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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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