Delorme's intertwining conditions for sections of homogeneous vector bundles on two and three dimensional hyperbolic spaces

TL;DR

This paper explicitly characterizes intertwining conditions for the Paley-Wiener space on certain semi-simple Lie groups, aiding the analysis of invariant differential operators on hyperbolic spaces.

Contribution

It provides a complete explicit description of Delorme's intertwining conditions for specific rank-one groups, based on a new criterion derived from the Paley-Wiener theorem.

Findings

Explicit intertwining conditions for $SL(2,\mathbb{R})^d$ and $SL(2,\mathbb{C})$

A new criterion for the Paley-Wiener space for real rank one groups

Foundation for future study of invariant differential operators

Abstract

The description of the Paley-Wiener space for compactly supported smooth functions $C^\infty_c(G)$ on a semi-simple Lie group $G$ involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for $G=\mathbf{SL}(2,\mathbb{R})^d$ ($d\in \mathbb{N}$) and $G=\mathbf{SL}(2,\mathbb{C})$. Our results are based on a defining criterion for the Paley-Wiener space, valid for general groups of real rank one, that we derive from Delorme's proof of the Paley-Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.