The notion of cusp forms for a class of reductive symmetric spaces of split rank 1

Abstract

We study a notion of cusp forms for the symmetric spaces $G/H$ with $G=\mathrm{SL}(n,\mathbb{R})$ and $H=\mathrm{S}(\mathrm{GL}(n-1,\mathbb{R})\times \mathrm{GL}(1,\mathbb{R}))$. 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 representations of $G/H$ coincides with the space of cusp forms.