$K$-invariant cusp forms for reductive symmetric spaces of split rank one

TL;DR

This paper characterizes cusp forms on reductive symmetric spaces of split rank one, showing their relation to discrete series representations and establishing multiplicity one results for $K$-spherical discrete series.

Contribution

It proves the equivalence between cusp forms and $K$-finite matrix coefficients of discrete series, and shows all $K$-spherical discrete series occur with multiplicity one.

Findings

Cusp forms coincide with the closure of $K$-finite matrix coefficients of discrete series.

Existence of no $K$-spherical discrete series is equivalent to cusp forms characterization.

All $K$-spherical discrete series have multiplicity one in the Plancherel decomposition.

Abstract

Let $G/H$ be a reductive symmetric space of split rank $1$ and let $K$ be a maximal compact subgroup of $G$. In a previous article the first two authors introduced a notion of cusp forms for $G/H$. We show that the space of cusp forms coincides with the closure of the $K$-finite generalized matrix coefficients of discrete series representations if and only if there exist no $K$-spherical discrete series representations. Moreover, we prove that every $K$-spherical discrete series representation occurs with multiplicity $1$ in the Plancherel decomposition of $G/H$.