Abstract. For reductive symmetric spaces G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra-Schwartz functions are absolutely convergent. Using these integrals we introduce a notion of cusp forms and investigate its relation with representations of the discrete series for G/H.

In the present paper we further the study of the compression cone of a real spherical homogeneous space Z = G/H. In particular we provide a geometric construction of the little Weyl group of Z introduced recently by Knop and Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra Lie(H) along one-parameter subgroups in the Grassmannian of subspaces of Lie(G). The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone. Contents 1 Introduction 2 Notation and assumptions 3 Adapted points 4 A description of h z 5 Limits of subspaces 6 The compression cone 7 Limit subalgebras and open P -orbits 8 Limits of h z 9 Adapted points in boundary degenerations 10 Admissible points 11 The little Weyl group 12 Spherical root system 13 Reduction to quasi-affine spaces Appendix: Faces of the compression cone * JJK was funded by the Deutsche Forschungsgemeinschaft grant 262362164

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We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the σ -minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals.

Let Z = G/H be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on L 2 (Z). It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of L 2 (Z), have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup of G. Then each irreducible representation of K occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of H.