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Rapidly decreasing functions on general semisimple groups. (English) Zbl 0587.22006
In their paper, ibid. 57, 271-355 (1986; see the preceding review), the authors extended Harish-Chandra’s Plancherel theorem for semisimple Lie groups with finite center to a class of reductive Lie groups which contains all connected semisimple Lie groups. In contrast to Harish- Chandra’s argument which relied on the analysis of the Schwartz space of rapidly decreasing smooth functions on the group, their argument worked with the space of compactly supported smooth functions on the group.
In this paper the authors introduce the analogues of Schwartz spaces of rapidly decreasing smooth functions on groups of their class, and extend their Plancherel theorem to these spaces. There are two versions of these results. One deals with the ”relative case”, where the functions transform by an unitary character under the action of the center and which are, mod center, rapidly decreasing in the sense of Harish-Chandra. The other case is the ”global” one, where functions decrease rapidly in all directions, including that of the possibly infinite center. For groups of Harish-Chandra class, the global Schwartz space is the same as Harish-Chandra’s, and the authors’ argument gives a direct route to Harish-Chandra’s Plancherel theorem bypassing the machinery of Eisenstein integrals and wave packets.
Reviewer: D.Miličić

MSC:
22E46 Semisimple Lie groups and their representations
43A32 Other transforms and operators of Fourier type
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References:
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