio.discala@polito.it

A.L.: Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy

E-mail address: loi@unica.it

G.R.: Nevski prospekt 113/4-53, 191024 St Petersburg, Russian Federation

   E-mail address: guy.roos@normalesup.org