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 G. 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, a maximal compact subgroup of G. Our techniques involve derivatives of holomorphic families of continuous representations and Harish-Chandra modules