Geometric and Harmonic Analysis on Homogeneous Spaces
and Applications
Sousse, December 12-16, 2011

Abstracts
Ahmad Abouelaz: Nazarov's Inequality on the Lattice Zn.
In this talk,the following theorem: Let S be a nite subset in Zn and Σ be
( 1we1 )provena subset in − 2 , 2 . Then for all f ∈ l1 (Zn ) , we have∑|f (m)|

2≤ γ (S, Σ) {

m∈Zn

∫

∑

|f (m)| +2m∈Zn S(∫|Ff (x)|2

+

1−

Σ

Tn Σ

|Ff (x)|2 dx

)2
n (
∏
sin (πxi )
πxi

i=1

)
dx, }

where γ (S, Σ)∑is the Nazarov 's constant and Ff is the discrete Fourier transform given
by Ff (x) = m∈Zn f (m) exp (−2iπxm) , ∀x ∈ Tn .
As a consequence,let S be a nite subset in Zn and Σ be
( 1 1 )n we get the following1 result:a subset in − 2 , 2 . Then for all f ∈ L (Tn ) ∩ L2 (Tn )∫|f (x)| dx ≤ γ (S, Σ) {
2

Tn

∫

∑
m∈Zn S(|f (x)|2+2fb(m) +
1−

Σ

∫

Tn Σ

|f (x)|2 dx +

)2
n (
∏
sin (πxi )
i=1

πxi

)
dx},

∫

where fb(m) = Tn f (x) exp (−2iπmx) dx.
We also prove other results on the Radon transform on the torus, in addition we show
the Mandelbrojt 's theorem (Uncertainty principle) on the lattice Zn . (This is a joint
work with T. Kawazoe).

Mouadh Akriche: Real algebraic surfaces bered in curves ofgenus 2 (The classication of singular bers).
Our aim is to classify the bers of real smooth algebraic surfaces bered by curvesof genus 2. As a rst step, we explain how one can reduce the situation by using the1real stable bration associated to these surfaces of general type. Then, We discuss thegeometry (and topology) of (general) singular bers of elliptic monodromy.

Walid Aloulou: Pre-Gerstenhaber algebras up to homotopy.
In this lecture, we study the concepts of algebra up to homotopy for a structure denedby two operations · and [, ]. Having determined the structure of Gerstenhaber algebras
up to homotopy (G∞ algebra), we generalize this construction and we dene a structureof pre-Gerstenhaber algebra up to homotopy (pre G∞ algebra).
Given a structure of Zinbiel and dierential graded pre-Lie algebras and working overthe corresponding operads, we will give an explicit construction of the associate preGerstenhaber algebra up to homotopy. This is a joint work with Didier Arnal and Ridha
Chatbouri.

Didier Arnal: Canonical Coordinates for a class of solvablegroups.
For class R, type I solvable groups of the form NH, N nilpotent, H abelian, we construct
an explicit layering with cross-sections for coadjoint orbits. We show that any ultranelayer has a natural structure of ber bundle. The description of this structure allows us
to build explicit local canonical coordinates on the layer. Joint work with Bradely Currey
and Bechir Dali.

Salma Azaouzi: A generalized analogue of Hardy's
uncertainty principle on compact extensions of Rn.
Let K be a compact subgroup of automorphisms of Rn . We prove in this paper a
generalized analogue of Hardy's uncertainty principle on the semi-direct product K n
Rn extending earlier partial results on Euclidean motion groups where K = SO(n).
As a consequence, a complete analogue of classical Hardy's theorem is obtained. The
representation theory and the Plancherel formula play an important role in the proofs.
This is a joint work with Mounir Elloumi and Ali Baklouti.

Mabrouk Ben Ammar: Deformation of sl(2) andosp(1|2)-Modules of Symbols.
We consider the sl(2)-module structure on the spaces of symbols of dierential operators acting on the spaces of weighted densities. We compute the necessary and sucient
integrability conditions of a given innitesimal deformation of this structure and weprove that any formal deformation is equivalent to its innitesimal part. We study also
the super analogue of this problem getting the same results. This is a joint work with
Imed Basdouri.

Salem Bensaîd: From the trigonometric to the rational
Dunkl setting.
I will summarize the main results obtained with B. Orsted on this topic. The relationship with spherical functions on Riemannian and pseudo-Riemannian symmetric spaces
will be elucidated.

Abdellatif Bentaleb: On extension and renement of the
Poincaré inequality.
The aim of this paper is to analyze the heat semigroup (Nt )t≥0 generated by the usual
Laplacian operator ∆ on Rd equipped with the d−dimensional measure Lebesgue. We
obtain and study, via a method involving some semigroup techniques, a large family offunctional inequalities that does not exist in the literature and with the local Poincaré andreverse local Poincaré inequalities as particular cases. As an consequence, we establish
in parallel a new functional and integral inequalities related to the Hermite semigroup.

Michel Duo: Frobenius Lie subalgebras of simple Liealgebras.
A Frobenius Lie algebra is a Lie algebra for which the coadjoint action has an open
orbit. I present results on Frobenius Lie subalgebras of a simple complex Lie algebra
which contain a Cartan subalgebra, with a special emphasis on the "Ooms spectrum"(the Ooms spectrum is an equivalent for Frobenius Lie algebras of the set of exponents
of a simple Lie algebra). This is a joint work with M. S. Khalgui and P. Torasso.

Saîd Fahlaoui: Sharp estimates for some inequalities related
to the ultraspherical semigroup.
This paper studies the ultraspherical semigroup (exp(tL(α) ))t≥0 , α > 0, generated by
the operator Lα f (x) := (1 − x2 )f ′′ − αxf ′ acting on the Hilbert space αL2 ((−1, +1) , µα )with weight the ultraspherical probability measure µα (dx) = cα (1 − x2 ) 2 −1 dx. We obtain
new inequalities which interpolates in a sharp wary between the Poincaré inequality andthe logarithmic Sobolev inequality. We also prove a extension and renement of the
Poincaré and Jensen inequalities.

Jacques Faraut: Analysis on symmetric cones andmultivariate Meixner-Pollaczek polynomials.
We consider the homogeneous space Ω = G/K associated to a Gelfand pair (G, K),
where G is a Lie group, and K a compact subgroup. The algebra D(Ω) of G-invariant
dierential operators on Ω is commutative. Fix a C ∞ and K -invariant function ψ0 on Ωsuch that, for every D ∈ D(Ω), Dψ0 ∈ L2 (Ω). Dene the linear form ℓ on D(Ω) byℓ(D) = (Dψ0 | ψ0 )L2 (Ω) ,and the inner product⟨D1 , D2 ⟩ = ℓ(D1 D2∗ ).

We are interested in the construction of an orthogonal basis in D(Ω). In the case ofa symmetric cone Ω, and choosing ψ0 (u) = e−tru , the multivariate Meixner-Pollaczek
polynomials provide such an orthogonal basis. Joint work with Masato Wakayama.

Mahmoud Filali: Approximable interpolation sets.
In a group, I0 -sets and Sidon sets are interpolation sets for the almost periodic functions and the Fourier-Stieltjes functions, respectively. Both of these sets have been extensively studied.
Extending and unifying these concepts, we introduce in this talk the notion of approximable interpolation sets for general algebras of functions on a locally compact group. We
characterize the approximable interpolation sets for for the right uniformly continuousfunctions and for the weakly almost periodic functions. Joint work with Jorge Galindo.

Ahmad Fitouhi: On some q−integral transforms andapplications.
We study in this talk some q−integral transforms and some of their applications in

q−analysis.

Hidenori Fujiwara: Plancherel formula for the restriction ofunitary representations of nilpotent Lie groups.
Let G = exp g be a connected and simply connected nilpotent Lie group with Liealgebra g. Let π be an irreducible unitary representation of G, K = exp k an analytic

subgroup of G, K̂ the unitary dual of K and p : g∗ → k∗ the restriction mapping. Let Ω
be the coadjoint orbit of G associated with π , µ̃ a nite measure on Ω equivalent to theG−invariant measure on Ω and θ : k∗ → K̂ the Kirillov mapping for K . Now we regardµ̃ as a measure on g∗ , take the image measure µ = (θ ◦ p)∗ (µ̃) on K̂ and denote m(σ)
for σ ∈ K̂ , the number of K−orbits contained in p−1 (θ−1 (σ)) ∩ Ω. Then the restriction
π|K to K decomposes into irreducibles as follows:
π|K ≃

∫ ⊕

m(σ)σdµ(σ).
K̂

According to this decomposition and by use of intertwining distributions, we try todescribe a Plancherel formula for the restriction π|K . (Joint work with Ali Baklouti andJean Ludwig)

Abdelhamid Hassairi: Cauchy-Stieltjes Kernel Families withcubic pseudo-variance functions.
We extend the study of Cauchy-Stieltjes kernel families to cover generating measureswith support that is unbounded on one side. This extension leads to the characterizationof a class of cubic pseudo-variance functions corresponding to free-innitely divisible
distributions without the rst moment. A joint work with Wlodzimierz Bryc, Universityof Cincinnati, USA.
Bibliography.
H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support.
Indiana Univ. Math. J., 42(3):733773, 1993.
W. Bryc and A. Hassairi. One-sided Cauchy-Stieltjes Kernel Families. Journal of Theoretical Probability, 24: 577-594, 2011.
W. Bryc. Free exponential families as kernel families. Demonstratio Mathematica,
XLII(3):657672, 2009. arxiv.org:math.PR:0601273.
W. Bryc and M. Ismail. Approximation operators, exponential, and q-exponential
families. Preprint. arxiv.org/abs/math.ST/0512224, 2005.
G. Letac. Lectures on natural exponential families and their variance functions, volume
50 of Monograas as de Matematica [Mathematical Monographs]. Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 1992.
G. Letac and M. Mora. Natural real exponential families with cubic variance functions.
Ann. Statist., 18(1):137, 1990.
H. Maassen. Addition of freely independent random variables. J. Funct. Anal., 106(2):409438,
1992.

Junko Inoue: The norm of the Lp-Fourier transform oncompact extensions of Rn.
b theThis is a joint work with Ali Baklouti. Let G be a unimodular Lie group, Gpunitary dual of G, 1 < p ≤ 2 and q = p/(p − 1). We are concerned with the L -Fourier
b . When the group G is a compact extension of Rn , wetransform F p (G) : Lp (G) → Lq (G)show that the norm of the Lp -Fourier transform on G is equal to that on Rn . That is,∥F p (G)∥ = Anp , where Ap = (p1/p /q 1/q )1/2 .

Hideyuki Ishi: The Wallach set for a homogeneous boundeddomain.
The Wallach set is dened for a symmetric bounded domain as the set of parameters
admitting the analytic continuation of holomorphic discrete series representations. In
this talk, we generalize the denition for a homogeneous bounded domain, and discuss
the unitary representation corresponding to the parameter in the Wallach set. Recent
results about the isotropy subgroup will play an important role for the observation of
the representation.

Imed Kédim: On the rigidity Conjecture of deformations onsolvable Lie groups.
Let G be an exponential solvable Lie group and H a connected Lie subgroup of G.
Given any discontinuous 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 as H is not compact (except for the case when it is trivial).
To emphasize this specic issue, we present in this talk dierent questions related to
the geometry of the parameter and the deformation spaces of any discrete subgroup Γacting properly discontinuously and xed point freely on G/H for an arbitrary H . We
focus attention of some rigidity conjectures substantiated in the more general context ofsolvable Lie groups and discuss some solutions.

Samir Kabbaj: Equations Fonctionnelles Classiques.
Dans ce travail, on s'intéresse à la résolution de l'équation fonctionnelle
∫ ∫
G

K

f (xtk.y)dµ(t)dk = f (x)f (y) ∀x, t, y ∈ G et ∀k ∈ Koù G est un groupe localement compact, K un groupe compact opérant continûment
sur G muni de sa mesure de Haar normalisée dk, µ est une mesure bornée sur G et f
est l'inconnue à déterminée, une fonction continue sur G à valeurs dans C. On montre
d'abord que les solutions de l'équation proposée, dans le cas d'une mesure µ idempotente,coincident avec les caractères d'une sous algèbre de L1 (G). Comme résultat principal, on
montre que les solutions de l'équation ci-dessus, dans le cas d'une mesure K −invariante,donnent naissance à des coecients de sous représentations irréductibles de la K −régulière gauche de G.

Chifune Kai: The linearity of order isomorphisms onregular convex cones.
When we have a regular closed convex cone C in a (nite-dimensional) vector space
V , the associated partial order is introduced in V . A bijection on a subset of V (notnecessarily continuous) is called an order isomorphism if f and f −1 preserves the partial
order. We show that if C has no half-line factor, every order isomorphism on the interiorof C is a restriction of a linear map on V .

Khalid Koufany: Hua operators, Poisson transform andrelative discrete series on line bundle over boundedsymmetric domains.
Let Ω = G/K be a bounded symmetric domain and S = K/L its Shilov boundary. We
consider the action of G on sections of a homogeneous line bundle over Ω and the corresponding eigenspaces of G-invariant dierential operators. The Poisson transform maps
hyperfunction-valued sections of a line bundle over S to the eigenspaces. We characterize
the image in terms of twisted Hua operators. For some singular parameters the Poissontransform is of Szegö type whose image is in a relative discrete series; we compute thecorresponding elements in the discrete series. This is a joint work with Genkai Zhang.

Sami Kouki: Duo conjecture for solvable Lie groups.
Let G be an exponential solvable Lie group, g its Lie algebra and π a unitary irreducible representation of G wich is square intégrable modulo the center, associated by
the Kirillov-Bernat map to a G-orbit Ω. Let H be a closed connected subgroup of G
with Lie algebra h and p : g∗ → h∗ the restriction map. We say that the representation π
is H -admissible if its restriction to the subgroup H splits in irreducible representations
with nite multiplicities. We shall prove the following conjecture due to Duo:The representation π is H -admissible, if and only if, the restriction of p to Ω is properon the range p(Ω). In the case in hand, these two conditions are equivalent to g = h + z,
where z is the center of g.

Jean Ludwig: The C ∗-algebra of a thread-like Lie group.
We describe via Fourrier transform the C ∗ algebra of a thread-like Lie group as analgebra of operator elds dened over its dual space.

Nobuaki Obata: Quantum Probabilistic Spectral Analysis
of Large Graphs.
In recent years, quantum probability has been applied to spectral analysis of graphs,
in particular, to the study of spectral properties of the adjacency matrices of large,
growing, or random graphs. The basic techniques of quantum probability consists of (i)quantum decomposition; (ii) various concepts of independence and quantum central limit
theorems; (iii) partition statistics and moment-cumulant formulae. In this talk we review
how the above basic tools are applied to the spectral analysis of graphs. In particular,
we focus on relation between product structures of graphs (Cartesian, comb, star, free,etc) and various concepts of independence. We also report some recent achievements on
the Manhattan product of digraphs for future direction.
[1] A. Hora and N. Obata: Quantum Probability and Spectral Analysis of Graphs
Springer, 2007.
[2] N. Obata: Lecture Notes (Tokyo, Sendai, Wroclaw, Cheongju, etc) available at
http://www.math.is.tohoku.ac.jp/ obata/

Sanjay Parui: Generalized Hermite expansions of functions
arising from Hardy conditions.
Consider a function f on RN which is O(N ) nite. Assume that both f and its Dunkl
1
2
Fourier transform Dk (f ) is bounded by e− 2 a|x| for some 0 < a < 1. We will show that
the Generalized-Hermite coecients of f will have exponential decay.

Khemais Maktouf: Sur la restriction de la représentation
métaplectique aux sous-groupes compacts.
In order to study restriction of discrete series to closed subgroups of solvable groups
over a p-adic eld, it is necessary to compute the restriction of the metaplectic representation to anistropic torus of the metaplectic group. For this, we rst give the restriction

of the metaplectic representation to the maximal compact subgroups. We give an explicit
description of the occuring irreducible representations or characters. This is a joint workwith Pierre Torasso.

Dominique Manchon: Pre-Lie and post-Lie algebras : some
applications.
Since the pioneering work of A. Cayley in the middle of the Nineteenth century, vector
elds on the ane space are known to be closely related with rooted trees. The modern
interpretation of this phenomenon relies on the notion of pre-Lie algebra. The structureof the pre-Lie operad also enters into the picture in an essential way. Replacing the anespace by a more general smooth manifold together with a transitive Lie group action, thecorrect corresponding algebraic framework is given by post-Lie algebras. Rooted trees
must then be replaced by (formal Lie brackets of) planar rooted trees. We will explain
this correspondence and describe the structure of the post-Lie operad (joint work inprogress with K. Ebrahimi-Fard, A. Lundervold, H. Munthe-Kaas and J-E. Vatne).

Ratnakumar P.K.: Non linear Schrodinger equation and thespecial hermite operator.
We discuss a well posedness result for the nonlinear Schrödinger associated to thespecial Hermite operator on Cn . We consider a class of nonlinearities, which includes
power type non lineariies, in particular. The results relies of the Strichartz estimate for
the Schroedinger propagator for the special Hermite operator. This is a joint work with
Vijay Sohani.

Atsumu Sasaki: An application of the classication of visible
linear actions to nilpotent orbits.
The notion of (strongly) visible actions on complex manifolds has been introduced
by T. Kobayashi to give an unied explanation of various multiplicity-free theorems.
Recently, we give a classication of strongly visible actions on complex linear spaces. In
this talk, we treat the case where a complex manifold is a nilpotent orbit in a complex
Lie algebra. We explain that the action of a compact real form of its adjoint group on
a spherical nilpotent orbit is strongly visible. The result on strongly visible actions oncomplex linear spaces plays an important role to our proof.

Sunduram Thangavelu: Riesz transforms and multipliers for
the Grushin operator.
We show that Riesz transforms associated to the Grushin operator −∆ − |x|2 ∂t2 arebounded on Lp (Rn+1 ), 1 < p < ∞. We also prove a Hormander-Mihlin type multipliertheoem and study Bochner-Riesz means associated to the Grushin operator.

Khalifa Trimeche: Hypergeometric convolution structure on
Lp -spaces and application for the Heckman-Opdam Theory.
We study for the Heckman-Opdam theory, the trigonometric Dunkl intertwining operator and its dual, a hypergeometric convolution structure on Lp -spaces, and the Kunze
and Stein phenomenon relative to this theory.

Taro Yoshino: Topological Blow-up.
Suppose that a Lie group G acts on a manifold M . The quotient space X := G\Mis locally compact, but not Hausdor in general. Our aim is to understand such a nonHausdor space X . The space X has the crack S . Roughly speaking, S is the causal
subset of non-Hausdorness of X , and especially X \ S is Hausdor.
We introduce the concept of `topological blow-up' as a `repair' of the crack. The`repaired' space X̃ is locally compact and Hausdor space containing X \ S as its open
subset. Moreover, the original space X can be recovered from the pair of (X̃, S).