A unified approach to concrete Plancherel theory of homogeneous spaces
Let \(G\) be a Lie group, \(H\subset G\) a closed subgroup, and \(\tau= \text{Ind}^G_H1\) the quasi regular representation supposed to be type I. Then we have the abstract Plancherel theorem \(\tau= \int^\oplus_{\widehat G}n_\pi \pi d\mu(\pi)\) for the homogeneous space \(G/H\). According to this formula, we can decompose the Dirac measure in the distribution sense [cf. \textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016); \textit{P. Bonnet}, ibid. 55, 220-246 (1984; Zbl 0531.43001)]. Furthermore we can write down in many cases this decomposition more explicitly to get a concrete Plancherel formula. The search for such a formula has been carried out in two mutually exclusive cases according as the multiplicity function \(n_\pi\) is finite or infinite \(\mu\) -- a.e. [cf. a series of the author's works: Progr. Math. 82, 135-145 (1990; Zbl 0758.22005); Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012); ibid. 159, 351-377 (1993; Zbl 0798.22005); the reviewer's two works: Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008); Representation theory of Lie groups and Lie algebras, World Scientific, Singapore, 140-150 (1992), \textit{G. Grélaud}, J. Reine Angew. Math. 398, 92-100 (1989; Zbl 0666.43004)]. In this paper, the author considers Strichartz homogeneous spaces \(G=V\times R\), a semidirect product of a real algebraic group \(R\) by a normal vector subgroup \(V\) with a subgroup \(H=U\times S\) of the same type satisfying \(S\subset R\), \(U\subset V\), and shows that the two cases separated by the property of multiplicities should be treated in a unified way, namely that the distinction between the two cases is not really a matter of Plancherel theory but of orbital constructs or of other invariants associated to the homogeneous space \(G/H\). Several interesting remarks are given relating to Penney's and Bonnet's Plancherel formula. Unfortunately there are some typographical errors.
Penney's and Bonnet's Plancherel formula
scientific article; zbMATH DE number 1135212
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