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The Infinitesimal Characters of Discrete Series for Real Spherical Spaces
Geometric and Functional Analysis  (IF2.148),  Pub Date : 2020-08-04, DOI: 10.1007/s00039-020-00540-6
Bernhard Krötz, Job J. Kuit, Eric M. Opdam, Henrik Schlichtkrull

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.