The Penney-Fujiwara Plancherel formula for Gelfand pairs
Let \(G=H\ltimes N\) be a semidirect product, where \(H\) is compact and \(N\) is simply connected nilpotent. The author studied in [J. Math. Pures Appl., IX. Ser. 59, 337-374 (1980; Zbl 0441.22007)] the harmonic analysis of such a group by using the orbit method and the Mackey machinery. This paper is concerned with the quasi-regular representation \(\tau\) of the homogeneous space \(G/H\). When \((G,H)\) is a Gelfand pair, an explicit description is given for the Penney's distributions to get a concrete form of Penney's Plancherel formula for \(\tau\) [\textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016)]. We must use complex polarizations but the Penney's distributions take exactly the same form as in the case of real polarizations. One of the other main results is a distribution-theoretic form of Frobenius reciprocity for Gelfand pairs. The author has been pursuing an extensive research on the subject [see, e.g., Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012), etc.]; some of the results of this paper were announced in [Representation theory of Lie groups and Lie algebras, World Scientific Press, Singapore, 120-139 (1992)].
scientific article; zbMATH DE number 944412
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