Discrete series representations with non-tempered embedding

TL;DR

This paper provides an example of a semisimple symmetric space where a specific irreducible representation exhibits different multiplicities in the spaces of square-integrable and smooth functions, highlighting nuanced representation behavior.

Contribution

It introduces a concrete example demonstrating the disparity in multiplicities of an irreducible representation in different function spaces on a symmetric space.

Findings

Multiplicity is 1 in L^2(G/H)

Multiplicity is 2 in C^∞(G/H)

Highlights differences in representation theory in symmetric spaces

Abstract

We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.