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

In this paper we make a detailed comparison between the Paley–Wiener theorems of J. Arthur and P. Delorme for a real reductive Lie group . 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- -finite distributions supported on , a maximal compact subgroup of . Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules.