A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on $G/K$

TL;DR

This paper extends the Paley-Wiener-Schwartz theorem to sections of homogeneous vector bundles on symmetric spaces, characterizing the Fourier transform's range for these sections.

Contribution

It develops a topological Paley-Wiener-Schwartz theorem for distributional sections of vector bundles on symmetric spaces, generalizing existing results for smooth functions.

Findings

Characterization of the Fourier transform range for sections

Extension of Paley-Wiener-Schwartz theorem to vector bundle sections

Application of Delorme's theorem to this setting

Abstract

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type $X = G/K$. We prove a characterisation of their range. In fact, from Delorme's Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener-Schwartz theorems for sections.