# Baklouti AliUniversity of Sfax | US · Department of Mathematics

Baklouti Ali

Prof. Doctor

## About

111

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Introduction

Representation theory of Lie groups, Geometric structures and non-commutative harmonic analysis

## Publications

Publications (111)

Let [Formula: see text] be a compact subgroup of [Formula: see text] and [Formula: see text] the semi-direct product group. Let [Formula: see text] be a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. We establish a geometrical criterion of the proper action of [For...

Let G = exp ( g ) G = \operatorname {exp}(\mathfrak {g}) be a connected and simply connected real nilpotent Lie group with Lie algebra g \mathfrak g , H = exp ( h ) H = \operatorname {exp}(\mathfrak {h}) an analytic subgroup of G G with Lie algebra h \mathfrak h , χ \chi a unitary character of H H and τ = ind H G χ \tau = \text {ind}_H^G \chi t...

In [Formula: see text], Auslander conjectured that every crystallographic subgroup [Formula: see text] of the affine group [Formula: see text] acting properly discontinuously on [Formula: see text] is virtually solvable, i.e. contains a solvable subgroup of finite index. One of the major difficulties to prove such a conjecture is to generate a crit...

Given a Müntz–Szàsz sequence of positive real numbers \((\lambda _k)_k\) and a bounded interval \(I\subset {\mathbb {R}}\), Müntz–Szàsz theorem for completeness of the monomials \(\{x^{\lambda _k}\}_k\) in \(L^2(I)\) can be extended to a class of compact extensions of Heisenberg groups. The idea is to define infinitely many coordinate functions of...

We remember Takaaki Nomura’s outstanding contribution to international collaborations of mathematicians as well as his significant works on representation theory of Lie groups and non-commutative harmonic analysis on homogeneous spaces.

Once we leave the exponential picture, we know the fundamental results by Auslander-Kostant and Pukanszky. Here we study some intertwining operators and real polarizations. First, let be a connected and simply connected nilpotent Lie group with Lie algebra and a complex positive polarization of at .

Let π be a unitary representation of a connected Lie group G acting on a Hilbert space Hπ. Denote by Hπ∞ the Fréchet space of smooth vectors for π, and by Hπ−∞ the space of continuous, anti-linear functionals on Hπ∞.

When we study problems in representation theory, it is often very useful to construct an intertwining operator between two equivalent unitary representations. A prototype arises in the following situation. Let be an exponential solvable Lie group with Lie algebra and two (real) polarizations of at verifying the Pukanszky condition.

Let be an exponential solvable Lie group. In this chapter we characterize bounded, topologically irreducible Banach-space representations of G using triples ( Ω, τ, ∥∥), where is a coadjoint orbit of G, τ is a topologically irreducible representation of the algebra L1(ℝn,ω) for a certain n∈ℕ∗ and a weight ω on ℝn, and ∥∥ is a so-called extension no...

When investigating a concrete problem in the representation theory of solvable Lie groups, it is easy to realize that we have enough tools only for the nilpotent case. Here we shall consider Penney’s Plancherel formula for exponential solvable Lie groups. Let G be an exponential solvable Lie group, H an analytic subgroup of G and χ a unitary charac...

The aim of this chapter is to present two conjectures which are interesting and still open (cf. Baklouti and Ludwig (Monatsh Math 134:19–37, 2001), Baklouti et al. (Bull Sci Math 129:187–209, 2005), Corwin and Greenleaf (J Funct Anal 108:374–426, 1992), Fujiwara (J Lie Theory 7:121–146, 1997)). Here we briefly explain them. Let be a connected and s...

One of the most important problems in the disintegration theory of group representations is to write down explicit unitary operators that intertwine unitary representations and their decomposition into irreducibles. The answer to this question can help to solve several important problems, such as the solvability of invariant differential operators...

Throughout the chapter G denotes an exponential solvable Lie group. Let A and H be two closed connected subgroups of G and π and σ unitary and irreducible representations of G and H respectively. Branching laws and explicit formulas of the disintegration into irreducibles of the representations indHGσ and π|A in terms of the representations’ spectr...

The purpose of the book is to discuss the latest advances in the theory of unitary representations and harmonic analysis for solvable Lie groups. The orbit method created by Kirillov is the most powerful tool to build the ground frame of these theories. Many problems are studied in the nilpotent case, but several obstacles arise when encompassing e...

This book collects a series of important works on noncommutative harmonic analysis on homogeneous spaces and related topics. All the authors participated in the 6th Tunisian-Japanese conference "Geometric and Harmonic Analysis on homogeneous spaces and Applications" held at Djerba Island in Tunisia during the period of December 16-19, 2019. The aim...

Let G be a separable unimodular locally compact group of type I, and let N be a unimodular closed normal subgroup of G of type I, such that G/N is compact. Let for \(1<p\le 2\), \(\Vert {\mathscr {F}}^p(G)\Vert \) and \( \Vert {\mathscr {F}}^p(N )\Vert \) denote the norms of the corresponding \(L^p\)-Fourier transforms. We show that \(\Vert {\maths...

For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ℝn, which largely extends the classical Beurling’s theorem. We then define its analog for compact extensions of ℝn and also for Heisenberg groups.

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $\sigma $ such that $\sigma |_S = \operatorname{id}_S$. Let $G$ be the Heisenberg group and $H$ a non-trivial connected close...

We recall the notion of Poisson characteristic variety of a unitary irreducible representation of an exponential solvable Lie group, and conjecture that it coincides with the Zariski closure of the associated coadjoint orbit. We prove this conjecture in some particular situations, including the nilpotent case.

Let G be an exponential solvable Lie group, H an analytic subgroup of G and \(\chi \) a unitary character of H. We study some problems related to the induced representation \(\tau = \text {ind}_H^G \chi \) of G when \(\tau \) has multiplicities either finite or infinite of discrete type. In particular, we are interested in the Plancherel formula fo...

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group [Formula: see text] acting on a homogeneous space [Formula: see text], where [Formula: see text] stands for a closed subgroup of the Heisenberg motion group [Formula: see text]. That is, the parameter space admits a locally...

This book presents a number of important contributions focusing on harmonic analysis and representation theory of Lie groups. All were originally presented at the 5th Tunisian–Japanese conference “Geometric and Harmonic Analysis on Homogeneous Spaces and Applications”, which was held at Mahdia in Tunisia from 17 to 21 December 2017 and was dedicate...

Let G G be a connected and simply connected nilpotent Lie group, K K an analytic subgroup of G G and π \pi an irreducible unitary representation of G G whose coadjoint orbit of G G is denoted by Ω ( π ) \Omega (\pi ) . Let U ( g ) \mathscr U(\mathfrak g) be the enveloping algebra of g C {\mathfrak g}_{\mathbb C} , g \mathfrak g designating the Lie...

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also...

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in G^, the unitary dual of G, sends smooth vectors of the representation space of π to 𝒰(𝔤)*, the dual vector space of 𝒰(𝔤). The convex hull of the image of the generalized moment map for π is called its generalized moment set, denoted by J(π)....

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]...

Let \(G = \mathbb {G}_n^r\) be the \((n+1)\)-dimensional reduced threadlike Lie group, H an arbitrary closed subgroup of G and \(\Gamma \subset G\) a non-abelian discontinuous group for G / H. Unlike the setting where \(\Gamma \) is abelian, we show that the stability property holds on the related parameter space.

This book provides the latest competing research results on non-commutative harmonic analysis on homogeneous spaces with many applications. It also includes the most recent developments on other areas of mathematics including algebra and geometry.
Lie group representation theory and harmonic analysis on Lie groups and on their homogeneous spaces fo...

Let $G=\mathbb{H}^{n}\rtimes K$ be the Heisenberg motion group, where $K=U(n)$ acts on the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$ by automorphisms. We formulate and prove two analogues of Hardy’s theorem on $G$ . An analogue of Miyachi’s theorem for $G$ is also formulated and proved. This allows us to generalize and prove...

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space G/H, determining explicitly which homogeneous spaces G/H allow nontrivial continuous deformations of co-compact discontinuous groups. When G is assumed to be exponential solvable and H ⊂ G is a...

In this article, we study the rigidity properties of deformation parameters of the natural action of a discontinuous subgroup Γ G, on a homogeneous space G/H, where H stands for a closed subgroup of a Euclidean motion group G:= On(R)Rn. That is, we prove the following local (and global) rigidity theorem: the parameter space admits a rigid (equivale...

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 which are the universal coverings of Euclidean motion groups SO(n)⋉Rn. Towards such an objective, we rephrase the completeness condition in terms of an integral again...

Let G = ℍr2n+1 be the (2n +1)-dimensional reduced Heisenberg group, and let H be an arbitrary connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, we show that resulting deformation space J (Γ,G,H) of the natural action of Γ on G/H is endowed with a smooth manifold structure and is a disjoint union of open smooth manifolds....

So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of Rd with the Heisenberg group H2d+1 In this setting, we formulate and prove in this paper some other uncertaint...

Let 2n+1r be the reduced Heisenberg Lie group, G = 2n+1r × 2n+1r be the (4n + 2)-dimensional Lie group and ΔG = {(x,x) G : x2n+1r} be the diagonal subgroup of G. Given any discontinuous subgroup Γ G for G/ΔG, we provide a layering of the parameter space (Γ, G, ΔG), which is shown to be endowed with a smooth manifold structure, we also show that the...

We prove in this paper an L2-version of Beurling's theorem for an arbitrary exponential solvable Lie group G with a non-trivial center, which encompasses the setting of nilpotent connected and simply connected Lie groups. When G has a trivial center, the uncertainty principle may fail to hold and an example is produced. The representation theory an...

This book includes selected papers presented at the MIMS (Mediterranean Institute for the Mathematical Sciences) - GGTM (Geometry and Topology Grouping for the Maghreb) conference, held in memory of Mohammed Salah Baouendi, a most renowned figure in the field of several complex variables, who passed away in 2011. All research articles were written...

Let
$K$
be a compact subgroup of automorphisms of
$\mathbb R ^n$
. We prove in this paper a generalization of Hardy’s uncertainty principle on the semi-direct product
$K\ltimes \mathbb R ^n$
, and we solve the sharpness problem. As a consequence, a complete analogue of classical Hardy’s theorem is obtained. The representation theory and the P...

This paper aims to prove that the norm of the L p -Fourier transform of the semidirect product ℝ n ⋊ K $\mathbb {R}^n\rtimes K$ is A p n , where 1 < p ≤ 2 $1 < p \le 2$ , q = p / ( p - 1 ) $q=p/(p-1)$ , A p = p 1 2 p q - 1 2 q $A_p=p^{\frac{1}{2p}}q^{\frac{-1}{2q}}$ , and K stands for a compact subgroup of automorphisms of ℝ n . An extremal functio...

Let K be a compact subgroup of automorphisms of ℝn
. We formulate and prove an analogue of Miyachi’s theorem for the semi-direct product K ⋉ ℝn
. This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L
p
− L
q
analogue of Morgan theorem for any compact extension of ℝn
. These upshots are proved using the represen...

The purpose of this paper is to formulate and prove an L p -L q analog of Miyachi’s theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy’s and Cowling-Price’s uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty pr...

Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a deformation of a discontinuous subgroup Γ of G for the homogeneous space G/H may be locally rigid. When G is nilpotent, connected and simply connected, the question whether the Weil-Kobayashi local rigidity fails to hold is posed by Baklouti [Proc. Japan Ac...

Let G be a real solvable exponential Lie group with Lie algebra g and let f 2 g . We take two polarizations pj,j = 1,2, at f which meet Pukanszky's condition. Let Pj := exp(pj),j = 1,2, be the associated subgroups in G. The linear functional f defines unitary characters j(exp(X)) := e ihf,Xi, X 2 pj, of Pj. Let j := ind G Pj j, j = 1,2, be the corr...

A local rigidity Theorem proved by Selberg and Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a
non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear noncompact semi-simple Lie group G except some few cases: G is no...

We formulate in this note some analogues of certain classical uncertainty principles in the setting of some solvable Lie groups. Some sharpness problems are also treated. The orbit method and the Plancherel theory turn out to be an important ingredient to prove such analogues. Some other Lie groups cases are also discussed.

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup & Gamma; for the homogeneous space script:X sign = G/H and any deformation of Γ, the deformed discrete subgroup may fail to be discontinuous for script:X sign. To understand this phenomenon in the case when G is a two-step nilpotent Lie group, we provide a st...

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space script T sign(Γ, H2n+1, H) of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly dis...

We present in this note an analogue of the Selberg-Weil-Kobayashi local rigidity Theorem in the setting of exponential Lie groups and substantiate two related conjectures. We also introduce the notion of stable discrete subgroups of a Lie group $G$ following the stability of an infinitesimal deformation introduced by T. Kobayashi and S. Nasrin (cf....

Let G be a connected and simply connected nilpotent Lie group, and m the dimension of the generic coadjoint orbits of G. Then it is proved in [1] that the operator value Fourier transform norm satisfies ||F p(G)|| ≤ A pdim G-m/2(1 < p < 2), where A p = p 1/2p q -1/2p and q being the conjugate of p. When the assumption on G to be simply connected is...

We outline an approach of the Kirillov map for exponential groups based on deformations, and suggest a possible way towards an alternative proof of the Leptin-Ludwig bicontinuity theorem [14] along these lines.

Following the notion of stability introduced by T. Kobayashi and S. Nasrin in [14], we show in the context of a threadlike
Lie group G that any non-Abelian discrete subgroup is stable. One consequence is that any resulting deformation space ℐ(Γ,G,H) is a Hausdorff space, where Γ acts on the threadlike homogeneous space G/H as a discontinuous subgro...

In this paper, we define an analog of the L
p
-L
q
Morgan’s uncertainty principle for any exponential solvable Lie group G (p, q ∈ [1,+∞]). When G is nilpotent and has a noncompact center, the proof of such an analog is given for p, q ∈ [2,+∞], extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular rest...

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
\fancyscriptM=G/H{\fancyscript{M}=G/H} and any deformation of Γ, the deformed discrete subgroup may utterly destroy its proper discontinuous action on
\fancyscriptM{\fancyscript{M}} as H is not compact (ex...

We establish analogues of Hardy’s uncertainty principle for the Fourier transform on ℝ for locally compact Abelian groups
and for some classes of solvable Lie groups, such as diamond groups and exponential Lie groups with non-trivial centre.

We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

Let H be a closed connected subgroup of an exponential solvable Lie group G. We consider the deformation space T (¡;G;H) of a discontinuous subgroup ¡ of G for the homogeneous space G=H. When H contains (G;G) or (¡;¡) is uniform in (G;G), we exhibit a description of the space T (¡;G;H). This requires a precise algebraic description of the parameter...

Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie group G. We consider the deformation space $\mathcal{T}(Γ, G,H)$ of a discontinuous subgroup Γ of G for the homogeneous space G/H. When H contains [G, G], we exhibit a description of the space {T}(Γ, G,H) which appears to involve GL k() as a direct produc...

The aim of the present note is to discuss the explicit determination of the deformation space of the action of a discontinuous subgroup for a homogeneous space in the setting where the basis group is the Heisenberg group.

Let G be a nilpotent connected and simply connected Lie group, K an analytic subgroup of G and Π a unitary and irreducible representation of G. We study in this paper a variant of the Frobenius reciprocity for the restricted representation Π|K of Π on K. It consists in proving that generically, the multiplicity of any isotopic component involved in...

Let G be a connected simply connected nilpotent Lie group. In [A. Baklouti, N. Ben Salah, The Lp−Lq version of Hardy's Theorem on nilpotent Lie groups, Forum Math. 18 (2006) 245–262], we proved for 2⩽p,q⩽+∞ the Lp−Lq version of Hardy's Theorem known as the Cowling–Price Theorem. In the setup where 1⩽p,q⩽+∞, the problem is still unsolved and the ups...

In (Kaniuth and Kumar in Math. Proc. Camb. Phil. Soc. 131, 487–494, 2001) Hardy’s uncertainty principle for
\mathbbRn{\mathbb{R}}^n was generalized to connected and simply connected nilpotent Lie groups. In this paper, we extend it further to connected
nilpotent Lie groups with non-compact centre. Concerning the converse, we show that Hardy’s the...

Let H be an arbitrary closed connected subgroup of the connected, simply connected Heisenberg G = H2n+1. We present in this paper a complete description of the deformation space and the moduli space of adiscontinuous abelian subgroup Γ of G for the homogeneous space G/H. The topological features of deformations, namely the topological stability, th...

Let H and K be closed connected subgroups of a connected, simply connected solvable Lie group G. We define the notion of weak and finite proper action of K on the homogeneous space X = G/H and prove that they are equivalent to the notion of (CI)-action of K on X in the sense of Kobayashi. We show also that the action of K on X is proper if and only...

We study the L
p
-Fourier transform for a special class of exponential Lie groups, the strong ✱-regular exponential Lie groups. Moreover, we
provide an estimate of its norm using the orbit method.

We provide in this paper a counterexample to the Benson-Ratcliff
conjecture about a cohomology class invariant on coadjoint
orbits on nilpotent Lie groups. We prove that this invariant
never vanishes on generic coadjoint orbits for some restrictive
classes. As such, it does separate up to invariant factor,
unitary representations associated to gene...

We show in this paper that the multiplicities of mixed representations are uniformly infinite or uniformly finite and bounded, in the setting of completely solvable Lie groups extending then the situation of nilpotent Lie groups. Necessary and sufficient conditions for these multiplicities to be finite are provided.

Let p, q be such that 2 ≤ p, q ≤ +∞. We prove in this paper the Lp -Lq version of Hardy's Theorem for an arbitrary nilpotent Lie group G extending then earlier cases and the classical Hardy theorem proved recently by E. Kaniuth and A. Kumar. The case where 1 ≤ p, q ≤ +∞ is studied for a restricted class of nilpotent Lie groups.

Let G be an exponential solvable Lie group and H an analytic subgroup of G. Let x be a unitary character of H and τ = Ind
H
G
χ. We provide a necessary and a sufficient condition for the representation τ to split finitely on its isotopic components
for adapted triplets. For the case in which H is maximal, we provide a concrete and smooth intertw...

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G, H, K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is als...

The purpose of the present work is to describe a dequantization procedure for topological modules over a deformed algebra. We define the characteristic variety of a topological module as the common zeroes of the annihilator of the representation obtained by setting the deformation parameter to zero. On the other hand, the Poisson characteristic var...

Let G be a connected simply connected nilpotent Lie group, K an analytic subgroup of G and π an irreducible unitary representation of G. Let Dπ(G)K be the algebra of differential operators keeping invariant the space of C∞ vectors of π and commuting with the action of K on that space. In this paper, we assume that the restriction of π to K has fini...

Soit G un groupe de Lie nilpotent connexe et simplement connexe, K un sous-groupe fermé connexe de G et π une représentation unitaire et irréductible de G. Nous étudions l'algèbre Dπ(G)K des opérateurs différentiels qui laissent invariant l'espace des vecteurs C+∞ de π et qui commutent avec l'action de K sur cet espace. Nous montrons alors que la c...

Soient G = exp {\frak g} un groupe de Lie résoluble exponentiel et H = exp {\frak h} un sous-groupe connexe de G. Soient un caractère unitaire de H et = . Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalen...

The purpose of the present work is to describe a dequantization procedure for topological modules over a deformed algebra. We define the characteristic variety of a topological module as the common zeroes of the annihilator of the representation obtained by setting the deformation parameter to zero. On the other hand, the Poisson characteristic var...

Let $G$ be an exponential solvable Lie group, $H$ and $A$ two closed connected subgroups of $G$ and $\sigma$ a unitary and irreducible representation of $H$. We prove the orbital spectrum formula of the Up-Down representation $\rho(G,H,A,\sigma)=\operatorname{Ind}_H^G\sigma_{\vert A}$. When $G$ is nilpotent, the multiplicities of such representatio...

Let G be an exponential solvable Lie group, let H be a closed subgroup of G, and let p be an irreducible unitary representation of G. As is well known, if H is proper, then the down-up representation r(G,H,p)=Ind H G (p| H ) (i.e., the representation of G induced from H by the restriction to H of the representation p) is reducible. We prove the orb...

Let be a nilpotent connected and simply connected Lie group, and an analytic subgroup of G. Let , be a unitary character of H and let . Suppose that the multiplicities of all the irreducible components of τ are finite. Corwin and Greenleaf conjectured that
the algebra of the differential operators on the Schwartz-space of τ which commute with τ is...

Let G be a connected and simply connected nilpotent Lie group with Lie algebra g and unitary dual b G. The moment map for 2 b G sends smooth vectors in the representation space of to g . The closure of the image of the moment map for is called its moment set. N. Wildberger has proved that the moment set for coincides with the closure of the convex...

We give in this paper a desintegration into explicit irreducible of the restrictions to connected closed subgroups of a unitary and irreducible representations on nilpotent Lie group. We describe also an intertwining operator which does not take into account the multiplicities of the desintegration.

We give in this paper a desintegration into explicit irreducible of the restrictions to connected closed subgroups of a unitary and irreducible representations on nilpotent Lie group. We describe also an intertwining operator which does not take into account the multiplicities of the desintegration.

We prove the Penney-Fujiwara Plancerel Formula associated to a monomial representation of a nilpotent Lie group. We give also a short proof of a theorem due to Corwin and Greenleaf about the algebra of differential operators on certain nilpotent homogeneous space.

We show that every irreducible unitary representation π of an exponential Lie group G = exp g is characterized by its generalized moment set.