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

TL;DR

This paper investigates the $L^2$-boundedness of Poisson transforms on certain vector bundles over quaternionic hyperbolic spaces, characterizing their $L^2$-range via spectral projections related to the Casimir operator.

Contribution

It provides a detailed description of the $L^2$-range of Poisson transforms on specific vector bundles over quaternionic hyperbolic spaces, extending understanding of harmonic analysis in this setting.

Findings

Characterization of the $L^2$-boundedness of Poisson transforms.

Description of the image of $L^2$ section spaces under spectral projections.

Connection between Poisson transforms and eigenfunctions of the Casimir operator.

Abstract

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)/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.