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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.

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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