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Spinnler, F. (2015). Star-exponential of normal j-groups and adapted Fourier transform (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Mathématiques, Bruxelles.
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Star-exponential of normal j-groups and
transform
adapted Fourier
F lorian S pinnler A
vril2015
Thèse présentée en vue de l'obtention du grade de docteur en sciences
Faculté des sciences Université catholique de Louvain
Faculté des sciences
Université libre de Bruxelles
Star-exponential of normal j-groups and adapted Fourier transform
Florian Spinnler
April 2015
3
Aknowledgements
I would like to express my deepest gratitude to my advisors, Mélanie Bertelson (ULB) and Pierre Bieliavsky (UCLouvain), for supervising this thesis and answering my many mathematical and less mathemat- ical questions. I am indebted to Pr. Bieliavsky for his patience and unconditional willingness to introduce me to his mathematical universe.
Thank you to ail 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 privilège to share an office with Stéphane Korvers, and thank him for ail the good laughs (and serions 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. Spécial 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 hâve 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 hâve been completed without the financial sup
port of the F.R.I.A. (Belgium), of which I had the great privilège 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.
Contents
0.1 Notations... 7
0.2 Introduction... 9
1 Quantization of HBD’s 17 1.1 Pyatetskii-Shapiro theory... 17
1.1.1 Homogeneous bounded domains ... 17
1.1.2 Theorem of Pyatetskii-Shapiro on HBDs...18
1.1.3 Structure resuit for normal j-algebras ... 23
1.1.4 Coordinates on elementary normal j-groups S ... 31
1.2 Symplectic symmetric space structure... 33
1.2.1 Symplectic symmetric spaces... 33
1.2.2 Coadjoint orbits ...34
1.2.3 Symplectic symmetric space structure... 39
1.2.4 Explicit realization as coadjoint orbit... 41
1.3 Quantization of normal j-groups ... 46
1.3.1 Kirillov’s method of orbits... 46
1.3.2 Explicit results for elementary normal j-groups . . 50
1.3.3 Quantization of symmetries for the polarized sym plectic symmetric space structure of S... 56
1.3.4 Weyl quantization map for compactly supported fonctions on S ... 57
1.3.5 Normal j-groups ... 59
1.3.6 Coadjoint orbits ...60
1.3.7 Quantization of normal j-groups ... 62
2 Star-exponential of normal j-groups 67 2.1 BG and Moyal products on normal j-groups... 67
2.1.1 BG product and symbol map... 67
2.1.2 Formai star-products and star-representations ... 75
2.2 Two approaches to the computation... 78
2.2.1 The star-exponential as a distributional trace ... 78
2.2.2 The star-exponential as a star-representation ... 89
5
2.2.3 BCH property ... 92
2.3 Tempered structures... 92
2.3.1 Fréchet vector spaces and algebras... 92
2.3.2 Fréchet algebras ...95
2.3.3 Non-formal star products on normal j-groups and UDF for actions of normal j-groups...97
2.3.4 Weights on Lie groups... 97
2.3.5 Admissible tempered pairs on elementary normal j-groups... 98
2.3.6 Schwartz space for tempered pairs... 105
2.4 Schwartz spaces and multiplier property... 111
2.4.1 Four isomorphic Schwartz spaces...111
2.4.2 The multiplier algebra... 118
3 Adapted Fourier transform 125 3.1 Unitary dual of normal j-groups... 125
3.2 The modified star-exponential... 126
3.3 Properties of the AFT...132
3.4 Non commutative Fourier transform... 133
4 OverView and future work 139 4.1 Homogeneous Kàhler manifolds... 139
4.2 Tempered structures and Schwartz space... 140
Tempered structures and Schwartz space... 140
4.3 Other Hamiltonian spaces... 140
4.4 Représentation of normal j-groups ... 141
4.5 Non commutative Fourier transform... 141
4.6 Weyl algebra... 143
0.1. NOTATIONS 7
0.1 Notations
The following notations will be commonly used throughout this text:
• 0, É : Lie algebras of the Lie groups G, K
• 0*: real vector space dual of 0
• U{
q)\ universal envelopping algebra of 0
• Ad, Coad: adjoint and coadjoint actions of G on 0,0* respectively
• O^: orbit of ^ 6 0* under the coadjoint action of G
• X^: fundamental vector field corresponding to X € 0 for the action of a Lie group G on a smooth manifold M
• S: an elementary normal j-group of arbitrary dimension 2d + 2
• <S(S): the (one of the 3 isomorphic ones) Schwartz space on S
• T>{X): smooth, compactly supported functions on the manifold X.
• the non formai star-product on functions on S.
• -k^-. the formai Moyal product on functions on
• the non formai Weyl product on Schwartz functions on
• TM\ the tangent bundle of the manifold M
• [7(t)]: an element of ^7(0) I with 7 a path in M.
• /*j; : TxM Tf(^^)N: the differential of the map / : M —>■ X at a point X € M.
• {V,cü): the vector space V endowed with the constant symplectic form U
• H, B: the upper half plane and open unit disk in C
• xp : G ^ Q*: a tempered structure on G.
• 1-L: a Hilbert space, which will be specified
• C{'H): algebra of bounded operators on V.
• Û{'H): trace-class and Hilbert-Schmidt operators on H
respectively.
0.2. INTRODUCTION 9
0.2 Introduction
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 Fréchet algebra A
q(along with a countable family | • of semi- norms defining the topology of .4o) and a strongly continuons action a : X Ao A
qwhich is assumed to be isométrie in each seminorm, Rieffel in [74] developped a way to construct new Fréchet algebras, called deformation quantizations of the original algebra A
q. This procedure thus yielded a strict analog of the deformation quantization program introduced in Reference [6]. A central element in this deformation pro
cedure is the intégral expression (/*° g){x) = / f{y)g{z)dydz for the Weyl product of Schwartz functions on where 5(x, y, z) 4(u;(x, y) + oj{y, z) -|- x)) is four times the symplectic area of the tri
angle ^x,y,z (for the canonical symplectic form eu® on R^-^). Consider the action a and, for two smooth vectors a and b oî a, define the functions a(a) := [R^^ —>■ A
q: x ax(a)| and a(b) := [R^-^ ^ .4o : æ i->- cxx{b)].
Using .4o-valued oscillatory intégrais, Rieffel gives a meaning to the ex
pression (a(a) a{b)){x) := f ay{a)az{b)dydz. The formula a b := (o:(a) Q;(^))(0) then defines an associative product on A
q, or at least does so on the dense subspace of smooth vectors of a. Such a formula was called a universal deformation formula. The only ingré
dients needed are an isométrie action of R^^ on a Fréchet 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 of R^-^, was a non formai associa
tive product on some appropriate subspace of analogous to *o on R^^. Results in this respect hâve been building up since the early 2000’s with Works by Bieliavsky k. Massar in [16] (2001), Bieliavsky k Maeda in [15] (2002). A universal deformation formula for tempered actions (Définition 2.3.8) of Lie groups of a certain type on Fréchet alge
bras is provided in [13]. It relies on an intégral formula that is shown to produce an associative, non commutative product on functions on cer
tain Lie groups of a spécial type, namely on the Iwasawa factor AN of the Iwasawa décomposition 517(1, n) = KAN (cf. [46] for general défi
nition and [50] for the explicit construction). The connected component of the identity of the ax + b group is a spécial case of this construction.
In this work [13] by Bieliavsky k Gayral (2014), actions of Kàhlerian
Lie groups are considered. These groups can ail be obtained by taking
semi-direct products of the Iwasawa factors AN of SU{l,n). A group containing only one AN component will be called elementary normal j-group in what follows and denoted by S.
For example, the product *0 of [13] and [15] on the ax + b group reads / e“ e“/ \
as follows in the coordinates (a, /)
h->- ( _
q1 € Sl2{^) — SU{1,1):
{fl
*9f
2){a,l) 1
L R2
xR2 Ae e fi{ai,h)h{a
2, h)daidhdaidl
2, ( 1 ) where S = — sinh(2(ai — «2))^ — sinh(2(a2 ~ o-))h ~ sinh(2(a — oi))/2 and A = 4y^cosh(2(ai — 02)) cosh(2(oi — a)) cosh(2(a — 02))-
The importance of the phase function S (the analogous of the sym- plectic area for R^^), already apparent in the work of Weinstein in [81]
(1994), is also made explicit in [10] (2000), where it is shown how a sym- plectic symmetric space structure can be added on the Lie groups under considération. The phase function is then identified with the symplectic area of géodésie triangles of the symmetric space.
In [16] and [11], a canonical symplectic symmetric structure is con- structed on S, that is in some sense a contraction of the Hermitian symmetric space structure on SU{l,n)/U{n); this canonical symmetric structure is invariant under a group G
qof symplectic automorphisms, which is detailed in Section 1.2.4. Moreover, the group G
qhas a sub- group isomorphic to S. In this text, we will explicitly compute the coad
joint orbits of G, the central extension of Go, and show that particular orbits of G are acted upon simply transitively by the subgroup of G iso
morphic to S. The Kirillov-Kostant-Souriau symplectic form on an orbit O can be transported to the group S, which then becomes a symplectic group in the sense of Médina 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 A : S —)• s*. The coadjoint orbits considered in this text admit a real polarization, i.e. an invariant Lagrangian foliation (under G) denoted by £. A family of Weyl-type quantizers is then pro- vided on the symmetric space S ~ G, parametrized by functions m on Q ~ G/£. A spécial choice of mo then yields a quantization map for S associating compactly supported functions on S to trace class operators on a Hilbert space, and the composition of symbols of this quantization map in turn provides the product of [13].
The BG product *g, of which Equation 1 is a particular case, admits
a formai development in 9 denoted by , and this star-product is shown
0.2. INTRODUCTION 11
to be Q-covariant ([42]):
Ax Ay - Xy Ax = TA[x,y]- (2)
In other words, the map g End{C°°{0),*^) : X [u i-)- ^Ax w]
is a Lie algebra représentation, and the question arises whether one can compute and find explicit expressions for the corresponding Lie group représentation. 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 représentation program, i.e. the realization of Lie group représentations 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 décades. In the references [9], [17] and [18] (1999-2004) for instance, the principal and discrète sériés représentations of Sl2{^) arise as properties of the Moyal product on the Poincaré disk or one-sheeted hyperboloid, which are both coadjoint orbits of 5/2 (IK)-
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) on O. For X € g and denoting by the functions on g* given by [^ i->- ^(X)], it is shown that the Weyl Moyal star product in these canonical Darboux coordinates
satisfîes iX^^ = i[X,Y]^^ (where i such
that = —1), thus yielding a représentation X ^ Ix oî the Lie algebra g, with lx{f) := iX^^ / for / G C^{0).
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 be the unitary irreducible représentation 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 représentation can be intertwined by partial Fourier transform operators such that J~P O Ix O = dp^{X), where the differential dp^ is defined by dp^{X) \= ^p(exp(fX))|t=o-
This thesis provides explicit expressions for the group représenta
tions arising from the covariance property of the moment map under
the product These représentations, 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
a modifiée! version of the star-exponential, is then used to define, for each coadjoint orbit O, an adapted Fourier transform FŸ on the group.
A functional transform close to this adapted Fourier transform is often called group Fourier transform or non commutative Fourier trans
form 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 : L‘^{G) —>■
L^(g*,dm(^)), by patching together the expressions of J’P, where O runs over ail coadjoint orbits of the normal j-group G. Here dm{^) 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 définitions and results about homogeneous complex domains.
The theorem of Gyndikin, Pyatetskii-Shapiro, Vinberg, (cf. [43](1964), [69](1969)) establishing the classification of homogeneous bounded do
mains is quoted and illustrated. The classification relies on a one-to-one correspondance between homogeneous bounded domains and so called normal j-groups, which arise as subgroups of the symmetry group of ho
mogeneous Kahler manifolds (cf. [44] (1967)). It turns out that normal j-groups hâve a very constrained structure, namely they can ail be ob
tained via semi-direct products of so-called elementary normal j-groups, which in the classification of Pyatetskii-Shapiro arise as transitive sub
groups of the automorphism groups of unit balls in C".
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 (5 is a Lie group having S as a subgroup acting simply transitively on G/K. A unitary irreducible représenta
tion 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 C (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
tt; G/K —)■ O/C, the symmetry at the base point
s^
kcan then be lifted on functions on the leaf space Q ~ O/C. For a fonction g) on Q and a point q Ç. Q, one defines E*if(q) := (p(n(seK(s(q)))). As in [13] and [14], the Weyl-type quan- tizer is then defined on G/K by fî(^) := U(g) o E* o U{g)~^, where gK = ^ € G/K and C/ is a unitary irreducible représentation of G.
The Weyl-type quantizer is then transported to S via the simply tran-
0.2. INTRODUCTION 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 R, : T>{S) —)■ is defined on elementary normal j-group through the assignment / s-> Jg f{g)0.{g)d^g and then extended to normal j-groups through a method introduced in [14](2014) and de- tailed here.
Chapter two introduces the pseudo differential calculus and universal deformation formula of [13]. The intertwining operator T
qbetween the Weyl product *o and the BG product -kQ of [13] is explicitly given, as well as the formai Moyal star-product, which can be obtained as power sériés expansion of *o. Covariant star products are then introduced and illustrated with the example of the three dimensional Heisenberg group i/3. On coadjoint orbits of this group, the quantization method of chapter one yields precisely the Weyl product and its formai counterpart the Moyal product, which are shown to be covariant for the moment maps.
For an elementary normal j-group S, the Moyal product is then shown to be covariant for the moment map of the action S x Cl —> O. The star- exponential : G x O on S can then be computed in two ways. The fîrst is by computing the distributional trace
5,(0 = Tv{U{g)m)- (3)
The second is by explicitly solving the differential équation
ôt^t(exp(X))(0 = {-^\x n Et{eMX))){0 (4) for the function : Sx G —> C, with initial condition E\ = l. The group multiplier property is then shown to be satisfied: for every g, g' € G,
£g £g' = 5,,/
holds. In this text this second way of explicitely finding the star- exponential plays a minor rôle. 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 of tempered structure on a Lie group, introduced in [13],
is then given and illustrated. It is a global diffeomorphism ij) : G ^
0* ~ such that the multiplication law and inversion law on the
Lie group are tempered functions, i.e. can be bounded by polynomials
in the coordinates given by the chart ip. A function 5 : S —>■ M can give lise to a tempered structure, by defining the map
, G-, Q* : g ^ [X ^ dS{X){g)]. (5) Given a tempered structure '0, a simple criterion is provided for the existence of a function S such that (called the primitive of tp).
For a tempered structure a Schwartz space S^{S) can be defined on S. Another Schwartz space 5‘^(S) can be defined when ip has 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 ^ on S x §, and the function Scan underlying the BG product *0, and the restricting the corresponding Schwartz space on S x S to a Schwartz space of one variable »S^(S) on S.
Yet another Schwartz space 5 2 (S) is used in [14]. It is proven (albeit in the two dimensional case, essentially to simplify notations) that ail these Schwartz spaces are identical and ail isomorphic as Fréchet vector spaces to the canonical Schwartz space and so also isomorphic to the Schwartz space on solvable Lie groups introduced in [26] (2010); the proof makes use of the tameness of the pair (S x S, Scan)-
The left action of S on the Fréchet algebra (5(S), •) is tempered, and this fact allows one to use the deformation formula of [13] to produce an algebra (<S(S),*0) deforming (5(S),-). The star-exponential is then shown to belong to the multiplier space of (<S(S),*0).
In the third chapter, two adapted Fourier transforma for normal j- groups are suggested, yielding results analogous to those given in [38]
for the case of the ax + h group. First, by using a slightly altered version of the star-exponential, defined by
£f\g') := Tx{Uc{g)K"meM))^ (6) where K is the formai dimension operator of the représentation 11^, introduced in [34]. This modified star-exponential was introduced in [38] (2008). Defining the first Fourier transform by
Jf' : 5(S) ^ : / ^ b' ^ jj{g)S°\g')dg], (7) where we use the identification g' i->- Coad^/ eE*, the following properties are proven: the operator intertwines the convolution product on the group and the BG product *0 on the orbit O^. An inversion formula also holds, which is given by
f{g) = ^ / J^):FŸ^Î{9')dp{g'). (8)
0.2. INTRODUCTION 15
A Plancherel formula is then shown to hold as well;
[ \f{9)\‘^d9= f \J"?‘{f){9')\‘^dfi{g'), (9)
7s JO‘
thus realizing explicitly the Plancherel measure on G as described in e.g.
[35]. The general results are then provided for normal j-groups.
A simple rewriting of the Fourier transform yields a map T2 : L'^{G) —> L^(g*,dm(^)), where dm(^) is the measure on g* 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 The formulas are de- tailed 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 ex-
ponential 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 ail 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.
Chapter 1
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 auto- morphisms 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 canon- ical 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 com
plex domains.
1.1.1 Homogeneous bounded domains An open connected subset of C-^ is called a domain.
Définition 1.1.1. A domain A of the affine space is said to be a homogeneous bounded domain (or HBD for short) if it is bounded in and if the group Aut{A) of biholomorphic automorphisms of A acts
17
transitively on A, where holomorphic refers to the canonical complex structure on A C C^.
Définition 1.1.2. A domain A is said to be a symmetric domain if for every x G A there exists G Aut{A) such that
1. al = idA, 2. ax{x) = X, and
S. (ax)irx • -^A ^ T
xA ! Vx • ^ Vx'
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 ofC:
D := {2; G C such that \z\ < 1}.
The complex structure is given by multiplication by i in the tangent space at each point; using the Schwarz lemma (cf. [51]), it is shown that the group of biholomorphic automorphisms o/B is the set of transformations
with a G B and ^ G M. This group is isomorphic to 517(1,1) through the identification of the pair {6, à) with the element of SU(1,1) given by the product of
1 f ei 0 \ f 1 -a \
,/r^ V 0 e=f )[â 1 J
1.1.2 Theorem of Pyatetskii-Shapiro on HBDs
In the case of the homogeneous bounded domain B given in the previous section (the unit disk of C) it is well know that there exists a biholomor
phic map called the Cayley transform mapping the upper half-plane Hl to the unit disk, which is explicitly given by
IHI —>• B : 2:
Z +1
where BI := {2 G C such that '^{z) > 0}, with Cl(2) the imaginary part
of the complex number 2 G C.
1.1. PYATETSKII-SHAPIRO THEORY 19
A convex cône of R” is a subset V of R” such that ax + /3ÿ € V, Vx, y EV, a > 0 and /3 > 0.
A generalization of the upper half-plane to the case of (unbounded) domains of C-^ is given in the following définition.
Définition 1.1.4. (cf. Reference [37]) Let V be a convex cône in R"
not containing any straight Unes. A F-hermitian form is a map F :
£m ^ £171 £71
1. F{\x + py, w) = XF{x, w)+jj.F{y, w) \f x,y,w E and X,pEC, 2. F{u,v) = F{v,u),
S. F{u,u) E V, the closure ofV, and 4. F{u, u) = 0 if and only if u = 0.
Définition 1.1.5. A Siegel domain of type II associated to the cône V and the V-hermitian form F is the domain in consisting of points (z, n) € C" X such that
Im{z) — F{u, u) E V. (1-1)
This domain will be denoted by D{V,F).
To illustrate this, take the case m = 0 and choose the cône V Rg in R. When the form F is trivial then the Siegel domain D{V, F) is the upper half plane IHI = {2 € C|/m(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 structure h = g+iuj. The following conditions are équivalent:
1. du) = 0,
2. X/J = 0, where V is the Levi-Civita connexion induced by the Riemannian metric g on M,
3. in local real coordinates {z^, • • • , 2”, 2^, • • • , the coefficients h^-^
of h take the form h^-^ = for some positive function ip.
Définition 1.1.7. A Kahler manifold is a complex manifold (M,J) endowed with a positive definite Hermitian product h = g + iui satisfying either one of the équivalent conditions of Theorem 1.1.6.
A map : M ^ M \s said to be an automorphism of the Kahler manifold if
• is holomorphic, invertible with holomorphic inverse, and
• ^*/i = h.
The Kahler manifold is said to be homogeneous if its automorphism group acts transitively.
Consider now a quadruple {9,i,j,p), where g is a real Lie algebra, t is a Lie subalgebra of g, j : g —> g is a linear map such that
fX + X ۔
for ail A" 6 g, and p G g* A g*, where g* is the real dual vector space of 0'
Définition 1.1.8. The data (g,t,j, p) is said to be a Kahler algebra if 1- ji Ç É,
â. [K,jX]-j[K,X]€t,
3. \JXJY]-j\jX,Y]-j[X,jY] - [A, y] € t, 4. p{K,X)=0forKGÎ,XeQ,
5. p{jX,jY) = piX,Y),
6. p{jX, A) > 0 for X ^ î, and
7. p([A,F],Z) + p([y,Z],A) + p([Z, A],y) = 0, for ail X,Y,Z € g,K G î.
Remark 1.1.9. Every homogeneous Kahler manifold M détermines a
Kahler algebra. If we dénoté by G the transitive group of automorphisms
of M and by K the stabilizer subgroup of a fixed point
xqG M, then
M ~ G/K as homogeneous Kahler manifolds. Let us dénoté the Lie
algebras of G and K by g and ê respectively. Using the isomorphism
Q/i ~ T
xqM yields an operator j := K
xq■ g/£ —>■ g/É ihat can be lifted
to g into an operator satisfying the conditions of Définition 1.1.8 above.
1.1. PYATETSKII-SHAPIRO THEORY 21
Conversely, every Kàhler algebra détermines at least one homogeneous Kàhler manifold (cf. Reference [SS]). One is the homogeneous space G/K where G and K are connected Lie groups having g and f as Lie algebras. The complex structure at a point Jgx of the quotient G/K is then defined using left translations from the tangent space j : T
skG/K ->
T
xqg/K as follows:
JgK '■ TgxG/K ->■ TgliG/K : XgK 1-^ Lgi,eK ° j ° L g-^*gK^gK- For X G 0, consider the operator adx defined by
adx :0^0:F^[X,y].
A particular and important class of Kàhler Lie algebra consists of so called normal j-algebras. In the définition of Kàhler Lie algebras, they correspond to the case where î = 0 and adx has only real eigenvalues
vag
0,
Définition 1.1.10. A normal j-algebra (cf. [69]) is a real solvable Lie algebra b such that
1. the operator adx has only real eigenvalues for any X G b,
2. there exists an endomorphism j : b ^ b such that = —Idb and [X, Y] + j \jX, Y] + j [X, jY] -\jX,jY] = 0\fX,Yeb, S. there exists a linear form w : b ^ M such that
Lü{\jX,X])>0 ifX^OandLü{\jX,jY])=u{[X,Y])VX,Yeb.
A subalgebra b' ofb such that j b' C b' and b' is a normal j-algebra, will be said to be a normal j-subalgebra of b.
A Kàhler manifold will be said to be normal Kàhler manifold if it admits a transitive group of automorphisms G such that the Lie algebra of G is 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 C” is biholomorphic with a homogeneous Siegel domain of type II (cf.
Définition 1.1.5).
Moreover, the Siegel domains of type II hâve the following important
property:
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-group G, 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 décomposition of the Lie algebra g of G as detailed in the proof of Lemma 1.1.16.
Example 1.1.13. Consider the Poincaré disk D. It is biholomorphic via the Cayley map to the upper half-plane H. Consider the following matrix group
§ := {M{a,l) := e“ e“Z
0 e-“ a^l Ç. M}.
Then, the diffeomorphism
</) : S H : M(a, l) ^ + 6^“/, shows the existence of a simply transitive action
S X m —^ O.
It turns ont that normal j-algebras can be obtained through succes
sive semi-direct products of so-called elementary normal j-algebras.
Définition 1.1.14. Consider a complex hermitian vector space (0,j) with scalar product h{Xi,X2) and an élément ro of length 1. Dénoté by rg the space {F G g such that h{ro,Y) = 0}. The space (Q,j,h) is said to be an elementary normal j-algebra if it is endowed with a Lie algebra structure such that
1- r-o] = ro,
2. \jrQ,Y] = \Y, VY
gand
3. [F, F'] = (7m(h(F, F')))ro, VF , F' G
where Im{z) dénotés the imaginary part of z
eC.
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 bail by Theorem 1.1.11 and Proposition 1.1.12
above.
1.1. PYATETSKII-SHAPIRO THEORY 23
1.1.3 Structure resuit for normal j-algebras
Normal j-algebras as described in Définition 1.1.10 hâve a very con- strained structure, given by the following Lemma.
Lemma 1.1.16. Let (b,j) be a normal j-algebra and let j be a one- dimensional idéal of b. Then the Lie algebra b can be decomposed as follows :
b = 3 © y © ja © b', where
1. b' is a normal j-subalgebra ofb,
2. ©J3 is an elementary normal j-algebra as given in Définition 1.1.U,
3- [3 © ja, b'] = 0, and 4. [b',V]ÇV.
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 orig
inal formulation of the Lemma and its proof can be found in [69].
We start by rewriting basic définitions and results, and then proceed with the proof. The main points are
1. the existence of a one-dimensional idéal in every normal j-algebra and
2. the explicit construction of an elementary normal j-algebra from the data of this one-dimensional idéal.
For the sake of completeness, we first introduce some standard ter- minology and two useful Lemmas.
Définition 1.1.17. Let
qbe a Lie algebra. One recursively defines the ideals by
• g^°^ := g, and
The Lie algebra g is said to be solvable if there exists k € N such that g^'') = 0.
We now list two Lemmas that will be used in the proof of 1.1.16.
Lemma 1.1.18. Let g be a Lie algebra and a an idéal 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 définitions and observe that for any surjective Lie algebra morphism
tt: gi -> g2, one bas
>r(0f>) = (02)<‘>. □
Lemma 1.1.19. Every solvable Lie algebra g has a codimension 1 idéal gi. Since g is solvable, gi is solvable by Lemma 1.1.18.
Proof. Let n dénoté the dimension of g. Since g is solvable, the derived idéal is a proper idéal, g' [g, g] C g. Any vector subspace b of g of dimension n — 1, containing g', is then a codimension one idéal since
[fl, b] Ç [g, g] Ç g' Ç b. □
Remark 1.1.20. Let g be a solvable Lie algebra and g\ be a codimension 1 idéal. Then [g,g[*^] Ç g^^^Vfc S N.
Proof. By induction, using the Jacobi identity. □ Lemma 1.1.21. Let g be a solvable algebra such that adx has only real
eigenvalues. Then g has a one-dimensional idéal in Rg, the last non zéro idéal in the sequence g D g^^^ D • • • D i?g D 0.
Proof. We prove the following : there exists a linear form A : g —>• R and a vector ro € Rg such that
[A,ro] = A(A:)ro
for every X 6 g. We proceed by induction on the dimension of g. If g has dimension 1, it is itself a one-dimensional idéal. In dimension 2, g is either
• abelian and any one dimensional vector subspace of g is a one- dimensional idéal, or
• isomorphic to the Lie algebra [X, F] = Y, and in this case the subspace RF is a one-dimensional idéal; moreover, Rg = RX 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, g has a codimension 1 idéal, gi. By the induction hypothesis, there exist A : gi ^ R and ro € Rg^ such that [X, ro] = A(X)ro, VX G gi. In that case, the vector space
V\ := {v G iîgi such that [X,v\ — A(X)u, VX G gi}
1.1. PYATETSKII-SHAPIRO THEORY 25
is not trivial since it contains at least the vector subspace Mro. We first want to show that [g, V^] Ç V^. By définition this is équivalent to show that [X, [y, u]] = A(X)[y, v], for ail u € V
aï€ 0i and y € g.
By the Jacobi identity on g, one has
[x,[y,u]] = [[x,y],u] + [y,[x,u]]
= A([x,y]> + A(x)[y,u], where we used that gi is an idéal of g. We will prove that
A([Jf,y]) = OVX€gi,y eg, which will yield the following property:
[X, [y,u]] = A(x)[y,u]), for ail y € g and X £ gi, therefore showing that
[0,Vx]cyA. (1.3)
Lemma 1.1.22. For a fixed element v G iîgj, let W be the subspace generated by the (ady(u))j>o. Then
tr{adx\^) = XiX).dim{W), for ail X G gi-
Proof. Observe first that W is non trivial by the induction hypothesis since it contains at least Mro and finite dimensional since iîgj is a finite dimensional idéal. By induction, we now show that
adx{adi^{v)) = X{X)ad^y{v) + aéyiv), (1.4) i<j
for any j G N, j > 0. For j = 0 it is immédiate since adx{v) = X{X)v by définition of u; and for j = I one has the identity adxadyiv) — X{X)ady{v) + A([X, y])u, 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 fc = j + 1, one has
adxady~^{v) = adyadxadiy{v) + ad^x,Y]<^diy{v)
= adyi^X{X)adiy{v) + ady{v)) i<j
+ A([X, y])ad^(î;) + ady(v) by Eq. 1.4 i<j
= A(X)ad-^^(-y) + c, ady{v).
i<j+l
This shows that the matrices of ad\ are upper triangular in the basis {(ady(n))}j>0 and that the diagonal éléments are ail equal to X{X).
This proves the daim. □
Since gi is an idéal in g, [X, T] € 0i and we can apply Lemma 1.1.22.
The following holds:
X{[X,Y]) = tr{adixx]\w)
= tr{{adxadY)\yy - (adyodx)|^)
= 0 ,
by the symmetry and linearity of the trace. This observation, combined with Equation 1.2, yields the invariance [g. V
a] C V\.
Since gi is a codimension 1 idéal in g, one has g = gi © K5 for some element 5 in g that is not in gi. Now, since the subspace V\
is g-invariant, it is in particular invariant under S: [5, VÂ] Ç V^. By hypothesis, ads has only real eigenvalues, and therefore there is a vector
vq g
V\ and 0 € M such that [5, uo] = dvo- This vector satisfies [c5 + X, uo] = {c6 + A(X))uo for ah € gi and c € R, and in particular the one dimensional subspace R
uqis a one dimensional idéal of g, with an associated linear form given by 9S* + X where S* is the element of g*
such that S*(S) = 1 and S*{X) = 0 VX € gi.
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 dimen
sional idéal. Dénoté this idéal by 3, and set (cf. Definitionl.1.14)
b U,
where j is the endomorphism of axiom 2. in Définition 1.1.10 and U is the orthogonal complément of the vector subspace 3 of b for the hermitian scalar product
{x, y) := u{\jx, y]) -|- iuj{[x, y]). (1.5) Let ro be a generator of 3 normalized such that the following holds:
b'ro,ro] = ro (1.6)
Let U G U-, then, by définition of U, {ro, u) = 0. The vector space U is invariant under j:
(ro,ju) = -i{ro,u) = 0.
1.1. PYATETSKII-SHAPIRO THEORY 27
Since the endomorphism j is a vector space isomorphism, this shows that U EU 4^ ju E U.
Consider the element ro of 3 satisfying Equation 1.6. We hâve that uj{ro) = ti;([jro, ro]) > 0 by axiom 3. of normal j-algebras (Définition
1.1.10).
The subspace 3 + {/ is a subalgebra of b, because it is the centralizer of the element
tq. Indeed, looking at the imaginary part of Equation 1.5 and using orthogonality, for any u EU, one has that
0 = u;([ro, uj) by orthogonality
=
u;(
qro) for some a, since 3 is an idéal
= auj{ro).
On the other hand, u;(ro) > 0. Therefore, a = 0 and [3, U] — 0. This proves the daim that 3 + 17 is the centralizer of ro-
Consider the operator adjro- We will show that adjrgU Ç U. By the Jacobi identity and [jro, ro] =
vqone has
[ro, [jro, u]] = - [ro, u] + [jro, [ro, u]] = 0 since [ro, tt] = 0
Therefore, [jro, u] E } + U, the centralizer of ro. But [jro, u] E j} + U as well:
[jro, u] = [jro,ju'] for u' EU such that ju' = u, since jU = U
= [ro,u'] + j[jro,u'\ + ^[ro,^^'] by axiom 2. of j-algebras
= jb>o, u'] E jii + U) = J3 -F U,
where the last line follows from [3,17] = 0, the previous observation that [jro, u] E ^ + U and the invariance jU = U. Hence, jjro, ^t] ^ (ji Y U) n {} + U) = U. Since u = ju' for some u' E U, we hâve also proved that [jro,ju'[ = j\jro,u'\, or in other words that
adjrç, ° j\u — j O 0,djrQ \jj. (f - 7 ) We now show that the operator adjro\jj can only hâve 0 or 5 as eigenvalues. Let A
qbe an eigenvalue of adjr^ and let an element uo EU be the corresponding eigenvector. One has
[j’’o,iio] = Aouo, and
[jro, juo] = )^Qjuo, by Eq. 1.7.
Therefore,
b^'o, [jwo,ao]] = [\jro,juo],uo] + [jro, [juo,uo\]
— 2Ao[juo, î^o]-
( 1 . 8 )
Since 3 + is a subalgebra, \juo,
uq] = r + u for some r E j,u € U. But then,
2Ao(r + u) = 2Aob'wo,uo] = [jro, [juoi'Wo]]
= b'^o, r + u]
= r + [jro,u],
because brO)?'] = for ail r € 3. We know that broj^^] € U, and considering the 3 component of this last equality one finds (2Ao —l)r = 0.
This implies that either A
q= 5 or that r = 0. In the latter case, 2Ao uj{[juo, uo]) = w(b>o, [juo, uo]]) by Eq. 1.8
= c^([-ro,ib^o, wo]]) by axiom 3.
= —o;([ro, jw]) = 0 by orthogonality of U and 3 But again by axiom 3. of Définition 1.1.10, a;(buO)Wo]) > 0. Therefore, in the case r = 0, A
q= 0 must hold.
Define b' as follows :
b' := [X G U such that ad^^X = 0 for some m G N}, and define the subspace V oi U as follows :
V := {X G U such that {adjro ~ ~ ^ for some m G N}.
We will now prove that b' is a j-subalgebra, the first daim 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 éléments X,Y G b'. As can be proven by using an induction argument and the identity (^) + {^i) =
) the following holds:
m / s
ü<rp„ix, r] = E U (1.9)
p=0
then, assuming adj^^X = 0 and = 0 for some integers k\ and A:2, the choice m = fci + ^2 satisfies adip^^[X, E] = 0. Indeed, if
m — P = kl + k2 — P < kl,
preventing the vanishing of the left term ad^^^X in the bracket, then P > k2, implying the vanishing of the right term ad^ro^ in the bracket.
The argument is the same when ki and k2 are reversed.
1.1. PYATETSKII-SHAPIRO THEORY 29
To prove that [b', V] Ç V, the fourth daim of Lemma 1.1.16, we first introduce some notations. Consider the following subspaces
b'i^ := {X G bKadjroŸX = 0, V/ > k}, and Vk := {X G V\(adjro ~ = 0 V/ > k}.
One has the set equalities b' = Ujt>ob^ and V = Ufc>ol4, as well as the indusions 14 C I4+1 and bjj, C bji.^^
Let X and Y G Vfcj- Then, adjr^X G b^ and {adjro — ^)Y G Vk,-i.
Remark 1.1.23. The operator {adjro ~ è) follows on the bracket [X,Y]:
{adjro - = Kro^,î"] + [^, {adjro ~
(it is an instance of a so-called generalized dérivation, cf. Reference [52])
By this observation, we see that adjro^ > ^^2-1]- Actually we can show an analog of Equation 1.9 to obtain
{adj„ - ^riX,Y] = (“)l«<Ç."’’X,(ad,., - im. (1.10)
p=0
Now, using the same argument as above, the choice m — k\ + k2 yields the resuit adj).^[X,Y] = 0, VX e b'i^^,Y e Vk2-
We now want to prove that 3 © E © is a j-subalgebra, before proving that it is an elementary normal j-algebra. We already know that [3, E] = 0 and [73, V] Ç V. We now show that [F, V] Ç 3.
Consider the Jordan décomposition of b as direct sum of generalized eigenspaces for the action of the operator adjro-'-
b = b[o] © bjij © b[i].
where
• ^’lo] = n ©
• bjij = F, and
• b[i] =3.
Now observe that {adjrç, — 1) acts as follows on a bracket [u, ra] € [V, V] : {adjro — 1)[
î;, ta] = [adjrgV^w] + [v,adjrow] — [v,w]
= [adjroV,w] + [v^adjrow] - ]^[v,w] - ^[v,w]
= [(adjro - + [-i;, {adjro ~ \)w]-
Using the same argument as above, this shows that, if there exist k\
and /c2 such that (adj>o ~ =
{adjro~
—0 , then there exists m such that {adjro ~ ^)^[aiw\ = 0. But b[ij = 3 is one dimensional, thus proving that [u, ta] G 3 for ail v,w G V. We hâve now shown that 3 © F © J3 is a subalgebra. Using that adjro °j=j° adjro onU D V, we see that if {adjro ~ 5)”^^ = 0 some v
eV then {adjro ~ = 0 as well, thus proving that jV Ç V. Therefore, 3 © U © J3 is a j-subalgebra.
Remark 1.1.24. The operator adjro semi-simple on U.
Proof. We first show that the operator adjro is semi simple on b'. For two éléments X,Y E b', one has that üj{\jrQ,[X,Y\[) = 0 because b' is a subalgebra, b' C U and U := (3)-*" for the scalar product (■, •) of Equation 1.5. But, by the Jacobi identity,
0 = uj{\jro, [X, y]]) = u;([L7'ro, X], >^]) + b>o, i"]])- Now, since 7 : b' —> b' is a vector space isomorphism, one has
iü{[\jro,X],Y])-bu{[X,\jro,Y]]) = 0, VX,y G b'
^u{[\jro,jX],Y])-bu{\jX, \jro,Y]]) = 0, VX,y G b'
^ u;(bb>o, ^], 1^]) + i^{\jX, [jro, F]]) = 0 by Eq. 1.7.
This shows that adjro skew-Hermitian on b' for the scalar product (•, ■), and ail skew-Hermitian operators are semisimple. This shows that adjro actually diagonalizable on b', and therefore that adjro{X) = b'ro)-^] = 0) for ail X G b'. This fact, along with the fact [ro,{7] = 0, proves the third daim of Lemma 1.1.16.
Now, for two éléments v,w
eV, we hâve shown that [u,tü] G 3. We now show that the operator adjro ~ è semisimple on V. One has:
ad■jro - is skew-Hermitian on V 2
^aj{\j{[jro,v] - ^v),w]) -\-ai{[jv, [jro,'w] - ^u;]) = 0, Vu,u; G U
^{[[jro, jv],w]) u{\jv, b>o,HD - ^‘*^(b^>'^]) - = 0
<^(b^o, [jv, lü]]) = uj{\jv, w]), \/
v,
wEV
1.1. PYATETSKII-SHAPIRO THEORY 31
But we hâve shown that [V, 1^] C i — b(i). Therefore, \jro,\jv,w]] —
[jv, w]; this ends the proof. □
This proves that \jro, u] = ^v, Vu G V. Therefore, 3 0 F© satisfies property 2. of Définition 1.1.14.
Since [u, w] = aro for some a G R, the third property of Définition 1.1.14 follows from u;([u, ru]) = uj{aro) = a, where the last equality cornes from the fact that ro has length one for the scalar product (•, ■).
Therefore, [v,w] = a;([u, i(;])ro = Im{{v,w))ro. This ends the proof of
Lemma 1.1.16. □
1.1.4 Coordinates on elementary normal j-groups § Let s be an elementary normal j-algebra as given in Définition 1.1.14.
An elementary normal j-algebra is a one dimensional extension of a Heisenberg algebra g F © Rro, where F is a 2n-dimensional real vector space endowed with a constant symplectic form
uqand the only non-zero brackets are given by
[v,w] = iüo{v,w)ro.
The one-dimensional extension by Rr^is given by the brackets
• [^o> ^o] = ro, and
• [»'o> ^] = ^ S
and the complex structure j is chosen so that T
q= jro and is compatible with the conditions of Définition 1.1.10.
Let s be this elementary normal j-algebra and cj) the algebra isomor- phism given by
0 : s -> s : ajro + v + ^
vq i->- —jro + u + /3ro. ex (1-11)
The model for an elementary normal j-algebra will be (s, where [•,is the bracket of Définition 1.1.14 transported by (p. To avoid confusion, we will use the following notations :
• (t>{jro) = H, and
• 0(ro) - E.
Remark 1.1.25. The transported bracket [•, on the vector space s = Rif © y © M.E is given by
. [H,E] = 2E,
• [H, v] — V, and
• [v, w] =
u!
q{
v, w)E, where v,w Ç. V.
On S, the connected and simply connected Lie group with Lie algebra 5, one choses the following coordinates as in [13]:
çl) : s —>• S : aH + v + lE exp(aiL) exp(?;) exp(Z£^). (L12) The élément 4>{aH + f + lE) will be denoted by (a, v, l).
Let S be an elementary normal j-group, with Lie algebra s given as in Remark 1.1.25, and coordinate chart defined by Equation 1.12. Within this chart, one has an identification S ~ M x x M and the product law is given by
(a, V, v', l') = (a + a , e~“ v + v', l + l' + 2^~°‘ ^0))
where w is a given constant symplectic form on the vector space In this chart, the inverse of the element (a, v, l) is given by
= (—a, —e“v, —e^“Z).
Given a vector field Y on the manifold underlying the Lie group S, one dénotés by 4>J (g) the intégral curve at time t of the vector field Y starting at g ior t — 0.
Since, for any X € s ail the eigenvalues of the endomorphism s 3 Y H-)- [X, y] are real, the Lie group S is exponential (cf. for example [66]), i.e. the exponential map
exp : 5 S : X (f>f {g) ■= exp(X)
is a global diffeomorphism. Here X is the left invariant vector field on S associated to the element X € s:
Xg := [5exp(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 = aH + V + PE be an element ofs, and dénoté by [a{t),v{t),l{t)) the
1.2. SYMPLECTIC SYMMETRIC SPACE STRUCTURE 33
image ex.p{tX) G S. Multiplying [a{t),v{t),l{t)) with [a{s),v{s),l{s)) and differentiating at s = 0 yields the following expression:
(
à{t) - a,
v{t) = -av{t) + y,
i{t) = -2al{t) + \üj{v{t),y) +
where we imposed the conditions â(0) = a, 'û(O) = y € and Z(0) = 13. The solution to this Cauchy System with initial condition (û(0), u(0), Z(0)) = 0 € M X X R is given by
a{t) = at
< v{t)
with adéquate limit when a = 0.
1.2 Symplectic symmetric space structure on coadjoint orbits
1.2.1 Symplectic symmetric spaces
The reference we use for définitions and résulta about symplectic sym
metric spaces is [8].
Définition 1.2.1. A symplectic symmetric space is a connected sym
plectic manifold (M, w) endowed with smooth map s : M x M M : (x,y) s{x,y) := Sx(y) such that
1. • for ail X € M, Sx ■ M ^ M is involutive and admits x as isolated fixed point, and
• SxOSyOSx = Sg^(y), for ail x,y € M;
2. the symplectic form is invariant under the symmetries Sx-' S*xU} = U,
for ail X G M.
Symplectic symmetric spaces give rise to the following data (cf. Réf
érencés [46], [8]).
Définition 1.2.2. A symplectic involutive Lie algebra (or siLa for short), is a triple (g,a,uj) where g is a Lie algebra, uj G A^g* is a closed two-cocycle for the Chevalley cohomology associated to the trivial rep
résentation of g on M and a is an involutive Lie algebra isomorphism of g. The ±l-eigenspaces of a are denoted by t (corresponding to -hl) and P (corresponding to -1) respectively, and the following properties are required to be satisfied:
1. t C Rad(co), and
2. <^|pxp non-degenerate.
Here Rad(o;) is the radical of the two-form ui. A siLa is said to be exact if there exists an element ^ G g* such that 6^ = u>, where ô is the
differential operator of the Chevalley cohomology.
Let G and K be connected Lie groups having g and ê as Lie algebras, respectively (with K < G). The homogeneous space M := G/K is then naturally endowed with a symplectic symmetric space structure s ■. M X M ^ M, where the symmetries are defined in terms of the group involution cr as follows (assuming the Lie algebra involution a lifts to a Lie group involution such that C C K, where H’^ is the subgroup of G consisting of fixed points of a)\
Sgxg'K := ga{g~'^g')K. (1.16) The elementary normal j-group S will be endowed with a symplectic symmetric space structure giving rise to an exact siLa.
1.2.2 Coadjoint orbits
The elementary normal j-group S acts on itself by conjugation : C : S X S ;—>• S : (5, /i) i-> ghg~^.
The differential of this action in the second argument is called the adjoint action and will be denoted by
Ad : S X
5—> s : (5, AT) !->■ [ÿexp(fA')5“^].
Differentiating this action in the first argument yields the Lie bracket of
5:
|Adexp(tX)>^|0=[X,y].
We will use the notation adxiX) — [X, T] to dénoté the differential
of the adjoint action in the first argument.
1.2. SYMPLECTIC SYMMETRIC SPACE STRUCTURE 35
Given an element ^ of the dual s* of s, the coadjoint action of S on
^ is defined by
Coadg(0 := [>" ^ ^(Ad3-i(X))] € s*.
Orbits of the coadjoint action are called coadjoint orbits. Using the expression of the exponential map, the adjoint and coadjoint actions can now be explicitly given. For g = (a, v,l) Ç. S and X = aH + y + ^E G s, one has
Ad(a,^,,) (aff + y + j3E) := [gexp(tX)g-^]
= l(a,v,l).(at,^(l - ^(1 - e-^‘^%(-a, -e“n, -e^»/))]
= aff + e“(y — av) + e^“(—2aZ + /3 + u(v, y))E.
(1.17) Remark 1.2.3. An élément^ of the dual s* will be written as follows :
^ = yff* +iü{w,-) + iyE*, where E M.,w € V and
1. H*{H) = l, 2. E*{E) — 1, and 3. üü{w.,-){y) :=ui{w,v),
and ail other relations are equal to 0.
This leads to the following expression for the coadjoint action:
Coad^a,v,l){l^H* + u{w, ■) + lyE*) =
(jj, + uj{w, v) + 2i>ï)ff* + üj{e~°'w — ue~°‘v, •) + ue~‘^°‘E*.
Let ^ = gff* + üj{vü, •) + vE* as above, and let us dénoté by the orbit of under the coadjoint action given in Equation 1.18. From there, one concludes that there are three types of orbits.
1. Z//O:
. z^>0: Oç = {C€s*|C(^)>0},
• 1/ < 0 : dç = {C G 5*|C(^) < 0},
2. V = 0, w ^ 0: Oç = {aff* + ru){w, -)| a € R, r € R^}, and
3. u = 0,w = 0: 0^ =
One therefore has the following Lemma.
Lemma 1.2.4. An elementary normal j-group S of dimension 2d + 2 has exactly two coadjoint orbits of dimension 2d + 2, namely O
e* and O-E* ■
We will Write eE* to allow for an arbitrary sign, e = ±1. One therefore obtains two diffeomorphisms
: (a, v, l) ^ 2elH* - ee-“w(n, •) + ee-^^E\ (1.19) one for each value of e. We will make use of the notation to dénoté OeE*-
Kirillov-Kostant-Souriau symplectic structure on coadjoint or- bits
Given an element ^ € s*, consider the bilinear form : 5 X s R : (X, F) ^{[X, Y]).
Définition 1.2.5. Given a coadjoint orbit O of the Lie group S, and an element X £ s, dénoté by X^ the fundamental vector field associated to X, acting on functions on O in the following way:
(X^/)(0 := ^/(Coadexp(iX) OL=o- (l‘20) and the fundamental vector field of an arbitrary smooth action of G on a manifold M are defined and denoted in a similar manner in what follows. We also use the notation
Xç . [Coadexp(tx)
Every coadjoint orbit O is naturally endowed with a symplectic struc
ture cj®, invariant under the coadjoint action (cf. Référencés [48], [49]
and Lemma 1.2.7 below), called the Kirillov-Kostant-Souriau symplectic form (KKS for short).
Définition 1.2.6. Let X,Y € s and let X^ and Y^ be the associated fundamental vector fields (cf. Equation 1.20). The symplectic form uj^
on O is defined on fundamental vector fields by:
(1.21)
1.2. SYMPLECTIC SYMMETRIC SP ACE STRUCTURE 37 Let g E S and let us fix a coadjoint or bit O. We will also dénoté by Coadÿ the restriction of the coadjoint action of the group element g on the orbit O.
Let Coadp* ; TO —)• TO dénoté the differential of the map Coad^ : 0-^0, where TO is the tangent bundle of O. We will use the following fact in Lemma 1.2.7 below:
Coadp*^ . [Coadp Coadgj^p^j_^^
= [Coadg Coadexp(tx) Coad^-i Coad^ (1-22) [Coadexp(fAdgX) Coadg ^],
where the last equality follows from the définition of AdgX.
Lemma 1.2.7. For any g ES the following invariance property holds:
Coad* .
Proof. Let ^ E s*.
{Coad;u^)^{X^,Yf)
= ^Coad, ç ( Coads*ç Xf, Coad,*ç Yf)
= ‘^Coad,ç((AdgX)gg^^ç,(Ad,F)g„,^^ç) byEq. 1.22 := (CoadpO([AdpX,Adgy])
= ^oAd^-i(Ad3[X,F])
= ^(X,y)=o;^(Xf,yf).
□
Lemma 1.2.8. The diffeomorphism <f)(, '. S of Equation 1.19 is S-equivariant for the left action of S on itself and the coadjoint action
of S on :
Mg-g') = Coadg {(piig')), for ail g, g' E S.
The coadjoint orbit O, if it is not a point, is therefore a non trivial homogeneous (for the coadjoint action of S) symplectic manifold.
Définition 1.2.9. (cf. Référencé [1]) Let {M,
lo) be a symplectic man
ifold and a : G X M M be a smooth, symplectic, transitive action of
a Lie group G on M; let g he the Lie algebra of G. A moment map (or momentum mapping or momentum map) is a smooth map
A:M^g*, (1.23)
such that, if we dénoté by A the map:
 : g ^ C~(M) : X ^ Ax := [p ^ H
p) (^)l, the following then holds for every X £
q:
d(Â(X)) := dXx = ixooJ, (1.24) where ixoui := co(X^, •).
The quadruple (M,
lo, G, A) is called a Hamiltonian G-space or Hamil- tonian space for short when the group G is understood; it is said to be a strongly Hamiltonian G-space or strongly Hamiltonian space when G is understood and the following additional condition is satisfîed for ail X,Y€
q:
{Ax, Ay} = A[x,y] (1.25)
Left-invariant symplectic structure on elementary normal j- groups and moment map for the left action
Let be the KKS symplectic structure on one of the two coadjoint orbits of maximal dimension O^.
Remark 1.2.10. Let be the diffeomorphism of Equation 1.19. The two-form cUj := is a symplectic structure on S which is left- invariant:
r *, ,S _ s
