The Schwartz space of a general semisimple Lie group. II: Wave packets associated to Schwartz functions.

*(English)*Zbl 0741.22006This 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.

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.

Reviewer: V.F.Molchanov (Tambov)

##### MSC:

22E30 | Analysis on real and complex Lie groups |

22E46 | Semisimple Lie groups and their representations |

43A80 | Analysis on other specific Lie groups |