Geometric and Harmonic Analysis on Homogeneous Spaces
and Applications
Hammamet, December 18-21, 2013

Abstracts
Lobna Abdelmoula : The Selberg-Weil-Kobayashi local
rigidity Theorem for exponential Lie groups.
A local rigidity Theorem proved by A. Selberg and A. Weil for Riemannian symmetricspaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H ,asserts that there are no continuous deformations of a cocompact discontinuous subgroupΓ for G/H in the setting of a linear non-compact semi-simple Lie group G except somefew cases : G is not locally isomorphic to SL2 (R) for H compact or G is not locallyisomorphic to SO(n, 1) or SU (n, 1) for G × G and H = ∆G . When in large contrast
G is assumed to be exponential solvable and H ⊂ G a maximal subgroup, we provean analogue of such a Theorem stating that the local rigidity holds on the parameter

space R(Γ, G, H) if and only if G is isomorphic to the two-dimensional group of anetransformations of the line ax + b. Remarkably, we do also drop the assumption on Γ tobe uniform for G/H . This is a joint work with Ali Baklouti and Imed Kedim.

Luigi Accardi : Innite dimensional Lie algebras andrenormalized higher powers of white noise.
The problem of dening, in an operational way, renormalized powers of free quantum
elds (white noises in mathematical language) has been a central one in mathematicalphysics for the past 50 years (see the paper [3] for a survey).
In 1999 a new, Liealgebraic, approach to this problem was proposed by Accardi, Lu
and Volovich and has led, through the contributions of a multiplicity of authors, to theconstruction of the renormalized square of (quantum) white noise (RSWN) and to its
didentication with the theory of unitary representations of the current algebra over Rof sl(2, R).
The success and the non-triviality of the theory in the quadratic case naturally rose
the problem of extending these results to powers higher than 2.
This program has been systematically pursued by Accardi and Boukas in the past 14
years and has led to the conclusion that the role of sl(2, R), for higher powers is an innite
dimensional Lie algebra 'strictly related' the VirasoroZamolodzhikov hierarchy of Liealgebras, introduced in the theory of strings for totally dierent purposes (the relation1

is that the two algebras have the same closure in an appropriately dened topology, see
the survey [1]).
This research has brought to light an interesting connection between renormalizationand central extensions of Lie algebras as well as several new results on central extensionsof innite dimensional Lie algebras (see the survey [2]).
The present talk will be a survey of the main new ideas emerged in this eld withemphasis on open problems, whose solution will require the collaboration of experts ofquantum probability with experts of Lie algebras.
References.

[1] L. Accardi, A. Boukas : Quantum probability, renormalization and innite dimensional ∗Lie algebras, SIGMA (Symmetry, Integrability and Geometry : Methods andApplications) 5 (2009), electronic journal, Special issue on : Kac-Moody Algebras andApplications.
http ://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html[2] L. Accardi, A. Boukas : Renormalization and central extensions p-Adic Numbers,
Ultrametric Analysis and Applications 4 (2) (2012) 89101, Special issue dedicated to
Igor V. Volovich. DOI : 10.1134/S20700466120200K.
[3] L. Accardi, R. Roschin : Renormalized squares of Boson elds, IDAQP (Innite
Dimensional Anal. Quantum Probab. Related Topics) 8 (2) (2005) 307326.

Ahmad M. A. Alghamdi : New Characterization of Lie
Triple Systems.
This work is a joint project with Dr. Amir Baklouti. We are studying the concept of Lie
Triple Systems. A Lie triple system is a vector space with trilinear map which satises
the condition of antisymmetry, Jacobi identity and derivative identity. We are trying
to get a new characterization of such concept by using the so-called Casimir operator
as a tool. A Casimir element (operator) is a distinguished element of the centre of theuniversal enveloping algebra of a Lie algebra. In this talk, I shall discuss and focus in theuse of this operator and mention some properties of this distinguished element.

Souhail Bejar : Deforming discontinuous subgroups of
Euclidean motion groups.
Let G be a Lie group and H a connected closed subgroup of G. Given any discon-tinuous subgroup Γ for the homogeneous space M= G/H and any deformation of Γ,

the deformed discrete subgroup may utterly destroy its proper discontinuous action on

M whenever H is not compact. To understand this specic issue in the setting whereG := On (R) n Rn stands for the Euclidean motion groups, we provide an explicit description of the parameter and the deformation spaces of any discrete Γ acting properly
discontinuously and xed point freely on G/H for an arbitrary H . Remarkably, it happensthat H turns to be compact whenever Γ is innite, which means that the proper action
resists to any deformation of Γ but probably the free action does not. Furthermore theparameter space has a quite complicated topological features imposing a distinction of
two types of stability. The situation where Γ is a crystallographic subgroup is extensively
studied. This is a joint work with Ali Baklouti.

Agrebaoui Boujemâa : Jeu de taquin and diamond cone forclassical Lie (super)algebras.
Each irreducible module of a complex semisimple Lie super algebra g is characterized
by its highest weight λ and is generated by the action of g on the unique (up to constant)λhighest weight vector vλ , with weight λ. The sum of all such modules S , has a structureof a commutative associative algebra, denoted S and called the shape algebra of g. The
classical Weyl theory for sln (C) and its extension to the case of sp(2n), and so(2m+1) (see[FH]) gives a cobinatorial basis for the shape algebra in terms of semistandard tableaux.
This notion of semistandard tableaux was extended to the case of Lie superalgebrassl(m, n) and spo(2n, 2m + 1) (see [CW]). For sl(m, n), semistandard tableaux give abasis for the sum of all simple tensor modules (see [BR]).
Consider now the nilpotent factor n of g. The reduced shape algebra Sred is the union
λof all maximal monogenic modules S |n . It is the quotient of S by the ideal generated by
all vλ − 1.
A quasistandard tableau is a semistandard one from which it is impossible to extract
any trivial subtableau by using the Schützenberger's jeu de taquin. An explicit combinatorial model for a basis, called the diamond cone, of Sred in terms of quasistandard
tableaux is given in the case of sln (C) and sp(2n)(see [ABW, AK]).
We extended these notions to the case ofso(2m + 1) (see [AAB1]), sl(m, n) (see[AAK2]) and to spo(2n, 2m + 1) (see [AAB2]).

Références[AAB1]

B. Agrebaoui, D. Arnal, A. Ben Hassine, Diamond module for the Lie algebra so(2n + 1, C).

arXiv :1208.3349v1 (2012), 41 p.

[AAB2]

B. Agrebaoui, D. Arnal, A. Ben Hassine,
preprint.

Orthosymplectic jeu de taquin and diamond cone ,[AAK2]

B. Agrebaoui, D. Arnal, O. Khli, Diamond cone for sl(m, n). arXiv :1211.4158v1 (2012),

35 p.

[ABW]

D. Arnal, N. Bel Baraka, N. Wildberger, Diamond representations of sl(n). Ann. Math.

Blaise Pascal, 13 (2006), n. 2, 381-429.

Le cône diamant symplectique. Bull. Sci. Math. 134 (2010), no. 6, 635-663.

[AK]

D. Arnal, O. Khli, [BR]

A. Berelee, A. Regev, [CW]

S.J. Cheng and W. Wang, 

Hook Young diagrams with applications to combinatorics and torepresentations of Lie superalgebras. Adv. in Math. 64 (1987), no.2, 118175.
Dualities and Representations of Lie Superalgebras. Graduate

Studies in Mathematics, 144. American Mathematical Society, Providence, RI, 2012.

[FH]

W. Fulton and J. Harris, Representation theory, Readings in Mathematics. Springer-Verlag,

New York, 1991.

Didier Arnal(1) and Bradely Currey(2) : Regularity andcross-section for multiply generated abelian group actions : I
and II.
In two lectures, we present recent joint work with Béchir Dali and Vignon Oussa.
Let g be the real span of a nite set of commuting endomorphisms of a real vector
space V and put G = exp g. To each G−orbit in V , we attach a subgroup of a torus,
with the property that the orbit is regular if and only if the subgroup is closed. We
construct an layering of V into G−invariant semi-algebraic subsets in which the rotation
subgroups are described by explicit G−invariant rational functions. We show that oneof the following holds : either every orbit in the generic layer is regular, or there is aco-null, G−invariant Gδ -subset in which every orbit is not regular. We show that in thea.e. regular case, the functions describing the rotation subgroups for the generic orbits
are constant. In each layer, if every orbit is regular, we present an explicit construction of
a topological section for the orbits in the layer. We discuss applications to the harmonic
analysis of V o G, and provide various examples.
(1) : Inst. de Mathématiques de Bourgogne, Université de Bourgogne, Dijon, France,
E-mail address : didier.arnal@u-bourgogne.fr(2) : Dept. of Mathematics and Computer Science, St. Louis University, St. Louis, MO
63103 U.S.A., E-mail address : curreybn@slu.edu

Salma Azaouzi : A generalized analogue of Hardy's
uncertainty principle on compact extensions of Rn. (Phd
Thesis)A classical theorem of Hardy proved in 1933 in the case of the real line, states that an
integrable function f and its Fourier transform fˆ cannot both have arbitrary Gaussian
αkxk2
βkξk2
decay unless f is identically zero. More precisely, if both f (x)eand fˆ(ξ)e
are in∞n
L (R ) for some α, β > 0 then the following conclusions hold :1. f = 0 whenever αβ > 1/4.
2. The function f is a constant multiple of e

−αkxk2

when αβ = 1/4.

3. When αβ < 1/4, there are innitely many linearly independent functions satisfying
both conditions.
There has been much eort to prove Hardy-like theorems for various classes of nonabelian connected Lie groups. Specically, analogues of Hardy's theorem have been shownfor the Euclidean motion groups (Sundari 1998, Sarkar and Thangavelu 2005, BakloutiKaniuth 2010).

\
λ > 0 and σ ∈ SO(n− 1), then we have an irreducible unitary representan
tion πλ,σ of the motion group SO(n) n R . Then functions satisfying the conditions22|f (a, x)| ≤ ϕ(a)e−αkxk and kπλ,σ (f )kHS ≤ ψ(σ)e−βλ , where ϕ ∈ L2 (SO(n)) and\ψ ∈ l2 (SO(n− 1)), are proved to be trivial on SO(n) n Rn in the only case of αβ > 41 .
n
Now if we let K be a compact subgroup of automorphisms of R . Can we replace SO(n)1by an arbitrary compact subgroup such as K ? Then in the case of αβ =, can we4Letcharacterize the functions satisfying similar assumptions as above ? Answering to thesequestions, I obtained a generalization of Hardy's uncertainty principle on the semi-directnproduct K n R (that we denote by G), with a complete characterization of the function
f in the three cases αβ > 41 , αβ = 14 and αβ < 41 .
To prove such a result, we have rst of all to know explicitly the unitary dual of G.
We can ensure that using the Mackey theory. Once the dual is identied and Fouriertransform dened, we estimate the Hilbert-Shmidt norm of the Fourier transform of
the function. These two aspects are comparatively more complicated than the case of

K = SO(n). Moreover, having proved Hardy's theorem on G, I take up other uncertaintypqprinciples. Namely, Cowling-Price, Miyachi, L −L Morgan, Beurling, Bonami-DemangeJaming and Gelfand-Shilov theorems, which proofs require an explicit knowledge of theb.
unitary dual of G and of the Hilbert-Shmidt norm of the Fourier transform on G

Abdessatar Barhoumi : White Noise Meixner Processes andRelated ?-Lie Algebras.
By using an appropriate one-mode type interacting Fock spaces, ΓM (H ), we derive achaotic decomposition property for the Hilbert space of quadratic integrable functionals
with respect to the Meixner white noise measure ΛM . The constructed decomposition is20∗
used to dene a nuclear triple FM,θ (E ) ⊂ L (E , ΛM ) ⊂ F(E ) of test and generalizedM,θ

functions, with θ being a suitable Young function. Moreover, a general characterization
theorems are proven for the fundamental nuclear spaces. For the applications, we introduce a new renormalized products for the generators of the renormalized higher powersof white noise ?-Lie algebra and the Virasoro-Zamolodchikov-w∞ ?-Lie algebra. Then weshow that these new renormalized products lead to a nuclear realizations of these famous
Lie algebras in terms of quantum Meixner white noise operators.

Nizar Ben Fradj : Cohomology of osp(n|2) Acting on the
Modules of Linear Dierential Operators on the Weighted
Densities on the Supercircle S 1|n and classication of thesemodules.
We compute the frst dierential cohomology of the orthosymplectic Lie superalgebraosp(n|2) with coecients in the superspace of linear dierential operators acting on the1|nspace of weighted densities on the Supercircle S. We give, in contrast to the classicalnnsetting, a classication of these modules. for n = 1. We also prove that Dλ,µ and Dρ,ν2−nare isomorphic for ρ =− µ and ν = 2−n− λ. := Homdiff (Fnλ , Fnµ ), where Fnλ is the22n,k+ 1

2
K (n)-module of λ-densities on R1|n . It is ltered : · · · ⊂ Dn,k
· · · , whereλ,µ ⊂ Dλ,µ1
k ∈ 2 N. We prove that, in ontrast to the classical setting, the K (n)-modules Dn,kλ,µ and2−n2−nn,kDρ,ν are isomorphic if and only if ρ = 2 − µ and ν = 2 − λ. This work is the simplest

superization of a result by Gargoubi and Ovsienko [Modules of Dierential Operators on
the Real Line, Functional Analysis and Its Applications, Vol. 35, No. 1, pp. 1318, 2001.]

Abdellatif Bentaleb : Some sharp interpolation inequalitiesof the Poincaré inequality.
The main purpose of this short note is establish and analyze study, via a method
involving some semigroup techniques, a new family of sharp integral inequalities relatedαβ
to the Jacobi probability measure µ(dx) = cx (1 + x) dx; α, β > −1 : These estimations
are a natural extension and renement of the spectral gap and reverse spectral gap

inequalities. New integral inequalities for the uniform measure on a compact interval arealso derived with the classical Poincaré inequality as particular case.

Aline Bonami : Bergman projection and Bloch classes inbounded symmetric domains of tube type.
n
Let D be an irreducible bounded symmetric domain of tube type in C . The class ofBloch functions is well known in this context as the space of bounded Hankel operators
1or as the dual of the Bergman space A . Contrarily to what happens in the unit ball,p
Bloch functions do not belong to all Lebesgue spaces L (D) for p < ∞ in rank r>1.
We will give both necessary and sucient conditions on p for such an embedding. This
question is equivalent to local boundedness properties of the Bergman projection in the
tube domain over a symmetric cone that is conformally equivalent to D .
This is a joint work with G. Garrigós and C. Nana, to appear in Journ. Geo. Anal.
We will also mention recent work of Cyrille Nana in the non tube case.

Radhouane Daher : On a Theorem of Titchmarsh.
In this paper, we prove an analog of Titchmarsh's theorem in certain sitting. Moreprecisely for the circle group, Bessel-Jacobi hypergroups, non compact symmetric spaces,
the sphere, ....we obtain an analog of one classical Titchmarsh theorem on descriptionof the image under the Fourier transform of class of functions satisfying the Lipschitz
condition. The results are obtained in various collaborationa with Mohamed ELHAMMA.

Kahar El-Hussein : Existence Theorems and Left Ideal ofGroup Algebra on the Nilpotent Cartan-Lie Group G5.
Let G5 be the 5-dimensional nilpotent Cartan-Lie group. Thanks to its nest structure,
this group can be shown as a semi-direct product of three real vector groups. The purposeof this paper is to generalize the Fourier transform and to prove the Plancherel theorem.
Moreover we prove some existence theorems for the invariant dierential opera- tors on
G5 . To this end we give a classication of all left ideals of group algebra L1 (G) of G5 .

Saïd Fahlaoui : On extension and renement of the Poincaré
inequalityt∆
The aim of this paper is to analyze the heat semigroup (Nt )t>0 = {e }t>0 generateddby the usual Laplacian operator ∆ on R equipped with the d−dimensional Lebesgue
measure. We obtain and study, via a method involving some semigroup techniques, a large

family of functional inequalities that does not exist in the literature and with the local
Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence,
we establish in parallel a new functional and integral inequality related to the OrnsteinUhlenbeck semigroup.

Jacques Faraut : Multivariate Markov-Krein type formulafor projections of orbital measures.
The orthogonal group U = O(n) (resp. the unitary group U (n)) acts on the space

V = Sym(n, R) of n×n symmetric matrices (resp. Herm(n, C) of Hermitian matrices)∗by the transformations X 7→ uXu (u ∈ U ). For A ∈ V ,the orbital measure µA is dened by :Z

Z
f (X)µA (dX) =f (uAu∗ )du.

V
Uk
We consider the projection µA of µA on the subspace Vk of k × k matrices. The measureµ1A on V1 ' R satises the following Markov-Krein type formulaZ

1

1d µA (dt) =n
R (z − t) 2

n
Y

1
d

2
j=1 (z − λj )(z ∈ C \ R),where λ1 , . . . , λn are the eigenvalues of A, d = 1 in case of Sym(n, R), and d = 2 incase of Herm(n, C). By using this formula it is possible to determine the density of the1measure µA (F. Fourati, 2011, Journal of Lie Theory). For d = 2 this density is a spline
function.

k
We will present a multivariate analogue of this formula involving the measure µA fork ≥ 1.

Ahmad Fitouhi : q-Bessel-Hahn-Exton transform andapplications.
We prove the positivity of the q-generalized translation associated with the q-bessel
function published recently in constructive approximation and we give some applictions
such denite positive functions q-theory.

Hidenori Fujiwara : Some problems for induced andrestricted representations of exponential solvable Lie groups.
We discuss in framework of the orbit method some problems for in- duced and restricted representations of exponential solvable Lie groups ; intertwining operators andpolynomial conjectures. (Joint works with Jean Ludwig and Ali Baklouti)

Sonia Ghaour : Deforming discontinuous subgroups of
reduced Heisenberg groups.
r
Let G = H2n+1 be the 2n+1-dimensional reduced Heisenberg group and H an arbitrary
connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H , weshow that resulting deformation space T (Γ, G, H) of the natural action of Γ on G/His endowed with a smooth manifold structure and is a disjoint union of open smoothmanifolds. Unlike the setting of simply connected Heisenberg groups, we show that thestability property holds and that any discrete subgroup of G is stable in the sense of
A. Baklouti, following the notion of stability introduced by Kobayashi-Nasrin. On theother hand, a local (and hence global) rigidity theorem is obtained. That is, the relatedparameter space R(Γ, G, H) admits a rigid point if and only if Γ is nite. This is a joint
work with Ali Baklouti and Fatma Khlif.

Abdelhamid Hassairi : On Cauchy-Stieltjes Kernel families.
We extend the classication of the Cauchy-Stieltjes kernel families with compactly
supported generating measures to cover measures with support unbounded on one side.
We illustrate the need for such an extension by showing that cubic pseudo-variancefunctions correspond to free-innitely divisible laws without the rst moment. We thenexplore properties of Cauchy-Stieltjes families that have no counterpart in exponential
families. We relate the variance function of the iterated Cauchy-Stieltjes family to thepseudo-variance function of the initial Cauchy-Stieltjes family. We also investigate when
the domain of means can be extended beyond the "natural domain".
Laboratory of Probability and Statistics, Sfax University

Junko Inoue : Holomorphically induced representations ofsome exponential solvable Lie groups.
We are concerned with holomorphically induced representations ρ of exponential Liegroups G starting from weak polarizations or general complex subalgebras of gC , thecomplexication of the Lie algebra of G. We shall discuss problems of non-triviality anddescription of decomposition of ρ into irreducible representations, treating some concrete
examples.

Hideyuki Ishi : Generalized highest weight unitary
representations of a split solvable Lie group.
We generalize the notion of highest weight unitary repre- sentation to a real Lie groupwhich is not necessarily reductive. Such representations are classied for a connected andsimply connected split solvable Lie group, where our argument is based on Fujiwara's
structure theorem of totally complex positive polarizations.

Eberhard Kaniuth : Power boundedness in Fourier andFourier-Stieltjes algebras of locally compact Groups.
n
An element a of a Banach algebra A is called power bounded if sup{ka k : n ∈ N} < ∞.
Elements with spectral radius r(a) < 1 are power bounded, and conversely the spectral
radius of any power bounded element is ≤ 1. Accordingly, the algebra A is said to havethe power boundedness property if every element with spectral radius ≤1 is actually

power bounded.
Let G be a locally compact group, and let B(G) denote the Fourier-Stieltjes algebraand A(G) the Fourier algebra of G. Recall that when G is abelian, then B(G) and A(G)b and L1 (G)b , the measure algebra and the L1 -algebra of the dual
are isomorphic to M (G)b of G, respectively. This talk will mainly focus on a structure theorem for powergroup G
bounded elements in B(G) and the question of when the algebras B(G) and A(G) do
have the power boundedness property. I will also touch the extension problem for powerbounded functions and list some open problems.

Imed Kédim : On discontinuous subgroups acting on
solvable homogeneous spaces.
Let G be a Lie group, H a closed subgroup of G and Γ a discrete subgroup of G. If
the action of Γ on G/H is discontinuous, we consider the parameter space
ϕ is injective and ϕ(Γ)
R(Γ, G, H) := ϕ ∈ Hom(Γ, G)

is a discontinuous subgroup for G/H

The action of G on Hom(Γ, G) by composition on the left of the inner automorphisms
leaves the parameter space invariant. The deformation space is the quotient spaceτ (Γ, G, H) = R(Γ, G, H)/G.
There are many natural questions about the structure and the topological and geometricfeatures of τ (Γ, G, H). In this lecture, we present some recent developments, focusing on
Hausdorness and rigidity problems in the context of a solvable Lie groups.

Sami Kouki : Sur la restriction des séries discrètes decertains groupes résolubles algébriques.
Dans cet exposé, nous étudions la restriction des représentations, irréductibles et unitaires, de carré intégrable modulo le centre du groupe de Lie G produit semi-direct dugroupe de Heisenberg par un tore maximal du groupe symplectique à un de ses sousgroupes algébriques connexes H . La détermination de la fonction de multiplicité est en
rapport avec la conjecture de Guillemin-Sternberg connu sous le slogan "La quantica-b est une série discrète H -admissible, i.e.
tion commute à la réduction". En eet, si π ∈ G
sa restriction Ã H est de la forme

π|H =

X

mπ (σ)σ,bσ∈H
avec mπ (σ) < ∞, alors nous calculons cette multiplicité en terme de volume de variété
réduite associée. Autrement dit, le principe "quantication commute à la réduction"
reste valable dans ce cas.

Jean Ludwig : Intertwining operators for irreducible
representations of exponential Lie groups.
Let G be a real solvable exponential Lie group with Lie algebra g and let f∈ g∗ .

We take two polarizations ∂j , j = 1, 2, at f which meet the Pukanszky condition. Let

Pj := exp pj , j = 1, 2, be the associated subgroups in G. The linear functional f denes
ihf,Xiunitary characters χf (exp X) := e
, X ∈ pj , of Pj . Let τj := indG
Pj χf , j = 1, 2,
be the corresponding induced representations, which are unitary and irreducible. It is
well known that τ1 and τ2 are unitarily equivalent. The description of the intertwining
operator of such an equivalence is given via an abstract integral Ip2 ,p1 . The main problem
is to show that this formal integral converges. In this talk we present a method, which
has been developed in the early 90th by Hidenori, Didier Arnal and myself and publishedin 1996, to avoid this problem.

Nobuaki Obata : Spectral analysis of digraphs and coupledoscillators.
Spectrum of a graph or a digraph (directed graph) contains useful information for the
analysis of (deterministic or stochastic) dynamics on it. We will report some new aspectsof spectral analysis, in particular, of a Manhattan street network, i.e., a digraph where
the underlying graph is the n × m lattice and each edge is given direction in such a waythat left and right directed horizontal lines are placed alternately, and so are up anddown directed vertical lines. We will discuss some results on spectral analysis for the
preliminary case of 2 × n lattice and synchronization of coupled oscillators.
References :[1] F. Comellas, C. Dalfó, M. A. Fiol and M. Mitjana : The spectra of Manhattan
street networks, Linear Algebra Appl. 429 (2008), 1823-1839.
[2] N. Obata : Spectra of Manhattan products of directed paths, Interdisciplinary
Information Sciences 18 (2012), 43-54. DOI : 10.4036/iis.2012.43.
[3] N. Obata and J. Rodriguez : in preparation (2013).

Christine Oerman : Multi-temporal Wave Equations on
Riemannian Symmetric Spaces.
In 1976, Semenov, Tjan and Shansky introduced a formulation of the classical Cauchy
problem for Riemannian symmetric spaces. In this case, the partial dierential equation
becomes a system of equations involving invariant dierential operators on the symmetric space and multiple 'time' variables along with initial conditions involving the Weylgroup harmonic polynomials. In 1999, Helgason studied thismulti-temporal systemon
noncompact symmetric spaces using Fourier analysis. I will discuss recent results, obtained in collaboration with F. Gonzalez, for the cases of at and compact symmetricspaces. I will begin by giving solution forms for the system involving the Fourier andRadon transforms. I will then talk about some properties of these solutions and the issue
of uniqueness. In addition, I will present an energy form for this system as well as someof its properties, including its relation to the Fourier transform and a Plancherel-type
result.

Takayuki Okuda : Classication of semisimple symmetricspaces with proper SL(2,R)-actions.
We give a complete classication of irreducible symmetric spaces for which there exist
proper SL(2,R)-actions as isometries, using the criterion for proper actions by T. Kobayashi [Math. Ann. '89] and combinatorial techniques of nilpotent orbits. In particular, weclassify irreducible symmetric spaces that admit surface groups as discontinuous groups,
combining this with Y. Benoist's theorem [Ann. Math. '96].

Detlev Poguntke : Topology on the unitary dual ofcompletely solvable Lie groups.
In his seminal doctoral dissertation (1962) A.A. Kirillov gave birth to the so-called orbit
method in the case of a simply connected nilpotent Lie group G : The unitary dual Ĝ is∗∗in bijective correspondence with the orbit space g /G, where g is the real linear dual of∗
the Lie algebra g of G, and G acts on g via the coadjoint representation. This result waslater extended to exponential Lie groups (i.e., the exponential map is a dieomorphism),
mainly by P. Bernat. It is natural to expect that these correspondences are actuallydieomorphisms, often called Kirillov conjecture. It took quite a while to establish thisconjecture in the nilpotent case : I. Brown (1973), later K.I. Joy gave a dierent proof.
Concerning the exponential case, in a monography (1994) H. Leptin and J. Ludwig
showed that the Kirillov conjecture is here true as well. The splendid fundamental idea
is the introduction of the notion of Variable Groups, i.e., one considers families of(exponential) Lie groups, where the group law depends continuously on a parameter.
Unfortunately, the mentioned proof is hard to comprehend, this applies in particular tothe philosophy behind various cases and subcases. Even worse, there are errors in the
proof, some of minor weight, but others are more serious : at least one case is missing.
Those observations led me to a reconsideration of that proof. I think that in the (special)case of completely solvable Lie groups I can present a satisfactory picture (still, of course,
resting on the notion of Variable Groups)  and this will be the subject of my talk. The
general case is work in progress.

Dominique Manchon : The double shue structure of the
Ohno-Okuda-Zudilin q-multiple zeta values.
Several q-analogues of multiple zeta values have been explored in the recent years.
The model recently proposed by Y. Ohno, J.-I. Okuda and W. Zudilin shows particularly
good algebraic properties, and readily extends to arguments running over all the integers,
regardless to the sign. We exhibit the iterated Jackson integral representation of theseqMZVs, and describe the q -double shue relations thus obtained. Renormalization at thelimit q>1 will also be addressed. Joint work with Jaime Castillo-Medina and Kurusch
Ebrahimi-Fard.

Salah Mehdi : Representation theoretic dierential
operators.
We present several algebraic and geometric results on representations of Lie groups
related to invariant dierential operators. We will emphasis on Dirac operators in thecontext of coherent families of modules and translation functors.

Ayman Rahali : Contribution to the theory of unitary
representations of Lie groups with co-compact nilradical.
(Phd Thesis)b of G, we mean the set
Let G be a locally compact group. By the unitary dual Gof all equivalence classes of irreducible unitary representations of G equipped with theFell topology. The rst representation-theoretic question concerning the group G is theb. Apart this question, a signicant importance is attached to thedescription of the set G
b. If G is a Lie group with Lie algebra g, then thedetermination of the topology of Gb and the space g∗ /G of G-coadjoint orbits
investigation of the relationship between G
turns out to be a deep mathematical problem. In this direction, it is well-known that fora simply connected nilpotent Lie group or, more generally, for an exponential solvableb is homeomorphic to the orbit space g∗ /G.
Lie group G, the unitary dual G
In the present PhD thesis, rst we consider (G, K) as a compact Riemannian symmetricpair, and let G0 be the associated Cartan motion group. Under some assumptions onc0 )gen of all equivalence classes
the pair (G, K), we give a precise description of the set (Gof generic irreducible unitary representations of G0 . We also determine the topology of‡the space (g0 /G0 )gen of generic admissible coadjoint orbits of G0 and we show that thec0 )gen and (g‡0 /G0 )gen is a homeomorphism. Furthermore, in the casebijection between (Gc0 is homeomorphic
where the pair (G, K) has rank one, we prove that the unitary dual G‡
to the space g0 /G0 of all admissible coadjoint orbits of G0 .
Also we describe the space of admissible coadjoint orbits of the Heisenberg motion group
Gn = Tn n Hn , and we determine the topology of this space, where Hn is the (2n +1)-dimensional Heisenberg group, and Tn is the n-dimensional torus acting on Hn by
cn of Gn and its
automorphisms. We show that the bijection between the unitary dual G
admissible coadjoint orbit space is a homeomorphism.
Finally, we show that every irreducible unitary representation of the Euclidean motionngroup Mn = SO(n)nR , n ≥ 2, is characterized by a particular element in its generalized
moment set.

Atsumu Sasaki : A characterization of non-tube type
Hermitian symmetric spaces by visible actions.
Our object of this talk is the complex manifolds which is the line bundle over thecomplexication of a non-tube type Hermitian symmetric space. This manifold is a homogeneous space but not a symmetric one. In this talk, we discuss a generalizationof Cartan decomposition for this non-symmetric homogeneous space, in particular, theconstruction of an abelian part. Further, we explain that the above decomposition gives
rise to a new example of visible actions.

Taro Yoshino : On Topological Blow-up.
Consider a Lie group G (or more generally, a topological group) acts continuously on amanifold M (or more generally, a locally compact Hausdor space). The quotient space

X := G\H is locally compact, but not always Hausdor. In this talk, we introduce amethod to understand the topology on such a non-Hausdor space X . More precisely,
for a given locally compact (not necessarily Hausdor ) space X , we construct a locallyYcompact Hausdor space Y , and a map τ : X → 2 . Then, the pair (Y, τ ) has a complete
information on the topology on X . In particular, (Y, τ ) describes convergence of sequencesor lters on X .