Poisson transforms for differential forms

TL;DR

This paper introduces a new Poisson transform that maps differential forms on flag manifolds to those on symmetric spaces, generalizing classical Poisson transforms through finite-dimensional Lie group representations.

Contribution

It constructs a family of degree-preserving Poisson transforms with coclosed images, extending classical Poisson transform theory to differential forms on symmetric spaces.

Findings

Explicit construction of Poisson transforms for differential forms

Generation of a family of degree-preserving transforms

Relation to classical Poisson transform on density bundles

Abstract

We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite dimensional representations of reductive Lie groups. Moreover, we will explicitly generate a family of degree-preserving Poisson transforms whose restriction to real valued differential forms has coclosed images. In addition, as a transform on sections of density bundles it can be related to the classical Poisson transform, proving that we produced a natural generalization of the classical theory.