Tempered reductive homogeneous spaces (Q907272)

Let \(G\) be an algebraic semisimple Lie group and \(H\) a reductive subgroup. The main result of the paper under review is Theorem 4.1, which offers a criterion for detecting whether the unitary representation of \(G\) in \(L^2(G/H)\) is tempered or not. This new criterion is easy to check. As an application, in a subsequent paper, the authors will give a complete classification for the pairs \((G, H)\) such that \(L^2(G/H)\) is non-tempered.