The Infinitesimal Characters of Discrete Series for Real Spherical Spaces

Abstract

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.