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In this paper we study the Plancherel formula for a new class of homogeneousspaces for real reductive Lie groups; these spaces are fibered overnon-Riemannian symmetric spaces, and they exhibit a phenomenon of uniforminfinite multiplicities. They also provide examples of non-temperedrepresentations of the group appearing in the Plancherel formula. Severalclasses of examples are given.

A Plancherel formula for L^2(G/H) for almost symmetric subgroups