The Plancherel decomposition for a reductive symmetric space. I: Spherical functions (Q2574955)

Let \(G\) be a real reductive Lie group of Harish-Chandra's class, \(H\) an open subgroup of the group \(G^{\sigma}\) of fixed points for an involution \(\sigma\) of \(G\) and \(X=G/H\) the corresponding reductive symmetric space. In the paper under review the authors establish the Plancherel formula for \(K\)-finite spherical Schwartz functions on \(X\), with \(K\) a \(\sigma\)-invariant maximal compact subgroup of \(G\). They also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass-Selberg relations satisfied by the associated \(C\)-functions.