Cusp forms for reductive symmetric spaces of split rank one

TL;DR

This paper defines cusp forms on reductive symmetric spaces of split rank one using minimal parabolic subgroups and explores their connection with discrete series representations.

Contribution

It introduces a new class of cusp forms for these spaces and analyzes their relation to discrete series representations.

Findings

Identified a class of minimal parabolic subgroups with convergent cuspidal integrals.

Defined cusp forms in the context of reductive symmetric spaces of split rank one.

Established links between cusp forms and discrete series representations.

Abstract

For reductive symmetric spaces G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra - Schwartz functions are absolutely convergent. Using these integrals we introduce a notion of cusp forms and investigate its relation with representations of the discrete series for G/H.