Multiplicities in the Plancherel decomposition of a semisimple symmetric space

Abstract

Let G=H be a semisimple symmetric space, where G is a connected semisimple Lie group provided with an involution ?; and H = G ? is the subgroup of xed points for ?: Assume moreover that G is linear (for the purpose of the introduction, the assumptions on G and H are stronger than necessary). Then G has a ?-stable maximal compact subgroup K; the associated Cartan involution commutes with ?: Let g = h + q and g = k +p be the decompositions of the Lie algebra g induced by ? and , then h is the Lie algebra of H and k is the Lie algebra of K.