The multiplicities of certain K-types in spherical representations

AbstractLet G be a connected semisimple Lie group with finite center,G0 its Lie algebra. G0 = K0 ⊕ P0 a Cartan decomposition of G0 and let K be the corresponding maximal compact subgroup. We denote the complexification of P0 by P. For an irreducible spherical representation, X, of G we compute dim HomK(PX). The result is given explicitly in terms of the Langlands parameters of X