The notion of cusp forms for a class of reductive symmetric spaces of split rank one

TL;DR

This paper investigates cusp forms on specific symmetric spaces G/H with G=SL(n,R) and H=S(GL(n-1,R) x GL(1,R)), classifies relevant parabolic subgroups, and links cusp forms to discrete series representations.

Contribution

It classifies minimal parabolic subgroups for convergence of cuspidal integrals and establishes the equivalence between cusp forms and the closure of discrete series representations.

Findings

Classification of minimal parabolic subgroups for convergence

Definition of cusp forms in this context

Equivalence of cusp forms with discrete series closure

Abstract

We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms.