The Continuous Spectrum in Discrete Series Branching Laws

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.