Cuspidal discrete series for projective hyperbolic spaces

TL;DR

This paper extends the understanding of cuspidal discrete series in projective hyperbolic spaces, demonstrating the existence of infinitely many cuspidal series and characterizing non-cuspidal series, with implications for the Abel transform of Schwartz functions.

Contribution

It generalizes previous results to broader hyperbolic spaces and analyzes the conditions under which the Abel transform preserves Schwartz functions.

Findings

Existence of infinitely many cuspidal discrete series in certain hyperbolic spaces.

Spherical discrete series are the only non-cuspidal discrete series in projective spaces.

Conditions identified for the Abel transform of Schwartz functions to remain Schwartz.

Abstract

We have in [1] proposed a definition of cusp forms on semisimple symmetric spaces $G/H$, involving the notion of a Radon transform and a related Abel transform. For the real non-Riemannian hyperbolic spaces, we showed that there exists an infinite number of cuspidal discrete series, and at most finitely many non-cuspidal discrete series, including in particular the spherical discrete series. For the projective spaces, the spherical discrete series are the only non-cuspidal discrete series. Below, we extend these results to the other hyperbolic spaces, and we also study the question of when the Abel transform of a Schwartz function is again a Schwartz function.