A comparison of Paley-Wiener theorems for real reductive Lie groups

TL;DR

This paper compares two major Paley-Wiener theorems for real reductive Lie groups, showing their equivalence and providing alternative formulations using Hecke algebras, with techniques involving representation derivatives and Harish-Chandra modules.

Contribution

It demonstrates the equivalence of Arthur's and Delorme's Paley-Wiener theorems and offers an alternative formulation via Hecke algebras, advancing understanding in harmonic analysis on Lie groups.

Findings

Proves the equivalence of the two theorems

Provides an alternative Hecke algebra formulation

Uses derivatives of holomorphic families in the proof

Abstract

In this paper we make a detailed comparison between the Paley-Wiener theorems of J. Arthur and P. Delorme. We prove that these theorems are equivalent from an a priori point of view. We also give an alternative formulation of the theorems in terms of the Hecke algebra of bi-K-finite distributions supported on K. Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules.