The Continuous Spectrum in Discrete Series Branching Laws

TL;DR

This paper investigates the conditions under which the restriction of discrete series representations from a reductive Lie group to a symmetric subgroup decomposes with finite multiplicities and demonstrates that multiplicities are constant along continuous parameters.

Contribution

It introduces a new convolution-based technique for analyzing multiplicities in the restriction of discrete series representations, applicable to both continuous and discrete spectra.

Findings

Finite multiplicities in restrictions under certain conditions

Multiplicities are constant along continuous parameters

Develops a new convolution technique for studying multiplicities

Abstract

If $G$ is a reductive Lie group of Harish-Chandra class, $H$ is a symmetric subgroup, and $\pi$ is a discrete series representation of $G$, the authors give a condition on the pair $(G,H)$ which guarantees that the direct integral decomposition of $\pi|_H$ contains each irreducible representation of $H$ with finite multiplicity. In addition, if $G$ is a reductive Lie group of Harish-Chandra class, and $H\subset G$ is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of $\pi|_H$ is constant along `continuous parameters'. In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction $\pi|_H$ via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.