Associated symmetric pair and multiplicities of admissible restriction of Discrete Series

Abstract

Let $(G, H)$ be a symmetric pair for a real semisimple Lie group $G$ and $(G, H_0)$ its associated pair. For each irreducible square integrable representation $\pi$ of $G$ so that its restriction to $H$ is admissible, we find an irreducible square integrable representation $\pi_0$ of $H_0$ which allows to compute the Harish-Chandra parameter of each irreducible $H-$subrepresentation of $\pi$ as well as its multiplicity. The computation is based on the spectral analysis of the restriction of $\pi_0$ to a maximal compact subgroup of $H_0.$