Abstract. In the first part of the paper we generalize a descent technique due to Harish-Chandra to

Abstract. Let Z be an algebraic homogeneous space Z = G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.

Abstract. We show that every irreducible representation in the discrete automorphic spectrum of GL n (A) admits a non vanishing mixed (Whittaker-symplectic) period integral. The analog local problem is a study of models first considered by Klyachko over a finite field. Locally, we show that for a p-adic field F every irreducible, unitary representation of GL n (F ) has a Klyachko model.

In this paper we provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of essentially self-dual representations, that is representations which are isomorphic to the twist of their own contragredient by some Hecke character. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin2n to GL2n.

We show the uniqueness and disjointness of Klyachko models for GL n over a non-Archimedean local field. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum of GL n .