  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 $H$.
