Poisson transforms adapted to BGG-complexes

TL;DR

This paper introduces a novel construction of Poisson transforms for vector bundle valued differential forms on homogeneous parabolic geometries, linking BGG-sequences with twisted de Rham sequences, and explicitly designs compatible transforms for real hyperbolic space.

Contribution

It presents a new method for constructing Poisson transforms using finite dimensional representations, connecting BGG-sequences with de Rham sequences, and designs explicit transforms for hyperbolic space.

Findings

Constructed Poisson transforms relating BGG-sequences and de Rham sequences.

Explicitly designed a family of compatible Poisson transforms for real hyperbolic space.

Established a framework using finite dimensional representations of reductive Lie groups.

Abstract

We present a new construction for Poisson transforms between vector bundle valued differential forms on homogeneous parabolic geometries and the corresponding Riemannian symmetric space, which can be described in terms of finite dimensional representations of reductive Lie groups. In particular, we use these operators to relate the BGG-sequences on the domain with twisted deRham sequences on the target space. Finally, we explicitly design a family of Poisson transforms between standard tractor valued differential forms for the real hyperbolic space which are compatible with the BGG-complex.