Salem Bensaïd | Searcharxiv

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↗

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↗