The Infinitesimal Characters of Discrete Series for Real Spherical Spaces

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 L2(Z). It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of L2(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.