Paley-Wiener spaces for real reductive Lie groups (Q855590)

The Paley-Wiener theorem for \(K\)-finite compactly supported smooth functions on a real reductive Lie group \(G\) of the Harish-Chandra class is due to \textit{J. Arthur} [Acta Math. 150, 1--89 (1983; Zbl 0514.22006)] in general, and to \textit{O. A. Campoli} [Rev. Union Mat. Argent. 29, 197--221 (1980; Zbl 0457.22008)] for \(G\) of split rank one. In the paper [Ann. Math. 164(3), 879--909 (2006)] the authors established a similar Paley-Wiener theorem for smooth functions on a reductive symmetric space. In the paper under review the authors show that Arthur's theorem is a consequence of their result if one considers the group \(G\) as a symmetric space for \(G\times G\) with respect to the left times right action. Also the authors formulate a Paley-Wiener theorem for \(K\)-finite generalized functions (in the sense of distribution theory) on \(G\), and prove that it is a special case of the Paley-Wiener theorem for symmetric spaces established in [arXiv math.RT/0511585].