Ellipticity and discrete series

TL;DR

This paper provides an elementary explanation of why the existence of a discrete series representation in a real reductive group implies the presence of a compact Cartan subgroup, with potential generalizations.

Contribution

It offers a new elementary approach to understanding the relationship between discrete series and compact Cartan subgroups, potentially extendable to real spherical spaces.

Findings

Discrete series implies compact Cartan subgroup existence

Elementary proof approach introduced

Potential for generalization to spherical spaces

Abstract

We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real spherical spaces.