A characterization of the $L^2$-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

We study the $L^2$-boundedness of the Poisson transforms associated to the homogeneous vector bundles $Sp(n,1){\times }_{Sp(n)\times Sp(1)}{V}_{\tau }$ over the quaternionic hyperbolic spaces $Sp(n,1)\mathrm{/}Sp(n)\times Sp(1)$ associated with irreducible representations $\tau$ of $Sp(n)\times Sp(1)$ which are trivial on $Sp(n)$. As a consequence, we describe the image of the section space $L^2(Sp(n,1){\times }_{Sp(n)\times Sp(1)}{V}_{\tau })$ under the generalized spectral projections associated to a family of eigensections of the Casimir operator.