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The Schwartz space of a general semisimple Lie group. II: Wave packets associated to Schwartz functions. (English) Zbl 0741.22006
This paper is the second in a series concerning the Schwartz space \({\mathcal C}(G)\) for a semisimple Lie group \(G\) with infinite center. The first (by the author and J. A. Wolf) is published in Adv. Math. 80, No. 2, 164-224 (1990; Zbl 0711.22005), the third (by the author) is to appear in Adv. Math. The Plancherel formula expands \(f\in{\mathcal C}(G)\) in terms of the distribution characters of tempered representations [see, for example, the paper of the author and J. A. Wolf in Compos. Math. 57, 271-355 (1986; Zbl 0587.22005)]. So \(f=\sum f_ H\), where \(H\) ranges over a set of representatives for conjugacy classes of Cartan subgroups and \(f_ H\) corresponds to representations associated with \(H\). A new feature of the infinite center case is that, for \(f\in{\mathcal C}(G)\), \(f_ H\) is not necessarily in \({\mathcal C}(G)\). So there must be some mating conditions between \(f_ H\).
Let \(K\) be a relative maximal compact subgroup of \(G\). A function \(f\) is called \(K\)-compact if its \(K\)-types lie in a compact subset of \(\hat K\). The set \({\mathcal C}(G)_ K\) of such functions is dense in \({\mathcal C}(G)\). In the first paper the authors defined Eisenstein integrals and used them to form wave packets. These are formed from a single continuous family of representations. It is shown that every function in \({\mathcal C}(G)_ K\) is a finite sum of these wave packets. Some conditions are obtained which are necessary for a wave packet to occur in the decomposition of a function in \({\mathcal C}(G)\). In particular, some necessary conditions are obtained for such a wave packet to be in \({\mathcal C}(G)\). In the third paper, these conditions are shown to be sufficient.

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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