A characterization of the L2-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

Abstract

We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles Sp(n,1)×Sp(n)×Sp(1)Vτ over the quaternionic hyperbolic spaces Sp(n,1)/Sp(n)×Sp(1) associated with irreducible representations τ of Sp(n)×Sp(1) which are trivial on Sp(n). As a consequence, we describe the image of the section space L2(Sp(n,1)×Sp(n)×Sp(1)Vτ) under the generalized spectral projections associated to a family of eigensections of the Casimir operator.