On Poisson transform for spinors

Let $(\tau ,V_{\tau })$ be a spinor representation of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)$ and let $(\sigma ,V_{\sigma })$ be a spinor representation of $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n-1)$ that occurs in the restriction ${\tau }_{\mid \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n-1)}$. We consider the real hyperbolic space $H^n(\mathbb{R})$ as the rank one homogeneous space ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_0(1,n)\mathrm{/}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)$ and the spinor bundle $\mathrm{\Sigma }{H}^n(\mathbb{R})$ over $H^n(\mathbb{R})$ as the homogeneous bundle ${\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_0(1,n){\times }_{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)}{V}_{\tau }$. Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on $\mathrm{\Sigma }{H}^n(\mathbb{R})$ which can be written as the Poisson transform of $L^p$-sections of the bundle $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n){\times }_{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n-1)}{V}_{\sigma }$ over the boundary $S^{n-1}\simeq \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\mathrm{/}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n-1)$ of $H^n(\mathbb{R})$, for $1<p<\mathrm{\infty }$.