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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.

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