Searcharxiv⌕ Search

arXiv subjects

Khalid Koufany

Publications and source records attributed to Khalid Koufany.

16 recordsLinked to original sources
arXiv2026

Cholesky-Vinberg and Log-Vinberg means on the Vinberg cone

We study two means on the five-dimensional Vinberg cone $Ω$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $Ω$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $Ω$.

math.FA↗
arXiv2026

Mostow-Type Decompositions and Geometric Means for Symmetric Cones

Motivated by Mostow's decomposition theorem for positive definite matrices and by subsequent matrix factorizations involving geometric means, we establish a Mostow-type decomposition for arbitrary symmetric cones. As a consequence, we derive an analogous decomposition for the automorphism group of the cone. We also introduce a natural Hadamard metric on the symmetric cone by pulling back an $\ell^2$-product metric, together with its midpoint operation and associated Karcher mean.

math.MG↗
arXiv2025

Strichartz's conjecture for the spinor bundle over the real hyperbolic space

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 generalized spectral projections associated to the spinor bundle over $H^n(\mathbb R)$, thereby extending Strichartz's conjecture from the scalar case to the spinor setting.

math.RT↗
arXiv2024

On Poisson transforms of differential forms on real hyperbolic spaces

This paper is concerned with the Poisson transform of differential forms on the hyperbolic space $H^n(\mathbb R)$. Consider an integer $p$ such that $1\leqslant p\leqslant n$ and let $q$ be either $p-1$ or $p$. For $1<r<\infty$, we prove that the Poisson transform is a topological isomorphism from the space of $L^r$-differential $q$-forms on the boundary $\partial H^n(\mathbb R)$ onto a Hardy-type subspace of $p$-eigenforms of the Hodge-de Rham Laplacian on $H^n(\mathbb R)$.

math.RT↗
arXiv2024

A characterization of the $L^2$-range of the generalized spectral projections related to the Hodge-de Rham Laplacian

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 the boundary $\partial H^n(\mathbb R)$. This gives a positive answer to a conjecture of Strichartz on differential forms.

math.RT↗
arXiv2023

The source operator method: an overview

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 for differential forms and for spinors, and also for the construction of Juhl's operators corresponding to the restriction from the sphere $S^n$ to $S^{n-1}$.

math.RT↗
arXiv2022

On Poisson transform for spinors

Let $(τ,V_τ)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(σ,V_σ)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $τ_{\mid \mathrm{Spin}(n-1)}$. 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 spinor bundle $ΣH^n(\mathbb R)$ over $H^n(\mathbb R)$ as the homogeneous bundle $\mathrm{Spin}_0(1,n)\times_{\mathrm{Spin}(n)} V_τ$. Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on $ΣH^n(\mathbb R)$ which can be written as the Poisson transform of $L^p$-sections of the bundle $\mathrm{Spin}(n)\times_{\mathrm{Spin}(n-1)} V_σ$ over the boundary $S^{n-1}\simeq \mathrm{Spin}(n)/\mathrm{Spin}(n-1)$ of $H^n(\mathbb R)$, for $1<p<\infty$.

math.RT↗
arXiv2021

Symmetry Breaking Differential Operators for Tensor Products of Spinorial Representations

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $ρ$ and $ρ'$ be the corresponding representations of the spin group ${\rm Spin}(n)$. The group $G= {\rm Spin}(1,n+1)$ is a (twofold) covering of the conformal group of $\mathbb R^n$. For $λ, μ\in \mathbb C$, let $π_{ρ, λ}$ (resp. $π_{ρ',μ}$) be the spinorial representation of $G$ realized on a (subspace of) $C^\infty(\mathbb R^n,\mathbb S)$ (resp. $C^\infty(\mathbb R^n,\mathbb S')$). For $0\leq k\leq n$ and $m\in \mathbb N$, we construct a symmetry breaking differential operator $B_{k;λ,μ}^{(m)}$ from $C^\infty(\mathbb R^n \times \mathbb R^n,\mathbb{S}\,\otimes\, \mathbb{S}')$ into $C^\infty(\mathbb R^n, Λ^*_k(\mathbb R^n) \otimes \mathbb{C})$ which intertwines the representations $π_{ρ, λ}\otimes π_{ρ',μ} $ and $π_{τ^*_k,λ+μ+2m}$, where $τ^*_k$ is the representation of ${\rm Spin}(n)$ on the space $Λ^*_k(\mathbb R^n) \otimes \mathbb{C}$ of complex-valued alternating $k$-forms on $\mathbb{R}^n$.

math.RT↗
arXiv2020

The compression semigroup of the dual Vinberg cone

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

math.GR↗
arXiv2018

Conformally covariant bi-differential operators for differential forms

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 group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.

math.RT↗
arXiv2017

Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

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 $V=\mathbb{R}$).

math.RT↗
arXiv2011

Hua operators, Poisson transform and relative discrete series on line bundle over bounded symmetric 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 differential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.

math.RT↗
arXiv2006

Jordan algebras, geometry of Hermitian symmetric spaces and non-commutative Hardy spaces

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 contributions for studing (non commutative) Hardy spaces of holomorphic functions on Lie semi-groups which is a part of the so called Gelfand-Gindikin program.

math.RT↗
arXiv2004

Primitive du cocycle de Maslov généralisé

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.} {\bf 83} (2004), 99-114]. It generalizes the Souriau index in the case of the Lagrangian manifold. A variation of this construction yields a generalization of the Arnold-Leray-Maslov index. This primitive is used to generalize the symplectic rotation number.

math.DG↗
arXiv2004

Hilbert's metric on symmetric cones

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

math.MG↗