Geometric and Harmonic Analysis on Homogeneous Spaces
Kerkennah, November 2-7, 2009

Abstracts
Ahmed ABOUELAZ
The explicit inversion formulas for the Radon transformin the torus Tn and its dual Zn
Abstract. We establish the explicit inversion formulas for the Radon transform in thetorus Tn and also on the lattice Zn , by using certain arithmetical geometrical techniques.

Boujemaa AGREBAOUIOn cohomology and deformations of the Lie superalgebra ofcontact vector fields on S 1|m
Abstract. The classical deformation theory of Lie algebras and modules over Lie algebras
traditionally deals with one parameter deformations (see M. Gerstanhaber, A. Nijenhuis
and R. W. Richardson.) It is, however, natural to consider, as in other deformation
theories, multi-parameter deformations. This viewpoint was initiated by A. Fialowski
and D. Fuks. The Constructions of deformations of the natural embedding of Vect(S 1 )of vector fields in the circle S 1 inside the Poisson algebra of Laurent series on Ṫ ∗ (S 1 ) (resp.
the Lie algebra of pseudodifferential operators ) are considered by V. Ovsienko and C.
Roger . The multiparameter deformations of the Lie derivative action of the Lie algebra
Vect(Rn ) of vector fields on Rn on the space of symmetric andantisymmetric tensor fields
was considered by B.Agrebaoui F.Ammar, M.Ben ammar,N.Ben Fraj, V.Ovsienko and
P. Lecomte.
The first step of any approach to the deformation theory consists in the study of
infinitesimal deformations. Given a Lie algebra (or superalgebra) g and a g-module V ,the infinitesimal deformation is defined, up to equivalence,by the cohomology classes
c1 , . . . , cn in H 1 (g, End(V )).
The second step is to compute the integrability conditions of infinitesimal deformations.
For the vector fields Lie superalgebra case, the first examination apperies in the endof the seminal paper on modular forms by P.B. Cohen, Y. Manin, and D. Zagier. In
my talk, I’ll speak about a series of recent papers with N.Ben Fraj, S.Omri and S.
Mansour, where the pseudodifferential operator module-valued first cohomology groups1cohomology of the Lie superalgebra K(n) of contact vector fields on S 1|n (or R1|n ) is
considered and multiparameter deformation has been studied.
The main tools of this lecture are in the papers:- On the Cohomology of the Lie Superalgebra of contact vector fields on S ( 1|2). Nonlinear Mathematical Physics Vol 13 (2006) Number 4 523-534 (with N.Ben Fraj et Salem
Omri).
- On the cohomology of Lie superalgebra of contact vector fields on S 1|m Accepté pourpublication au journal Communications in Algebra (with Sabeur Mansour).
-Deforming the Lie Superalgebra of Contact Vector Fields on S1|2. Accepted for publication in journal of Geometry and Physics (with Sabeur Mansour).

Didier ARNAL
Diamond cones and quasistandard Young tableaux
Abstract. In this lecture, we shall recall the notion of shape algebra S for a semisimple
Lie algebra g. Then we define the diamond cone C of g as a quotient of S.
Diamond cone was introduced by N.J. Wildberger for sl(3). If n is the nilpotent partof the Iwasawa decomposition of g, the diamond cone is a n module, built from the collection of all maximal, locally nilpotent n modules.
To understand the structure of the n module C, we look for an explicit, combinatorial
basis for C. Generally speaking, such a basis is given by particular Young tableaux: thequasistandard Young tableaux. We shall present here explicit results for the following
cases:
g = sl(m) (with N. Bel Baraka and N.J. Wildberger),g = sp(2m) (with O. Khlifi),
rank(g) = 2 (with B. Agrebaoui and O. Khlifi),
g = sl(m, 1) (this generalization is due to O. Khlifi).

Najib BEN SALEM
Some applications of harmonic analysis associatedwith the Jacobi-Dunkl operators
Abstract. The Jacobi-Dunkl operators are differential-difference operators of the form:
0Aα,β f (x) − f (−x)
df (x) +()
Λα,β (f )(x) =dx
Aα,β (x)2where11Aα,β (x) = 22p (sin h2 (x))α+ 2 (cos h2 (x))β+ 2 , x ∈ R, in the non compact case
and
1
1
π π
Aα,β (x) = 22p (sin2 (x))α+ 2 (cos2 (x))β+ 2 , x ∈ [− , ], in the compact case.
2 2Here α ≥ β − 12 , α 6= − 12 and ρ = α + β + 1.
In the first case, the differential-difference equation½ V
α,β υ(x) = −λυ(x), λ ∈ C
υ(0),
= 1,
admits a unique C ∞ solution ψλα,β on R given by
½ α,β
id
α,β
ψλ (x) = ϕα,βµ (x) + λdx ϕµ (x)if λ \ {0} ∈ Cα,βψ0 (x), = 1,
where ϕµ is the Jacobi function and λ2 = µ2 + ρ2 .
While, in the second case, the differential-difference equation½ V
α,β υ(θ) = −iλn υ(θ), n ∈ Z
υ(0)
= 1,
admits a unique C ∞ solution ψnα,β on ] − π2 , π2 [ given by
(
α,β
α,β
d
P|n|
(cos(2θ))if λ ∈ Z\{0}ψnα,β (θ) = P|n|(cos(2θ)) + λin dθα,βψ0 (θ) = 1,pα,βwhere P|n|is the Jacobi polynomial of degree |n| and order (α, β), λn = 2sgn(n) |n|(|n| + ρ).
With the help of the properties of Jacobi functions and polynomials and using harmonic
analysis associated with these operators, we study some applications, especially, we deal
with Poisson and conjugate Poisson integrals, Hilbert transforms, also we prove theboundedness of such operators.

Radhouane DAHER
Further results in Dunkl harmonic analysis
Abstract. We obtain new inequalities for Dunkl transform. There are expressed as gauge
on the size of the transform in terms of a suitable integral modulus of continuity of thefunction. We prove also an analogue of Titchmarch’s theorem and Jackson’s theorem for
Dunkl transform. (This a joint work with Prof M. Maslouhi.)

Sami DHIEB
When the deformation space T (Γ, H2n+1 , H) is a manifold
Abstract. Let H2n+1 be the 2n+1-dimensional Heisenberg group and H a connected Lie
subgroup of G. Given any discontinuous subgroup for G/H, we know a precise layering
into open sets of the resulting deformation space T (Γ, H2n+1 , H). We study in this talkwhen this space is endowed with a smooth manifold structure. (Joint work with Ali
Baklouti and Khaled Tounsi)

Michel DUFLO
Weyl’s functional calculus and equivariant differential forms
Abstract. Let A1 , A2 , ..., Ad be d Hermitian matrices of size n. Weyl’s functional calculus
is a compactly supported distribution W on Rd which associates to a smooth function
f of dvariables a matrix W (f ) := f (A1 , ..., Ad ). Forty years ago, Edward Nelson gave aformula for W , explicitly describing it as the derivative of a probability measure on Rd
supported on the joint numerical range of the Ai . We show how this formula fits in thesetting of Hamiltonian geometry and equivariant differential forms.

Ahmed FITOUHI
An essay toward a unified theory of the Lp version of
Hardy’s inequality and q-analogues
Abstract. In this paper, we give an Lp version of Hardy’s uncertainty principles for alarge class of integral transforms. As applications, we discuss an Lp version of Hardy’s
theorem for the generalized Fourier transform associated to the Sturm-Liouville operator
and for the Jacobi-Dunkl transform.

Hidenori FUJIWARA
Monomial representations with multiplicities of discrete type
Abstract. Let G = exp g be an exponential solvable Lie group with Lie algebra g andb the unitary dual of G. Let H = exp h be a closed connected subgroup of G and χ aGunitary character of H. We construct the induced representation τ = indG
H χ of G andconsider the canonical irreducible decomposition of τ :
Z ⊕
τ'

b
G

m(π)πdµ(π)

b and the multiplicity function m(.). We discuss some topics
with a Borel measure µ on G
concerning τ when the data (G, H, χ) satisfy a certain condition. This a joint work with
Ali Baklouti and Jean Ludwig.

Junko INOUE
Estimate for the norm of the Lp -Fourier transformon nilpotent Lie group
Abstract. Let G be a connected nilpotent Lie group, G̃ be its universal covering group,
G = G̃/Γ, where Γ is a discrete subgroup, and let 1 < p ≤ 2, q be such that 1q + p1 = 1.
We discuss the Lp -Fourier transform F p (G) on G and obtain an estimate for the norm as
follows: Let Λ = log Γ, h = R-span(Λ), H̃ = exp h, H = H̃/Γ. Then we have kF p (G)kq ≤Aνp , where ν = 12 (2 dim(G/H) − m), m is the dimension of the generic coadjoint orbits,
1

1

1

and Ap = (p p /q q ) 2 . (This is a joint work with Ali Baklouti.)

Hideyuki ISHI
Equivariant holomorphic line bundles over a bounded
homogeneous domain
Abstract. Let D be a bounded homogeneous domain and G a split solvable Lie groupacting on D simply transitively. We shall see that G-equivariant holomorphic line bundles
L over D are parametrized by one-dimensionel representations χ of G as L = Lχ . We
shall also determine all χ for which the representation πχ of G defined naturally on thespace Γhol (Lχ ) is unitarizable. The classification of such unitary representation πχ willbe also given.

Philippe JAMING
Distributions that are convolvable with Poisson kernels
Abstract. In this talk, we will characterise all distributions on a homogeneous Lie groups
that can be extended via a convolution with a Poisson (type) kernel. To do so, we willtake an appropriate definition of convolution of distributions and show that distributions
that are convolvable with Poisson kernels are derivatives of appropriately weighted L1
functions. Moreover, the result of this convolution has all the expected properties (harmonicity, Fatou type theorem...). (This is joint work with E. Damek, J. Dziubanski and
S. Prez-Esteva.)Chifune KAI
The representative domain of a homogeneous bounded domain
Abstract. The representative domain introduced by S. Bergman gives a nice realization
for a homogeneous bounded domain, which is a generalization of the Harish-Chandra
realization for a symmetric bounded domain. We show that the representative domain
coincides with the image of the Cayley transform introduced by R. Penney and T. Nomura. As an application, we see that a homogeneous bounded domain is symmetric if
and only if its representative domain is convex.

Takeshi KAWAZOE
Hardy space for Jacobi analysis and its applications
Abstract. Let (R+ , ∆) denote a space equipped with the measure ∆(x)dx, where∆(x) = (2shx)2α+1 (2chx)2β+1 ,x ∈ R+ , α ≥ β ≥ −1/2, and a suitable convolution associated with Jacobi function
φλ (x). Since ∆(x) has an exponential growth order, (R+ , ∆) is not of homogeneous type.
A real Hardy space H 1 (∆) on (R+ , ∆) is induced by using radial maximal operator asa usual manner. Then, H 1 (∆) is related with the classical weighted H 1 Hardy spaces
on R, especially, it can be characterized in terms of weighted Tribel-Lizorkin spaces
on R. Hence, we can obtain an atomic decomposition of H 1 (∆) and the interpolation
space between H 1 (∆) and L1 (∆) that coincides with Lp (∆). As an application, we shallR∞
consider Fourier multiplier for Jacobi transform fb(λ) = 0 f (x)φλ (x)∆(x)dx.

Imed KEDIM
On discontinuous subgroups acting on homogeneous spaces
Abstract. In this lecture, we give a short and basic historical introduction to the theoryof deformations associated to Clifford-Klein forms. We present an algebraic formula ofthis spaces in the setting of completely solvable Lie groups. As an application, we describe
this space in the case of the Heisenberg group.(Based on joint works with Ali Baklouti).

Mohamed Salah KHALGUIClasse des algèbres de Lie quasi-réductives
Abstract. Nous étudions une classe d’algèbres de Lie, appeles algèbres de Lie quasiréductives, liée à l’étude des séries discrètes des groupes de Lie réels et ayant des relations
avec les espaces préhomogènes au sens de Kimura-Sato.

Fatma KHLIF
Deforming discontinuous subgroups forthreadlike homogeneous spaces
Abstract. Let G be an exponential solvable Lie group and H a connected Lie subgroupof G. Given any discontinuous subgroup Γ for the homogeneous space M = G/H andany deformation of Γ, the deformed discrete subgroup may utterly destroy its proper
discontinuous action on M as H is not compact (except the case when it is trivial).
To understand this specific issue, we provide an explicit description of the parameterand the deformation spaces of any abelian discrete Γ acting properly discontinuously
and fixed point freely on G/H for an arbitrary H of a threadlike nilpotent Lie group G.
The topological features of deformations, such as the local rigidity and the stability arealso discussed. Whenever the Clifford-Klein form Γ\G/H in question is assumed to becompact, these spaces are cutely determined and unlike the case of Heisenberg groups,
the deformation space fails in general to be a Hausdorff space. In such a case, this space
is shown to admit a smooth manifold as its open dense subset. (This is a joint work with
Ali Baklouti).

Jean LUDWIG
Simple modules of the L1 group algebra of SL2 (R)Abstract. We determine the simple modules of the algebra L1 (SL2 (R)) up to equivalence
and we show that these modules are the finite rank submodules of the Lp -principal
series and of the discrete series representations of SL2 (R). This is a joint work with A.
PASQUALE.

Khemais MAKTOUF
The Plancherel formula for solvable p-adic groups
Abstract. In this talk, we prove a Plancherl formula for almost connected solvable padic groups (p 6= 2). To do this, we give a necessary and sufficient condition for a unitary
irreducible representation to be admissible in terms of Kirillov-Duflo orbit method. Thenwe establish a character formula around each semi-simple element.

Dominique MANCHON
A deformation approach of the Kirillov map for exponential groups
Abstract. We consider a particular set of ideals of the C ∗ -algebra of the tangent groupoidof an exponential Lie group G, such that the evaluation ev1 at t = 1 maps this set onto

the unitary dual of G, whereas the evaluation ev0 at t = 0 identifies it with the set of
coadjoint orbits. The composition ev1 ◦ ev0−1 is the Kirillov map. We will indicate how analternative proof of the Leptin-Ludwig bicontinuity theorem should be expected along
these lines. Joint work in progress with A. Baklouti and S. Dhieb.

Sho MATSUMOTO
Orthogonal matrix integrals and zonal spherical functions name
Abstract. We consider about integrals of polynomial functions of Haar-distributed orthogonal matrix entries. Benoit Collins and Piotr Sniady proved that the integrals areevaluated by using special functions on symmetric groups. The functions are called Weingarten functions. Our main result is to give an expansion of the Weingarten function withrespect to zonal spherical functions of a finite Gelfand pair. Moreover, we derive a connection between Weingarten functions and Jucys-Murphy elements. This talk is a partof the joint work with Benoit Collins.

Carine MOLITOR-BRAUN
Flat orbits, primary ideals and spectral synthesis
Abstract. Let G = exp g be a connected, simply connected, nilpotent Lie group and let ω
be a continuous symmetric weight on G with polynomial growth. Let l ∈ g∗ be such thatthe co-adjoint orbit of l is flat. In the weighted group algebra L1ω (G) we characterize all
the twosided closed ideals whose hull is {πl }, where πl denotes the element of Ĝ associatedto the co-adjoint orbit Ad∗ (G)(l) by the Kirillov map. These ideals are parametrized bya set of G-invariant, translation invariant spaces of complex polynomials dominated by
the weight ω and are realized as kernels of specially built induced representations. This
results among others from the fact that, if the co-adjoint orbit of l is flat, every closed
two-sided ideal of L1ω (G) with hull {πl } is necessarily L∞ (G/G(l))-invariant, where G(l)denotes the stabilizer of l in G. (This is a joint work with J. Ludwig).

Atsumu SASAKI
Visible actions on complex vector spaces
Abstract. I plan to explain that the multiplicity-free space has a strongly visible actions.
Let V be a vector space over C and GC be a connected complex reductive Lie group. Givena holomorphic representation of GC on V , we define a repesentation π on the polynomial
ring C[V ] of GC . We show that the polynomial representation π is multiplicity-free if andonly if the action of a maximal compact subgroup of GC on V is strongly visible.

Mohamed SELMI
Séparation des représentations par les sur-groupes quadratiques
Abstract. Let π be an unitary irreducible representation of a Lie group G. π defines amoment set Iπ , subset of the dual g§ of the Lie algebra of G. Unfortunately, Iπ does notcharacterize π.
However, we sometimes can find an overgroup G+ for G, and associate, to π, a representation π + of G+ in such a manner that Iπ+ characterizes π, at least for genericrepresentations π. If this construction is based on polynomial functions with degree at
most 2, we say that G+ is a quadratic overgroup for G.
In this paper, we prove the existence of such a quadratic over-group for many diﬀerent
classes of G.

Nobukazu SHIMENO
Matrix-valued commuting diﬀerential operators
associated with root systems
Abstract. We obtain matrix-valued commuting diﬀerential operators whose coeeficientsare elliptic functions, by generalizing explicit expressions of matrix-valued commuting
diﬀerential operators on homogeneous vector bundles on certain rank 2 symmetric spaces.

Mohamed SIFI
Three results in Dunkl analysis
Abstract. In this talk, we establish first a geometric Paley–Wiener theorem for the
Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the
Lp → Lp norm of Dunkl translations in dimension 1. Finally we describe more precisely
the support of the distribution associated to Dunkl translations in higher dimension.
(This is a joint work with B. Amri and J. Ph. Anker).

Kais SMAOUIOn Beurling’s theorem for nilpotent Lie groups
Abstract. In the case of the real line, the Beurling theorem asserts that for any non
trivial function f ∈ L1 (R) ∩ L2 (R), the function f (x)fb(y) is never integrable on R2
with respect to the measure e|xy| dxdy. We prove an analogue of this theorem for an
arbitrary simply connected nilpotent Lie group extending then earlier cases and theclassical Cowling-Price theorem proved recently by A. Baklouti and N. Ben Salah.

Sundaram THANGAVELU
A Paley-Wiener theorem for eigenfunction expansions
Abstract. We prove a Paley-Wiener theorem for certain eigenfunction expansions which
includes Fourier series on compact Lie groups, spherical harmonic expansions, Hermiteand Laguerre series. We characterise functions having finite expansions in terms of growthestimates of their holomorphic extensions. We use the holomorphically extended heat
kernels to measure the growth.

Khalifa TRIMECHE
Harmonic analysis associatedwith the Cherednik operators and the Heckman-Opdam theory
Abstract. By using the trigonometric Dunkl intertwining operators and their duels
introduced by the author, we define and study the hypergeometric translation operators
associated with the Cherednik operators T1 , ..., Td and the Heckman-Opdam theory. Next
with the help of these translation operators we define and study the hypergeometricconvolution product of functions and distributions associated with the operators T1 , ..., Td
and the Heckman-Opdam theory.

Yoshinori YAMASAKI
Zeta regularized products and higher depth
determinants of Laplacians
Abstract. In this talk, as analogues of the Milnor gamma functions, we introduce ”higher
depth determinants” of Laplacians via zeta regularized products. In particular, we explicitly calculate them in the case of compact Riemann surfaces with negative constantcurvature and higher dimensional spheres.