1
Abstracts
1.1
1.1.1

Quadratic overgroups for Diamond Lie groups

Lobna Abdelmoulab
Let G be a connected and simply connected solvable Lie group. The moment map for π in G,∗
unitary dual of G, sends smooth vectors in the representation space of π to g . The closureof the image of the moment map for π is called its moment set, denoted by Iπ . Generally, theb does not charcterize π, even for generic representations. However, wemoment set Iπ , π ∈ Gbsay that G is moment separable when the moment sets differ for any pair of distinct irreducible
unitary representations. In the case of an exponential solvable Lie group G, D. Arnal and M.
Selmi exihibited an accurate construction of an overgroup G+ , containing G as a subgroup andc+ in such a manner that Φ(G)b into G
b is moment separable and IΦ(π)
an injective map Φ from Gbcharacterizes π, π ∈ G. In this work, we provide the existence of a quadratic overgroup forthe diamond group, which is the semi-direct product of Rn with 2n + 1-dimensional Heisenberg
group for some n ≥ 1. This is a joint work with Yasmine Bouaziz.

1.2
1.2.1

Diamond cone for classical Lie superalgebras

Boujemaâ Agerbaoui

Let g be a semisimple Lie algebras and n+ its nilpotent factor in the Iwasawa decomposition. Each simple finite dimensional g− modules is caracterized by its highest weight λ ∈ Λ =nX{ ai wi , ai ∈ N} where wi are fundamental weights. We denote it by Sλ . In classical Lie
i=1

algebras, each λ =nXai wi is the shape of a young diagram Y (λ) (i columns of hight i). The

i=1

fillings of Y (λ) to semistandard Young tableaux indexing a basis SST (λ) of Sλ . The space1

CHAPTER 1. ABSTRACTS

S• :=

X

λ

S , is a quotient of S(

2nXSwi ) called the shape algebra and SST (Λ) =

i=0

λ∈Λ

[

SST (λ) is aλ∈Λbasis.
Representations of classical Lie superalgebras, initiated by V. Kac, are closed to those of classicalLie algebras, simple finite dimensional modules are highest weight modules Sλ . The modules
realized as submodules of the tensor algebra of the natural module have basis indexed by thefamily of semistandard Young tableaux of shape λ. The shape[ algebra is defined analogously.
00λ•
If λ ≤ λ, S |n+ . The reduced shape algebra Sred :=Sλ |n+ has a basis indexed by theλ∈Λ

so called quasistandard tableaux QST (Λ) and respecting the stratification that is QST (λ0 ) ⊂QST (λ). Each T 0 ∈ QST (λ0 ) is extracted[from T ∈ SST (λ) through the use of the ’jeu detaquin’ giving a bijection pλ : SST (λ) →QST (λ0 ). The diamond cone is by definition theλ0 ≤λcollection of all quasistandard tableaux. A program to construct the diamond cone was completed for sl(m), sp(2n), so(2n + 1), the rank 2 semisimple Lie algebras and sl(m, 1), sl(m, n)and spo(2n, 2m + 1) to (see [ABW, AAB, AAB1, AAK, AAK1, AK, ?]).
REFERENCES[AAB ]Agrebaoui, B., Arnal, D., Hassine, A. B. (2015). Jeu de taquin and diamond cone for Lie(super) algebras. Bulletin des Sciences Mathématiques, 139 (1), 75-113.
[AAB1 ] B. Agrebaoui, D. Arnal, A. Ben Hassine, "Diamond module for the Lie algebra so(2n +1, C)"; arXiv:1208.3349v1 (2012),41 p.
[AAK ]B. Agrebaoui, D. Arnal, O. Khlfi, "Diamond cone for sl(m, n)" arXiv.org arXiv: 1211.4158v
12012),30 p.
[AAK1 ]B. Agrebaoui, D. Arnal, O. Khlfi, "Diamond representations for rank two semisimple Liealgebras"; J. Lie Theory 19 (2009), no. 2, 339-370.
[ABW ]D. Arnal, N. Bel Baraka, N. Wildberger, "Diamond representations of sl(n)"; Ann. Math.
Blaise Pascal, 13 (2006), no. 2,381-429.
[AK ]D. Arnal, O. Khlfi, "Le cône diamant symplectique"; Bull. Sci. Math. 134 (2010), no. 6,
635-663.
[AAK ]O. Khlfi, "Diamond cone for sl(m/1)"; Bulletin des Sciences Mathématiques Volume 134,
Issue 7, October November 2010, Pages 724-746.

1.3
1.3.1

Invariants réels d’un pinceau en courbe de genre 2

Mouadh Akriche

Je donne les invariants déterminants d’une manière unique la classe réelle d’un pinceau en
courbes de genre 2. Avant de donner la preuve de notre résultat principal, j’expliquerai les invariants de la classe complexe de tel pinceau dans le contexte analytique (Celui de Y. Namikawa
et K. Ueno puis dans le contexte algébrique (celui de E. Viehweg). (Travail en collaboration
avec S. Moulahi).

CHAPTER 1. ABSTRACTS

1.4
1.4.1

3

Hom-Gerstenhaber algebras up to homotopy

Walid Aloulou

We study the concept of algebra up to homotopy for a structure defined by two operations.
An important example of such a structure is the Gerstenhaber algebra(commutatitve and Lie).
The notion of Gerstenhaber algebra up to homotopy (G8-algebra) is known. In this paper, wegive a Hom-Gerstenhaber algebra defined by a structure of commutative and Hom-associative
algebra and a structure of a Hom-Lie algebra. We will give an explicit construction of theassociated Hom-Gerstenhaber algebra up to homotopy, this is a bicoalgebra (Hom-coLie andHom-coassociative) equipped with a codifferential which is a coderivation for the two coproducts
allowing the construction of HomG8-algebra.

1.5
1.5.1

On Beurling’s Theorem for compact extensions of Rn

Salma Azaouzi

Based on a note of N. Wiener, a classical theorem of Hardy proved in 1933 states that an
integrable function on the real line and its Fourier transform fˆ cannot both have arbitrary22
Gaussian decay unless f is identically zero. More precisely, if both f (x)eαx and fˆ(x)eβx arebounded for some α, β > 0 then the following conclusions hold:1. f = 0 whenever αβ > 1/4.
2

2. The function f is a constant multiple of e−αx when αβ = 1/4.
3. When αβ < 1/4, there are infinitely many linearly independent functions satisfying bothconditions.
One year after, G. Morgan expanded the Hardy theorem by using generalized Gaussian functions. He proved that if we take p0 and q 0 real numbers such that p0 > 2 and 1/p0 + 1/q 0 = 1
p0
q0
and f a measurable function on Rn such that eαk.k f and eβk.k fˆ are bounded, then f is null
1
1
1
almost everywhere whenever (αp0 ) p0 (βq 0 ) q0 > sin( π2 (q 0 − 1)) q0 .
A few years later appeared another sharp interpretation of Wiener remark, the so-called Heinsenberg inequality which statement is:ZZ
kf k422
2
inf (x − α) |f (x)| dx (ξ − β)2 |fˆ(ξ)|2 dξ ≥.
α,β4During the years after, many authors have shown several formulating ways of the uncertainty
principle. One beautiful result among them is the Beurling-Hörmander Theorem published in
1991, which states that if an integrable function f satisfies the condition :Z Z|fb(y)||f (x)|e|xy| dxdy < ∞.
R R

Then f = 0 almost everywhere. This result was generalized in the case of nilpotent Lie groups(A. Baklouti, N. BenSalah, K. Smaoui, 2004 : the case of connected simply connected nilpotent
Lie groups, A. Baklouti, N. BenSalah, 2008 : the case of 2-NPC nilpotent Lie groups, K.
Smaoui, 2011 : the case of nilpotent Lie groups, AMA. Ghamdi, A. Baklouti, 2015 : the case of

CHAPTER 1. ABSTRACTS

4

exponential solvable Lie Groups). Now let K be a compact subgroup of automorphisms of Rn ,I will propose in this presentation a formulation of Beurling’s Theorem for compact extensionsof Rn which are the semi-direct products K n Rn .

1.6
1.6.1

Deforming a locally Euclidean geometry.

Souhail Bejar

Let G be a Lie group and H a connected closed 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 whenever H is notcompact. We treat in this talk the case where G := On (R) n Rn stands for the Euclidean motion
groups. We study the deformation spaces of any discrete Γ acting properly discontinuously and
fixed point freely on G/H for an arbitrary H. Remarkably, it happens that H turns to becompact whenever Γ is infinite, which means that the proper action resists to any deformationof Γ but probably the free action does not. Furthermore the parameter 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.

1.7
1.7.1

Spectral synthesis for coadjoint orbits of nilpotent LiegroupsIngrid Beltita

For every connected and simply connected nilpotent Lie group G, and every unitary irreducible
representation π of G, we study the space Iπ of primary ideals of L1 (G) with hull {π}. We showthat there exists a finite dimensional space of polynomials on G whose G-invariant subspaces
are in bijection with Iπ . This talk reports on joint work with Jean Ludwig.

1.8
1.8.1

Variants of Müntz-Szàsz analogues for Euclidean spingroups.

Sabria Ben Ayed

Given a strictly increasing sequence of positive integers (nk )k , the Müntz-Szàsz theorem for completeness of the monomials {xnk } in L2 ([0, 1]) can be extended to Euclidean spin groups, being
the universal coverings of Euclidean motion groups SO(n) n Rn . Towards such an objective, we
rephrase the condition of the completeness in terms of an integral against the monomials {xnk }of the coordinates functions associated to the Garding vectors of G, whose Fourier transform
are shown to admit an analytic continuation to the whole complex plane with an exponential
domination. This allows us to formulate and prove several variants of Müntz-Szàsz’s Theorem
in these settings. These upshots are proved using the Plancherel theory related to the group
Fourier transform.

CHAPTER 1. ABSTRACTS

1.9
1.9.1

5

Turán type inequality for q-special functions.

Mariem Ben Said

In this talk I would like to present some Turán type inequalities for modified q-Bessel function(q-MBF). In order to obtain the main results we apply the methods developed in the case ofclassical modified Bessel functions.
Moreover, we establish Turán type inequalities for q-Bessel Macdonald functions (q-BMF) byusing a q-version of Schwartz inequality.

1.10
1.10.1The Hardy space H 1 in the rational Dunkl setting.

Néjib Ben SalemThis paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H 1 atomically and by means of the heat maximal
operator. We also obtain a Fourier multiplier theorem for H 1 . These results are proved here in
the one-dimensional case and in the product case.

1.11
1.11.1Superior order Poincaré-type inequality related to the Jacobi semigroup.

Abdellatif Bentaleb

The aim objective of this note is to study the heat Jacobi semigroup generated by the operator
Lf (x) := (1 − x2 )f 00 + [(β − α) − (α + β + 2)x]f 0 , α, β > −1, acting on the Hilbert space
L2 ([−1, +1], µ) with respect to the normalized the Jacobi probability measure µ(dx) = c(1 −nYx)α (1 + x)β dx. We use some basic properties of the semigroups {exp(t (L − k(k + α + β +k=01)))}t≥0 , n ≥ 0, to analyze a large family of geometric inequalities that does not exist in theliterature and with which reinforced the (integral) Poincaré inequality.
Mathematics Subject Classification (2000). Primary 42A76, 47D07, 31A35; Sacondary
60J25, 60E15, 60J45.
Keywords and phrases. Jacobi operators, heat semigroup, Spectral gap, Poincaré inequality,
Superior order Poincaré-type inequality.

CHAPTER 1. ABSTRACTS

1.12
1.12.1

6

Modules of bilinear differential operators over the orthosymplectic superalgebra osp(1|2).

Taher Bichr
T. Bichr*

J. Boujelben†

K. Tounsi‡

Let Fλ , λ ∈ C, be the space of tensor densities of degree λ on the supercircle S 1|1 . We consider
the superspace Dλ1 ,λ2 ,µ of bilinear differential operators from Fλ1 ⊗ Fλ2 to Fµ as a module
over the orthosymplectic superalgebra osp(1|2). We prove the existence and the uniqueness ofa canonical conformally equivariant symbol map from Dkλ1 ,λ2 ,µ to the corresponding space ofsymbols. An explicit expression of the associated quantization map is also given.

1.13
1.13.1

Bilinear differential operators: Projectively equivariant
symbol and quantization maps.

Jamel Boujelbène

We study the space of bilinear differential operators on weighted densities as a module oversl(2; R). We introduce the corresponding space of symbols and we prove the existence and theuniqueness of canonical projective equivariant symbol and quantization.

1.14
1.14.1

Poisson transform on homogeneous vector bundles.

Abdelhamid Boussejra

We study the Lp -range of the Poisson transform on the homogeneous vector bundles over thecomplex and the quaternionic hyperbolic spaces.

1.15
1.15.1

Deforming discontinuous subgroups of 3−step nilpotent
Lie groups.

Mariem Boussoffara

Let G be a Lie group, H a closed subgroup of G and a discret subgroup of G. We consider
the deformation space T (Γ, G, H) of a discontinuous group Γ for the homogeneous space G/H.
There are many natural questions about the structure and the topological and geometric featuresof T (Γ, G, H). We interprete this phenomenon in the case where G is a 3−step nilpotent Liegroup we provide an explicit description of the parameter and the deformation spaces of any
discrete group Γ acting properly discontinuously and fixed point freely on G/H. We give astratification of the deformation space which depends upon the dimensions of G−adjoint orbits.
The topological features of deformations, such as the stability and the Hausdorff property ofT (Γ, G, H) are also discussed.

CHAPTER 1. ABSTRACTS

1.16

1.16.1

7

The Philosophy of Mathematical Modeling and Numerical Simulation: Surprises from the Wave Equation and
its Finite Difference Models.

James B. Cole

1. Philosophy of Modeling:
Contrary to what one might suppose, the best model is not the one that most accurately
fits the data. Rather, a good model is simple (has few parameters - Occam’s razor), has
the essential mathematical behavior of the system it is supposed to model, and predicts
phenomena that have yet to be observed.
2. The Wave Equation and its Finite Difference Models:
The wave equation is based on a simplified model of a stretched string and is derived
using elementary Newtonian mechanics, yet it is invariant under the Lorentz (but not the
Galilean) transformation - and hence "predicts" special relativity. The second-order finite
difference (FD) model of the wave equation is less accurate than a fourth-order one, but
it is a much better model of the wave equation. Its numerical stability condition implies
a finite upper limit on signal propagation, but it can also tell us something about thegeometry of space-time. We introduce what is called the nonstandard (NS) FD modelof the wave equation and show that its stability condition depends on the dimensionality
and connectivity of the space-time grid. A physical light signal in a medium, in which the
speed of light is lower than its vacuum, speed has a precursor wave that propagates withthe vacuum speed of light. When an FD model is used to compute signal propagation in a
medium with a propagation velocity lower than that of free space, a precursor is also seen,
the velocity of which depends on the geometry of the space-time grid. On a space-time
grid, the finite size of the cells gives rise to dispersion in the FD model. Thus if physicalspace-time were also quantized, the FD model predicts dispersion. It is unclear, however,whether or not this could be measured if it exists.
3. Green’s Function Methods for Broadband Solutions
NS-FDTD is exact for any combination of wave number (k) and angular frequency (w)that satisfy sin(w∆t/2)/ sin(kh/2) = u, where 0 < u ≤ 1 is a real constant and h =∆x = ∆y = ∆z, thus we can solve the inhomogeneous finite difference model of the wave
equation using a discrete Green’s function that (in the one-dimensional form) satisfies0τ −τ(d2τ − u2 d2χ )gχ,χ
0 = δχ,χ0 δτ,τ 0 .

Although the Green’s function for a differential equation with boundary conditions isusually difficult or impossible to find analytically, can be found using NS-FDTD. To theextent that the NS-model is exact model of the wave equation this Green’s function by
convolution with the source gives an exact solution to the inhomogeneous wave equation.

CHAPTER 1. ABSTRACTS

1.17
1.17.1

8

Solvability on Riemanien symmetric spaces.

Radhouane Daher

We study the exixtence of fundamental solutions for certain invariant linear differential operators
on simply connected riemannian symmetric spaces S. We prove that an invariant differential
operator on S admits a fundamental solution if and only if its partial Fourier coefficients satsfy
a condition of slow growth.

1.18
1.18.1

Note on the cortex of some exponential Lie groups.

Béchir DaliIn this paper, we built a family of 4d-dimensional two-step nilpotent Lie algebras (gd )d≥2 so thatthe cortex of the dual of each gd is a projective algebraic set. This example consists a generalization of the 8-dimensional two-step nilpotent Lie algebra given in ”Irreductible representationof locally compact group that cannot be Hausdorff separated from the identity representation”(Bekka and Kaniuth) and in which the authors show that the cortex of the dual of the given
Lie algebra is a quadric. In our case, we show that the cortex of the dual g∗d of each gd is a
projective algebraic set given by:Cor(g∗d ) = {` = (zi , yi , xi ) ∈ g∗d :zi = 0, y2d−1

 X

y2i

i=1d−1d−1
Yy2j−1 = 0 }.

j=1

We also give some remarks on the cortex of the Lie group Rn n R.

1.19
1.19.1On the restriction of discrete series of a real reductive
Lie group to a subgroup locally isomorphic to SL(2, R).

Michel Duflo

We consider a reductive real connected Lie group G, and a discrete series representation π of G
(this means that π is unitary, irreducible, and has square integrable coefficients). We determine
the subgroups H of G,locally isomorphic to SL(2, R), such that the restriction of π to H isadmissible (this means that, as a representation of H, it is a –possibly infinite– direct sum ofirreducible representations of H occurring with finite multiplicities). This is a joint work with
Esther Galina and Jorge A. Vargas.

CHAPTER 1. ABSTRACTS

1.20
1.20.1

9

Asymptotic spherical analysis

Jacques Faraut

If G is a locally compact group and K a compact subgroup, the pair (G, K) is called a Gelfand
pair if the convolution algebra L1 (K\G/K) of K-biinvariant integrable functions on G is commutative. A K-biinvariant continuous function ϕ on G is said to be spherical ifZϕ(xky)dk = ϕ(x)ϕ(y)K(dk is the normalized Haar measure of the compact group K.)
An Olshanski spherical pair (G, K) is the inductive limit of an increasing sequence (Gn , Kn )of Gelfand pairs. A K-biinvariant continuous function ϕ on G is said to be spherical ifZϕ(xky)dk = ϕ(x)ϕ(y).
lim
n→∞ K
n

The spherical dual Ω of the Olshanski spherical pair (G, K) is the set of spherical functions ofpositive type. What is the relation of the spherical dual Ω with the sequence Ωn of the sphericalduals of the Gelfand pairs (Gn , Kn ) ? We have no general answer, but we will give answers forsome examples.

1.21
1.21.1

Study of some deformation spaces of connected nilpotent
Lie groups.

Sonia Ghaouar

We study some problems bound to the theory of deformation of the discontinuous subgroups on
the solvable homogeneous spaces such as topological problems like the stability and the rigidity,
and geometrical problems. We treat the cases of reduced Heisenberg Lie group, the product of
reduced Heisenberg Lie groups and the reduced threadlike Lie group.

1.22
1.22.1

Intertwining operators for representations induced fromsubgroups of nilpotent Lie groups.

Amira Ghorbel

Let G be a connected and simply connected two-step nilpotent Lie group and let Γ be a cocompact discrete subgroup of G. A new decomposition into irreducibles of the quasi-regular
representation RΓ = IndG
Γ 1 of G is given. We describe also an intertwining operator which does
not take into account the multiplicities of the decomposition. Finally, we give an explicit intertwining operator between the two representations RΓ1 and RΓ2 induced from two representationequivalent subgroups Γ1 and Γ2 .

CHAPTER 1. ABSTRACTS

1.23
1.23.110

Beta-hypergeometric probability distribution on symmetric matrices.

Abdelhamid Hassairi

We first give some properties based on independence relations between matrix beta randomvariables of the first kind and of the second kind which are satisfied under a condition on theparameters of the distributions. We then use results on Jordan algebras and their symmetriccones to introduce a class of matrix-variate beta-hypergeometric distributions containing thebeta ones as a particular case. We show that with these distributions, the proprieties established
for the beta are satisfied without any condition on the parameters. The results involve many
remarkable properties of the zonal polynomials with matrix arguments and the use of randommatrix continued fractions.

1.24
1.24.1

Lp -Fourier transforms on compact extensions of unimodular Lie groups.

Junko Inoueb be the unitary dual of G. We are concerned
Let G be a unimodular Lie group of type I and Gpppwith the L -Fourier transform F = F (G) on G for exponents 1 < p ≤ 2 and q = p/(p − 1),b by the Hausdorff-Young theorem
which is defined as a continuous operator F p : Lp (G) → Lq (G)generalized by Kunze for non-abelian groups. In this talk, we study the norm kF p (G)k of the
Lp -Fourier transform. We treat the case when G is a compact extension of a unimodular Liegroup N of type I and obtain that the norm satisfies kF p (G)k ≤ kF p (N )k. This is a joint workwith Ali Baklouti.

1.25
1.25.1On the space of holomorphic equivalence classes of bounded
homogeneous domains.

Hideyuki Ishi

Piatetskii-Shapiro constructed a family of mutually non-biholomorphic 7-dimensional bounded
homogeneous domains parametrized by a closed bounded interval. His construction made use ofa family of homogeneous embeddings from 2-dimensional unit ball into the Siegel disk of rank 4.
Inspired by this method, we shall discuss how to parametrize holomorphic equivalence classesof n-dimensional bounded homogeneous domains. Since bounded homogeneous domains are in
one-to-one correspondence with normal j-algebras up to natural isomorphisms, the problem isreduced to a close investigation of the root space decomposition of normal j-algebras.

CHAPTER 1. ABSTRACTS

1.26
1.26.1

11

Intertwining operators for holomorphic multiplier representations ans their infinitesimal generators.

Souheyl Jendoubi

Given a Lie group G acting two domains D1 and D2 in Cn and let ζ, η be holomorphic functions
on D1 such that η(D1 ) ⊂ D2 and ζ : D1 →. We determine necessary and sufficient conditions on
ζ, η in order that the weighted composition operator Wζ,η induced by ζ and η be an intertwining
operator for certain reducible holomorphic Lie group representations having the form(j)
(Tg(j) F )(z) = h(j)g ∈ G, j = 1, 2,
g (z)F kg (z) ,
(j)

(j)

where hg : Dj → are holomorphic functions, called the automorphic factors and kg : Dj → Dj
are biholomorphic automorphisms of Dj . We give a solution of the this problem in the case when
D1 is homogeneous. Furthermore, we examine conditions on ζ, η to ensure that Wζ,η is also an(j)intertwining operator for the infinitesimal representation of Tg given bydρ(j) (v)F (z) =Tg(j)F (z) = H (j) (v)F (z) + b(j) (v)(z)F (z),d =0where (g )∈R is a smooth one-parameter subgroup of G, v ∈ G = Te G, H (j) (v) is an holomorphic
vector field and b(j) (v) is an holomorphic function.
As an analogy of change of coordinates on a Riemannian surface, the intertwining identity allow
us to establish a relationship between the structure constants for pairs of the following Lie bracketdddvector fields, [H (j) (v) , H (j) (w)], [ dz, [H (j) (v) , H (j) (w)] ] and [ dz
, [ dz, [H (j) (v) , H (j) (w)] ] ](z) −
 d

d
(j)
(j)
[ dz , H (v)] , [ dz , H (w)] .

1.27
1.27.1Inequalities in Clifford-Fourier Analysis.

Rim JdayRecently, there has been an interest on generalizations of the Fourier transform. Specially there
is a focus on the so-called Clifford-Fourier transform introduced by Sommen. In this context,
many research are devoted to find similar results to the classical Fourier transform for the
Clifford-Fourier transform. In 2011, De Bie and Xu clarify the Clifford-Fourier transform byseveral properties and an integral expression given by:Z−m
F − (f )(x) = (2π) 2
K − (y, x)f (y)dy,
Rm

where

π
K − (y, x) = exp(i Γx ) exp(−ihy, xi).
2
In this paper, according to the work of De Bie and Xu, we add further properties to the theClifford-Fourier transform: we provide the Riemann-Lebesgue lemma, in addition we prove that−(m−2)
the the Clifford-Fourier transform is bounded in C(Rm , (1 + |x|) 2 ) and we state a versionof Young’s inequalities for the the Clifford-Fourier transform.

CHAPTER 1. ABSTRACTS

1.28
1.28.1

12

On a Solutions of D’Alembert’s Functional Equation.

Samir Kabbaj

The aim of this conference is to give the solutions of the following functional equations:n−1
Xf (xσ k (y)) = nf (x)f (y)

(I)

k=0

where x, y ∈ G which is a compact group, n ∈ N∗ and σ is a continous authormophism of G,
such that σ n = IG . We express the solutions in terms of unitary characthers of G.
Z{f (x + y − t) + f (x − y + t)}dµ(t) = f (x) + f (y)(II)
G

where (G, +) is a locally compact abelian Hausdorff group, and µ is a regular compactly supported complex-valued Borel measure on G such that µ(G) = 12 .
ZZf (xty)dµ(t) + f (σ(y)tx)dν(t) = 2f (x)g(y)
(III)
G

G

where G is a locally compact group, σ is a continious involution of G and µ and ν are a complex
bounded and σ-invariant measures. In the end, I will investigate the superstability of thisfunctional equation.
Keywords: Fourier transform, Unitary representation of a compact group, Cauchy, D’Alembertand Welson Functional Equation, Quadratic and Additive Functions, Hyers-Ulam Stability.

1.29
1.29.1

On discontinuous groups acting on solvable homogeneous
spaces.

Imed Kédim

Let G be a Lie group, H a closed subgroup of G and Γ a discrete subgroup of G. If the actionof 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 theparameter 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 geometric featuresof τ (Γ, G, H). In this lecture, we present some recent developments, focusing on Hausdorffness,stability and rigidity problems in the context of some Lie groups.

CHAPTER 1. ABSTRACTS

1.30
1.30.1

13

Hamiltonians for polynomial potentials and representations of nilpotent groups.

Khalid Koufany

Let H be a Hamiltonians of polynomial potential. We associate to H a nilpotent group G, andconsider a unitary representation π of G induced from a subgroup K ≤ G such that H = π(D)where D is a left-invariant differential
operator. The representation π can be decomposed in aR⊕
direct sum of irreducibles, π =πλ dµ(λ). The analysis of the operators πλ (D) gives solutionsof the Shrodinger equation −Hu = ∂u/∂t. Our talk will mainly focus on the quartic and the
Hénon-Heiles Hamiltonians.

1.31
1.31.1

The group algebra L1 (G).

Jean Ludwig

The multiplication in a locally compact group G with Haar measure dx determines the convolution product in the involutive Banach algebra L1 (G) = L1 (G, dx) and this algebra determines
also the group G up to isomorphism. The structure of this algebra is mostly unknown. We recall
some properties of this algebra, properties which have been discovered during the last 50 years.

1.32
1.32.1

Twistings of algebra and bialgebra structures.

Abdenacer Makhlouf

In the last years, many concepts and properties from classical algebraic theories have been
extended to the framework of Hom-structures. In this talk,we deal with a recent generalization
involving two linear maps. We mainly discuss constructions and representations of BiHomassociative algebras, BiHom-Lie algebras and BiHom-bialgebras, as well as some constructionsof twisted tensor products and smash products.

1.33
1.33.1

Transfer group for renormalized multiple zeta values.

Dominique Manchon

How to extend multiple zeta values to integer arguments of any sign such that the quasi-shuffle
relations are still verified? We will show that the set of solutions to this problem is nonempty,
and is a principal homogeneous space under the left action of a pro-unipotent group. Joint workwith K. Ebrahimi-Fard, J. Singer and J. Zhao.

CHAPTER 1. ABSTRACTS

1.34
1.34.1

14

An Integral Expression for the Dunkl Kernel in the Dihedral Setting.

Mostafa Maslouhi

We establish an integral expression for the Dunkl kernel in the context of Dihedral group of
an arbitrary order. As a consequence, an explicit expression of the Dunkl kernel Ek (x, y) is
given when one of its argument x or y is invariant under the action of a known reflection in theDihedral group. We obtain also a generating series for the homogeneous components Em (x, y),m ∈ Z+ , of the Dunkl kernel from which we derive new sharp estimates for the Dunkl kernel
when the parameter function k satisfies mrRe(k) > −ν, ν an arbitrary nonnegative integer.

1.35
1.35.1

Matrix-valued commuting family of differential operators
associated with symmetric spaces.

Shimeno Nobukazu

I present commuting family of matrix-valued differential operators whose coefficients are givenby elliptic functions. These operators are generalization of radial parts of invariant differential
operators on certain homogeneous vector bundles on Riemannian symmetric spaces of rank two.
I also discuss about eigenfunctions.

1.36
1.36.1

Ideals in L1 (G) associated with compact orbits in theunitary dual of a nilpotent Lie group G.

Detlev Poguntke

For each closed ideal I in L1 (A), A a locally compact abelian group, one may consider, as aninvariant, its hull h(I) in the Pontryagin dual A∧ . A closed subset Ω of A∧ is called a set ofsynthesis if there is exactly one ideal whose hull is Ω. Besides several positive results it turned
out that even spheres in Rn are not sets of synthesis for n ≥ 3 (L. Schwartz, N. Varopoulos).
With each closed subset of A∧ one can associate a largest ideal whose hull is Ω, namely the kernelk(Ω), and also a smallest ideal, say j(Ω), whose hull is Ω. The quotient k(Ω)/j(Ω) “measuresthe distance” from being a set of synthesis. For spheres this quotient is nilpotent, there is m ∈ Nwith [k(S n−1 )/j(S n−1 )]m = 0.
Some 30 years ago the interest in this circle of questions seemed to decline, but it was revitalized
by Jean Ludwig in a non-commutative context, in particular for a simply connected nilpotent
Lie group G. Replacing the Pontryagin dual by the unitary dual G∧ one may as well associatewith each closed subset Ω of G∧ the kernel k(Ω) which is the largest ideal with hull Ω. Using
Dixmier’s functional calculus J. Ludwig established also the existence of a smallest ideal j(Ω).
Even a singleton is a set of synthesis only very rarely, but J. Ludwig could show that k(Ω)/j(Ω)
is nilpotent in this situation. It is the purpose of this talk to contribute such claims in case oforbits under a compact group, or even a semidirect product of a compact group with a unipotent
group.

CHAPTER 1. ABSTRACTS

1.37
1.37.1

15

On the Chabauty space of a locally compact group.

Firas Sadki

Let G be a locally compact group. We denote by Sub(G) the hyperspace of closed subgroups of
G endowed with the Chabauty topology. In this talk we study the continuity of the following
map
CentG : G −→ Sub(G); g 7−→ CentG (g);where CentG (g) is the centralizer of g in G.

1.38

1.38.1

Admissible representations, multiplicity-free representations, and visible actions on non-tube type Hermitiansymmetric spaces.

Atsumu Sasaki

In this talk, we give a new characterization for a non-compact Hermitian symmetric space
to be of tube type (or non-tube type) by multiplicities in some branching laws and visible
actions. Further, we give an example of a kind of the Cartan decomposition for non-symmetrichomogeneous spaces.

1.39
1.39.1

Cohomology of sl(2) acting on the space of n-ary differential operators on R.

Rabeb Sidaouin1
We compute the cohomological space Hdiff (sl(2), Dλ,µ ) where µ ∈ R, λ = (λ1 , . . . , λn ) ∈ R
and Dλ,µ is the space of multilinear differential operators from Fλ1 ⊗ . . . ⊗ Fλn to Fµ . The
structure of these spaces was conjectured in [M. Ben Ammar et al. in International Journal of
Geometric Methods in Modern Physics Vol. 9, No. 4 (2012) 1250033 (15 pages).]
Keywords: Cohomology, Weighted Densities.

1.40
1.40.1

Trajectories of Quadratic Differentials for Generalized Laguerre and Jacobi Polynomials.

Faouzi Thabet

αnThe motivationof this lecture is the large-degree analysis of the behavior of the Laguerre (Ln )αn ,βnand Jacobi Pn
polynomials, when the parameters are complex and depend on the degree nlinearly. We show that the Cauchy transforms of the limits (weak) of the roots-counting measuresof these polynomials satisfy quadratic algebraic equations of the form : p(z) C 2 (z)+q(z)C(z)+r =0, with deg p = 1, 2, deg q = 1, and r ∈ C∗ . We investigate the existence of solution of theseequations as Cauchy transform of compactly supported positive measures. Any connected curveCHAPTER 1. ABSTRACTS

16

of the support of these measures (if exist) coincides with a horizontal trajectory of some quadratic
differential. In this lecture, we discuss the trajectories of a family of quadratic differentials, andwe emphasize the connection with the distribution of zeros of generalized Laguerre & Jacobi
polynomials.
Keywords : Orthogonal polynomials. Cauchy transform of a Borelian measure. Quadratic
differentials. Trajectories and orthogonal trajectories of a quadratic differential. Homotopy
classes.

1.41
1.41.1

Vertical and horizontal rigidity.

Taro Yoshino

Ali Baklouti conjectured that no Clifford-Klein form Γ\G/H is locally rigit if G is a 1-connectednilpotent Lie group and H is a closed subgroup. In this talk we introduce the concept of verticalrigidity and horizontal rigidity. Here, one can easily check that a Clifford-Klein form is locallyrigid if and only if it is vertical and horizontally rigid