Fourier transform on the Schwartz space of a reductive symmetric space
This work is part of a series of papers by the same authors, culminating in \textit{P. Delorme}'s proof of the Plancherel formula for a non-Riemannian reductive symmetric space \(G/H\) [Ann. Math., II. Ser. 147, 417-452 (1998; Zbl 0906.22015)]. Here \(G\) is a reductive Lie group in Harish-Chandra's class and \(H\) is an open subgroup of the fixed point group of some involutive automorphism of \(G\). The problem is to decompose the left regular representation of \(G\) on \(L^2(G/H)\) into irreducible representations, a result already proved by Harish-Chandra for the group case (i.e. when \(G/H\) is a reductive Lie group), and for the case of Riemannian symmetric spaces (i.e. when \(H\) is compact). Among the basic tools is a study of Eisenstein integrals associated to parabolic subgroups, which are shown to be holomorphic near the imaginary axis (after suitable normalization), and a proof of the Maass-Selberg relations. The Fourier transform \(\mathcal F\) on \(G/H\) is then defined by means of normalized Eisenstein integrals, and the authors introduce an operator \(\mathcal I\) defined by wave packets, a natural candidate for inverting \(\mathcal F\). Then \({\mathcal F}\circ{\mathcal I}\) is the identity and \({\mathcal I}\circ{\mathcal F}\) is an orthogonal projection, which leads to a new proof of Harish-Chandra's result for the group case. As a final step towards the general Plancherel formula, a criterion is given here for \(\mathcal I\) to be onto, implying that it would indeed provide the required inverse of \(\mathcal F\).
scientific article; zbMATH DE number 1230015