The Plancherel formula for homogeneous spaces with polynomial spectrum
Let \(G\) be a connected Lie group, and \(H\) a closed subgroup of \(G\). The author continues to study Penney's Plancherel formula [cf. \textit{R. Penney}: J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016); \textit{H. Fujiwara}: Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008)], for the quasi-regular representation \(\tau = \text{ind}^ G_ H 1\). As a sequel to the author's previous one [Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012)], where almost all of the irreducible representations that appear in the spectrum of \(\tau\) were monomial, this paper treats the case that the spectrum of \(\tau\) consists of irreducible representations induced from a finite dimensional representation. Attention is focused on Strichartz homogeneous spaces wherein \(G\) and \(H\) are semidirect products of normal abelian subgroups by reductive Lie groups [cf. \textit{R. Strichartz}: Trans. Am. Math. Soc. 296, 387-409 (1986; Zbl 0618.43005)], especially on Strichartz symmetric spaces. It also finds place to discuss criteria for finite multiplicity and for multiplicity-free, and finally a generalization of Frobenius reciprocity.
scientific article; zbMATH DE number 509121