The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces
It is an interesting problem to give a concrete form to the abstract Plancherel formula for monomial representations of Lie groups. In this paper, the author studies quasi-regular representations of two types: abelian symmetric spaces and finite multiplicity completely solvable homogeneous spaces. Let \(G\) be a connected Lie group. Given a connected closed subgroup \(H\) of \(G\), one examines the quasi-regular representation \(\tau=\text{ind}^ G_ H1\). Assuming that \(\tau\) is of type I, one decomposes \(\tau\) into irreducibles \(\tau=\int^ \oplus_{\widehat G}m(\pi)\pi d\mu(\pi)\) with multiplicity function \(m(\pi)\) and a Borel measure \(\mu\) on the unitary dual \(\widehat G\) of \(G\). According to this decomposition, Penney gave the abstract Plancherel formula for \(\tau\), i.e. decomposition of the Dirac distribution \(\delta_ \tau:{\mathcal H}^ \infty_ \tau\ni\psi\mapsto\overline{\psi(e)}\in\mathbb{C}\) for \(\tau\), \({\mathcal H}^ \infty_ \tau\) denoting the space of \(C^ \infty\)-vectors of \(\tau\) given by \(\delta_ \tau=\int^ \oplus_{\widehat G}\sum^{m(\pi)}_{k=1}a^ k_ \pi d\mu(\pi)\) with distributions \(a^ k_ \pi\) \((1\leq k\leq m(\pi))\) in the antidual space of \({\mathcal H}^ \infty_ \pi\) [cf. \textit{R. Penney}, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016)]. In the two cases mentioned above, the orbit method shows how to decompose \(\tau\) into irreducibles and the author computes Penney's distributions \(a^ k_ \pi\) to get a concrete Plancherel formula for \(\tau\). This generalizes the work of the reviewer in the nilpotent case [Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008)].
scientific article; zbMATH DE number 32705