On Poisson transform for spinors

Abstract

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 spinor bundle $\Sigma H^n(\mathbb{R})$ over $H^n(\mathbb{R})$ as the homogeneous bundle ${\mathrm{Spin}}_0(1,n){\times }_{\mathrm{Spin}(n)}{V}_{\tau }$. Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on $\Sigma H^n(\mathbb{R})$ which can be written as the Poisson transform of $L^p$-sections of the bundle $\mathrm{Spin}(n){\times }_{\mathrm{Spin}(n-1)}{V}_{\sigma }$ over the boundary $S^{n-1}\simeq \mathrm{Spin}(n)/\mathrm{Spin}(n-1)$ of $H^n(\mathbb{R})$, for $1<p<\infty$.