Star-exponential of normal j-groups and adapted Fourier transform

(1)

Fourier transform

Florian Spinnler April 2015

(2)

(3)

Aknowledgements

I would like to express my deepest gratitude to my advisors, M´elanie Bertelson (ULB) and Pierre Bieliavsky (UCLouvain), for supervising this thesis and answering my many mathematical and less mathematical questions. I am indebted to Pr. Bieliavsky for his patience and unconditional willingness to introduce me to his mathematical universe.

Thank you to all members of the Jury for being a part of it.

Axel de Goursac has been of invaluable help during many phases of this thesis. For these many hours of computations, discussions, questions and answers, enlightening comments, thank you very much. I would also like to thank Professor Yoshiaki Maeda for his precious help and insights during many discussions that we had over four years. I had the privilege to share an office with St´ephane Korvers, and thank him for all the good laughs (and serious discussions too) that we had.

I am thankful to the administrative staffs of both the ULB and UCLouvain mathematics departments for their very friendly and pro- fessional help. Special thanks to Carine, Cathy and Martine.

My friends, other math graduate students from ULB and UCLou- vain, members of my family and many other people close to me have often been very helpful, showing me their constant support, faith in my abilities, or simply sharing a good coffee, and I thank them for that.

Thanks also to my chemist friends for their warm welcome last summer (pun intended), when the halls of the math departments were somewhat empty.

This thesis could not have been completed without the financial support of the F.R.I.A. (Belgium), of which I had the great privilege to benefit from 2010 to 2014.

Finally, I would like to thank my parents, brother and sister. Time spent together is often a shelter from outside worries, and has helped me go through these four years of the PhD program with greater confidence.

(4)

(5)

This thesis is the author’s contribution on star-exponentials and adapted Fourier transforms, two aspects of the field of deformation quantization, and in this text more specifically of strict deformation quantization, which was introduced by M.A. Rieffel in the early 90’s. Given the data of a Fr´echet algebra A₀ (along with a countable family | · |_j of semi- norms defining the topology of A₀) and a strongly continuous action α:R²ⁿ× A₀ → A₀ which is assumed to be isometric in each seminorm, Rieffel in [74] developped a way to construct new Fr´echet algebras, called deformation quantizations of the original algebra A₀. This procedure thus yielded a strict analog of the deformation quantization program introduced in Reference [6]. A central element in this deformation procedure is the integral expression (f ?⁰g)(x) =R

e~ⁱS(x,y,z)f(y)g(z)dydz for the Weyl product of Schwartz functions on R²ⁿ, where S(x, y, z) = 4(ω(x, y) +ω(y, z) +ω(z, x)) is four times the symplectic area of the tri- angle ∆_x,y,z(for the canonical symplectic formω⁰onR^2N). Consider the action α and, for two smooth vectorsaand bof α, define the functions α(a) := [R^2N → A₀ :x7→ αx(a)] and α(b) := [R^2N → A₀ :x7→αx(b)].

UsingA₀-valued oscillatory integrals, Rieffel gives a meaning to the expression (α(a)?⁰α(b))(x) := R

e^~ⁱ^S(x,y,z)α_y(a)α_z(b)dydz. The formula a ?A₀ b := (α(a)?⁰α(b))(0) then defines an associative product on A₀, or at least does so on the dense subspace of smooth vectors of α. Such a formula was called a universal deformation formula. The only ingre- dients needed are an isometric action of R^2N on a Fr´echet algebra and the formula of the Weyl product.

A natural question, following this 1994 work of Rieffel, was be to look for analogous deformation formulas for actions of non abelian Lie groups. The first step, as in the case ofR^2N, was a non formal associative product on some appropriate subspace of C^∞(G), analogous to?₀ on R^2N. Results in this respect have been building up since the early 2000’s with works by Bieliavsky & Massar in [16] (2001), Bieliavsky &

Maeda in [15] (2002). A universal deformation formula for tempered actions (Definition 2.3.8) of Lie groups of a certain type on Fr´echet algebras is provided in [13]. It relies on an integral formula that is shown to produce an associative, non commutative product on functions on certain Lie groups of a special type, namely on the Iwasawa factor AN of the Iwasawa decomposition SU(1, n) =KAN (cf. [46] for general definition and [50] for the explicit construction). The connected component of the identity of the ax+b group is a special case of this construction.

In this work [13] by Bieliavsky & Gayral (2014), actions of K¨ahlerian Lie groups are considered. These groups can all be obtained by taking

(10)

semi-direct products of the Iwasawa factors AN of SU(1, n). A group containing only one AN component will be called elementary normal j-group in what follows and denoted byS.

For example, the product?_θof [13] and [15] on theax+bgroup reads as follows in the coordinates (a, l)7→

e^a e^al 0 e^−a

∈Sl₂(R)'SU(1,1):

(f1?_θf2)(a, l) = 1 (πθ)²

Z

R²×R²

Ae⁻²ⁱ^θ ^Sf1(a1, l1)f2(a2, l2)da1dl1da1dl2, (1) whereS=−sinh(2(a₁−a₂))l−sinh(2(a₂−a))l₁−sinh(2(a−a₁))l₂ and A= 4p

cosh(2(a₁−a₂)) cosh(2(a₁−a)) cosh(2(a−a₂)).

The importance of the phase functionS (the analogous of the symplectic area forR^2N), already apparent in the work of Weinstein in [81]

(1994), is also made explicit in [10] (2000), where it is shown how a symplectic symmetric space structure can be added on the Lie groups under consideration. The phase function is then identified with the symplectic area of geodesic triangles of the symmetric space.

In [16] and [11], a canonical symplectic symmetric structure is constructed on S, that is in some sense a contraction of the Hermitian symmetric space structure on SU(1, n)/U(n); this canonical symmetric structure is invariant under a group G₀ of symplectic automorphisms, which is detailed in Section 1.2.4. Moreover, the group G0 has a subgroup isomorphic toS. In this text, we will explicitly compute the coadjoint orbits of ˜G, the central extension of G₀, and show that particular orbits of ˜Gare acted uponsimply transitively by the subgroup of ˜Giso- morphic toS. The Kirillov-Kostant-Souriau symplectic form on an orbit O can be transported to the groupS, which then becomes a symplectic group in the sense of Medina in [60] (2009). Since the coadjoint action is hamiltonian, the moment map can also be transported to S which then becomes a so called Hamitlonian Lie group with a moment map for the left action, given by λ:S→ s^∗. The coadjoint orbits considered in this text admit a real polarization, i.e. an invariant Lagrangian foliation (under ˜G) denoted byL. A family of Weyl-type quantizers is then provided on the symmetric space S ' O, parametrized by functions mon Q' O/L. A special choice of m0 then yields a quantization map forS associating compactly supported functions onSto trace class operators on a Hilbert space, and the composition of symbols of this quantization map in turn provides the ?_θ product of [13].

TheBG product ?_θ, of which Equation 1 is a particular case, admits a formal development inθdenoted by?^M_θ , and this star-product is shown

(11)

to be g-covariant ([42]):

λ_X ?^M₀ λ_Y −λ_Y ?^M₀ λ_X = θ

iλ_[X,Y_]. (2) In other words, the map g→End(C^∞(O), ?^M_θ ) :X7→[u7→ _θⁱλX ?^M_θ u]

is a Lie algebra representation, and the question arises whether one can compute and find explicit expressions for the corresponding Lie group representation. It has to be noted that the existence of a covariant star- product is guaranteed on coadjoint orbits with real polarizations, cf. for example Masmoudi in [59] (1995).

The star representation program, i.e. the realization of Lie group representations via star-products on their coadjoint orbits, on the dual of their Lie algebra g^∗, or on symmetric spaces on which they act, has seen many contributions in the last three decades. In the references [9], [17] and [18] (1999-2004) for instance, the principal and discrete series representations of Sl2(R) arise as properties of the Moyal product on the Poincar´e disk or one-sheeted hyperboloid, which are both coadjoint orbits ofSl2(R).

This program was entirely completed in the case of exponential Lie groups in the work of Arnal & Cortet in [4] (1990). Using an induction argument on the dimension of the Lie algebra g, canonical coordinates are constructed on g^∗, such that when restricted to each orbit O,→g^∗, they yield a global Darboux chart (p, q) onO. ForX ∈g and denoting by XÂC the functions on g^∗ given by [ξ 7→ ξ(X)], it is shown that the Weyl Moyal star product in these canonical Darboux coordinates satisfies iXÂC ?^M₀ iYÂC−iYÂC ?^M₀ iXÂC = i[X, Y]ÂC (where i such thati² =−1), thus yielding a representationX 7→lX of the Lie algebra g, withl_X(f) :=iXÂC?^M₀ f forf ∈C_c^∞(O).

The link with Kirillov’s method of orbits (cf. works by Kirillov in [48] and [49]) and the description of the unitary dual is then estab- lished as follows. Let ρÔ be the unitary irreducible representation of G corresponding to the coadjoint orbit O. On each orbit O, again using the global coordinates (p, q), it is then shown that this representation can be intertwined by partial Fourier transform operators F^p such that F^p ◦lX ◦(F^p)⁻¹ = dρÔ(X), where the differential dρÔ is defined by dρÔ(X) := _dt^dρ(exp(tX))|_t=0.

This thesis provides explicit expressions for the group representations arising from the covariance property of the moment map under the product?^M_θ . These representations, called star-exponential, are then shown to belong to the multiplier algebra of a certain Schwartz space S(O) of functions on the orbit O. The star-exponential, more precisely

(12)

a modified version of the star-exponential, is then used to define, for each coadjoint orbit O, anadapted Fourier transform F₁^O on the group.

A functional transform close to this adapted Fourier transform is often calledgroup Fourier transformornon commutative Fourier transform in some recent mathematical physics literature (cf. Raasakka in [72],[70],[71] (2011,2012,2013) or Oriti & al. in [41](2013)); we also give the formula of this non commutative Fourier transform F₂ : L²(G) → L²(g^∗, dm(ξ)), by patching together the expressions of F₁^O, where O runs over all coadjoint orbits of the normal j-group G. Here dm(ξ) is the corresponding measure on g^∗ obtained from the Liouville measures on the coadjoint orbits.

The first chapter of this thesis introduces many well known and less known definitions and results about homogeneous complex domains.

The theorem of Gyndikin, Pyatetskii-Shapiro, Vinberg, (cf. [43](1964), [69](1969)) establishing the classification of homogeneous bounded domains is quoted and illustrated. The classification relies on a one-to-one correspondence between homogeneous bounded domains and so called normal j-groups, which arise as subgroups of the symmetry group of homogeneous K¨ahler manifolds (cf. [44] (1967)). It turns out that normal j-groups have a very constrained structure, namely they can all be obtained via semi-direct products of so-calledelementary normal j-groups, which in the classification of Pyatetskii-Shapiro arise as transitive subgroups of the automorphism groups of unit balls in Cⁿ.

Coordinates identical to those introduced in [13] are then defined on these elementary factors, denoted by S. In addition to their Lie group structure, they are then endowed with a symplectic symmetric space structure, which is explicitly realized as the symmetric structure on a quotient space ˜G/K, where ˜G is a Lie group having S as a subgroup acting simply transitively on ˜G/K. A unitary irreducible representation of ˜G is then computed using Kirillov’s method of orbits, and then restricted to S. The symmetric space ˜G/K is realized as a coadjoint orbit O of ˜G, which is endowed with an invariant Lagrangian foliation L (where Lagrangian refers to the Kirillov-Kostant-Souriau symplectic form on coadjoint orbits) and a simply transitive action of S<G. Us-˜ ing a global section s of the bundle π : ˜G/K → O/L, the symmetry at the base point seK can then be lifted on functions on the leaf space Q ' O/L. For a function ϕ on Q and a point q ∈ Q, one defines Σ^∗ϕ(q) := ϕ(π(s_eK(s(q)))). As in [13] and [14], the Weyl-type quantizer is then defined on ˜G/K by Ω(ξ) := U(g) ◦Σ^∗ ◦U(g)⁻¹, where gK = ξ ∈ G/K˜ and U is a unitary irreducible representation of ˜G.

The Weyl-type quantizer is then transported to S via the simply tran-

(13)

sitive action (we refer to Gayral & al. in [38](2008), where a Weyl-type quantizer is also defined on the ax+b group). The quantization map sending compactly supported functions to bounded operators on the Hilbert space H, Ω_θ : D(S) → L(H), is defined on elementary normal j-group through the assignmentf 7→R

Sf(g)Ω(g)d^Lgand then extended to normal j-groups through a method introduced in [14](2014) and detailed here.

Chapter two introduces the pseudo differential calculus and universal deformation formula of [13]. The intertwining operator T_θ between the Weyl product ?₀ and the BG product ?_θ of [13] is explicitly given, as well as the formal Moyal star-product, which can be obtained as power series expansion of ?0. Covariant star products are then introduced and illustrated with the example of the three dimensional Heisenberg groupH3. On coadjoint orbits of this group, the quantization method of chapter one yields precisely the Weyl product and its formal counterpart the Moyal product, which are shown to be covariant for the moment maps.

For an elementary normal j-groupS, the Moyal product is then shown to be covariant for the moment map of the actionS× O → O. The star- exponential E : G× O on S can then be computed in two ways. The first is by computing the distributional trace

E_g(ξ) = Tr(U(g)Ω(ξ)). (3)

The second is by explicitly solving the differential equation

∂tEt(exp(X))(ξ) = i

θλ_X?_θEt(exp(X))

(ξ) (4)

for the functionE_t:S×O →C, with initial conditionE₁ ≡1. The group multiplier property is then shown to be satisfied: for everyg, g⁰∈G,

E_g?θE_g⁰ =E_gg⁰

holds. In this text this second way of explicitely finding the star- exponential plays a minor role. A partial Fourier operator is introduced to reduce the Equation 4 to a PDE, and the expression computed by solving Equation 3 solves this PDE.

The notion oftempered structure on a Lie group, introduced in [13], is then given and illustrated. It is a global diffeomorphism ψ : G → g^∗ 'R^dim(G) such that the multiplication law and inversion law on the Lie group are tempered functions, i.e. can be bounded by polynomials

(14)

in the coordinates given by the chart ψ. A function S:S→Rcan give rise to a tempered structure, by defining the map

ψ^S :G→g^∗ :g7→[X7→dS( ˜X)(g)]. (5) Given a tempered structure ψ, a simple criterion is provided for the existence of a function S such that ψ=ψ^S (called theprimitive ofψ).

For a tempered structure ψ, a Schwartz space S^S(S) can be defined on S. Another Schwartz spaceS^S(S) can be defined whenψhas a primitive S. Reproducing and expanding a proof already given in [13], it is shown that these two spaces coincide. Another Schwartz space on S can be defined, using a tempered structure ψ on S×S, and the function Scan

underlying the BG product ?_θ, and the restricting the corresponding Schwartz space onS×Sto a Schwartz space of one variableS¹(S) onS. Yet another Schwartz spaceS¹²(S) is used in [14]. It is proven (albeit in the two dimensional case, essentially to simplify notations) that all these Schwartz spaces are identical and all isomorphic as Fr´echet vector spaces to the canonical Schwartz space S(R^dim(G)), and so also isomorphic to the Schwartz space on solvable Lie groups introduced in [26](2010); the proof makes use of thetameness of the pair (S×S, Scan).

The left action ofSon the Fr´echet algebra (S(S),·) is tempered, and this fact allows one to use the deformation formula of [13] to produce an algebra (S(S), ?_θ) deforming (S(S),·). The star-exponential is then shown to belong to the multiplier space of (S(S), ?_θ).

In the third chapter, two adapted Fourier transforms for normal j- groups are suggested, yielding results analogous to those given in [38]

for the case of theax+bgroup. First, by using a slightly altered version of the star-exponential, defined by

E˜_g^O(g⁰) := Tr(U(g)K¹²Ω_θ,(g⁰)), (6) where K is the formal dimension operator of the representation U, introduced in [34]. This modified star-exponential was introduced in [38](2008). Defining the first Fourier transform by

F₁^O :S(S)→C^∞(O) :f 7→[g⁰7→

Z

S

f(g) ˜E_g^O(g⁰)dg], (7) where we use the identificationg⁰ 7→Coadg⁰E^∗, the following properties are proven: the operator ∆⁻¹²F₁^O intertwines the convolution product on the group and the BG product ?θ on the orbit O. An inversion formula also holds, which is given by

f(g) = X

=±1

Z

O

E˜_g(g⁰)F₁^Of(g⁰)dµ(g⁰). (8)

(15)

A Plancherel formula is then shown to hold as well:

Z

S

|f(g)|²dg = X

=±1

Z

O

|F₁^O(f)(g⁰)|²dµ(g⁰), (9) thus realizing explicitly the Plancherel measure on ˆGas described in e.g.

[35]. The general results are then provided for normal j-groups.

A simple rewriting of the Fourier transform F₁ yields a map F₂ : L²(G)→L²(g^∗, dm(ξ)), where dm(ξ) is the measure ong^∗ obtained by patching together the Liouville measures on the coadjoint orbits, and the Fourier inversion formula and Plancherel formula are then provided for this non commutative Fourier transform F₂. The formulas are detailed both in the case of elementary normal j-groups as well as in the general normal j-groups case.

This thesis, beyond providing explicit expressions for the star exponential and the adapted Fourier transform, aims at clarifying some auxiliary questions that are inevitably met by someone navigating these waters. The results on primitive of tempered structure, the details of the structure lemma for normal j-groups, the isomorphism of all Schwartz spaces at play, are examples of such small questions that were settled here. Many other questions remain, and the fourth and last chapter of this text intends on clarifying exactly where these questions lie and what could be the leads to solve them.

(16)

(17)

Quantization of

homogeneous bounded domains

This chapter introduces the theory of homogeneous bounded domains, as well as fundamental results on the existence of transitive groups of automorphisms of a certain type. These groups, called normal j-groups, can be obtained by successive semi-direct products of so-called elementary normal j-groups. The normal j-groups are then endowed with a canonical symplectic symmetric space structure. This allows, via Kirillov’s method of orbits for elementary normal j-groups, to define a Weyl-type quantizer for these elementary building blocks. The semi-direct product structure of normal j-groups is then used to define a Weyl-type quantizer and Weyl quantization map.

1.1 Pyatetskii-Shapiro theory of complex do- mains

This section introduces basic notions and examples of homogeneous complex domains.

1.1.1 Homogeneous bounded domains An open connected subset of C^N is called a domain.

Definition 1.1.1. A domain ∆ of the affine space C^N is said to be a homogeneous bounded domain (or HBD for short) if it is bounded in C^N and if the groupAut(∆) of biholomorphic automorphisms of∆ acts

17

(18)

transitively on ∆, where holomorphic refers to the canonical complex structure on ∆⊂C^N.

Definition 1.1.2. A domain∆is said to be a symmetric domainif for every x∈∆there exists αx∈Aut(∆) such that

1. α²_x= id∆, 2. αx(x) =x, and

3. (αx)?x: Tx∆→Tx∆ :vx7→ −v_x.

Every bounded symmetric domain is homogeneous but the converse (a problem posed by E. Cartan in 1935) does not hold (cf. the original couterexample by Pytatetskii-Shapiro in Reference [68] - in Russian).

Example 1.1.3. The simplest example of homogeneous bounded domain is the unit disk of C:

D:={z∈Csuch that |z|<1}.

The complex structure is given by multiplication byiin the tangent space at each point; using the Schwarz lemma (cf. [51]), it is shown that the group of biholomorphic automorphisms ofDis the set of transformations

z7→e^iθ z−a 1−az,

witha∈Dandθ∈R. This group is isomorphic toSU(1,1)through the identification of the pair(θ, a) with the element ofSU(1,1)given by the product of

1 p1− |a|²

e^iθ² 0 0 e^−iθ²

!

1 −a a 1

1.1.2 Theorem of Pyatetskii-Shapiro on HBDs

In the case of the homogeneous bounded domainDgiven in the previous section (the unit disk ofC) it is well know that there exists a biholomorphic map called the Cayley transform mapping the upper half-plane H to the unit disk, which is explicitly given by

H→D:z7→ z−i z+i,

where H:={z∈Csuch that =(z) >0}, with =(z) the imaginary part of the complex number z∈C.

(19)

Aconvex cone ofRⁿ is a subsetV of Rⁿ such that αx+βy∈V,∀x, y∈V, α≥0 andβ ≥0.

A generalization of the upper half-plane to the case of (unbounded) domains of C^N is given in the following definition.

Definition 1.1.4. (cf. Reference [37]) Let V be a convex cone in Rⁿ not containing any straight lines. A V-hermitian form is a map F : C^m×C^m→Cⁿ such that

1. F(λx+µy, w) =λF(x, w) +µF(y, w)∀x, y, w∈C^m andλ, µ∈C, 2. F(u, v) =F(v, u),

3. F(u, u)∈V, the closure of V, and 4. F(u, u) = 0 if and only if u= 0.

Definition 1.1.5. A Siegel domain of type II associated to the cone V and the V-hermitian formF is the domain inC^n+m consisting of points (z, u)∈Cⁿ×C^m such that

Im(z)−F(u, u)∈V. (1.1) This domain will be denoted by D(V, F).

To illustrate this, take the casem= 0 and choose the cone V :=R⁺₀ inR. When the formF is trivial then the Siegel domainD(V, F) is the upper half plane H={z∈C|Im(z)>0}.

Let us introduce some terminology and important results on complex manifolds.

Theorem 1.1.6 (cf. Reference [44]). Let (M, J) be a complex manifold endowed with a Hermitian structureh=g+iω. The following conditions are equivalent:

1. dω= 0,

2. ∇J = 0, where ∇ is the Levi-Civita connexion induced by the Riemannian metric g onM,

3. in local real coordinates{z¹,· · ·, zⁿ, z¹,· · · , zⁿ}the coefficientsh_αβ of h take the formh_αβ = ^∂_∂²^`og(ϕ)

zα∂_zβ for some positive function ϕ.

(20)

Definition 1.1.7. A K¨ahler manifold is a complex manifold (M, J) endowed with a positive definite Hermitian product h=g+iω satisfying either one of the equivalent conditions of Theorem 1.1.6.

A map ψ : M → M is said to be an automorphism of the K¨ahler manifold if

• ψ is holomorphic, invertible with holomorphic inverse, and

• ψ^∗h=h.

The K¨ahler manifold is said to behomogeneous if its automorphism group acts transitively.

Consider now a quadruple (g,k, j, ρ), where gis a real Lie algebra, k is a Lie subalgebra of g,j:g→gis a linear map such that

j²X+X∈k

for all X ∈g, andρ∈g^∗∧g^∗, where g^∗ is the real dual vector space of g.

Definition 1.1.8. The data(g,k, j, ρ) is said to be a K¨ahler algebra if 1. jk⊆k,

2. [K, jX]−j[K, X]∈k,

3. [jX, jY]−j[jX, Y]−j[X, jY]−[X, Y]∈k, 4. ρ(K, X) = 0 for K∈k, X ∈g,

5. ρ(jX, jY) =ρ(X, Y),

6. ρ(jX, X)>0 for X /∈k, and

7. ρ([X, Y], Z) +ρ([Y, Z], X) +ρ([Z, X], Y) = 0, for all X, Y, Z ∈g, K ∈k.

Remark 1.1.9. Every homogeneous Kähler manifold M determines a Kähler algebra. If we denote byGthe transitive group of automorphisms of M and by K the stabilizer subgroup of a fixed point x0 ∈ M, then M ' G/K as homogeneous Kähler manifolds. Let us denote the Lie algebras of G and K by g and k respectively. Using the isomorphism g/k 'Tx0M yields an operator j := J?x0 :g/k → g/k that can be lifted to ginto an operator satisfying the conditions of Definition 1.1.8 above.

(21)

Conversely, every K¨ahler algebra determines at least one homogeneous K¨ahler manifold (cf. Reference [33]). One is the homogeneous space G/K where G and K are connected Lie groups having g and k as Lie algebras. The complex structure at a point J_gK of the quotient G/K is then defined using left translations from the tangent spacej :TeKG/K→ Tx0G/K as follows:

J_gK :T_gKG/K→T_gKG/K: X_gK 7→L_g?eK ◦j◦L_g⁻¹_?gKX_gK. ForX∈g, consider the operator ad_X defined by

ad_X :g→g:Y 7→[X, Y].

A particular and important class of K¨ahler Lie algebra consists of so called normal j-algebras. In the definition of K¨ahler Lie algebras, they correspond to the case where k = 0 and adX has only real eigenvalues

∀X ∈g,

Definition 1.1.10. A normal j-algebra (cf. [69]) is a real solvable Lie algebra b such that

1. the operator ad_X has only real eigenvalues for any X ∈b, 2. there exists an endomorphism j:b→b such that j²=−Id_b and

[X, Y] +j[jX, Y] +j[X, jY]−[jX, jY]≡0∀X, Y ∈b, 3. there exists a linear form ω:b→R such that

ω([jX, X])>0 if X6= 0 andω([jX, jY]) =ω([X, Y])∀X, Y ∈b.

A subalgebrab⁰ of b such thatjb⁰ ⊂b⁰ and b⁰ is a normal j-algebra, will be said to be a normal j-subalgebra of b.

A K¨ahler manifold will be said to be normal K¨ahler manifold if it admits a transitive group of automorphismsGsuch that the Lie algebra of Gis a normal j-algebra.

The importance of normal j-algebras lies in the combination of the following Theorem 1.1.11 as well as Proposition 1.1.12.

Theorem 1.1.11 (cf. [43]). Every homogeneous bounded domain in Cⁿ is biholomorphic with a homogeneous Siegel domain of type II (cf.

Definition 1.1.5).

Moreover, the Siegel domains of type II have the following important property:

(22)

Proposition 1.1.12 (cf. Reference [43]). Every Siegel domain of type II admits a simply transitive solvable group of automorphisms, the Lie algebra of which is a normal j-algebra.

Given a normal j-groupG, which are solvable exponential Lie groups the Lie algebra of which are normal j-algebras, the corresponding Siegel domain can be constructed explicitly as is done in [47], and it relies on the decomposition of the Lie algebra g of G as detailed in the proof of Lemma 1.1.16.

Example 1.1.13. Consider the Poincar´e disk D. It is biholomorphic via the Cayley map to the upper half-plane H. Consider the following matrix group

S:=

M(a, l) :=

e^a e^al 0 e^−a

|a, l∈R . Then, the diffeomorphism

φ:S→H:M(a, l)7→e^2ai+e^2al, shows the existence of a simply transitive action

S×H→H.

It turns out that normal j-algebras can be obtained through successive semi-direct products of so-called elementary normal j-algebras. Definition 1.1.14. Consider a complex hermitian vector space (g, j) with scalar product h(X₁, X₂) and an element r₀ of length1. Denote by r^⊥₀ the space {Y ∈g such that h(r0, Y)≡0}. The space (g, j, h) is said to be an elementary normal j-algebra if it is endowed with a Lie algebra structure such that

1. [jr₀, r₀] =r₀,

2. [jr0, Y] = ¹₂Y, ∀Y ∈r₀^⊥, and

3. [Y, Y⁰] = (Im(h(Y, Y⁰)))r0, ∀Y , Y⁰∈r₀^⊥, where Im(z) denotes the imaginary part of z∈C.

Remark 1.1.15. It is shown in [69] that the elementary normal j- algebra of dimension 2n+ 2 is the Lie algebra of the group associated to the open unit ball of Cⁿ⁺¹ by Theorem 1.1.11 and Proposition 1.1.12 above.

(23)

1.1.3 Structure result for normal j-algebras

Normal j-algebras as described in Definition 1.1.10 have a very constrained structure, given by the following Lemma.

Lemma 1.1.16. Let (b, j) be a normal j-algebra and let z be a one- dimensional ideal of b. Then the Lie algebra b can be decomposed as follows :

b=z⊕V ⊕jz⊕b⁰, where

1. b⁰ is a normal j-subalgebra ofb,

2. z⊕V⊕jz is an elementary normal j-algebra as given in Definition 1.1.14,

3. [z⊕jz,b⁰] = 0, and 4. [b⁰, V]⊆V.

The purpose of this section is to provide a detailed proof of the Lemma 1.1.16, which plays a fundamental part in this thesis. The original formulation of the Lemma and its proof can be found in [69].

We start by rewriting basic definitions and results, and then proceed with the proof. The main points are

1. the existence of a one-dimensional ideal in every normal j-algebra and

2. the explicit construction of an elementary normal j-algebra from the data of this one-dimensional ideal.

For the sake of completeness, we first introduce some standard terminology and two useful Lemmas.

Definition 1.1.17. Let g be a Lie algebra. One recursively defines the ideals g^(k) by

• g⁽⁰⁾ :=g, and

• g^(k+1) = [g^(k),g^(k)]

The Lie algebra g is said to be solvable if there exists k∈ N such that g^(k) = 0.

We now list two Lemmas that will be used in the proof of 1.1.16.

(24)

Lemma 1.1.18. Let g be a Lie algebra and a an ideal of g. Then g is solvable if and only if g/a and a are solvable.

Proof. To prove this it is enough to apply the definitions and observe that for any surjective Lie algebra morphism π : g₁ → g₂, one has π(g^(k)₁ ) = (g2)^(k).

Lemma 1.1.19. Every solvable Lie algebraghas a codimension 1 ideal g₁. Since g is solvable, g₁ is solvable by Lemma 1.1.18.

Proof. Letndenote the dimension ofg. Sincegis solvable, the derived ideal is a proper ideal, g⁰ := [g,g] ⊂ g. Any vector subspace b of g of dimension n−1, containing g⁰, is then a codimension one ideal since [g,b]⊆[g,g]⊆g⁰ ⊆b.

Remark 1.1.20. Letgbe a solvable Lie algebra andg₁ be a codimension 1 ideal. Then [g,g^(k)₁ ]⊆g^(k)₁ ,∀k ∈N.

Proof. By induction, using the Jacobi identity.

Lemma 1.1.21. Let gbe a solvable algebra such that ad_X has only real eigenvalues. Thenghas a one-dimensional ideal inR_g, the last non zero ideal in the sequence g⊃g⁽¹⁾⊃ · · · ⊃Rg⊃0.

Proof. We prove the following : there exists a linear formλ:g→Rand a vector r₀ ∈R_g such that

[X, r0] =λ(X)r0

for every X ∈g.We proceed by induction on the dimension of g. If g has dimension 1, it is itself a one-dimensional ideal. In dimension 2, g is either

• abelian and any one dimensional vector subspace of g is a one- dimensional ideal, or

• isomorphic to the Lie algebra [X, Y] = Y, and in this case the subspace RY is a one-dimensional ideal; moreover,R_g =RY and the linear form is simply X^∗, the element of the dual such that X^∗(X) = 1 and X^∗(Y) = 0.

Now, by Lemma 1.1.19,ghas a codimension 1 ideal,g₁.By the induction hypothesis, there exist λ : g₁ → R and r0 ∈ Rg1 such that [X, r0] = λ(X)r0,∀X∈g₁. In that case, the vector space

V_λ :={v∈R_g₁ such that [X, v] =λ(X)v, ∀X∈g₁}

(25)

is not trivial since it contains at least the vector subspaceRr₀. We first want to show that [g, Vλ]⊆Vλ.By definition this is equivalent to show that [X,[Y, v]] =λ(X)[Y, v], for allv∈V_λ, X∈g₁ andY ∈g.

By the Jacobi identity ong, one has

[X,[Y, v]] = [[X, Y], v] + [Y,[X, v]]

=λ([X, Y])v+λ(X)[Y, v], (1.2) where we used that g₁ is an ideal of g. We will prove that

λ([X, Y])≡0∀X ∈g₁, Y ∈g, which will yield the following property:

[X,[Y, v]] =λ(X)[Y, v]), for all Y ∈g and X∈g₁, therefore showing that

[g, V_λ]⊂V_λ. (1.3)

Lemma 1.1.22. For a fixed element v ∈ Rg1, let W be the subspace generated by the (ad^j_Y(v))j≥0. Then

tr(ad_X

W) =λ(X).dim(W), for all X∈g₁.

Proof. Observe first that W is non trivial by the induction hypothesis since it contains at least Rr₀ and finite dimensional since R_g₁ is a finite dimensional ideal. By induction, we now show that

ad_X(ad^j_Y(v)) =λ(X)ad^j_y(v) +X

i<j

c_iadⁱ_Y(v), (1.4) for any j ∈N, j ≥ 0. For j = 0 it is immediate since ad_X(v) =λ(X)v by definition of v; and for j = 1 one has the identity ad_Xad_Y(v) = λ(X)adY(v) + λ([X, Y])v, coming from the Jacobi identity. We now fix a value of j and assume that Equation 1.4 holds for any k ≤ j.

Therefore, for k=j+ 1, one has

adXad^j+1_Y (v) =adYadXad^j_Y(v) +ad_[X,Y_]ad^j_Y(v)

=ad_Y λ(X)ad^j_Y(v) +X

i<j

c_i adⁱ_Y(v) +λ([X, Y])ad^j_Y(v) +X

i<j

diadⁱ_Y(v) by Eq. 1.4

=λ(X)ad^j+1_Y (v) + X

i<j+1

˜

ci adⁱ_Y(v).

(26)

This shows that the matrices of ad_X are upper triangular in the basis {(ad^j_Y(v))}j≥0 and that the diagonal elements are all equal to λ(X).

This proves the claim.

Sinceg₁is an ideal ing, [X, Y]∈g₁and we can apply Lemma 1.1.22.

The following holds:

λ([X, Y]) =tr ad_[X,Y_] W

=tr (adXadY)

W −(adYadX) W

= 0,

by the symmetry and linearity of the trace. This observation, combined with Equation 1.2, yields the invariance [g, V_λ]⊂V_λ.

Since g₁ is a codimension 1 ideal in g, one has g = g₁ ⊕RS for some element S in g that is not in g₁. Now, since the subspace Vλ

is g-invariant, it is in particular invariant under S: [S, V_λ] ⊆ V_λ. By hypothesis,ad_S has only real eigenvalues, and therefore there is a vector v₀ ∈ V_λ and θ ∈ R such that [S, v₀] = θv₀. This vector satisfies [cS + X, v0] = (cθ+λ(X))v0 for all X ∈g₁ and c ∈R, and in particular the one dimensional subspace Rv₀ is a one dimensional ideal of g, with an associated linear form given by θS^∗ +λwhere S^∗ is the element of g^∗ such that S^∗(S) = 1 and S^∗(X) = 0∀X∈g₁.

This ends the proof of Lemma 1.1.21.

Proof of Lemma 1.1.16

Proof. By Lemma 1.1.21 we know that there exists such a one dimensional ideal. Denote this ideal by z, and set (cf. Definition1.1.14)

b:=z⊕jz⊕U,

wherejis the endomorphism of axiom 2. in Definition 1.1.10 andUis the orthogonal complement of the vector subspace z of b for the hermitian scalar product

hx, yi:=ω([jx, y]) +iω([x, y]). (1.5) Let r₀ be a generator ofz normalized such that the following holds:

[jr0, r0] =r0 (1.6)

Letu∈U; then, by definition of U,hr₀, ui= 0. The vector space U is invariant under j:

hr₀, jui=−ihr₀, ui= 0.

(27)

Since the endomorphism j is a vector space isomorphism, this shows that u∈U ⇔ju∈U.

Consider the element r0 of z satisfying Equation 1.6. We have that ω(r₀) = ω([jr₀, r₀]) > 0 by axiom 3. of normal j-algebras (Definition 1.1.10).

The subspacez+U is a subalgebra ofb, because it is the centralizer of the elementr₀. Indeed, looking at the imaginary part of Equation 1.5 and using orthogonality, for any u∈U, one has that

0 =ω([r0, u]) by orthogonality

=ω(α r0) for some α, sincez is an ideal

=αω(r₀).

On the other hand, ω(r₀) > 0. Therefore, α = 0 and [z, U] = 0. This proves the claim that z+U is the centralizer of r0.

Consider the operatorad_jr₀. We will show that ad_jr₀U ⊆U.By the Jacobi identity and [jr₀, r₀] =r₀ one has

[r0,[jr0, u]] =−[r₀, u] + [jr0,[r0, u]] = 0 since [r0, u] = 0

Therefore, [jr0, u]∈z+U, the centralizer ofr0. But [jr0, u]∈jz+U as well:

[jr₀, u] = [jr₀, ju⁰] foru⁰ ∈U such that ju⁰ =u, sincejU =U

= [r0, u⁰] +j[jr0, u⁰] +j[r0, ju⁰] by axiom 2. of j-algebras

=j[jr0, u⁰]∈j(z+U) =jz+U,

where the last line follows from [z, U] = 0, the previous observation that [jr₀, u] ∈ z +U and the invariance jU = U. Hence, [jr₀, u] ∈ (jz+U)∩(z+U) = U. Since u = ju⁰ for some u⁰ ∈ U, we have also proved that [jr0, ju⁰] =j[jr0, u⁰],or in other words that

ad_jr₀ ◦j

U =j◦ad_jr₀

U. (1.7)

We now show that the operator adjr0

_U can only have 0 or ¹₂ as eigenvalues. Letλ₀ be an eigenvalue ofad_jr₀ and let an elementu₀∈U be the corresponding eigenvector. One has

[jr0, u0] =λ0u0, and

[jr0, ju0] =λ0ju0, by Eq. 1.7.

Therefore,

[jr₀,[ju₀, u₀]] = [[jr₀, ju₀], u₀] + [jr₀,[ju₀, u₀]]

= 2λ₀[ju₀, u₀]. (1.8)

(28)

Since z+U is a subalgebra, [ju₀, u₀] =r+u for somer∈z, u∈U. But then,

2λ₀(r+u) = 2λ₀[ju₀, u₀] = [jr₀,[ju₀, u₀]]

= [jr₀, r+u]

=r+ [jr0, u],

because [jr₀, r] = r for all r ∈ z. We know that [jr₀, u] ∈ U, and considering thezcomponent of this last equality one finds (2λ0−1)r = 0.

This implies that either λ₀= ¹₂ or thatr = 0. In the latter case, 2λ0 ω([ju0, u0]) =ω([jr0,[ju0, u0]]) by Eq. 1.8

=ω([−r₀, j[ju0, u0]]) by axiom 3.

=−ω([r₀, ju]) = 0 by orthogonality of U and z But again by axiom 3. of Definition 1.1.10, ω([ju0, u0])>0. Therefore, in the case r = 0,λ₀= 0 must hold.

Defineb⁰ as follows :

b⁰ :={X∈U such thatad^m_jr₀X= 0 for some m∈N}, and define the subspace V ofU as follows :

V :={X∈U such that ad_jr₀−1 2

m

X= 0 for some m∈N}.

We will now prove thatb⁰ is a j-subalgebra, the first claim of Lemma 1.1.16. It is j-invariant, because of Equation 1.7. Now, to show that it is closed under the Lie bracket, chose two elementsX, Y ∈b⁰.As can be proven by using an induction argument and the identity ^m_p

+ _p+1^m

=

m+1 p+1

, the following holds:

ad^m_jr₀[X, Y] =

m

X

p=0

m p

[ad^m−p_jr

0 X, ad^p_jr

0Y]; (1.9) then, assumingad^k_jr¹

0X = 0 andad^k_jr²

0Y = 0 for some integersk₁ andk₂, the choice m=k₁+k₂ satisfiesad^m_jr

0[X, Y] = 0. Indeed, if m−p=k₁+k₂−p < k₁,

preventing the vanishing of the left term ad^m−p_jr₀ X in the bracket, then p > k2, implying the vanishing of the right termad^p_jr

0Y in the bracket.

The argument is the same when k₁ andk₂ are reversed.

(29)

To prove that [b⁰, V]⊆V, the fourth claim of Lemma 1.1.16, we first introduce some notations. Consider the following subspaces

b⁰_k:={X∈b|(ad_jr₀)^lX= 0, ∀l≥k}, and V_k:={X∈V|(ad_jr₀−1

2)^lX = 0∀l≥k}.

One has the set equalities b⁰ =∪_k≥0b⁰_k and V =∪_k≥0Vk, as well as the inclusions V_k⊂V_k+1 and b⁰_k⊂b⁰_k+1.

LetX∈b⁰_k

1 and Y ∈V_k₂.Then,ad_jr₀X ∈b⁰_k

1−1 and (ad_jr₀−¹₂)Y ∈ Vk2−1.

Remark 1.1.23. The operator(ad_jr₀−¹₂) acts as follows on the bracket [X, Y]:

(adjr0 −1

2)[X, Y] = [adjr0X, Y] + [X,(adjr0− 1 2)Y]

(it is an instance of a so-called generalized derivation, cf. Reference [52])

By this observation, we see thatad_jr₀[X, Y]∈[b⁰_k

1−1, V_k₂]+[b⁰_k

1, V_k₂−1].

Actually we can show an analog of Equation 1.9 to obtain (ad_jr₀ −1

2)^m[X, Y] =

m

X

p=0

m p

[ad^m−p_jr

0 X,(ad_jr₀−1

2)^pY]. (1.10) Now, using the same argument as above, the choice m =k1+k2 yields the result ad^m_jr₀[X, Y] = 0, ∀X ∈b⁰_k

1, Y ∈V_k₂.

We now want to prove that z ⊕V ⊕jz is a j-subalgebra, before proving that it is an elementary normal j-algebra. We already know that [z, V] = 0 and [jz, V]⊆V. We now show that [V, V]⊆z.

Consider the Jordan decomposition ofbas direct sum of generalized eigenspaces for the action of the operator ad_jr₀.:

b=b_[0]⊕b_[1

2]⊕b_[1], where

• b_[0]=jz⊕b⁰,

• b_[1

2]=V, and

• b_[1]=z.

(30)

Now observe that (ad_jr₀−1) acts as follows on a bracket [v, w]∈[V, V] : (ad_jr₀−1)[v, w] = [adj_r₀v, w] + [v, ad_jr₀w]−[v, w]

= [adj_r₀v, w] + [v, ad_jr₀w]−1

2[v, w]−1 2[v, w]

= [(adjr0−1

2)v, w] + [v,(adjr0− 1 2)w].

Using the same argument as above, this shows that, if there existk1

and k₂ such that (ad_jr₀−¹₂)^k¹v= (ad_jr₀− ¹₂)^k²w= 0,then there exists m such that (ad_jr₀ −1)^m[v, w] = 0. But b_[1] = z is one dimensional, thus proving that [v, w]∈z for all v, w∈V. We have now shown that z⊕V⊕jz is a subalgebra. Using thatad_jr₀◦j=j◦ad_jr₀ onU ⊃V, we see that if (ad_jr₀−¹₂)^mv= 0 for somev∈V then (ad_jr₀−¹₂)^mjv= 0 as well, thus proving that jV ⊆V. Therefore,z⊕V ⊕jz is a j-subalgebra.

Remark 1.1.24. The operator adjr0 is semi-simple on U.

Proof. We first show that the operator adjr0 is semi simple on b⁰. For two elements X, Y ∈ b⁰, one has that ω([jr₀,[X, Y]]) = 0 because b⁰ is a subalgebra, b⁰ ⊂ U and U := (z)^⊥ for the scalar product h·,·i of Equation 1.5. But, by the Jacobi identity,

0 =ω([jr0,[X, Y]]) =ω([[jr0, X], Y]) +ω([X,[jr0, Y]]).

Now, since j:b⁰ →b⁰ is a vector space isomorphism, one has ω([[jr0, X], Y]) +ω([X,[jr0, Y]]) = 0, ∀X, Y ∈b⁰

⇔ω([[jr0, jX], Y]) +ω([jX,[jr0, Y]]) = 0, ∀X, Y ∈b⁰

⇔ω([j[jr0, X], Y]) +ω([jX,[jr0, Y]]) = 0 by Eq. 1.7.

This shows that adjr0 is skew-Hermitian on b⁰ for the scalar product h·,·i, and all skew-Hermitian operators are semisimple. This shows that ad_jr₀ is actually diagonalizable on b⁰, and therefore that ad_jr₀(X) = [jr0, X] = 0, for all X ∈ b⁰.This fact, along with the fact [r0, U] = 0, proves the third claim of Lemma 1.1.16.

Now, for two elements v, w∈V, we have shown that [v, w]∈z.We now show that the operator adjr0 −¹₂ is semisimple onV. One has:

ad_jr₀− 1

2 is skew-Hermitian onV

⇔ω([j([jr0, v]−1

2v), w]) +ω([jv,[jr0, w]−1

2w]) = 0, ∀v, w∈V

⇔ω([[jr0, jv], w]) +ω([jv,[jr0, w]])−1

2ω([jv, w])−1

2ω([jv, w]) = 0

⇔ω([jr₀,[jv, w]]) =ω([jv, w]), ∀v, w∈V

(31)

But we have shown that [V, V] ⊂ z = b₍₁₎. Therefore, [jr₀,[jv, w]] = [jv, w]; this ends the proof.

This proves that [jr₀, v] = ¹₂v, ∀v∈V.Therefore,z⊕V ⊕jzsatisfies property 2. of Definition 1.1.14.

Since [v, w] =αr0 for some α ∈R, the third property of Definition 1.1.14 follows from ω([v, w]) = ω(αr₀) = α, where the last equality comes from the fact that r₀ has length one for the scalar product h·,·i.

Therefore, [v, w] = ω([v, w])r0 =Im(hv, wi)r₀. This ends the proof of Lemma 1.1.16.

1.1.4 Coordinates on elementary normal j-groups S Let s be an elementary normal j-algebra as given in Definition 1.1.14.

An elementary normal j-algebra is a one dimensional extension of a Heisenberg algebra g := V ⊕Rr₀, where V is a 2n-dimensional real vector space endowed with a constant symplectic form ω0 and the only non-zero brackets are given by

[v, w] =ω₀(v, w)r₀.

The one-dimensional extension by Rr⁰₀is given by the brackets

• [r₀⁰, r₀] =r₀, and

• [r₀⁰, v] = ¹₂v, ∀v∈V,

and the complex structurejis chosen so thatr⁰₀=jr₀and is compatible with the conditions of Definition 1.1.10.

Letsbe this elementary normal j-algebra andφthe algebra isomorphism given by

φ:s→s:αjr₀+v+βr₀ 7→ α

2jr₀+v+βr₀. (1.11) The model for an elementary normal j-algebra will be (s,[·,·]^φ), where [·,·]^φ is the bracket of Definition 1.1.14 transported by φ. To avoid confusion, we will use the following notations :

• φ(jr₀) =H, and

• φ(r0) =E.

Remark 1.1.25. The transported bracket [·,·]^φ on the vector space s= RH⊕V ⊕RE is given by

• [H, E] = 2E,

(32)

• [H, v] =v, and

• [v, w] =ω0(v, w)E, where v, w∈V.

OnS, the connected and simply connected Lie group with Lie algebra s, one choses the following coordinates as in [13]:

φ:s→S:aH +v+lE 7→exp(aH) exp(v) exp(lE). (1.12) The elementφ(aH+v+lE) will be denoted by (a, v, l).

LetSbe an elementary normal j-group, with Lie algebrasgiven as in Remark 1.1.25, and coordinate chart defined by Equation 1.12. Within this chart, one has an identification S' R×R^2d×R and the product law is given by

(a, v, l)(a⁰, v⁰, l⁰) = a+a⁰, e^−a⁰v+v⁰, e^−2a⁰l+l⁰+1

2e^−a⁰ω(v, v⁰)

, (1.13) whereω is a given constant symplectic form on the vector spaceR^2d. In this chart, the inverse of the element (a, v, l) is given by

(a, v, l)⁻¹ = (−a,−e^av,−e^2al).

Given a vector field Y on the manifold underlying the Lie group S, one denotes by φ^Y_t (g) the integral curve at time t of the vector field Y starting atg fort= 0.

Since, for any X ∈ s all the eigenvalues of the endomorphism s 3 Y 7→ [X, Y] are real, the Lie group S is exponential (cf. for example [66]), i.e. the exponential map

exp :s:→S:X7→φ^X₁^˜(g) := exp(X)

is a global diffeomorphism. Here ˜X is the left invariant vector field on S associated to the elementX ∈s:

X˜_g := [gexp(tX)]. (1.14) The exponential map satisfies the one-parameter subgroup property

exp(sX) exp(tX) = exp (t+s)X ,

and it will now be used to compute the exponential map explicitly. Let X =αH+v+βE be an element ofs, and denote by a(t), v(t), l(t)

the

(33)

image exp(tX) ∈ S. Multiplying a(t), v(t), l(t)

with a(s), v(s), l(s) and differentiating at s= 0 yields the following expression:







˙

a(t) =α,

˙

v(t) =−αv(t) +y,

l(t)˙ =−2αl(t) +¹₂ω(v(t), y) +β,

where we imposed the conditions ˙a(0) = α, ˙v(0) = y ∈ R^2d and l(0) =˙ β. The solution to this Cauchy system with initial condition (a(0), v(0), l(0)) = 0∈R×R^2d×Ris given by







a(t) =αt

v(t) = _α^y 1−e^−αt and l(t) = _2α^β 1−e^−2αt

,

(1.15)

with adequate limit when α= 0.

1.2 Symplectic symmetric space structure on coadjoint orbits

1.2.1 Symplectic symmetric spaces

The reference we use for definitions and results about symplectic symmetric spaces is [8].

Definition 1.2.1. A symplectic symmetric space is a connected symplectic manifold (M, ω) endowed with smooth map s : M ×M → M : (x, y)7→s(x, y) :=sx(y) such that

1. • for all x ∈ M, sx : M → M is involutive and admits x as isolated fixed point, and

• s_x◦s_y◦s_x=s_s_x_(y), for all x, y∈M;

2. the symplectic form is invariant under the symmetriessx: s^∗_xω=ω,

for all x∈M.

Symplectic symmetric spaces give rise to the following data (cf. Ref- erences [46], [8]).