The infinitesimal characters of discrete series for real spherical spaces

TL;DR

This paper investigates the properties of discrete series representations in the regular representation of a real reductive group acting on a real spherical space, establishing their infinitesimal characters and occurrence patterns.

Contribution

It proves that all discrete series representations have real infinitesimal characters in a lattice and each irreducible K-type appears finitely often within these series.

Findings

Discrete series have real infinitesimal characters in a lattice.

Each K-type appears finitely many times in the discrete series.

Results extend to twisted discrete series with line bundle sections.

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 $H$.