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The Schwartz space of a general semisimple Lie group. III: $$c$$-functions. (English) Zbl 0781.22006
Let $$G$$ be a connected reductive Lie group (in particular, a semisimple Lie group with infinite center; this case is of main interest). This paper is the third in a series concerning the Schwartz space $${\mathcal C}(G)$$. In the first paper which is a joint paper with J. A. Wolf [ibid. 80, No. 2, 164-224 (1990; Zbl 0711.22005)] the authors defined Eisenstein integrals and used them to form wave packets. These are formed from single continuous families of representations. In the second paper [Trans. Am. Math. Soc. 327, No. 1, 1-69 (1991; Zbl 0741.22006)] some necessary conditions are obtained for such a wave packet to be in $${\mathcal C}(G)$$. In the paper reviewed these conditions are shown to be sufficient. Thus, a complete characterization of Schwartz class wave packets is obtained. When the derived group of $$G$$ has a finite center, this result reduces to Harish-Chandra’s description of the Schwartz space.

##### MSC:
 22E30 Analysis on real and complex Lie groups 22E46 Semisimple Lie groups and their representations 43A80 Analysis on other specific Lie groups
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