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,H0)(G,H0) its associated pair. For each irreducible square integrable representation ππ of G so that its restriction to H is admissible, we find an irreducible square integrable representation π0π0 of H0H0 which allows us to compute the Harish-Chandra parameter of each irreducible H-subrepresentation of ππ as well as its multiplicity. The computation is based on the spectral analysis of the restriction of π0π0 to a maximal compact subgroup of H0.