Khalid Koufany

• University of Lorraine

Let $H^n(\mathbb R)$ be the real hyperbolic space. In this paper, we present a characterization of the $L^2$-range of the generalized spectral projections on the bundle of differential forms over $H^n(\mathbb R)$. As an underlying result we show a characterization of the $L^2$-range of the Poisson transform on the bundle of differential forms on th...

Let $H^n(\mathbb R)$ denote the real hyperbolic space realized as the symmetric space $Spin_0(1,n)/Spin(n)$. In this paper, we provide a characterization for the image of the Poisson transform for $L^2$-sections of the spinor bundle over the boundary ${\partial H}^n(\mathbb R)$. As a consequence, we obtain an $L^2$ uniform estimate for the generali...

This is an overview on the source operator method, which leads to the construction of symmetry breaking differential operators (SBDO) in the context of tensor product of two principals series representations for the conformal group of a simple real Jordan algebra. This method can be applied to other geometric contexts: in the construction of SBDO f...

This is an overview on the source operator method which leads to the construction of symmetry breaking differential operators (SBDO) in the context of tensor product of two principals series representations for the conformal group of a simple real Jordan algebra. This method can be applied to other geometric contexts: in the construction of SBDO fo...

Let $(\tau,V_\tau)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(\sigma,V_\sigma)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $\tau_{\mid \mathrm{Spin}(n-1)}$. We consider the real hyperbolic space $H^n(\mathbb R)$ as the rank one homogeneous space $\mathrm{Spin}_0(1,n)/\mathrm{Spin}(n)$ and the...

We study the Poisson transform for differential forms on the real hyperbolic space $\mathbb H^n$. For $1<r<\infty$, we prove that the Poisson transform is an isomorphism from the space of $L^r$ differential $q$-forms on the boundary $\partial \mathbb H^n$ onto a Hardy-type subspace of $p$-eigenforms of the Hodge-de Rahm Laplacian, for $0\leq p<\fra...

We investigate the semigroup associated with the dual Vinberg cone and prove its triple and Ol’shanskiĭ polar decompositions. Moreover, we show that the semigroup does not have the contraction property with respect to the canonical Riemannian metric on the cone.

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\ccl(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group $\Spin(n)$. The group $G= \Spin(1,n+1)$ is a (twofold) covering of the conformal group of $\mathbb R^n$. For $\lambda, \mu\in \mathbb C$, let $\pi_{\rho, \la...

We investigate the semigroup associated to the dual Vinberg cone and prove its triple and Ol'shanskiȋ polar decompositions. Moreover, we show that the semigroup does not have the contraction property with respect to the canonical Riemannian metric on the cone.

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $C\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group $Spin(\mathbb R^n)$. The group $G=Spin(\mathbb R^{1,n+1})$ is the (twofold covering) of the conformal group of $\mathbb R^n$. For $\lambda, \mu\in \mathbb C...

The classical Rankin–Cohen brackets are bi-differential operators from \(C^\infty ({\mathbb {R}})\times C^\infty ({\mathbb {R}})\) into \( C^\infty ({\mathbb {R}})\). They are covariant for the (diagonal) action of \(\mathrm{SL}(2,{\mathbb {R}})\) through principal series representations. We construct generalizations of these operators, replacing \...

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$...

The classical Rankin-Cohen brackets are bi-differential operators from C 8 pRqˆCpRqˆpRqˆC 8 pRq into C 8 pRq. They are covariant for the (diagonal) action of SLp2, Rq through principal series representations. We construct generalizations of these operators , replacing R by R n , the group SLp2, Rq by the group SO 0 p1, n ` 1q viewed as the conforma...

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$).

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$).

Let G be a connected semisimple real-rank one Lie group with finite center. We consider intertwining operators on tensor products of spherical principal series representations of G. This allows us to construct an invariant trilinear form $\mathcal{K}_{\underline{\boldsymbol{\nu}}}$ indexed by a complex multiparameter ${\underline{\boldsymbol{\nu}}}...

Let M be a symmetric space of Cayley type. For any conformal diffeomorphism of M we study the relationship betwenn the conformal factor of f and a generalized Schwarzian derivative of f.

Let $\Omega=G/K$ 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 $\Omega$ and the corresponding eigenspaces of $G$-invariant differential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of...

Soit D\mathcal{D} un espace hermitien symétrique de type tube, de frontière de Shilov S. Nous décrivons une réalisation du revêtement universel [(S)\tilde]\widetilde{S} de S. Nous construisons ensuite sur [(S)\tilde]\widetilde{S} une primitive du cocycle de Maslov généralisé. C’est l’analogue de l’indice de Souriau pour la variété lagrangienne. Une...

Ce mémoire présente un point de vue basé sur la théorie des algèbres de Jordan pour faire une étude analytique, géométrique et topologique de certains espaces homogènes : espaces hermitiens symétriques, leurs frontières de Shilov et espaces symétriques causaux de type Cayley. En particulier, nous passons en revue des résultats sur l'indice de Masl...

We characterize the $L^p$-range, $1 < p < +\infty$, of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains. We prove that this range is a Hua-Hardy type space for harmonic functions satisfying a Hua system.

These notes were written following lectures I had the pleasure of giving on this subject at Keio University, during November and December 2004. The first part is about new applications of Jordan algebras to the geometry of Hermitian symmetric spaces and to causal semi-simple symmetric spaces of Cayley type. The second part will present new contribu...

Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)n/r/2|h(z,u)n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for th...

Let $\mathcal{D}$ be a Hermitian symmetric space of tube type, and let $S$ be its Shilov boundary. We give a realization of the universal covering $\widetilde{S}$ of $S$. Then we describe on $\widetilde{S}$ a primitive for the generalized Maslov cocycle as defined in [{\it Transform. Groups} {\bf 6} (2001), 303-320] and [{\it J. Math. Pures Appl.}...

Let Ω be a symmetric cone. In this note, we introduce Hilbert’s projective metric on Ω in terms of Jordan algebras and we apply it to prove that, given a linear invertible transformation g such that g(Ω) = Ω and a real number p, |p| > 1, there exists a unique element x ∈ Ω satisfying g(x) = x p .

For a symmetric cone $\Omega$ we compute its Riemannian distance in terms of the singular values of a generalized cross-ratio and prove that the semigroup of the compressions of $\Omega$ decreases the compounds distance.

. In this paper we study the minimal complex Lie semigroups associated with three classical series of groups by using a holomorphic continuation of a certain Cayley transform for the group. In particular we show, that for the symplectic group the odd part of the Hardy space on the double cover is isomorphic to the classical Hardy space on the Siege...

For the scalar holomorphic discrete series representations of SU(2,2) and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside SU(2,2). We construct a Cayley transform between the Ol’shanskiĭ semigroup having U(1,1) as Šilov boundary and an open dense subdomain of the Hermitian symmet...

35> C ae V is a convex cone if and only if x; y 2 C and ; ¯ ? 0 imply that x + ¯y 2 C. In the following, C will always denote a non-trivial closed convex cone and Omega will stand for an open non-trivial convex cone. we define 1. V C := C " GammaC , 2. ! C ?:= C Gamma C = fx Gamma y j x; y 2 Cg, 3. C ? := fy 2 V j 8x 2 C : (xjy) 0g. Then V C and !...

. --- For the scalar holomorphic discrete series representations of SU(2; 2) and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside SU(2; 2). We construct a Cayley transform between the Ol'shanskii semigroup having U(1; 1) as Silov boundary and an open dense subdomain of the Hermiti...

A une algèbre de Jordan euclidienne, on associe un cône symétrique et nous étudions le semi-groupe des éléments du groupe conforme de l'algèbre de Jordan qui préservent le cône symétrique. Pour cette étude, nous réalisons le groupe conforme comme le groupe des automorphismes holomorphes du domaine tube agissant sur la frontière de Shilov. Nous démo...