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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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