Cuspidal discrete series for semisimple symmetric spaces

TL;DR

This paper introduces cusp forms on semisimple symmetric spaces, analyzes real hyperbolic spaces in detail, and distinguishes between cuspidal and non-cuspidal discrete series, revealing that all spherical discrete series are non-cuspidal.

Contribution

It defines cusp forms in the context of semisimple symmetric spaces and characterizes the nature of discrete series in real hyperbolic spaces.

Findings

Existence of both cuspidal and non-cuspidal discrete series.

All spherical discrete series are non-cuspidal.

Detailed study of real hyperbolic spaces.

Abstract

We propose a notion of cusp forms on semisimple symmetric spaces. We then study the real hyperbolic spaces in detail, and show that there exists both cuspidal and non-cuspidal discrete series. In particular, we show that all the spherical discrete series are non-cuspidal.