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The Schwartz space of a general semisimple Lie group. IV: Elementary mixed wave packets. (English) Zbl 0781.22007
This paper is the fourth in a series [cf. the preceding review Zbl 0781.22006 for the other parts] concerning the Schwartz space $${\mathcal C}(G)$$ for a semisimple Lie group $$G$$ with infinite center. The Plancherel formula expands $$f \in {\mathcal C}(G)$$ in terms of the distribution characters of tempered representations. So $$f = \sum f_ H$$ where $$H$$ ranges over a set of representatives for conjugacy classes of Cartan subgroups of $$G$$ and $$f_ H$$ corresponds to representations associated with $$H$$. Let $$K$$ be a maximal compact modulo center subgroup of $$G$$. The set $${\mathcal C}(G)_ K$$ of functions whose $$K$$-types lie in a compact subset of $$\widehat{K}$$ is dense in $${\mathcal C}(G)$$. In the first paper the authors defined Eisenstein integrals and used them to form wave packets. For $$f\in {\mathcal C}(G)_ K$$, every $$f_ H$$ decomposes as a finite sum of wave packets. A new feature of the infinite center case is that $$f_ H$$ and corresponding wave packets are not necessarily in $${\mathcal C}(G)$$. So there must be some matching conditions between $$f_ H$$.
In this paper “elementary mixed wave packets” are defined and studied. These are finite sums of wave packets. They are proved to be in $${\mathcal C}(G)$$. (A part of the proof is deferred to another paper.) It is shown that every $$f\in {\mathcal C}(G)_ K$$ is a finite sum of elementary mixed wave packets.

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