Bounded multiplicity branching for symmetric pairs

TL;DR

This paper proves that for any simply connected non-compact semisimple Lie group, there exists an infinite-dimensional irreducible representation with uniformly bounded multiplicities upon restriction to all symmetric subgroups, and discusses which representations have this property.

Contribution

It establishes the existence of such representations with bounded multiplicities for all symmetric pairs and characterizes which irreducible representations possess this property.

Findings

Existence of infinite-dimensional irreducible representations with bounded multiplicities for all symmetric pairs.

Characterization of irreducible representations satisfying the bounded multiplicity property.

Abstract

We prove that any simply connected non-compact semisimple Lie group $G$ admits an infinite-dimensional irreducible representation $\Pi$ with bounded multiplicity property of the restriction $\Pi|_{G'}$ for all symmetric pairs $(G, G')$. We also discuss which irreducible representations $\Pi$ satisfy the bounded multiplicity property.