Plancherel formula for reductive symmetric spaces
Let \(G\) be a reductive Lie group of Harish-Chandra class, \(\theta\) a Cartan involution of \(G\), \(K\) the subgroup of the fixed point set of \(\theta\), \(\sigma\) an involution of \(G\) commuting with \(\theta\) and let \(H\) be an open subgroup of the fixed point subgroup of \(\sigma\). Based on many previous works, the author proves a Plancherel formula for the reductive symmetric space \(G/H\). Fixing a finite-dimensional unitary representation \(\tau\) of \(K\) and using the algebra \(D(G/H)\) of \(G\)-invariant differential operators on \(G/H\) to characterize functional spaces, one defines the normalized Fourier transformation on the space of \(\tau\)-spherical or \(K\)-finite Schwartz functions on \(G/H\). The author gives the inverse Fourier transformation making use of Eisenstein integrals. He then determines the image of the Schwartz space under the Fourier transformation and shows the Fourier inversion formula. The paper demands expert knowledge on representation theory for semisimple Lie groups [cf. \textit{P. Delorme}, J. Funct. Anal. 136, 422-509 (1996; Zbl 0865.43010); Acta Math. 179, 41-77 (1997; Zbl 0896.43001); \textit{J. Carmona} and \textit{P. Delorme}, J. Funct. Anal. 122, 152-221 (1994; Zbl 0831.22004); Transformation de Fourier pour les espaces symétriques réductifs (to appear in Invent. Math.)].
scientific article; zbMATH DE number 1187684